Graduate Texts in Mathematics
241
Editorial Board
S. Axler K.A. Ribet
Graduate Texts in Mathematics
T AKEun/ZARING. Introduction to
Axiomatic Set Theory. 2nd ed.
2 OxTOBY. Measure and Category. 2nd ed.
3 ScHAEFER. Topological Vector Spaces.
2nd ed.
4 HILTONISTAMMBACH. A Course in
Homological Algebra. 2nd ed.
5 MAc LANE. Categories for the Working
Mathematician. 2nd ed.
6 HuGHEs/PIPER. Projective Planes.
7 J.-P. SERRE. A Course in Arithmetic.
8 TAKEUTI/ZARING. Axiomatic Set Theory.
9 HuMPHREYS. Introduction to Lie Algebras and Representation Theory.
10 CoHEN. A Course in Simple Homotopy
Theory.
11 CoNWAY. Functions of One Complex
Variable I. 2nd ed.
12 BEALS. Advanced Mathematical
Analysis.
13 ANDERSON/FULLER. Rings and
Categories of Modules. 2nd ed.
14 GoLUBITSKY/GUILLEMIN. Stable
Mappings and Their Singularities.
15 BERBERIAN. Lectures in Functional
Analysis and Operator Theory.
16 WINTER. The Structure of Fields.
17 RosENBLATT. Random Processes. 2nd ed.
18 HALMOS. Measure Theory.
19 HALMOS. A Hilbert Space Problem
Book. 2nd ed.
20 HusEMOLLER. Fibre Bundles. 3rd ed.
21 HuMPHREYS. Linear Algebraic Groups.
22 BARNEs/MACK. An Algebraic
Introduction to Mathematical Logic.
23 GREUB. Linear Algebra. 4th ed.
24 HoLMES. Geometric Functional
Analysis and Its Applications.
25 HEWITT/STROMBERG. Real and Abstract
Analysis.
26 MANES. Algebraic Theories.
27 KELLEY. General Topology.
28 ZARISKIISAMUEL. Commutative
Algebra. Voi.I.
29 ZARISKI/SAMUEL. Commutative
Algebra. Voi.II.
30 JACOBSON. Lectures in Abstract Algebra
I. Basic Concepts.
31 JACOBSON. Lectures in Abstract Algebra
II. Linear Algebra.
32 JACOBSON. Lectures in Abstract Algebra
III. Theory of Fields and Galois
Theory.
33 HIRSCH. Differential Topology.
34 SPITZER. Principles of Random Walk.
2nd ed.
35 ALEXANDER!WERMER. Several Complex
Variables and Banach Algebras. 3rd ed.
36 KELLEY!NAMIOKA et al. Linear
Topological Spaces.
37 MoNK.Mathematical Logic.
38 GRAUERT/FRITZSCHE. Several Complex
Variables.
39 ARVESON. An Invitation to C*-Algebras.
40 KEMENYISNELLIKNAPP. Denumerable
Markov Chains. 2nd ed.
41 APOSTOL. Modular Functions and
Dirichlet Series in Number Theory.
2nd ed.
42 J.-P. SERRE. Linear Representations of
Finite Groups.
43 GILLMAN/JERISON. Rings of
Continuous Functions.
44 KENDIG. Elementary Algebraic
Geometry.
45 LoilvE. Probability Theory I. 4th ed.
46 LoilvE. Probability Theory II. 4th ed.
47 MmsE. Geometric Topology in
Dimensions 2 and 3.
48 SAcHs/Wu. General Relativity for
Mathematicians.
49 GRUENBERG/WEIR. Linear Geometry.
2nd ed.
50 EDWARDS. Fermat's Last Theorem.
51 KLINGENBERG. A Course in Differential
Geometry.
52 HARTSHORNE. Algebraic Geometry.
53 MANIN. A Course in Mathematical
Logic.
54 GRAVERIWATKINS. Combinatorics with
Emphasis on the Theory of Graphs.
55 BROWN/PEARCY. Introduction to
Operator Theory I: Elements of
Functional Analysis.
56 MASSEY. Algebraic Topology: An
Introduction.
57 CROWELL/Fox. Introduction to Knot
Theory.
58 KoBLITZ. p-adic Numbers, p-adic
Analysis, and Zeta-Functions. 2nd ed.
59 LANG. Cyclotomic Fields.
60 ARNOLD.Mathematical Methods in
Classical Mechanics. 2nd ed.
61 WHITEHEAD. Elements of Homotopy
Theory.
62 KARGAPOLOV/MERLZIAKOV.
Fundamentals of the Theory of Groups.
63 BoLLOBAS. Graph Theory.
(continued after index)
Joseph H. Silverman
The Arithmetic of
Dynamical Systems
With 11 Illustrations
~Springer
Joseph H. Silverman
Department of Mathematics
Brown University
Providence, RI 02912
USA
jhs@math.brown.edu
Editorial Board
S. Axler
Mathematics Department
San Francisco State University
San Francisco, CA 94132
USA
axler@sfsu.edu
K.A. Ribet
Mathematics Department
University of California, Berkeley
Berkeley, CA 94720-384
USA
ribet@math. berkeley.edu
Mathematics Subject Classification (2000): 11-01, 11099, 14G99, 37-01, 37F10
ISBN 978-0-387-69903-5
ISBN 978-0-387-69904-2 (eBook)
DOl 10.1007/978-0-387-69904-2
Library of Congress Control Number: 2007923502
Printed on acid-free paper.
© 2007 Springer Science+ Business Media, LLC
All rights reserved. This work may not be translated or copied in whole or in part without the
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Preface
This book is designed to provide a path for the reader into an amalgamation of two
venerable areas of mathematics, Dynamical Systems and Number Theory. Many of
the motivating theorems and conjectures in the new subject of Arithmetic Dynamics
may be viewed as the transposition of classical results in the theory of Diophantine
equations to the setting of discrete dynamical systems, especially to the iteration
theory of maps on the projective line and other algebraic varieties. Although there is
no precise dictionary connecting the two areas, the reader will gain a flavor of the
correspondence from the following associations:
Diophantine Equations
Dynamical Systems
rational and integral
points on varieties
rational and integral
points in orbits
torsion points on
abelian varieties
periodic and preperiodic
points of rational maps
There are a variety of topics covered in this volume, but inevitably the choice
reflects the author's tastes and interests. Many related areas that also fall under the
heading of arithmetic or algebraic dynamics have been omitted in order to keep the
book to a manageable length. A brief list of some of these omitted topics may be
found in the introduction.
Online Resources
The reader will find additonal material, references and errata at
http://www.math.brown.edu/-jhs/ADSHome.html
Acknowledgments
The author has consulted a great many sources in writing this book. Every attempt
has been made to give proper attribution for all but the most standard results. Much
of the presentation is based on courses taught at Brown University in 2000 and 2004,
and the exposition benefits greatly from the comments of the students in those
v
vi
Preface
courses. In addition, the author would like to thank the many students and mathematicians who read drafts and/or offered suggestions and corrections, including
Matt Baker, Rob Benedetto, Paul Blanchard, Rex Cheung, Bob Devaney, Graham
Everest, Liang-Chung Hsia, Rafe Jones, Daniel Katz, Shu Kawaguchi, Michelle
Manes, Patrick Morton, Curt McMullen, Hee Oh, Giovanni Panti, Lucien Szpiro,
Tom Tucker, Claude Viallet, Tom Ward, Xinyi Yuan, Shou-Wu Zhang. An especial
thanks is due to Matt Baker, Rob Benedetto and Liang-Chung Hsia for their help in
navigating the treachorous shoals of p-adic dynamics. The author would also like to
express his appreciation to John Milnor for a spellbinding survey talk on dynamical
systems at Union College in the mid-1980s that provided the initial spark leading
eventually to the present volume. Finally, the author thanks his wife, Susan, for her
support and patience during the many hours occupied in writing this book.
Joseph H. Silverman
January 1, 2007
Contents
Preface
v
Introduction
Exercises
1
7
1 An Introduction to Classical Dynamics
1.1 Rational Maps and the Projective Line . . . . . . .
1.2 Critical Points and the Riemann-Hurwitz Formula .
1.3 Periodic Points and Multipliers .
1.4 The Julia Set and the Fatou Set . . . . . . . . . . .
1.5 Properties of Periodic Points . . . . . . . . . . . .
1.6 Dynamical Systems Associated to Algebraic Groups
Exercises . . . . . . . . . . . . . . . . . . . . . . .
2
3
9
9
12
18
22
27
28
35
Dynamics over Local Fields: Good Reduction
2.1 The Nonarchimedean Chordal Metric ..
2.2 Periodic Points and Their Properties . . .
2.3 Reduction of Points and Maps Modulo p .
2.4 The Resultant of a Rational Map . . .
2.5 Rational Maps with Good Reduction .
2.6 Periodic Points and Good Reduction .
2. 7 Periodic Points and Dynamical Units .
Exercises . . . . . . . . . . . . . . .
43
43
Dynamics over Global Fields
3.1 Height Functions . . . . . . . . . . . .
3.2 Height Functions and Geometry . . . .
3.3 The Uniform Roundedness Conjecture .
3.4 Canonical Heights and Dynamical Systems
3 .5 Local Canonical Heights . .
3.6 Diophantine Approximation . . . . . . .
3.7 Integral Points in Orbits . . . . . . . . . .
3.8 Integrality Estimates for Points in Orbits .
3.9 Periodic Points and Galois Groups
81
81
vii
47
48
53
58
62
69
74
89
95
97
102
104
108
112
122
Contents
viii
3.10 Equidistribution and Preperiodic Points . . . . . . . . . . . . . . . 126
3.11 Ramification and Units in Dynatomic Fields . . . . . . . . .
129
Exercises . . . . . . . . .
135
4 Families of Dynamical Systems
4.1 Dynatomic Polynomials . . .
4.2 Quadratic Polynomials and Dynatomic Modular Curves . . . . . . .
4.3 The Space Ratd of Rational Functions . . . . . . .
4.4 The Moduli Space Md of Dynamical Systems . . . . . . .
4.5 Periodic Points, Multipliers, and Multiplier Spectra . . . .
4.6 The Moduli Space M 2 of Dynamical Systems of Degree 2
4.7 Automorphisms and Twists . . . . . . . . . . . . . . . . .
4.8 General Theory of Twists . . . . . . . . . . . . . . . . . . . . . . .
4.9 Twists of Rational Maps . . . . . . . . . . . . . . . . . . . . . . .
4.10 Fields of Definition and the Field of Moduli . . . . . . . . . . . .
4.11 Minimal Resultants and Minimal Models
. . . . . .
Exercises . . . . . . . . . . . . . . . . .
5 Dynamics over Local Fields: Bad Reduction
5.1 Absolute Values and Completions . . . .
5.2 A Primer on Nonarchimedean Analysis . . . . . . . . . . . . . . .
5.3 Newton Polygons and the Maximum Modulus Principle . . . . . . .
5.4 The Nonarchimedean Julia and Fatou Sets . . . .
5.5 The Dynamics of (z 2 - z)jp . . . . . . . . . . . . .
5.6 A Nonarchimedean Montel Theorem .
5. 7 Periodic Points and the Julia Set . . .
5.8 Nonarchimedean Wandering Domains
5.9 Green Functions and Local Heights .
5.10 Dynamics on Berkovich Space . . . . . .
Exercises . . . . . . . . . . . . . . . .
6 Dynamics Associated to Algebraic Groups
6.1 Power Maps and the Multiplicative Group
6.2 Chebyshev Polynomials . . . . . . .
6.3 A Primer on Elliptic Curves . . . .
6.4 General Properties of Lattes Maps .
6.5 Flexible Lattes Maps . . . . . . . .
6.6 Rigid Lattes Maps . . . . . . . . . . . . .
6.7 Uniform Bounds for Lattes Maps . . . . .
6.8 Affine Morphisms and Commuting Families . .
Exercises . . . . . . . . . . . . . . . . . . . .
147
148
155
168
174
179
188
195
199
203
206
218
224
239
240
242
248
254
257
263
268
276
287
294
312
325
325
328
336
350
355
364
368
375
380
Contents
7
ix
387
Dynamics in Dimension Greater Than One
7.1 Dynamics of Rational Maps on Projective Space.
7.2 Primer on Algebraic Geometry . . . . . . . . . .
7.3 The Weil Height Machine . . . . . . . . . . . . .
7.4 Dynamics on Surfaces with Noncommuting Involutions .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . .
388
402
407
410
427
Notes on Exercises
441
List of Notation
445
References
451
Index
473
Introduction
A (discrete) dynamical system consists of a set S and a function ¢ : S ----+ S mapping
the setS to itself. This self-mapping permits iteration
¢n = ¢ o ¢ o ... o ¢ = nth iterate of¢.
"--v-"
ntimes
(By convention, ¢ 0 denotes the identity map on S.)
For a given point o: E S, the (forward) orbit of o: is the set
The point o: is periodic if ¢n (o:) = o: for some n 2: 1. The smallest such n is called
the exact period of o:. The point o: is preperiodic if some iterate ¢m (o:) is periodic.
The sets of periodic and preperiodic points of¢ in S are denoted respectively by
Per(¢, S) = {o: E S: ¢n(o:) = o: for some n 2: 1},
PrePer(¢, S) = {o: E S: ¢m+n(o:) = ¢m(o:) for some n 2: 1, m 2: 0}
=
{o: E S: O¢(o:) is finite}.
We write Per(¢) and PrePer( ¢) when the set S is fixed.
Principal Goal of Dynamics
Classify the points o: in the set S according to the behavior of
their orbits 0 4> ( o:).
If S is simply a set with no additional structure, then typical problems are to describe the sets of periodic and preperiodic points and to describe the possible periods
of periodic points. Usually, however, the set S has some additional structure and
one attempts to classify the points in S according to the interaction of their orbits
with that structure. There are many types of additional structures that may imposed,
including algebraic, topological, metric, and analytic.
Example 0.1. (Finite Sets) LetS be a finite set and¢ : S ----+ Sa function. Clearly
every point of S is preperiodic, so we might ask for the number of elements in the
set of periodic points
2
Introduction
Per(¢,S) ={a E S: ¢n(a) =a forsomen ~ 1}.
An interesting class of sets and maps are finite fields S = lFP and maps
¢: lFP --t lFp given by polynomials ¢(z) E lFp[z]. For example, Fermat's Little Theorem says that
and
which gives two extremes for the set of periodic points. A much harder question is to
fix an integer d ~ 2 and ask for which primes p there is a polynomial ¢ of degree d
satisfying Per(¢,lFp) = lFp. Similarly, one might fix a polynomial ¢(z) E Z[z]
and ask for which primes p is it true that Per(¢, lFP) = lFp· In particular, are there
infinitely many such primes?
In a similar, but more general, vein, one can look at a rational function ¢ E lFP ( z)
inducing a rational map ¢ : JPl 1 (lFP) --t JPl 1 (JFP). Even more generally, one can ask
similar questions for a morphism¢ : V(lFp) --t V(lFp) of any variety VjlFP, for
example V = pN.
Example 0.2. (Groups) Let G be a group and let¢: G --t G be a homomorphism.
Using the group structure, it is often possible to describe the periodic and preperiodic
points of¢ fairly explicitly. The following proposition describes a simple, but important, example. In order to state the proposition, we recall that the torsion subgroup of
an abelian group G, denoted by G 10rs, is the set of elements of finite orderinG,
Gtors = {a E G : am = e for some m ~ 1},
where e denotes the identity element of G.
Proposition 0.3. Let G be an abelian group, let d
¢: G --t G be the d1h power map ¢(a)= ad. Then
> 2 be an integer, and let
PrePer(¢, G)= Gtors·
Proof The simple nature of the map ¢ allows us to give an explicit formula for its
iterates,
¢n(a) = adn.
Now suppose that a E PrePer(¢d, G). This means that ¢m+n(a) = ¢m(a) for
some n ~ 1 and m ~ 0, so adm+n = adm. But G is a group, so we can multiply
by a-dm to get adm+n_dm = e. The assumptions on d, m, and n imply that the
exponent is positive, so a E Gtors·
Next suppose that a E G10rs, say am = e, and consider the following sequence
of integers modulo m:
d, d 2 , d 3 , d4 , ... modulo m.
Since there are only finitely many residues modulo m, eventually the sequence has a
repeated element, say di = d1 (mod m) with i > j. Then
¢i(a) = adi = ad1 = ¢1 (a),
Hence a E PrePer(¢).
since am= e and di = dj (mod m).
D
Introduction
3
Example 0.4. (Topological Spaces) LetS be a topological space and let¢ : S--+ S
be a continuous map. For a given o: E S, one might ask for a description of the
accumulation points of 0 4> ( o:). For example, a point o: is called recurrent if it is an
accumulation point of 0 q, (o:). In other words, o: is recurrent if there is a sequence of
integers n 1 < n2 < n3 < · · · such that limi_,oo ¢ni ( o:) = o:, so either o: is periodic,
or it eventually returns arbitrarily close to itself.
Example 0.5. (Metric Spaces) Let (S, p) be a compact metric space. For example, S
could be the unit sphere sitting inside JR3 , and p( o:, /3) the usual Euclidean distance
from o: to j3 in JR3 . The fundamental question in this setting is whether points that
start off close to a given point o: continue to remain close to one another under repeated iteration of¢. If this is true, we say that ¢ is equicontinuous at o:; otherwise
we say that ¢ is chaotic at o:. (See Section 1.4 for the formal definition of equicontinuity.) Thus if¢ is equicontinuous at o:, we can approximate ¢n ( o:) quite well by
computing ¢n(/3) for any point j3 that is close too:. But if¢ is chaotic at o:, then no
matter how close we choose o: and /3, eventually ¢n ( o:) and ¢n (!3) move away from
each other.
Example 0.6. (Arithmetic Sets) An arithmetic set is a set such as Z or Q or anumber field that is of number-theoretic interest, but doesn't have a natural underlying
topology. More precisely, an arithmetic set tends to have a variety of interesting
topologies; for example, Q has the archimedean topology induced by the inclusion
Q c lR and the p-adic topologies induced by the inclusions Q C Qp. In the arithmetic setting, the map ¢ is generally a polynomial or a rational map. Here are some
typical arithmetical-dynamical questions, where we take ¢(z) E Q(z) to be a rational function of degree d 2 2 with rational coefficients:
• Let o: E Q be a rational number. Under what conditions can the orbit 0 q, (o:)
contain infinitely many integer values? In other words, when can 0 q, (o:) n Z
be an infinite set?
• Is the set Per(¢, Q) of rational periodic points finite or infinite? If finite, how
large can it be?
• Let o: E Per(¢) be a periodic point for ¢. It is clear that o: is an algebraic
number. What are the arithmetic properties of the field Q( o: ), or more generally
of the field generated by all of the periodic points of a given period?
What is in this book: We provide a brief summary of the material that is covered.
1. An Introduction to Classical Dynamics
We begin in Chapter 1 with a short self-contained overview, without proofs, of
classical complex dynamics on the projective line.
2. Dynamics over Local Fields: Good Reduction
Chapter 2, which starts our study of arithmetic dynamics, considers rational
maps ¢(z) with coefficients in a local field K, for example, K = Qp. The emphasis in Chapter 2 is on maps that have "good reduction modulo p." The good
Introduction
4
reduction property imples that many of the geometric properties of ¢ acting on
the points of K are preserved under reduction modulo p. In particular, the map ¢
is p-adically nonexpanding, and periodic points behave well when reduced modulo p. The remainder of the chapter gives applications exploiting these two key
properties of good reduction.
3. Dynamics over Global Fields
We move on in Chapter 3 to arithmetic dynamics over global fields such as Q
and its finite extensions. Just as in the study of Diophantine equations over global
fields, the theory of height functions plays a key role, and we develop this theory, including the construction of the canonical height associated to a rational
map. We discuss rationality ofpreperiodic points and formulate a general uniform
boundedness conjecture. Using classical results from the theory of Diophantine
approximation, we describe exactly which rational maps ¢ can have orbits containing infinitely many integer points, and we give a more precise result saying
that the numerator and denominator of ¢n (a:) grow at approximately the same
rate. We consider the extension fields generated by periodic points and describe
their Galois groups, ramification, and units.
4. Families of Dynamical Systems
At this point we change our perspective, and rather than studying the dynamics
of a single rational map, we consider families of rational maps and the variation
of their dynamical properties. We construct various kinds of parameter and moduli spaces, including the space of quadratic polynomials with a point of exact
period N (which are analogues of the classical modular curves X 1 ( N)), the parameter space Ratd of rational functions of degree d, and the moduli space Md
of rational functions of degree d modulo the natural conjugation action by PG1 2 .
In particular, we prove that M 2 is an isomorphism to the affine plane A 2 . We
also study twists of rational maps, analogous to the classical theory of twists of
varieties, and the field-of-moduli versus field-of-definition problem.
5. Dynamics over Local Fields: Bad Reduction
Chapter 5 returns to arithmetic dynamics over local fields, but now in the case of
"bad reduction." It becomes necessary to work over an algebraically closed field,
so we discuss the field Cp and give a brief introduction to nonarchimedean analysis and Newton polygons. Using these tools, we define the nonarchimedean Julia
and Fatou sets and prove a version ofMontel's theorem that is then used to study
periodic points and wandering domains in the nonarchimedean setting. This is followed by the construction of p-adic Green functions and local canonical heights.
The chapter concludes with a short introduction to dynamics on Berkovich space.
The Berkovich projective line IP' 8 is path connected, compact, and Hausdorff, yet
it naturally contains the totally disconnected, non-locally compact, non-Hausdorff
space IP' 1 ( Cp).
6. Dynamics Associated to Algebraic Groups
There is a small collection of rational maps whose dynamics are much easier to
understand than those of a general map. These special rational maps are associated to endomorphisms of algebraic groups. We devote Chapter 6 to the study of
Introduction
5
these maps. The easiest ones are the power maps M d(z) = zd and the Chebyshev
polynomials Td(z) characterized by Td(2cosB) = 2cos(dB). They are associated to the multiplicative group. More interesting are the Lattes maps attached to
elliptic curves. We give a short description, without proofs, of the theory of elliptic curves and then spend the remainder of the chapter discussing dynamical and
arithmetic properties of Lattes maps.
7. Dynamics in Dimension Greater Than One
With a few exceptions, the results in Chapters 1-6 all deal with iteration of maps
on the one-dimensional space lP' 1 , i.e., they are dynamics of one variable. In Chapter 7 we consider some of the issues that arise in studying dynamics in higher
dimensions. We first study a class of rational maps ¢ : lP'N --. lP'N that are not
everywhere defined. Even over C, the geometry of dynamics of rational maps is
imperfectly understood. We restrict attention to automorphisms ¢ : AN --. AN
and study height functions and rationality of periodic points for such maps. We
next consider morphisms ¢ : X --. X of varieties other than lP'N. In order to
deal with higher-dimensional dynamics, we use tools from basic algebraic geometry and Weil's height machine, which we describe without proof. We then study
arithmetic dynamics, heights, and periodic points on K3 surfaces admitting two
noncommuting involutions t 1 and t 2 . The composition¢ = t 1 o t 2 provides an
automorphism ¢ : X --. X whose geometric and arithmetic dynamical properties
are quite interesting.
What's missing: A book necessarily reflects the author's interests and tastes, while
space considerations limit the amount of material that can be included. There are
thus many omitted topics that naturally fit into the purview of arithmetic dynamics.
Some of these are active areas of current mathematical research with their own literature, including introductory and advanced textbooks. Others are younger areas that
deserve books of their own. Examples of both kinds include the following, some of
which overlap with one another:
• Dynamics over finite fields
This includes general iteration of polynomial and rational maps acting on finite
fields, see for example [41, 42, 102, 106, 109, 179, 220, 222, 275, 308, 336,
350, 385, 384, 400, 437, 444], and more specialized topics such as permutation
polynomials [275, Chapter 7] that are fields of study in their own right.
• Dynamics over function fields
The study of function fields over finite fields has long provided a parallel theory
to the study of number fields, but inseparability and wild ramification often lead
to striking differences, while function fields of characteristic 0 present their own
arithmetic challenges, e.g., they have infinitely many points of bounded height.
The study of arithmetic dynamics over function fields is in its infancy. For a handful of results, see [20, 61, 85, 107,207,279,354,356, 415].
• Iteration ofpower series
There is an extensive literature, but no textbook, on the iteration properties of
formal, p-adic, and Puiseux power series. Among the fundamental problems are
Introduction
6
the classification of nontrivial commuting power series (to what extent do they
come from formal groups?) and the description of preperiodic points. See for
example (246,266,267,268,269,270,271,272,273,274,280,281,28 2,283,
284,389,390].
• Algebraic dynamics
There is no firm line between arithmetic dynamics and algebraic dynamics, and
indeed much of the material in this book is quite algebraic. Some topics of an
algebraic nature that we do not cover include irreducibility of iterates [6, 17, 113,
115, 345, 346, 425], formal transformations and algebraic identities [55, 80, 79],
and various results of an algebro-geometric nature [140, 149, 163, 392].
• Lie groups and homogeneous spaces, ergodic theory, and entropy
This is a beautiful and much studied area of mathematics in which geometry,
analysis, and algebra interact. There are many results of a global arithmetic nature,
including for example hard problems of Diophantine approximation, as well as
an extensive p-adic theory. For an introduction to some of the main ideas and
theorems in this area, see [47, 247, 304, 423], and for other arithmetic aspects
of ergodic theory and entropy, including relations with height functions, ergodic
theory in a nonarchimedean setting, and arithmetic properties of dynamics on
solenoids, see for example [9, 31, 104, 130, 147, 160, 188, 229, 239, 241, 251,
277,278, 351,440,441,442, 447].
• Equidistribution in arithmetic dynamics
There are many ways to measure (arithmetic) equidistribution, including via
canonical heights, p-adic measures, and invariant measures on projective and
Berkovich spaces. In Section 3.10 we summarize some basic equidistribution conjectures and theorems (without proof). For additional material, see [15, 24, 28, 98,
99,168, 182,209,429,432,450].
• Topology and arithmetic dynamics on foliated spaces
This surprising connection between these diverse areas of mathematics has been
inverstigated by Deninger in a series of papers [124, 125, 126, 127, 128].
• Dynamics on Drinfeld modules
It is natural to study local and global arithmetic dynamics in the setting of Drinfeld modules, although only a small amount of work has yet been done. See for
example [180, 181, 182, 349, 395].
• Number-theoretic iteration problems not arising as maps on varieties
A famous example of this type of problem is the notorious 3x + 1 problem,
see [253] for an extensive bibliography. Another problem that people have studied is iteration of arithmetic functions such as Euler's 'P function; see for example [151, 342].
• Realizability of integer sequences
A sequence (an) of nonnegative integers is said to be realizable if there are a
set S and a function ¢ : S --+ S with the property that for all n, the map ¢ has an
periodic points of order n. See [158] for an overview and [12, 144, 157, 362, 363,
419] for further material on realizable sequences.
Exercises
7
Prerequisites: The principal prerequisite for reading this book is basic algebraic
number theory (rings of integers, ideals and ideal class groups, units, valuations and
absolute values, completions, ramification, etc.) as covered, for example, in the first
section of Lang's Algebraic Number Theory [258]. We also assume some knowledge of elementary complex analysis as typically covered in an undergraduate course
in the subject. No background in dynamics or algebraic geometry is required; we
summarize and give references as necessary. In particular, to help make the book
reasonably self-contained, we have included introduction/overview material on nonarchimedean analysis in Section 5.2, elliptic curves in Section 6.3, and algebraic
geometry in Section 7.2. However, previous familiarity with basic algebraic geometry will certainly be helpful in reading some parts of the book, especially Chapters 4
and 7.
Cross-references and exercises: Theorems, propositions, examples, etc. are numbered consecutively within each chapter and cross-references are given in full; for
example, Proposition 3.2 refers to the second labeled item in Chapter 3. Exercises
appear at the end of each chapter and are also numbered consecutively, so Exercise 5.7 is the seventh exercise in Chapter 5. There is an extensive bibliography, with
reference numbers in the text given in square brackets.
This book contains a large number of exercises. Some of the exercises are marked
with a single asterisk * , which indicates a hard problem. Others exercises are marked
with a double asterisk ** , which means that the author does not know how to solve
them. However, it should be noted that these ''unsolved" problems are of varying degrees of difficulty, and in some cases their designation reflects only the author's lack
of perspicacity. On the other hand, some of the unsolved problems are undoubtedly
quite difficult. The author solicits solutions to the ** marked problems, as well as
solutions to the exercises that are posed as questions, for inclusion in later editions.
The reader will find additional notes and references for the exercises on page 441.
Standard Notation: Throughout this book we use the standard symbols
Z, Q, JR., C, IFq, Zp, AN, and IP'N
to represent the integers, rational numbers, real numbers, complex numbers, field
with q elements, ring of p-adic integers, N -dimensional affine space, and N -dimensional projective space, respectively. Additional notation is defined as it is introduced
in the text. A detailed list of notation may be found on page 445.
Exercises
0.1. Let S be a set and ¢ : S --> S a function.
(a) If Sis a finite set, prove that¢ is bijective if and only if Per(¢, S) = S.
(b) In general, prove that if Per(¢, S) = S, then¢ is bijective.
(c) Give an example of an infinite set S and map ¢ with the property that ¢ is bijective and
Per(¢, S) -I S.
Exercises
8
(d) If¢ is injective, prove that PrePer( ¢, S) = Per(¢, S).
0.2. Let S be a set, let ¢ : S --+ S and 'lj; : S --+ S be two maps of S to itself, and suppose
that ¢ and 'lj; commute, i.e., assume that ¢ o 'lj; = 'lj; o ¢.
(a) Provethat'lj;(PrePer(¢)) C PrePer(¢).
(b) Assume further that 'lj; is a finite-to-one surjective map, i.e., 'lj;(S) = S, and for every x E
S, the inverse image 'lj;- 1 (x) is finite. Prove that 'lj;(PrePer(¢)) = PrePer(¢).
(c) We say that a point P E Sis an isolated preperiodic point of¢ ifthere are integers n >
m such that ¢n(P) = ¢m(P) and such that the set
is finite. Suppose that every preperiodic point of¢ is isolated. Prove that
PrePer( ¢) C PrePer( 'lj; ).
Conclude that if the commuting maps ¢ and 'lj; both have isolated preperiodic points,
then PrePer(¢) = PrePer('lj;).
0.3. Let ¢( z) = zd + a E Z[ z] and let p be a prime. Prove that Per(¢, lFP) = lFP if and only
ifgcd(d,p -1) = 1.
0.4. Let G be a group and let ¢ : G --+ G be a homomorphism.
(a) Prove that Per(¢, G) is a subgroup of G.
(b) Is PrePer( ¢,G) a subgroup of G? Either prove that it is a subgroup or give a counterexample.
0.5. Let G be a topological group, that is, G is a topological space with a group structure
such that the group composition and inversion laws are continuous maps. Let ¢ : G --+ G
be a continuous homomorphism. Exercise 0.4 says that Per(¢, G) is a subgroup of G, so its
topological closure Per(¢, G) is also a subgroup of G. Compute this topological closure for
each of the following examples. (In each example, d 2: 2 is a fixed integer.)
(a) G = C and ¢(a)= ad.
(b) G = R* and ¢(a) =ad.
(c) G = RN /ZN and ¢(a)= da mod zN.
0.6. (a) Describe Per(¢, Q) for the function ¢(z) = z 2 + 1.
(b) Describe Per(¢, Q) for the function ¢( z) = z 2 - 1.
(c) Let ¢(z) E Z[z] be a monic polynomial of degree at least two. Prove that Per(¢, Q) is
finite. (Hint. First prove that Per(¢, Q) C Z.)
(d) Same question as (c), but now ¢(z) E Q[z] has rational coefficients and is not assumed
to be monic.
0. 7. Let ¢( z) = z + 1/ z and let a E Q*. Prove that 0 <I> (a) n Z is finite. What is the largest
number of points that it can contain?
Chapter 1
An Introduction to Classical
Dynamics
Classically, the subject of discrete complex dynamical systems involves the study of
iteration of polynomial and rational maps on C or lP' 1 ( q. The local theory of analytic
iteration dates back to the 19th century, but the modem theory of global complex
dynamics starts with the foundational works ofFatou [166, 167] and Julia [223] in
1918-1920. From that beginning has arisen a vast literature.
Our goal in this book is to study number-theoretic questions associated to iteration of polynomial and rational maps. A standard technique in number theory is to
attempt to answer questions related to a number field K by first studying analogous
questions over each completion Kv of K. In particular, the archimedean completions of K lead back to classical dynamics over IR or C. In this chapter we describe
some of the fundamental concepts and theorems in complex dynamics. These theorems provide both tools and templates for our subsequent study of dynamics over
complete nonarchimedean fields in Chapters 2 and 5 and over number fields in Chapters 3 and 4. This chapter provides sufficient background and motivation for reading
the remainder of this book, but the reader wishing to learn more about the beautiful
subject of complex dynamics might look at one or more of the following texts:
•
•
•
•
A. Beardon [43], Iteration ofRational Functions.
L. Carleson and T. Gamelin [95], Complex Dynamics.
R. Devaney [132], An Introduction to Chaotic Dynamical Systems.
J. Milnor [302], Dynamics in One Complex Variable.
1.1
Rational Maps and the Projective Line
A rational function ¢>(z) E C(z) is a quotient of polynomials
¢(z) = F(z) = ao + a 1 z + · · · + adzd
G(z)
bo + b1 z + ... + bdzd
9
1. An Introduction to Classical Dynamics
10
with no common factors. 1 The degree of¢ is
deg¢ = max{degF,degG}.
So if we write¢ as above with at least one of ad and bd nonzero, then¢ has degree d.
We generally consider maps with deg ¢ ;::: 2.
A rational function of degree d induces a rational map of the complex projective
line JP> 1 (C),
(1.1)
The map ¢ is a ramified d-to-1 covering. The complex projective line is also known
as the Riemann sphere, since topologically, and indeed real analytically, JP> 1 (C) is
isomorphic to 8 2 • Two very important properties of rational maps (1.1) are that they
are continuous and open. 2
There are many equivalent (and useful) ways to construct the projective line,
including
and
JP>l(q ~ C 2 " {(o,on,
""
where [X, Y] ""[X', Y'] ifthere is au E C* such that X'= uX andY'= uY. In
terms of homogeneous coordinates, a rational map ¢ : JP> 1 __. JP> 1 is given by a pair
ofhomogeneous polynomials ¢(X, Y) = [F*(X, Y), G*(X, Y)] of degree d. These
are related to the description ¢(z) = F(z)/G(z) by the relations
F*(X, Y) = yd F(X/Y)
and
G*(X, Y) = YdG(X/Y).
A linear fractional transformation (or Mobius transformation) is a map of the
form
az+b
with ad - be ::j:. 0.
z r----t - cz+d
It defines an automorphism of JP> 1 , and composition of transformations corresponds
to multiplication of the corresponding matrices ( ~ ~). Two matrices give the same
transformation if and only if they are scalar multiples of one another, and it is not hard
to prove that these are the only automorphisms ofJP> 1 (C); see [198, Exercise 1.6.6 and
Remark 11.7.1.1]. So we have
We observe that given any two triples (a, ci, a") and (/3, /3', /3") of distinct points
in JP>\ there exists a unique automorphism f E PGL 2 (C) satisfying
f(a) = /3,
f(a') = /3',
f(a") = /3".
1It is permissible to have G(z) = 0, in which case F(z) = ao is a nonzero constant and <f>(z) maps
every point to oo. Similarly we may have F(z) = 0 and G(z) = bo f- 0. Note that by convention we
set deg 0 = -oo, so if <f>(z) is constant, then deg </> = 0.
2Let </> : X -> Y be a map between topological spaces. We recall that</> is continuous if U C Y open
implies </> - l ( U) C X open, and </> is open if V c X open implies </>( V) C Y open.
11
1.1. Rational Maps and the Projective Line
This is intuitively reasonable because PGL 2(C) is a three-dimensional space that
acts more-or-less simply transitively on JP> 1 x JP>1 x JP> 1.
If¢ : JP>1 - JP>1 is a rational map and f E PGL2(C), the linear conjugate of¢
by f is the map
¢! = f-1 0 ¢ 0 f.
Linear conjugation corresponds to a change of variables on JP> 1 as illustrated by the
commutative diagram
f
1
f
1
p1 ~ p1
Tworationalmaps¢and'¢are/inear/yconjugateif'¢ = ¢/ forsomef E PGLz(C).
This is clearly an equivalence relation. We also observe that
which shows that linear conjugation is a good operation to use when studying iteration.
A convenient metric on the projective line JP> 1(C) is the chordal metric, given in
homogeneous coordinates by the formula
p
IX1Y2- XzY1I
([X y;] [X y; ]) _
1' 1 ' z, 2 - v1Xll 2 + IYli 2JIXzl 2 + IYzl 2
If neither point is the point at infinity (i.e., neither point is equal to [1, 0]), then using
the substitution Zi = Xd~ to dehomogenize gives
lz1 - zzl
p(ZI,Z2)
= vlzll2 + 1ylzzl2 + 1
Notice that 0 S p S 1. Identifying JP> 1 (C) ~ C U { oo} with the sphere S 2 by drawing
lines from the north pole of the sphere (see Figure 1.1), the chordal metric p(z, w) is
related to the Euclidean distance between the corresponding points on the sphere by
the formula
1
p(z, w) = 2lz*- w*l·
(See Exercise 1.1.) In particular, the triangle inequality in IR 3 implies the triangle
inequality for the chordal metric
(1.2)
Rational maps ¢ : JP>1 (C) - JP> 1 (C) also have the Lipschitz property relative to
the chordal metric. This means that there is a constant C (¢) such that
p(ifl(a), ¢((3)) S C(¢)p(a, (3)
(See Exercise 1.3.)
for all a, (3 E JP> 1 (C).
(1.3)
12
1. An Introduction to Classical Dynamics
Figure 1.1: Identifying CC U oo with the Riemann sphere.
1.2
Critical Points and the Riemann-Hurwitz
Formula
Given a rational function ¢(z) = F(z)/G(z), the formal derivative
¢'( ) = G(z)F'(z)- F(z)G'(z)
G(z)2
z
is defined at any point a satisfying a -=/: oo and G(a) -=/: 0. (In order to compute
the derivative at the excluded points, make a linear change of variables. See Exercise 1.5.)
Looking locally around z = a, Taylor's theorem says that
¢(z) =¢(a)+ <P'(a)(z- a)+ O((z- a) 2 ).
The map ¢ : JP 1 (CC) - JP1 (CC) is ramified at a if ¢'(a) = 0, in which case the
point a is called a critical point or a ramification point of¢. (Algebraic geometers
talk about ramification points; dynamicists talk about critical points.) Note that¢ is
locally one-to-one around a noncritical point. As we will see, the orbits of the critical
points play an important role in determining the behavior of the dynamical system.
Assuming as above that a -I- oo and ¢(a) f:. oo, the ramification index of¢ at a
is
ea(¢) = orda(¢(z)- ¢(a)).
In particular, ¢ is ramified at a if and only if e 0 ( ¢) ?:: 2. Locally, there is a constant
c -I- 0 such that
¢(z) =¢(a)+ c(z- a)ea(<l>) + 0 ( (z- a)ea(<l>)+l) ,
and ¢ is locally ea (¢ )-to-1 in a neighborhood of a. It is not hard to see that eq, (a) :::;
deg(¢). If eq,(a) = deg(¢), then we say that¢ is totally ramified at a.
1.2. Critical Points and the Riemann-Hurwitz Formula
13
The next result gives a global relation satisfied by the locally defined ramification
indices. We sketch two distinct proofs of this important theorem.
Theorem 1.1. (Riemann-Hurwitz Formula for IP' 1)
2d- 2 =
L (ea(</>) -1).
aEll' 1
Proof (Algebraic Proof) After a change of variables using a linear fractional transformation, we may assume that oo is neither a ramification point nor the image of a
ramification point, and also that ¢( oo) = 0. This means that ¢( z) has the form
</>(z) = F(z) = azd- 1 + · ··
G(z)
bzd + .. ·
with ab =f. 0. Then
4>'( ) = G(z)F'(z)- F(z)G'(z) = -abz 2 d- 2 + ...
z
G(z) 2
G(z) 2
·
For any a =f. oo such that ¢(a) =f. oo, i.e., such that G(a) =f. 0, we have
</>(z) =</>(a)+ (z- a)ea(</>)1/!(z)
for a function 1/!(z) E C(z) satisfying 1/J(a) =f. 0, oo. Differentiating yields
</>'(z) = ea(</>)(z- a)ea(<I>)- 11/J(z) + (z- a)ea(</>)1/!'(z),
from which we deduce
orda 4>' (z) = ea (</>) - 1.
(Notice we are using strongly the fact that C has characteristic 0.) Summing over a
gives
L
a,</>(a)-,foo
(ea(</>)- 1) =
L
a,</>(a)-,foo
orda </>'(z) = deg( -abz 2 d- 2 + · · ·) = 2d- 2.
(Topological Proof) It is a well-known topological fact that if a sphere is triangulated
with V vertices, E edges, and F faces, then V - E + F = 2. We take a (sufficiently
small) triangulation of the sphere with the property that the image of every ramification point of 4> is a vertex. Let
{vi: 1 :S: i :S: V},
{ei: 1::; i::; E},
{/i: 1::; i::; F},
be the vertices, edges, and faces of the triangulation. The map 4> is a local isomorphism except around the ramification points, so
14
1. An Introduction to Classical Dynamics
also gives a triangulation of the sphere. To ease notation, we let V', E', and F' denote
the numbers of vertices, edges, and faces in this inverse image triangulation.
The map ¢ is exactly d-to-1 except over the ramification points, so the set
consists of exactly dE edges, and similarly the set
consists of exactly dF faces. In other words, E' = dE and F' = dF.
The vertices are a little more complicated. If v is a vertex, then ¢ - l ( v) consists of d points counted with appropriate multiplicities, where the multiplicity
of a E ¢ - l ( v) is precisely ea (¢). In other words,
L
d=
ea(¢),
aE¢- 1 (v)
or equivalently,
L (ea(¢)- 1).
d- #¢- 1 (v) =
aE¢- 1 (v)
Summing over the vertices Vi yields
v
dV-
L #¢-
v
L L
1 (vi) =
(ea(¢) -1).
i=l
Note that the lefthand side is dV- V'. Further, since we chose the vi's to include the
image of every critical point, we know that ea (¢) = 1 for every point a not in any
ofthe ¢- 1 (vi)'s, so we can extend the sum on the righthand side to include every
a E JP1 (C). This proves that
dV = V' +
L (ea(¢) -1).
aEJ1> 1 (1C)
Next we take an alternating sum to obtain
d(V- E +F)= V'- E' + F' +
L (ea(¢) -1).
aEJ1> 1 (1C)
Finally, we use the fact that
V - E + F = V' - E' + F' = 2
to obtain the Riemann-Hurwitz formula.
D
Corollary 1.2. A rational map ofdegree d has exactly 2d- 2 critical points, counted
with appropriate multiplicities.
15
1.2. Critical Points and the Riemann-Hurwitz Formula
The following weaker version of the Riemann-Hurwitz formula is often convenient for applications.
Corollary 1.3. Let ¢ : JP>1 (C) ~ JP> 1 (C) be a rational map ofdegree d ~ 1.
(a) Let a E JP> 1 (C). Then
e13 (¢) =d.
L
!3E</J- 1 (a)
(b) [Weak Riemann-Hurwitz Formula]
2d- 2 =
L (d- #¢- (a)).
1
aEIP1 (IC)
Proof. Making a change of coordinates reduces us to the case that a i= oo and
oo ~ ¢- 1 (a). Writing ¢(z) = F(z)/G(z) and ¢- 1 (a) = {/31 , ... ,/3r }, this means
that there is a factorization
Notice that the exponents are exactly the ramification indices ei = ef3; ( ¢ ), so
r
L e13(¢) =Lei= deg(¢) =d.
(3E¢- 1 (a)
i=l
This proves (a). We then use the Riemann-Hurwitz formula (Theorem 1.1) to compute
2d- 2 =
z::
(e13(¢)- 1) =
,8EIP1
z:: z::
(e13(¢) -1)
aEIP' 1 (1C) {3E</J- 1 (a)
=
L (d- #¢-l(a)).
D
aEIP1 (IC)
Remark 1.4. The Riemann-Hurwitz formula is an example of a local-global formula. The quantity 2d- 2 is a global quantity, given in terms of how many times the
map¢ covers the Riemann sphere JP>1 (C) and reflecting the topology of the sphere.
The general version of the Riemann-Hurwitz formula, which we now state, includes
global information about the genera of the curves under consideration.
Theorem 1.5. (Riemann-Hurwitz Formula) Let C1 and C2 be algebraic curves (Riemann surfaces) ofgenus 91 and gz, respectively, and let ¢ : C1 -+ Cz be a finite map
ofdegree d ~ 1. Then
2gl- 2 = d(2gz- 2) +
L (ep(¢)- 1).
PEC1
Proof. See [198, IV §2].
D
16
1. An Introduction to Classical Dynamics
In this general formulation, the quantity 291 - 2- d(292 - 2) is the global part
of the Riemann-Hurwitz formula. It reflects the topology of the two curves, since
the genus of a curve is an intrinsic global quantity. Of course, for rational maps
¢ : JP>1 ---t JP> 1 , these genera are 91 = 92 = 0.
The ramification indices, on the other hand, are purely local quantities, since they
can be computed in a neighborhood of a point. Thus the Riemann-Hurwitz formula
computes a difference of global topological quantities in terms of quantities that
can be computed purely locally. This helps to explain both its importance and its
usefulness.
The next two results illustrate some of the power of the Riemann-Hurwitz formula.
Theorem 1.6. Let ¢ : IP'1 (C) ---t IP' 1 (C) be a rational map of degree d 2 2, and let
E c IP' 1 (C) be afinite set satisfying ¢- 1 (E) =E. Then #E:::; 2. Further:
(a) If #E = 1, then there is an f E PGL2(C) such that
E = {f(oo)} and ¢f(z) E C[z].
(b) Jf#E = 2, then there is an f E PGL2(C) such that
E = {J(O), f(oo)} and ¢! (z) = zd or z-d.
Proof The assumption that E is finite and satisfies ¢- 1 (E) = E combined with
the fact that ¢ : IP' 1 (C) ---t IP' 1 (C) is surjective implies that ¢ acts as a permutation on E. Since E is finite, some power of¢ acts as the identity on E, say
<Pn(a) = a for every a E E. Replacing ¢ by ¢n, :we find that every point
a E E satisfies ¢- 1 (a) = {a}. Thus every a E E is a totally ramified fixed point
of¢, i.e., ¢(a) = a and eo. ( ¢) = d. Applying the Riemann-Hurwitz formula (Theorem 1.1) yields
2d-2= L(e,a(¢)-1)
o.EE
o.EE
Since d 2 2, we conclude that 2 2 #E, which completes the proof of the first part
of the theorem.
Suppose next that #E = 1, say E = {P}. We conjugate¢ by any f E PGL2(C)
satisfying f- 1(P) = oo. Then (¢f)- 1 {oo} = {oo}, so the only pole of ¢f(z) is
z = oo. This proves that ¢! is a polynomial.
Finally, suppose that #E = 2, say E = { P, Q}. We conjugate ¢ by some
f E PGL 2(C) satisfying f- 1 (P) = oo and f- 1 (Q) = 0. There are then two cases,
depending on whether¢! fixes or permutes the set {0, oo }. Thus
(¢f)- 1 { oo} = {oo}
and
(¢f)- 1 {0} = {0}
(¢f)- 1 {oo} = {0}
and
(¢f)- 1 {0} = {oo}
==> ¢! (z) = czd,
==> ¢f(z) = cz-d.
A further conjugation by 9(z) = az for an appropriate value of a removes the c,
which completes the proof of Theorem 1.6.
D
1.2. Critical Points and the Riemann-Hurwitz Formula
17
Definition. A rational map ¢ : JP' 1 ~ JP' 1 is a polynomial map if it has a totally
ramified fixed point, that is, if there is a point a E JP'1 such that ¢(a)= a and
ea(¢) = deg(¢).
Notice that the polynomial cp(z) = ao + a1 z + · · · + adzd has oo as a totally
ramified fixed point. This is typical. If ¢ has a as a totally ramified fixed point, then
after a change of variables using any f E PGL2 (C) satisfying f( oo) =a, one finds
that¢! (z) is in C[z]. See Exercise 1.9.
Definition. A set satisfying ¢- 1 (E) = E = ¢(E) is called a completely invariant
set for¢. We note that it suffices to require ¢- 1 (E) = E, since then the surjectivity
of¢: JP' 1 (C) ~ JP' 1 (C) automatically implies that ¢(E)= E. The classical term for
a finite completely invariant set is an exceptional set. Thus Theorem 1.6 says that an
exceptional set for ¢ contains at most two elements. Further, if the exceptional set
contains two elements, then¢ is conjugate to z±d, and if the exceptional set contains
one element, then ¢ is conjugate to a polynomial.
Theorem 1.7. Let ¢ : lP' 1 (C) ~ JP' 1 (C) be a rational map of degree d ~ 2, and suppose that cpn is a polynomial map for some n ~ 1. Then already ¢ 2 is a polynomial
map. Further, if¢ itself is not a polynomial map, then ¢ is linearly conjugate to the
function 1/zd.
Proof Let a be a totally ramified fixed point of cpn, so (cpn) - 1 (a) = {a}. Consider
the chain of maps
{a} ~ {¢a} ~ {¢2 a} ~ · · · ~ {¢n- 1 a} ~ {a}.
The fact that (cpn) - 1 (a) consists of the single point a implies that each set {¢i (a)}
in the chain is the complete inverse image of the next set {¢i+ 1 (a)} in the chain. It
follows that the set of points
E = {a, ¢(a), ¢ 2 (a), ¢ 3 (a), ... , ¢n- 1 (a)}
satisfies ¢- 1 (E) = E. Now Theorem 1.6 tells us that #E :::; 2, so there are two
cases to consider.
First, if #E = 1, then a = ¢(a), and we see that ¢ is a polynomial map, since it
has a as a totally ramified fixed point. Second, if #E = 2, then ¢ 2 (a) = a =1- ¢(a),
so ¢ 2 is a polynomial map with a and ¢(a) as distinct totally ramified fixed points.
In the second case, we choose a fixed point (3 of¢ and perform a linear conjugation to move a, ¢(a), (3 to 0, oo, 1 respectively. Then writing cp(z) = P(z)/Q(z) as
a quotient of polynomials with no common roots, we have
¢- 1 (0) = {oo} = {z: P(z) = 0}
and ¢- 1 (oo) = {0} = {z: Q(z) = 0}.
Thus P(z) has no zeros (in C), so it is constant, while Q(z) vanishes only at 0, so
has the form Q(z) = czd. Thus cp(z) = 1/czd, and the fact that ¢(1) = 1 gives
cp(z) = 1/zd.
D
Remark 1.8. There are easy counterexamples to Theorem 1. 7 for fields of positive
characteristic, but it remains true for separable maps; see Exercise 1.11. There is also
a higher-dimensional version of Theorem 1. 7; see Exercise 7.16.
1. An Introduction to Classical Dynamics
18
1.3
Periodic Points and Multipliers
If o: is a fixed point of¢, then the multiplier of¢ at o: is the derivative
Aa ( ¢) = ¢' (0:).
(If o: = oo, this formula needs to be modified; see Exercise 1.13.) In general, the
value of a derivative depends on a choice of coordinates, since ¢' (z) is specifically
the derivative with respect to the variable z. However, it turns out that the derivative
at a fixed point is coordinate-invariant.
Proposition 1.9. Let¢ E q z) be a rational map and let o: be a fixed point of¢. Let
f E PGL2 (<C) be a change of coordinates and set (3 = f- 1 ( o: ), so (3 is a fixed point
of the conjugate map¢!= f- 1 o ¢of. Then
¢'(o:) = (¢f)'(f3).
Proof Two applications of the chain rule yield
(¢f)'(w) =
u- 0¢0 J)'(w) = u- )'(¢(f(w))). ¢'(f(w)). f'(w).
1
(1.4)
1
Hence
u- ¢(0:)) . ¢' (0:) . !' ((3) evaluating (1.4) at w = (3 = f- o:),
since ¢(o:) = o:,
= u- )'(o:). ¢'(o:). J'(/3)
= u- )'(!(/3)). ¢'(o:). !'((3) since 0: = J(/3),
(¢f)' (/3) =
1(
1 )' (
1
1
= ¢'(o:)
since (f- 1 )'(f(z)) · j'(z) = (f- 1 o f)'(z) = 1.
D
Thus the multiplier Aa (¢) of a fixed point o: of¢ is well-defined, independent of
the choice of coordinates on lP' 1 .
We next consider points that are fixed by some iterate of¢. Recall that a point o:
is called a periodic point for ¢ if ¢n (o:) = o: for some n ~ 1. The smallest such n is
called the exact period of o:. 3 (A point of exact period 1 is a fixed point.) We set
Pern(¢) = { 0: E 1P' 1 (<C): ¢n(o:) = 0: },
Per~*(¢)= {o: E Pern(¢): o:hasexactperiodn} .
(The reason for this notation is that there is a weaker notion of formal period n, and
we will later write Per~ (¢) to denote the set of points having formal period n. See
Exercise 1.19 and Section 4.1.) It is easy to see that Pern(¢) is the disjoint union of
Per;: (¢) over all m I n; see Exercise 1.14.
3 In the literature one finds many terms for the smallest value of n satisfying </Jn (a) = a, including
exact period, least period, primitive period, and prime period. We will use the first three interchangeably,
but we eschew the fourth, since in arithmetic dynamics, the phrase "¢ has prime period n at a" should
mean that the integer n is a prime number!
1.3. Periodic Points and Multipliers
19
Let a E Per~* (¢) be a point of exact period n for ¢. In particular, a is a fixed
point of ¢n. We define the multiplier of¢ at a to be
Notice that this may be calculated using the chain rule as
In other words, Ao (¢) = (¢n )' (a) is the product of the values of¢' at each of the
n distinct points in the orbit of a.
Remark 1.1 0. The set of multipliers
{A.o(¢) :a E Per(¢)}
depends only on the conjugacy class {¢1 : f E PGL 2 (C)} of¢. It can thus be used
to define useful dynamical invariants of the map ¢.
Remark 1.11. There is a more intrinsic way to define the multiplier of a fixed point a
in terms of the space of differential one-forms 0~ on 1?1 at a. This space has dimension 1, since 1?1 is nonsingular of dimension 1, so ¢* : n~ --t n~ is an element
ofGL(O~) = C*. This element ofC* is A.o(¢). More concretely, taking any nonzero
w E 0~, we define A.o(¢) by the equation ¢*w = A. 0 (¢)w. For example, if w = dz
for some uniformizer z at a, then ¢*(dz) = d(¢(z)) = cjJ'(a)dz in 0~, and we
recover the earlier definition Ao (¢) = ¢' (a).
If IA.o(¢)1 < 1, then a small neighborhood of a will shrink each time it returns
to a, while if IA.o (¢)I > 1, then it will expand. This observation prompts the following definitions.
Definition. Let a be a periodic point for a rational function ¢ E q z), and let Ao (¢)
be the corresponding multiplier. Then a is called
superattracting
attracting
neutral
repelling
if A. 0 ( ¢) = 0,
if IAo (¢) I < 1,
ifiA.o(¢)1 = 1,
if IAo (¢) I > 1.
(Neutral periodic points are also sometimes called indifferent.)
Remark 1.12. A periodic point is superattracting if its orbit contains a critical point,
so Corollary 1.2 implies that a map of degree d can have at most 2d - 2 superattracting cycles. We will state a much stronger result below (Theorem 1.35(a)).
Remark 1.13. A neutral periodic point is called rationally neutral if its multiplier
is a root of unity, that is, if Ao (¢) k = 1 for some k ~ 1. Otherwise, it is called
irrationally neutral.
We next prove a useful formula giving a relation between the multipliers of the
fixed points of a rational map. For an application of this formula to p-adic dynamics,
see Corollary 5.19 and Proposition 5.20.
20
1. An Introduction to Classical Dynamics
Theorem 1.14. Let K be an algebraically closed field and let ¢(z) E K(z) be a
rational function ofdegree d ~ 2. Assume that
for all P E Fix(¢).
Then
L
PEFix(t/>) 1 -
1
A (¢) = 1.
p
Remark 1.15. The condition that Ap(</J) -=/= 1 is equivalent to the condition that the
fixed point P has multiplicity 1, so Theorem 1.14 holds provided ¢ has d + 1 distinct
fixed points. If some fixed point has Ap( ¢) = 1, then a similar, but more complicated, formula involving a higher-order index is true; see Exercise 1.17 or [302, §12].
Remark 1.16. More generally, we can apply Theorem 1.14 to an iterate </Jn to obtain
a relation among the multipliers of the n-periodic points of ¢ provided that those
multipliers are not equal to 1.
Example 1.17. Let ¢(z) = z 2 +c. The fixed points of </J(z) are
a=
1 + v'f-=-4c
(3 =
2
1- vf1- 4c
2
'
and
oo,
with corresponding multipliers Aa(¢) = 2a, Af3(¢) = 2(3, and Aoo(¢) = 0. (Note
that the map ¢ is totally ramified at oo, so in particular oo is a critical point.) The
sum in Theorem 1.14 is
1
1
1
1
1
--+--+--=
+
+1=1.
1-2a 1-2(3 1-0
-v1-4c v1-4c
Proof of Theorem 1.14. We prove the theorem under the assumption that K has characteristic 0, in which case the Lefschetz principle (see, e.g., [410, VI §6]) says that
we can embed a sufficiently large subfield of K into C. It thus suffices to prove
the theorem for K = C. (The case of characteristic p can be proven by reduction
modulo p from the characteristic 0 case or by developing the theory of residues in
characteristic pas described in Exercise 5.10.)
To simplify the proof, we make a change of variables (i.e., conjugate ¢ by an
element ofPGL 2 (C)) so that ¢(oo) -=/= oo. For any rational function 'ljJ(z) E C(z),
or more generally for any meromorphic function, the Cauchy residue theorem says
that
a:s('ljJ(z) dz) = 0.
L
PEI1'1 (1C)
For P = a E C in the affine plane, the residue is given by the usual formula
~s('ljJ(z) dz) = -21
z-o
.j
1r2
lz-ol=€
'ljJ(z) dz
for sufficiently small E > 0.
For P = oo, we substitute z = 1/w and use the same formula to compute the residue
with respect to w at a = 0.
21
1.3. Periodic Points and Multipliers
We apply the Cauchy residue formula to the rational function
1
'1/J(z)= ¢(z)-z'
whose poles are the fixed points of¢. For any a E Fix(¢) (note that a =!= oo) we
observe that
¢(z)- z = {¢(a)+ Aa(¢)(z- a)+ O(z- a) 2 } - z
Taylor expansion,
since ¢(a) =a,
= {a+ Aa(¢)(z- a)+ O(z- a) 2 } - z
= (z- a){ A0 (¢)- 1 + O(z- a)}.
Hence under the assumption that A0 (¢) =/= 1, the function </>(zl-z has a simple pole
at z = a and its residue is given by4
Res (
z=a
dz
cp(z)- Z
) = _1 {
dz
27ri }lz-ol=• (z- a){ A0 (¢)- 1 + O(z- a)}
1
= -:-Aa-:-(¢.,..,..)---1 .
The differential <1>(;)-z has no other poles in C. In order to compute its residue at oo,
we substitute z = w- 1 and used( w- 1 ) = -w- 2 dw to compute
1
dz ) =es
R (
d(w- ) ) =es
R (
-dw
) .
R es (
z=oo ¢(z)- z
w=O ¢(w- 1 ) - w- 1
w=O w(w¢(w-1)- 1)
The assumption that ¢(oo) =/= oo implies that the function w¢(w- 1 ) vanishes
at w = 0, so this last differential has a simple pole at w = 0 and its residue is equal
to
Res (
w=O
-dw
) = lim w (
-1
) = 1.
w(w¢(w- 1 ) -1)
w->0
w(w¢(w-1) -1)
Substituting these residue values into the Cauchy residue formula yields
0=
dz ) = L Res ( dz ) +Res ( dz )
L Res (cp(z)z
cp(z)- z
z=oo cp(z)- z
PE11' 1 (C)
P
oEFix(cf>)
=
L
oEFix(<f>)
z=a
1
A (¢)
0
-
1 + 1.
Finally, we move the sum to the other side, changing A - 1 to 1 - A, to complete the
proofofTheorem 1.14.
D
4 Ingeneral, if'!j>(z) has at most a simple pole at a, then Resa ('!j>(z) dz) equals limz-a (z -
a)'!j>(z).
22
1. An Introduction to Classical Dynamics
1.4 The Julia Set and the Fatou Set
The notion of equicontinuity is central to dynamics. We begin by recalling the classical definition of continuity. A function ¢ : (81, P1) - (82, P2) between metric
spaces is continuous at a E 8 1 if for every ~: > 0 there exists a 8 > 0 such that
P1(a,{3) < 8
P2(¢a,¢{3) < ~:.
::;::::}
We extend this notion to a collection of functions <P from 8 1 to 8 2 by requiring that
a single 8 work for every function in <P. This means that if {3 is chosen sufficiently
close to a, then ¢(a) and ¢({3) will be close to one another for every map ¢ in <P.
Definition. Let (81 , pi) and (82, P2) be metric spaces, and let <P be a collection of
maps from 8 1 to 8 2. The collection <Pis said to be equicontinuous at a point a E 8 1
if for every E > 0 there exists a 8 > 0 such that
P1(a,{3) < 8
::;::::}
P2(¢a,¢{3) < E for every¢ E <P.
The collection <P is equicontinuous on a subset U c 8 1 if it is equicontinuous at
every point of U.
For an individual map ¢ : 8 - 8 from a set 8 to itself, we say that ¢ is equicontinuous at a if the collection of iterates {¢n : n 2': 1} is equicontinuous at a.
Definition. Let ¢ be a map from a metric space to itself. The Fatou set of¢, denoted
by :F( ¢ ), is the maximal open set on which ¢ is equicontinuous. The Julia set of¢,
denoted by .:J(¢> ), is the complement of the Fatou set. Note that equicontinuity is
not, in general, an open condition (see Exercise 1.22), but that :F( ¢>) is an open set
by definition.
Informally, one might say that points in the Julia set .:J (¢) tend to wander away
from one another as ¢> is iterated, so ¢ behaves chaotically on its Julia set. Nearby
points in the Fatou set :F( ¢ ), on the other hand, tend to stay together, so the behavior of¢ on :F( ¢) is considerably easier to analyze. Much of complex dynamics
is devoted to studying the Fatou and Julia sets of rational maps. For further reading, see the textbooks cited at the beginning of this chapter and the two survey articles [74, 235).
Example 1.18. Let¢( z) E q z) be a rational map. If a is an attracting periodic point
of¢, then points close to a are all attracted to the points in the orbit of a, from which
it is easy to conclude that they are in the Fatou set. Similarly, it is easy to check that
repelling periodic points are in the Julia set. See Exercise 1.27. The behavior near
the neutral points is more complicated.
Example l.I9. Consider the map¢ : IP1(C) - IP1(C) given by ¢(z) = zd. Then
c/>n(z) = zdn, SO
lim ¢n(a) =
n--+oo
{0 iflal
< 1,
if Ia I >
00
1.
1.4. The Julia Set and the Fatou Set
23
It is easy to see that if Ia I =f. 1, then a E F (¢). In particular, the superattracting fixed
points 0 and oo are both in F(¢), and every point a with Ia I =f. 1 is attracted to one
of these fixed points.
However, if lal = 1, then l¢nal = 1 for all n 2: 1, but there are points f3
arbitrarily close to a satisfying q;n f3 --+ 0 and other points arbitrarily close to a
satisfying q;n f3 --+ oo. Therefore .J( ¢) is equal to the unit circle {a E C : Ia I = 1}.
We also observe that, aside from 0 and oo, the periodic points of¢ are the ( dn - 1)throots of unity
Per(¢)= {e21rik/W-l): k E Z,n 2: 1}.
If a has exact period n, then its multiplier is
so these periodic points are all repelling. Notice that Per(¢) is dense in .1(¢), and
indeed .J (¢) is the closure of the repelling periodic points of¢.
Example 1.20. The Julia set of the polynomial ¢( z) = z 2 - 2 consists of the closed
interval on the real axis between -2 and 2. See Exercise 1.28.
Example 1.21. The preceding two examples are actually quite misleading. We will
see in Section 1.6 what makes them so special. A more typical example of dynamical
behavior is exhibited by the polynomial ¢( z) = z 2 - 1. Its fractal-like Julia set
is a connected set, while the Fatou set consists of an infinite number of connected
components.
Another kind of behavior is seen for the polynomial ¢(z) = z2 + ~·The Julia
set for this map is totally disconnected, although it also has the property that every
point in .J( ¢) is the limit of a sequence of distinct points in .J(¢ ), i.e., the Julia set
is perfect.
Remark 1.22. It turns out that if the Julia set of a polynomial ¢ is neither a line
segment nor a circle, then ¢ is more-or-less determined by its Julia set. More precisely, let J C C be the Julia set of some polynomial. Then there is a polynomial ¢J(z) E C[z] associated to J with the following property: If ¢(z) E C[z]
satisfies .1(¢) = J, then¢ = f o ¢} for some n 2: 1 and some rotation of the
plane f with f(J) = J. See [394].
Remark 1.23. The above examples show that Julia and Fatou sets can be extremely
complicated, even for quadratic polynomials. The situation for polynomials of higher
degree becomes increasingly complex, and rational functions lead to additional difficulties. Finally, although possibly of less interest from an arithmetic viewpoint, the
dynamics of holomorphic or meromorphic functions C --+ C having an essential
singularity at oo (e.g., ¢(z) = ez) exhibit many further interesting dynamical properties.
We begin our description of the Fatou and Julia sets by showing that each is
invariant under both ¢ and ¢ - l .
Proposition 1.24. Let ¢ : lP' 1 (C) --+ lP' 1 (C) be a rational map of degree d 2: 2, and
let F and .J be the Fatou and Julia sets of¢, respectively.
1. An Introduction to Classical Dynamics
24
(a) The Fatou set F is completely invariant, i.e., ¢- 1 (F) = F =¢(F).
(b) The Julia set .J is completely invariant.
(c) The boundary .J of the Julia set is completely invariant.
a
ProofSketch. (a) Since¢ is surjective, it suffices to prove that ¢- 1 (F) =F. Suppose first that a E F, and let ¢((3) = a. For any point (3' that is close to (3, the
Lipschitz property (1.3) of¢ says that
p( a, ¢((3')) ~ C( ¢ )p((3, (3').
In particular, ¢((3') can be made arbitrarily close to a by taking (3' sufficiently close
to (3. Since a E F, we know that ¢n(a) stays close to i/Jn(¢(3'), and hence that
¢n+ 1 ((3) stays close to ¢n+l ((3'). Therefore (3 E F, which proves that ¢- 1 (F) c F.
Next suppose that a E F. We need to check that ¢(a) E F. If U is a small
neighborhood of a, then the open mapping property of¢ implies that ¢(U) is a
(small) open neighborhood of ¢(a). In particular, if (3 is sufficiently close to ¢(a),
then (3 will lie in ¢(U), say (3 = ¢((3') with (3' E U. Then the assumption that
a E F implies that the iterates ¢n (a) and ¢n ((3') remain close to one another, and
hence the same is true ofthe iterates ¢n- 1 (¢a) and 1/Jn- 1 ((3). Therefore ¢(a) E F,
which completes the proofofthe other inclusion F c ¢- 1 (F). (You should fill in
the details of this proof sketch and give a rigorous (8, E) proof. See Exercise 1.23.)
(b) The Julia set is the complement of the Fatou set, so the complete invariance
ofF implies the complete invariance of .J.
(c) Let .]0 be the interior of .J. The rational map¢ is continuous, so ¢- 1 (.1°)
is open, and the complete invariance of .J shows that it is contained in .J; hence
¢ - l ( .Jo) c .]0 • For the other inclusion, we use the fact that ¢ is an open map,
so ¢(.1°) is open. Again the complete invariance of .J shows that it is contained
in .J, and the fact that it is open shows that it is contained in .]0 • Hence
This proves that ¢- 1 ( . ]0 ) = .]0 , so the interior of .J is completely invariant. Finally,
the fact that .J and .]0 are completely invariant implies that the set difference 8.1 =
.J -. . _ .]0 is also completely invariant.
0
Proposition 1.25. For every integer n ;:::: 1,
and
Proof Let 1/J = 1/Jn. It is clear from the definition that F (¢) c F (1/J), since if we
know that iteration of¢ maintains closeness of points, then the same is certainly true
for ¢n.
To prove the opposite inclusion, for each 0 ~ i < n we consider the collection
of maps
<I>i = {1/Ji'l/Jk: k;:::: 0}.
For any fixed i, the map ¢i satisfies a Lipschitz inequality (1.3), so the Fatou
set F(<I>i) of<I>i contains the Fatou set F('ljJ) of'ljJ. Hence
25
1.4. The Julia Set and the Fatou Set
F(<I>o) nF(<I>l) n · · · nF(<I>n-d :J F('lj;).
But the intersection is clearly equal to F( ¢; ), since
{¢;1 : j ~ 0} =
u{
¢;i'l/Jk : k ~ 0}.
o::;i<n
This completes the proof that F( ¢;n) = F( ¢;). The corresponding fact for the Julia
set is then clear.
D
Remark 1.26. The idea of writing { ¢1 : j ~ 0} as the union
{¢;l:j~O}=
u
{¢i(¢n)k:k~O}
o::;i<n
will reappear later in the proof of Theorem 3.48 when we need to spread out the
ramification of a map.
The notions of equicontinuity and normality are closely related.
Definition. Let (81, PI) and (8 2, P2) be metric spaces, and let <!> be a collection of
maps from 81 to 82. The collection<!> is said to be normal, or a norma/family, on 8 1
if every infinite sequence of functions from <!> contains a subsequence that converges
locally uniformly on 8 1·
The following two important theorems from complex analysis are used extensively in the study of dynamical systems on IP' 1 ( q.
Theorem 1.27. (Arzela-Ascoli Theorem) Let U be an open subset of!P' 1 (C), and
let <!> be a collection of continuous maps U -+ IP' 1 (C). Then <!> is equicontinuous
on U if and only if it is a normal family on U.
Theorem 1.28. (Montel's Theorem) Let U be an open subset ofiP' 1 (C), and let<!>
be a collection of analytic maps U -+ IP' 1 (C). If U¢E<l> ¢(U) omits three or more
points ofiP' 1 (C), then<!> is a norma/family on U.
The next theorem summarizes a number of important properties of the Julia set.
Theorem 1.29. Let¢;( z) E q z) be a rational function of degree d ~ 2.
(a) The Julia set J (¢;) is nonempty.
(b) Leta E .:1(¢). Thenthebackwardorbit0¢(a) = {¢-n(a): n ~ l}ofais
dense in :1(¢;).
(c) Let U be a union of connected components of the Fatou set F( ¢;) that is completely invariant, i.e., ¢- 1 (U) = U = ¢;(U). Then J(¢;)
(d) The Julia set J( ¢;)contains no isolated points, i.e., J( ¢;) is a perfect set.
=au.
Proof (a) See [43, Theorem 4.2.1] or [95, Theorem Ill.1.2].
(b) See [43, Theorem 4.2.7] or [95, Theorem III.1.6].
(c) See [95, Theorem III. I. 7].
(d) See [43, Theorem 4.2.4] or [95, Theorem III.1.8].
D
1. An Introduction to Classical Dynamics
26
Theorem 1.29(a) says that the Julia set is always nonempty. The Fatou set, on
the other hand, may be empty. We state some conditions that characterize this phenomenon.
Theorem 1.30. Let ¢( z) E IC( z) be a rational function of degree d ;::: 2. Then the
following are equivalent:
(a) The Julia set .J( ¢) is equal to JJ!' 1(C).
(b) The Julia set .J (¢) has a nonempty interior.
(c) There exists a point a E JJ!' 1(C) whose (forward) orbit Oq,(a) is dense in JJ!' 1(C).
Further, if every critical point of¢ is strictly preperiodic, i.e., preperiodic, but not
periodic, then .J(¢) = JJ!' 1(C).
Proof See [43, Theorems 4.2.3, 4.3.2, 9.4.4] or [95, Theorems 111.1.9, V.1.2].
0
Example 1.31. Consider the map ¢( z) = 1 - 2 j z 2 . Its critical points are 0 and oo.
(See Exercise 1.25.) Applying¢ repeatedly to 0 yields
<P
<P
<P
<P
<P
0 -------+ 00 -------+ 1 -------+ -1 -------+ -1 -------+ . . . '
so the critical points are strictly preperiodic. It follows from Theorem 1.30 that
.J(¢) = JJ!'l(C).
Remark 1.32. Theorems 1.29 and 1.30 say that a rational map¢: JJ!' 1(C) ---+ JJ!' 1(C)
may have .J(¢) = JJ!' 1(C), but that it is not possible to have F(¢) = JJ!' 1(C). The
situation becomes exactly reversed ifC is replaced by an algebraically closed field K
having a nonarchimedean absolute value. For example, if K = Cp (the completion
of the algebraic closure ofQlp), then the Fatou set of¢: JJ!' 1(Cp) ---+ JJ!' 1(Cp) is always
nonempty, but it is possible, and indeed quite easy, to find maps with empty Julia set
and F(¢) = JJ!' 1(Cp)· We will discuss this in more detail when we study dynamics
over complete local fields in Chapters 2 and 5.
Theorem 1.33. Let ¢( z) E q z] be a polynomial ofdegree d ;::: 2.
(a) The Julia set .J( ¢) is connected if and only iffor every critical point a =J. oo,
the orbit 0 q, (a) is bounded in C.
oo, then the Julia
(b) If every critical point a of ¢ satisfies limn-->oo ¢n (a)
set .J (¢) is totally disconnected.
Proof See [95, Theorems III.4.1, III.4.2].
0
Remark 1.34. A quadratic polynomial has only one finite critical point. Thus Theorem 1.33 covers all cases, so we can divide the set of quadratic polynomials into
two classes according to the behavior of their finite critical point. Every quadratic
polynomial is linearly conjugate to a unique polynomial of the form ¢c ( z) = z 2 + c,
so we define a portion of the c-plane by
M = {c E C: ¢n(o) is bounded for n;::: 1}
= {c E C: .J(¢c) is connected}.
This set M is the famous Mandelbrot set, illustrated in Figure 1.2. It is a dynamically
determined subset of the moduli space of quadratic polynomial maps.
1.5. Properties of Periodic Points
27
Figure 1.2: The Mandelbrot set M.
1.5
Properties of Periodic Points
The dynamics of a rational map is influenced not only by the behavior of its critical
points, but also by the behavior of its periodic points. The next result describes some
of the properties of the periodic points and periodic cycles of a rational map.
Theorem 1.35. Let cp( z) E q z) be a rational function of degree d ;:: 2.
(a) The map cp has at most 2d- 2 nonrepelling periodic cycles in JP> 1 (C). If cp is a
polynomial map, then it has at most d - 1 nonrepelling periodic cycles in C.
(b) The Julia set J (cp) is equal to the closure of the repelling periodic points of cp.
(c) Let U C IP' 1 (C) be an open set such that J (cp) n U =f 0. Then there is an integer
n;:: 1 such that cpn(u n J(¢)) = J(¢).
Proof (a) The sharp bound of2d- 2 for rational maps is due to Shishikura, see [43,
Theorem 9.6]. Much earlier, Fatou gave a weaker bound that is sufficient for many
applications, see [95, Theorem III.2.7]. The bound for polynomial maps is due to
Douady, see [95, Theorem VI.1.2].
(b) This important result is due independently to Julia and Fatou. See [43, Theorem 6.9.2] or [95, Theorem III.3.1].
(c) See [95, Theorem 111.3.2].
D
The Fatou set :F( <P) is an open subset of!P' 1 (C), so it consists of one or more connected components. It is known that the number of components is equal to 0, 1, 2,
or oo; see [95, Theorem IV.1.2]. If U is a connected component of :F(cp), then
the open mapping property of <P implies that ¢(U) is also a connected component
of :F(cp). In this way we obtain a map
1. An Introduction to Classical Dynamics
28
Components(F( ¢)) -----> Components(F(¢)),
U r-----> ¢( U),
and we say that U is a periodic domain if ¢n (U) = U for some n 2: 1, a preperiodic
domain if some iterate ¢n(U) is periodic, and a wandering domain otherwise. The
following important result answers a long-standing question ofFatou and Julia.
Theorem 1.36. (Sullivan's No Wandering Domains Theorem) A rational map ¢ E
C( z) has no wandering domains.
Proof See [43, Chapter 8], [95, Theorem IV.l.3] or [426].
0
The possibility of wandering domains having been eliminated, the periodic components ofF(¢) are classified into several types. We begin with the necessary definitions, then state the classification theorem.
Definition. Let U be a connected component of the Fatou set F(¢), and assume
that U is forward invariant, i.e., ¢(U) = U.
• U is parabolic if the boundary of U contains a rationally neutral periodic
point (such that limn->oo ¢n(a) = (for every a E U.
• U is a Siegel disk if there is an analytic isomorphism f from the unit disk
{w E C : lwl < 1} to U such that¢! is a rotation of the unit disk.
• U is a Herman ring if there is an analytic isomorphism f from some annulus
{w E C : a < Iw I < b} to U such that ¢! is a rotation of the annulus.
Here a rotation is simply a map of the form w ~---+ ei!iw.
If U is a periodic connected component of period n of :F( ¢), then U is forwardinvariant for ¢n and we say that U is parabolic, a Siegel disk, or a Herman ring if it
is of the appropriate type for ¢n.
Theorem 1.37. Let U be a periodic connected component of the Fatou set of a
rational map¢ E C(z). Then U fits into exactly one of the following categories:
(a) U contains an attracting periodic point.
(b) U is parabolic.
(c) U is a Siegel disk.
(d) U is a Herman ring.
Proof See [43, Theorem 7 .I] or [95, Theorem IV.2.1]. We mention that it is nontrivial to prove that Siegel disks and Herman rings exist. In particular, the Fatou set of a
0
polynomial map cannot contain a Herman ring.
1.6
Dynamical Systems Associated to
Endomorphisms of Algebraic Groups
In this section we study certain rational maps whose dynamical properties are comparatively easy to analyze due to the existence of an underlying group structure.
1.6. Dynamical Systems Associated to Algebraic Groups
29
These maps thus provide a class of examples on which to make and test conjectures,
albeit with the caution that they are rather special and in some ways atypical. Additional material on this interesting collection of maps may be found in Chapter 6.
1.6.1
The Multiplicative Group
The dynamics of the polynomial ¢( z) = zd is extremely easy to analyze, since
we have the explicit formula ¢n (z) = zdn. However, there is a more intrinsic reason that zd is such a nice map; namely, it is an endomorphism of the multiplicative
group C* of nonzero complex numbers. In other words, the map
C* -----+ C*,
is a homomorphism. It is the existence of the underlying group C* that makes zd
relatively easy to analyze.
Remark 1.38. In fancier language, the map ¢(z) = zd is an endomorphism of the
algebraic group Gm, where for any field K, the set of points of Gm is the multiplicative group Gm(K) = K*. The full endomorphism ring of Gm is Z via the
identification
In even fancier language, Gm is a group scheme
Gm = SpecZ[X, YJI(XY -1)
called the multiplicative group scheme. It is characterized by the property that for
every ring R, the group of R-valued points ofGm is the unit group Gm(R) ~ R*.
1.6.2 Chebyshev Polynomials
The multiplicative group C* has a nontrivial automorphism given by the reciprocal
map z ~--+ z- 1 . We can take the quotient ofC* by this automorphism to obtain a new
space that turns out to be isomorphic to C via the map
[z]~---> z + z- 1 .
The inverse isomorphism takes w E C to the equivalence class of either of the roots
of z 2 - zw + 1 = 0.
This inversion automorphism commutes with the £ilh-power map, so when we
take the quotient, the £ilh-power map will descend to give a map on the quotient
space. In other words, there is a unique map Td that makes the following diagram
commute:
1. An Introduction to Classical Dynamics
30
C*
C*
1
1
C*
C*
The map Td is characterized by the equation
Td(z + z- 1 ) = zd + z-d
forallz E C*.
(1.5)
It is not hard to see that Td is a polynomial; indeed, it is a monic polynomial of de-
gree d with integer coefficients called the f1!h Chebyshev polynomial. Writing z = eit
with t E C, we see that the Chebyshev polynomial satisfies
Td(2 cost) = 2 cos(dt).
Using these formulas, it is not hard to prove that the Julia set of Td is given
by .J(Td) = [-2, 2], the closed real interval from -2 to 2, and to derive many further properties of the Chebyshev polynomials. See Exercises 1.30 and 1.31, as well
as Section 6.2.
Remark 1.39. Classically the Chebyshev polynomials are normalized to satisfy the
identity Td(cost) = cos(dt), in which case the Julia set becomes .J(Td) = [-1, 1].
The two normalizations are related by the simple formula Td(w) = ~Td(2w).
1.6.3
Rational Maps Arising from Elliptic Curves
There are three different types of (connected) algebraic groups of dimension one.
First is the additive group Ga(C) = c+, whose endomorphisms are the maps
z-+ dz. The dynamical systems associated to endomorphisms of the additive group
are relatively uninteresting. Second is the multiplicative group Gm (C) = C*, whose
endomorphism& z 1-t zd have interesting, but relatively easy to analyze, dynamical
properties. The same is true of the Chebyshev polynomials, which come from the
restriction of z 1-t zd to a quotient ofGm.
The third type of one-dimensional algebraic groups consists of a family of groups
called elliptic curves. We do not have space to go into the theory very deeply, so we
are content to make a few brief remarks here and later expand on this material in
Chapter 6. For further material on the analytic, algebraic, and arithmetic theory of
elliptic curves, see for example [1, 198, 254, 410,420, 412]. We note that the reader
may omit this section on first reading, since the material is not used other than as a
source of examples until Chapter 6.
An elliptic curve E(C) is the set of solutions (x, y) to an equation of the form
y 2 = x 3 +ax+b
1.6. Dynamical Systems Associated to Algebraic Groups
Addition of distinct points
31
Adding a point to itself
Figure 1.3: The addition law on an elliptic curve.
together with an extra point 0. We also require that .6. = 4a 3 + 27b2 =/:- 0, which
means that the cubic polynomial has distinct roots and ensures that the curve E(C)
is nonsingular.
Alternatively, E(C) is the curve in the projective plane IP'2 (C) defined by the
homogeneous equation
with the extra point being 0 = [0, 1, 0]. There is a natural automorphism on E(C)
given by [-1](X, Y, Z) = (X, -Y, Z). If P and Q are two points on E(C), then
the line through P and Q will intersect E (C) at a third point R, and we define an
addition law by setting P EB Q = [-1] R. (If P = Q, we take the "line through P
and Q" to be the tangent line to E(C) at P.) These operations satisfy
PEBO = P,
PEB[-1](P) = 0,
PEBQ = QEBP,
(PEBQ)EBR = PEB(QEBR),
so they make E(C) into an abelian group. (The first three equalities are obvious,
while the associativity is somewhat difficult to prove.)
An elementary calculation shows that the coordinates of P EB Q are given by rational functions of the coordinates of P and Q. In particular, if a and b are in some
field K, and if P and Q have coordinates inK, then P EB Q will also have coordinates inK. Thus E(K) = E(C) n IP'2 (K) will be a subgroup of E(C). Figure 1.3
illustrates the group law for points in E(IR).
Repeated addition in any abelian group gives an endomorphism of the group, so
for each d ;:::: 1 we obtain a map
[d] : E(C) ____. E(C),
d tenns
called the multiplication-by-d map. Setting
[O](P) = 0
and
we obtain an (injective) homomorphism
[-d](P) = [-1]([d]P),
32
1. An Introduction to Classical Dynamics
Z ______. End(E(C)),
d f----* [d].
If this map is not onto, then E is said to have complex multiplication, or CM for
short. For example, the curve E : y 2 = x 3 + x has complex multiplication, since it
has extra endomorphisms such as [i] : (x, y) f-+ ( -x, iy). Most curves do not have
CM.
The multiplication map [d] clearly commutes with the involution [-1], so it descends to a map on the quotient space E(C)/ { ±1 }. The quotient space is very easy
to describe,
(x, y) f----* x,
E(C)/{±1} ~ Jlll1(C),
so the multiplication-by-d map descends to give a rational map ¢E,d making both
squares in the following diagram commute:
E(C)
1
E(C)
{±1}
(xt1
~ E(C)
1
E(C)
-----+
{±1}
[dJ
1(xt
Jllll(q ~Jilll(q
The map ¢E,d is an example of a Lattes map. Thus ¢E,d is a rational function characterized by the formula
¢E,d(x(P)) = x([d](P))
for all P E E(C).
(1.6)
Notice the close similarity to the defining property (1.5) of the Chebyshev polynomials. More generally, it turns out that every endomorphism u E End( E) commutes
with [-1], even if E has complex multiplication, so every endomorphism u determines a rational function ¢E,u on Jlll1(C).
Example 1.40. Let E : y 2 = x 3 + ax + b be an elliptic curve as above. Then the
duplication map [2] : E(C) ----> E(C) leads to the rational function
¢E, 2(x) =
x 4 - 2ax 2 - 8bx + a 2
·
4x3 + 4ax + 4b
Example 1.41. Let E : y 2 = x 3 + x be the CM elliptic curve mentioned earlier.
Then the map [1 + i] : E(C) ----> E(C) defined by [1 + i](P) = P EB [i](P) leads to
the rational function
</JE,Hi(x) =
;i (X+~) ·
Proposition 1.42. Let E : y2 = x 3 + ax + b be an elliptic curve, let d ~ 2 be an
integer, and let ¢ E ,d : Jlll 1(q ----> Jlll1(q be the associated rational map as above.
Then
33
1.6. Dynamical Systems Associated to Algebraic Groups
•
•
•
•
•
L
•
•
w1 +w2
•
•
•
•
Figure 1.4: A lattice and fundamental domain associated to an elliptic curve.
PrePer(¢E,d) = x(E(C)tors),
where we recall that the torsion subgroup E(C) 1ors of E(C) is the set of elements of
finite order (cf Example 0.2).
Proof We leave the proof as an exercise; see Exercise 1.32.
D
The dynamics of the rational function cPE,d on JP> 1 (C) can be analyzed using the
group law on the larger space E(C). The utility of this approach is further enhanced
by the analytic uniformization of E(C), which we now briefly describe.
A lattice Lin Cis a set of all /£-linear combinations of two JR-linearly independent complex numbers w1 and w2 . Thus
The quotient space C / L is a torus obtained by identifying the opposite sides of the
fundamental domain
A lattice Land a fundamental domain are illustrated in Figure 1.4. Notice that C/ L
has a natural group structure induced by addition on C.
Let E(C) be an elliptic curve. Then one can prove that there exists a lattice L
and a complex analytic isomorphism
7/JE : C/ L ----+ E(C).
Further, the isomorphism 7/J E respects the group structure,
1. An Introduction to Classical Dynamics
34
In particular, if we set P = '1/JE(z) in formula (1.6) and use the relation
[d]('I/JE(z)) = '1/JE(dz),
then we obtain the useful formula
¢E,d(x('I/JE(z))) = x([d]('I/JE(z))) = x('I/JE(dz)).
To ease notation, we will let r(z) = x('I/JE(z)). The function r(z) is a slight
variant of the classical Weierstrass r-function. It is meromorphic on C with double
poles at the points of the lattice L and holomorphic elsewhere. Then our relation
becomes
(1.7)
¢E,d(r(z)) = r(dz),
and iteration is given by the simple formula ¢'E d(r(z)) = r(dnz).
The upshot of this discussion is that we ob~in a commutative diagram
that is of great assistance in studying the dynamics of the rational function ¢E,d·
For example, the map z ~--+ dz is clearly d2 -to-1 on C/ L, so we see that ¢E,d has
degree d 2 . Further, a point x = r( z) will be fixed by¢E,d if and only if z satisfies the
congruence dz = ±z (mod L), so the fixed points of <PE,d are precisely the points
(Note that r( -z) = r(z), so there are actually only d2 + 1 fixed points.)
Let r(() be a fixed point of ¢E,d with ( ~ ~L, which is equivalent to the assumption that r'(() ::/:- 0 or oo. (This excludes at most four such points modulo L.)
Differentiating ( 1.7) and evaluating at z = ( yields
¢£,d(r(())r'(() = dr'(d().
(1.8)
But we know that r( z) is a meromorphic function on C / L, or equivalently, it is a
meromorphic function on C with the periodicity property
r(z + w) = r(z)
forallw E L.
Differentiation gives the same relation p' ( z + w) = p 1 ( z) for the derivative. (Notice
the analogy with the function e2.,..iz, which is holomorphic on C and invariant under
the translations z ~--+ z + n for n E Z.) We evaluate at z = d(, use the fact that
d( ±( (mod L) to get r'(d() = r'(±() = ±r'((), and substitute into (1.8) to
obtain the relation
=
35
Exercises
¢'e,d(p(())p'(() = dp'(d() = ±dp'(().
Finally, canceling p' (() yields ¢'e d(p( ()) = ±d. In particular, (almost) every fixed
point of ¢E,d is expanding and th~s is in the Julia set.
But much more is true. The relation (1.7) shows immediately that
so ¢e,d = ¢E,dn. Thus the periodic points of ¢E,d of period dividing n are precisely
the fixed points of ¢E,dn, and their multipliers are (almost) all equal to ±dn. This
proves that (almost) every periodic point of ¢E,d is expanding, hence in the Julia set.
Finally, we observe that the set of points inC/ L satisfying
for some n = 1, 2, 3, ...
is dense inC/ L, so their image using the double cover p: C/ L-> IP' 1 (C) is dense
in IP' 1 (C). We have thus shown that .J(¢E,d) = IP' 1 (C). This proves the following
theorem, which was published by Lattes in 1918 and provided the first known examples of rational maps with empty Fatou set. (See also [300, §6] for a discussion
of earlier work by SchrOder and Bottcher on elliptic functions and their associated
rational maps, as well as an 1815 paper ofBabbage in which he uses semiconjugacy
to study the periodic points of certain maps.)
Theorem 1.43. (Lattes [260]) Let E be an elliptic curve, let d 2: 2 be an integer,
and let ¢ E ,d : IP' 1 (C) -> IP' 1 (C) be the rational map of degree d 2 characterized
by (1.6). Then
and
F(¢E,d) = 0.
Exercises
Section 1.1. Rational Maps and the Projective Line
1.1. Define a mapping from the xy-plane (which we identify with q to the unit sphere 5 2
in R 3 by mapping z E C to the point z* E 5 2 such that the line through z and z* goes through
the point (0, 0, 1). Notice that z* -> (0, 0, 1) as z -> oo, so if we set oo* = (0, 0, 1), we
obtain a bijection
.
z~z,
as illustrated in Figure 1.1. Prove that with this identification, the chordal metric on lP' 1 ( q is
given by p(z, w) = ~ lz* - w* 11.2. (a) Prove that the inversion map z >--> z- 1 is an isometry oflP' 1 (C) for the chordal metric
(i) by a direct calculation using the formula for p; and (ii) by using the identification
lP' 1 ( q e:! 5 2 described in Exercise 1.1.
(b) Let a, b E C satisfy lal 2 + lbl 2 = 1, and let f(z) = (az- b)/(bz +a). Prove that
p(f(z),f(w)) = p(z,w) for all z,w E lP'1 (C). (Hint. Although this can be proven
directly, an alternative method is to show that f corresponds to a rigid rotation of 5 2 ,
hence preserves chordal distances. For a nonarchimedean analogue, see Lemma 2.5.)
36
Exercises
1.3. Let¢ : IP' 1(IC) ---+ IP'1(IC) be a rational map.
(a) Prove that there is a constant C( ¢) such that¢ satisfies a Lipschitz inequality
p(¢(a), ¢({3)) ::; C( ¢ )p( a, {3)
for all a, {3 E IP' 1(IC).
(b) Find an explicit formula for the Lipschitz constant C(f) in the case of a linear map
f(z) = (az + b)/(cz +d) E PGL2(1C).
1.4. Let (a, a', a") and ({3, {3', {3") be triples of distinct points in IP'1. Prove that there exists
a unique automorphism f E PGL2(1C) satisfying
f(a) = {3,
f(a') = {3',
f(a") = {3".
(Hint. First do the case that one of the triples is (0, 1, oo ).)
Section 1.2. Critical Points and the Riemann-Hurwitz Formula
1.5. Let ¢(z) E C(z) be a rational function. In the text we defined the ramification properties
of¢ at a point a E IP'1(1C) provided that a =I= oo and ¢(a) =I= oo. In general, let f E PGL 2 (1C)
1 o ¢ o f, and let {3 =
1(a). If {3 =/= oo
be any linear fractional transformation, let ¢ 1 =
and ¢1 ({3) =I= oo, we say that¢ is ramified at a if(¢1)' ({3) = 0 and we define the ramification
index of¢ at a to be
ea(¢) = e13(¢1 ).
r
r
(a) Assuming that none of a, {3, ¢(a), ¢1 ({3) are equal to oo, prove that
(¢/)'({3) = u-1)'(¢(a)) ¢'(a).
(f-1 )'(a)
Conclude that the ramification points of a rational map ¢ : IP' 1 ---+ IP'1 are independent of
the choice of coordinates, i.e., the vanishing of¢' (a) is independent of the choice of f.
(b) Show that the ramification index ea(¢) is well-defined, independent of the choice of f.
(c) Under what conditions is it true that the derivative ¢' (a) is independent of the choice
off?
1.6. Let ¢(z) E C(z) be a rational function of degree d, say given by
¢(z) = F(z) = a0 + a1z + ... + adzd.
G(z)
b0 + b1z + · · · + bdzd
(a) Prove that¢ is ramified at z = 0 if and only if aob1 = a1b0.
(b) Prove that ¢ is ramified at z = oo if and only if adbd-1 = ad-1 bd.
(c) Findasimilarcriterionfore0 (¢) ~ 3. Samequestionfore 00 (¢) ~ 3.
1.7. Let¢ : IP' 1 ---+ IP' 1 be a rational function of degree d ~ 2. Using the standard spherical
measure on IP' 1(IC), normalized so total area is 1, prove that
1
i(¢n)'(z)ldJL(z),__,dn
asn-+oo.
JI>l(C)
Conclude that "on average," the map ¢ is expanding.
1.8. (a) Let C be a compact Riemann surface with g holes. Prove that if C is triangulated
using V vertices, E edges, and F faces, then V- E + F = 2- 2g. The number g is
called the genus of C.
Exercises
37
(b) Let ¢ : C1 ---+ C2 be a finite map of degree d of compact Riemann surfaces, and let g;
be the genus of C;. Give a topological proof of the Riemann-Hurwitz formula,
2gl-2=d(2g2-2)+
L ep(¢)-1.
PEC1
d::::
1.9. Let¢ E IC(z) be a rational map of degree
2.
(a) Prove that oo is a totally ramified fixed point of¢ if and only if¢ E C [z ].
(b) Let a E IP' 1 (C) with a -j. oo. Prove that a is a totally ramified fixed point of¢ if and
onlyif(z- a)d l(¢(z)- a) is in C[z].
(c) Let a E IP' 1 (C) be arbitrary. Prove that a is a totally ramified fixed point of¢ if and only
if there exists a linear factional transformation f E PGL 2 (C) such that f- 1 (a) = oo
and ¢ 1 (z) E C[z].
1.10. Let K be a field of characteristic p > 0 and let ¢(z) E K(z) be a rational function.
(a) Show that the following are equivalent:
(i) ¢(z) ~ K(zP).
(ii) The formal derivative ¢' (z) is not identically zero.
(iii) The field extension K (z) I K (¢( z)) is separable.
If these conditions hold, we say that¢ is separable.
(b) Assume that ¢ is separable. Recall that the ramification index is defined by the formula
ea (¢) = orda (¢(z) -¢(a)). Let ra (¢) = orda (¢' (z) ). If a is a critical point of¢,
then we say that ¢ is tamely ramified at a if p f ea (¢ ), and ¢ is wildly ramified at a
if p I ea (¢ ). Prove that
ra(¢) :::: ea(¢)- 1,
with equality if and only if¢ is either unramified or tamely ramified at a.
(c) Prove that
2d- 2::::
L ra(¢) :::: L (ea(¢)- 1).
(Hint. Mimic the algebraic proof of the Riemann-Hurwitz formula, Theorem 1.1.)
(d) Deduce that a separable map has at most 2d - 2 critical points.
(e) More generally, if the rational function ¢ is wildly ramified at t points, prove that ¢ has
at most 2d - 2 - (p - 1)t critical points.
1.11. Let K be a field of characteristic p > 0 and let ¢(z) E K(z) be a rational function.
(a) Suppose that¢ is separable and that ¢n E K[z] for some
1. Prove that ¢ 2 E K[z].
Thus Theorem 1.7 is true in characteristic p for separable maps.
(b) Let p = 2 and ¢(z) = 1 + z- 2 . Prove that ¢ 2 ~ K[z] and that ¢ 3 E K[z]. Thus
Theorem 1. 7 need not be true in characteristic p for inseparable maps.
(c) Suppose that ¢ 2 ~ K[z] and that ¢n E K[z] for some n:::: 3. Prove that¢ has the form
¢ = (azq + b)l(czq +d), where q is a power ofp.
n::::
1.12. Let ¢ : IP' 1 ---+ IP' 1 be a rational map of degree d :::: 2, and suppose that there is a point
a E IP' 1 and an E > 0 such that
ea(¢n):::: (1 +
ct
for infinitely many n :::: 1. Prove that a is a preperiodic point for ¢, and that the eventual
period of a is at most (2d - 2) I c.
Exercises
38
Section 1.3. Periodic Points and Multipliers
1.13. We defined the multiplier of 1> at a fixed point a to be the derivative Aa (1>) = 1>' (a).
This definition obviously must be modified if a = oo. In order to be compatible with Proposition 1.9, we must set Aoo(1>) = (¢f)'(J- 1(oo)) for all f E PGL 2 (1C). Show that taking f (z) = 1/z leads to the definition
Prove that Aoo (1>) is finite and may be computed without taking a limit by showing that the
fraction on the righthand size is a rational function in IC(z) with no pole at z = 0, so it may
be evaluated at z = 0.
1.14. Let 1> : lP' 1 ---+ lP'1 be a rational map.
(a) Let P E lP'1 be a point of exact period n for 1> and suppose that 1>k(P) = P. Prove
that n I k.
(b) Prove that Pern(1>) is the disjoint union of Per;,:' (1>) over all m I n.
1.15. Let 1> : lP' 1 ---+ lP' 1 be a rational map of degree at least 2.
(a) Let n > m ~ 0. Prove that { P E lP'1 : 1>n(P) = 1>m(P)} is a finite set.
(b) Suppose that 1>1 and 1>2 are rational maps of degree at least 2 and suppose that 1>1 and ¢ 2
commute with one another, i.e., 1>1 o 1>2 = 1>2 o ¢1. Prove that
PrePer( 1>1) = PrePer( 1>z).
(Hint. Use (a) and apply Exercise 0.2.)
1.16. Suppose that a is a periodic point of¢>. Prove that the orbit of a contains a critical point
of 1> if and only if the multiplier Aa ( 1>) vanishes.
1.17. This exercise generalizes the multiplier sum formula described in Theorem 1.14.
Let ¢( z) E iC( z) be a nonconstant rational map and let a E C be a fixed point of C. The
residue fixed-point index of 1> at a is the quantity
1
.
~(¢,a)= -2
7rZ
1
lz-al=< z
dz
)'
-1>(
z
where the integral is any sufficiently small loop around a.
{a) If.Aa(1>) # 1,provethat
1
~(¢>,a)= 1- Aa(¢>)'
(b) Prove that in all cases, the index ~( ¢>, a) is independent of the choice oflocal coordinate
at a.
(c) Prove the index summation formula
L ~(¢,a)=l.
aEFix(.P)
Exercises
39
(d) Suppose that Aa (¢) = ¢' (a) = 1 and that ¢" (a) #- 0. Prove that
c/J"'(a)/6
~(¢,a)=
(c/J"(a)/2)
2.
In other words, if the series expansion of¢( z) around z = a has the form
c/J(z) =a+ (z- a)+ A(z- a) 2 + B(z- a) 3 + ...
with A#- 0,
then~(¢, a)= BjA 2 .
(e) The polynomial ¢( z) = z + Az 2 + z 3 has fixed points {0, -A, oo}. The point at infinity
is superattracting, so Aoo(c/J) = 0. Use the formula in (d) and the summation formula to
compute LA (¢),and then check your answer by computing A-A (¢) directly.
1.18. Let cjJ(z) E C(z) be a rational function of degree d ~ 2.
(a) Prove that# Pern (¢) ::; dn + 1.
(b) Prove that# Pern( ¢) ---> oo as n ---> oo.
(c) Prove that Per~* (¢) is nonempty for infinitely many values of n.
(d) *More precisely, prove that ifPer~*(¢) is empty, then (n, d) is one of the pairs
(n,d) E {(2,2), (2,3), (3,2), (4,2)}.
1.19. Let ¢( z) E q z) be a polynomial of degree d ~ 2.
(a) Ifm In, prove that the polynomial c/Jm(z)- z divides the polynomial c/Jn(z)- z.
(b) Let f..t be the Mobius f..t function. Prove that for every n ~ 1, the rational function
<I>~(z) =
IJ
(c/Jm(z)- z)"(n/m)
min
is a polynomial. The polynomial <I>~ ( z) is a dynamical analogue of the classical cyclotomic polynomial flmln (zm- 1)~'(n/m).
(c) Compute the first few polynomials <I>~ (z) for the quadratic polynomial c/Jc(z) = z 2 +c.
(d) Let a be a root of<I>~(z). Prove that ¢n(a) =a. Thus roots ofct>;',(z) are in Pern(¢).
(e) Let c/Jc(z) = z 2 +c. Find a value of c such that <I>2(z) has a root whose exact period is
strictly smaller than 2?
( t) For n = 3 and n = 4, find all values of c such that <I>~ ( z) has a root whose exact period
is strictly smaller than n.
The roots of <I>~ ( z) are said to have formal period n. They behave in many ways as if they
have exact period n, although their actual period is smaller than n.
1.20. Find explicit formulas for the fixed points and corresponding multipliers of the function ¢( z) = az / (z 2 + b). Verify directly the formula in Theorem 1.14 (cf. Example 1.17).
Section 1.4. The Julia Set and the Fatou Set
1.21. Let (S1,p1) and (S2,p2) be metric spaces. A collection ofmaps <I> from S 1 to S 2 is
said to be uniformly continuous if for every E > 0 there exists a 8 > 0 such that
p1(a, (3) < 8
==}
p2(¢a, ¢(3) < E for all a, (3 E S1 and all¢ E <I>.
The family is said to be uniformly Lipschitz if there is a constant C = C (<I>) such that
p2 (¢(a), ¢((3)) ::; C · p1(a, (3)
for all a, (3 E S1 and all¢ E <I>.
40
Exercises
(a) If <.Pis uniformly continuous, prove that <.Pis equicontinuous at every point of 81.
(b) If <.P is uniformly Lipschitz, prove that <.P is uniformly continuous.
(c) Let <.P = { <f>n}n~ 1 and suppose that for every point a E 81, the limit
¢(a) = lim <f>n(a)
n->oo
exists. If <.P is equicontinuous at a E 81, prove that¢ is continuous at a. Give an example
to show that the equicontinuity assumption is necessary.
1.22. Give an example to show that equicontinuity is not an open condition. Thus in order to
ensure that the Fatou set is open, the definition of :F( Q>) must include the requirement that it
be open.
1.23. Let Q> : 8 --+ 8 be a surjective continuous map of a metric space with the additional
property that Q> maps open sets to open sets.
(a) Give a rigorous proof that the Fatou and Julia sets of Q> are completely invariant, that is,
Q>-1(:F) = :F = Q>(:F) and Q>-1(.7) = .J = Q>(.J).
(b) Is (a) true without the assumption that Q> is surjective?
(c) Is (a) true without the assumption that Q> maps open sets to open sets?
1.24. LetQ>(z) E C[z] be a polynomial map, andlet:Foo be the connected component of :F(Q>)
containing the point oo. Prove that Q>- 1(:Foo) = :Foo = Q>(:Foo)· In other words, for a polynomial map, the connected component of oo is completely invariant for Q>.
1.25. Let Q>(z) = 1- 2/z2 be the rational map from Example 1.31.
(a) Let f(z) = 1/ z. Prove that (Q>f)' (0) = 0 and conclude that oo is a critical point of Q>.
(b) Let f(z) = z/(z- 1). Prove that (q/)'(0) = 0 and conclude that 0 is a critical point
of Q>.
1.26. Let Q>(z) E Q(z) be a rational map of degree d ~ 2 with coefficients in Q, and let
.J(Q>; Q) = .J(Q>) nlP' 1(Q) be the set of points in the Julia set whose coordinates are algebraic
numbers.
(a) Is .J(Q>; Q) Galois-invariant?
(b) Is .J(Q>; Q) infinite? More generally, is .J(Q>; Q) dense in .J(Q>)?
(c) ** Does .J (Q>; Q) contain points that are not preperiodic? More generally, does .J (Q>; Q)
contain infinitely many nonpreperiodic points Pi such that the orbits of the Pi are disjoint?
Section 1.5. Properties of Periodic Points
1.27. Let a be a periodic point of a rational function Q> E C( z).
(a) If a is attracting, prove that a E :F( Q>).
(b) If a is repelling, prove that a E .J (Q>).
1.28. Let Q>(z) = z 2 - 2. Prove that the Julia set of Q> is the closed interval on the real axis
between -2 and 2. Find the periodic points of Q>, compute their multipliers, and show directly
that .J(Q>) is equal to the closure ofPer(Q>). (Hint. Write z = eit + e-it = 2cos(t), so
Q>(z) = 2 cos(2t). Prove that Q>n(z) = 2 cos(2nt) and use this formula to study the dynamics
of Q>.)
41
Exercises
1.29. Let ¢>(z) E C[z] be a polynomial of degree d ~ 1, and suppose that every a E Pern(¢>)
has multiplier A¢(a) =/= 1. Prove that the equation ¢>n(z) = z has simple roots, and deduce
that Per n ( ¢>) contains exactly ~ + 1 distinct points (including the point oo ).
Section 1.6. Dynamical Systems Associated to Algebraic Groups
1.30. This exercise describes algebraic properties of the Chebyshev polynomials Td(w).
(a) Prove that T2(w) = w2 - 2, T3 (w) = w3 - 3w, and T4 (w) = w 4 - 4w 2 + 2.
(b) Prove that Td(Te(w)) = Tde(w) for all d, e ~ 0.
(c) Prove that Td( -w) = (-1)dTd(w). Thus Td is an odd function if dis odd and it is an
even function if d is even.
(d) Prove that the Chebyshev polynomials satisfy the recurrence relation
where we use the initial values To (w) = 2 and T1 (w) = w.
(e) Prove that the generating function of the Chebyshev polynomials is given by
00
d
2- wX
LTd(w)X = 1-wX+X2'
d=O
(f) Prove that the d!' Chebyshev polynomial is given by the explicit formula
1.31. This exercise describes dynamical properties of the Chebyshev polynomial Td(w) for a
fixed d ~ 2.
(a) Let U be an open subinterval of [-2, 2]. Prove that there exists an integer n ~ 1 such
thatTd'(U) = [-2, 2].
(b) Prove that limn~= T:l(w) = oo for all points wE IC not lying in the interval [-2, 2].
(c) Prove that the Julia set .J(Td) is the closed interval [-2, 2].
(d) Find all of the periodic points ofTd and compute their multipliers. Observe that .J(Td)
is the closure of the (repelling) periodic points.
(e) Let F(w) E C[w] be a polynomial of degree d ~ 2 with the property that the interval
[-2, 2] is both forward and backward invariant for F. Prove that F(w) = ±Td(w).
1.32. (a) Let A be an abelian group, let Ators = {P E A : nP = 0 for some n ~ 1} be the
torsion subgroup of A, let d ~ 2 be an integer, and let ¢> : A ---> A be the map
¢>(P) = dP. Prove that
PrePer(¢>) = Ators·
(This is Proposition 0.3, but try reproving it without looking back at our proof)
(b) Let E be an elliptic curve, let d ~ 2 be an integer, and let ¢>E,d : 1P'1 ---> 1P'1 be the rational
map associated to multiplication-by-donE, as described in Section 1.6.3. Prove that
PrePer(¢>E,d) = x(E(C)tors)·
Chapter 2
Dynamics over Local Fields:
Good Reduction
The study of the dynamics of polynomial and rational maps over~ and C has a long
history and includes many deep theorems, some of which were briefly discussed in
Chapter 1. A more recent development is the creation of an analogous theory over
complete local fields such as the p-adic rational numbers Qp and the completion Cp
of an algebraic closure of Qp. The nonarchimedean nature of the absolute value
on Qp and Cp makes some parts of the theory easier than when working over C
orR But as usual, there is a price to pay. For example, the theory of nonarchimedean
dynamics must deal with the fact that Qp is totally disconnected and far from being
algebraically closed, while Cp is not locally compact.
In this chapter we begin our study of dynamics over complete local fields K by
concentrating on rational maps ¢ that have "good reduction." Roughly speaking, this
means that the reduction of¢ modulo the maximal ideal of the ring of integers of K
is a "well-behaved" rational map¢ over the residue field k of K. Thus studying the
dynamics of ¢ over k allows us to derive nontrivial information about the dynamics of ¢ over K. In Chapter 5 we take up the more difficult, but ultimately more
interesting, case of rational maps with "bad reduction."
2.1
The Nonarchimedean Chordal Metric
We begin by quickly recalling the definition and basic properties of absolute values,
especially those satisfying the ultrametric inequality.
Definition. An absolute value on a field K is a map
I·I:K----t~
with the following properties:
• lal ;?: 0, and Ia I= 0 if and only if a= 0.
43
44
2. Dynamics over Local Fields: Good Reduction
• la,BI = Ia I · I,BI for all a, ,B E K.
• Ia + ,61 :::; Ia I + I,BI for all a, ,B E K (triangle inequality).
A valuedfield is a pair (K, I · IK) consisting of a field K and an absolute value
on K, although we often omit the absolute value in the notation. A map of valued
fields i : K ---+ L is a field homomorphism that respects the absolute values,
foralla E K.
Definition. Let (K, I · I) be a valued field. If the absolute value satisfies the stronger
estimate
(2.1)
Ia + ,BI :::; max{lal, I,BI} for all a, ,BE K,
then the absolute value is nonarchimedean or ultrametric. The associated valuation
is the homomorphism
v: K*---+ R,
v(a) =-log Ia!.
It satisfies
v(a +,B) 2: min{ v(a), v(,B) }.
The valuation vis discrete if v(K*) is a discrete subgroup of JR, in which case the
associated normalized valuation (sometimes denoted ordv) is the constant multiple
ofv chosen to satisfy ordv(K*) = Z.
Example 2.1. The field !Q has the usual real absolute value
!aloe= max{ a, -a}.
For each prime pit also has a p-adic absolute value defined as follows. Every nonzero
rational number a has a unique factorization of the form
a
Then
= ± IT pep(a.)
p prime
lalp = p-ep(a.).
The p-adic absolute values are nonarchimedean.
Two absolute values are said to be equivalent if there is a constant r > 0 such
that lal1 = lal2 for all a E K. Ostrowski's theorem says that up to equivalence, the
real absolute value and the p-adic absolute values are the only nontrivial absolute
values on !Q; see [78, 1.4.2, Theorem 3] or [249, 1.2, Theorem 1].
Example 2.2. Let k be a field and let K = k(T) be the field of rational functions.
Then each a E k determines an absolute value on K associated to the valuation orda
that gives the order of vanishing of f(T) E k(T) at T = a. The degree map deg :
k(T)* ---+ Z is also a valuation. If k is algebraically closed, this yields the complete
set of absolute values on k(T) that are trivial on k, up to equivalence. More generally,
if k is not algebraically closed, there is a valuation corresponding to each monic
2.1. The Nonarchimedean Chordal Metric
45
irreducible polynomial in k [T]. In this book we are primarily interested in the number
field scenario, i.e., the field Q and its extensions and completions, but the reader
should be aware that it is also interesting to study the analogous case of function
fields, i.e., the field k[T] and its extensions and completions.
We now prove the elementary, but extremely useful, fact that if lal ::/:- I,BI, then
the ultrametric inequality (2.1) is actually an equality.
Lemma 2.3. Let K be a field with a nonarchimedean absolute value I · lv and
let a, ,6 E K. Then
Proof We suppose that lalv > I,Biv· The strict inequality
implies that the maximum on the right is Ia + ,Biv. so we find that lalv :::; Ia + ,6lv·
The opposite inequality is also true, since Ia + ,6lv:::; max{lalv, l,6lv} = lalv· 0
Recall that the chordal metric on IP' 1 ( iC), which we now denote by Poo, is defined
by the formula
for points P 1 = [X1, Y1] and P 2 = [X2, Y2] in IP' 1(iC). In the case of a field K having
a nonarchimedean absolute value I · lv. it is convenient to use a metric given by a
slightly different formula.
Definition. Let K be a field with a nonarchimedean absolute value I · lv. and let
P 1 = [X1, Y1] and P 2 = [X2, Y2] be points in IP' 1(K). The v-adic chordal metric
on IP' 1 (K) is
It is clear from the definition that Pv ( P1 , P 2 ) is independent of the choice of homogeneous coordinates for P 1 and P 2 .
The first thing to check is that Pv is indeed a metric. In fact, it is an ultrametric;
that is, it satisfies the nonarchimedean triangle inequality.
Proposition 2.4. The v-adic chordal metric has the following properties.
(a) 0 :S Pv (P1 , P2) :S 1.
(b) Pv(P1,P2) =OifandonlyifPl =P2.
(c) Pv(Pl, P2) = Pv(P2, PI).
(d) Pv(Pl, P3) :S max{pv(Pl, P2), Pv(P2, P3) }.
2. Dynamics over Local Fields: Good Reduction
46
Proof The lower bound in (a) and parts (b) and (c) of the proposition are obvious
from the definition. For the upper bound in (a), we use the nonarchimedean nature
of v to compute
IX1Y2- X2Y1Iv:::; max{IX1Y2Iv, IXzY1Iv}
:::; max{ IX1Iv, IY1Iv} max{ IXzlv, IYzlv} ·
The proof of (d) requires the consideration of several cases. The following useful
lemma makes the proof more transparent by allowing some freedom to change coordinates. It is the analogue of the fact that the classical chordal metric is invariant
under linear fractional transformations that define rigid rotations of the Riemann
sphere (cf. Exercise 1.2).
Lemma 2.5. Let
R = {a E K : lalv :::; 1}
be the ring of integers of K, and let f : JID 1 ----r JID 1 be a linear fractional transformation of the form
/([X Y]) = aX+ bY
eX +dY
'
with a, b, e, d E Rand ad - be E R*,
i.e., f E PGLz(R). Then
Pv(f(l\),J(Pz)) =pv(l\,Pz)
(N.B. It is crucial that the quantity ad - be is a unit.)
Proof Write each point as Pi = [Xi, Yi] with Xi, Yi E R and at least one of Xi
or Yi in R*. Then max{IXilv, IYilv} = 1, so
Pv(PI,Pz) = IX1Y2- X2Y1Iv·
Further, the identities
d(aXi + bYi)- b(cXi + dYi) =(ad- bc)Xi,
e(aXi + bYi)- a(cXi + dYi) =-(ad- be)Yi,
and the fact that max{IXilv, IYilv} = 1 and lad- belv = 1 immediately imply that
max{ laXi + bYi lv, leXi + d}i lv} = 1.
Thus
Pv(/(PI), J(Pz)) = l(aX1 + bYI)(cXz + dYz)- (aXz + bYz)(eXl + dYI)iv
= i(ad- be)(X1 Y2- XzYI)Iv
= Pv (P1, Pz),
where we have again made use of the fact that lad- belv = 1.
0
47
2.2. Periodic Points and Their Properties
We resume the proof of Proposition 2.4 and write each point as Pi = [Xi, Yi]
with Xi, Yi E Rand at least one of Xi or Yi in R*. Then
max{IXilv, IYilv} = 1
and
Pv(Pi, Pj) = IXi}j- XjYilv,
as usual. If IX2Iv > IY2Iv. we apply the map f = Y/X to the three points. This
preserves the chordal distance (Lemma 2.5) and allows us to assume that
IX2Iv :::; IY2Iv = 1.
Next we apply the map f = (Y2 ·X- X 2 · Y)/Y to the three points. Lemma 2.5
again tells us that the chordal distance is preserved (note that IY21v = 1), and we are
reduced to the case that P2 = [0, 1]. Finally, we compute
IX1Y3- X3Y1Iv:::; max{IX1Y31v, IX3Yllv}:::; max{IXllv, IX31v},
which is exactly the desired inequality when P2 is the point [0, 1].
2.2
0
Periodic Points and Their Properties
In Section 1.3 we described various properties of periodic points for rational maps
defined over C. Virtually all of these definitions make sense, mutatis mutandis, when
we work over any field with an absolute value. In this section we briefly recall the
relevant material.
Let K be a field with an absolute value I · I and let ¢( z) E K (z) be a nonconstant
rational map. The multiplier of¢ at a fixed point a E K is the derivative
Aa(cP) =¢'(a).
(See Exercise 1.13 for the case a= oo.) The multiplier is well-defined, independent
of the choice of coordinates on JP1; see Proposition 1. 9.
More generally, if a E IP'1(K) is a point of exact period n for ¢, then a is a fixed
point of q;n and we define the multiplier of¢ at a to be
The multiplier may be calculated using the chain rule as
The magnitude of the multiplier determines, to some extent, the behavior of¢ in
a small neighborhood of a periodic point a. The periodic point a is called
superattracting if Aa (¢) = 0,
attracting if IAa (¢)I < 1,
if IAa (¢)I = 1,
repelling if IAa (¢) I > 1.
neutral (or indifferent)
The neutral periodic points are further divided into two types. The rationally neutral
periodic points are those whose multiplier is a root of unity. The others are called
irrationally neutral.
48
2.3
2. Dynamics over Local Fields: Good Reduction
Reduction of Points and Maps Modulo p
One of the most important gadgets in the number theorist's toolbox is reduction
modulo a prime. Thus when studying the number-theoretic properties of an object,
we reduce it modulo a prime, analyze the properties of the hopefully simpler object,
and then lift the information back to obtain global information. A typical example
is provided by Hensel's lemma, which under certain circumstances allows us to lift
solutions of a polynomial congruence f(x) = 0 (mod p) to solutions in Zp. Then,
using information gathered from many primes, one is sometimes able to deduce results for a global field such as Ql.
Our principal objects of study are maps ¢ : JPl 1 --+ JPl 1 . In this section we study
the behavior of such maps under reduction modulo a prime. We work over a discrete
valuation ring, so we set the following notation:
K
a field with normalized discrete valuation v : K* ---* Z.
I · lv
= c-v(x) for some c > 1, an absolute value associated to v.
R
R*
={a E K: v(a) 2: 0}, the ring ofintegers of K.
={a E K: v(a) 2: 1}, the maximal ideal of R.
={a E K: v(a) = 0}, the group ofunits of R.
k
= Rjp, the residue field of R.
p
reduction modulo p, i.e., R --+ k, a f--t a.
Before studying reduction of maps, we consider the problem of reducing points
modulo p. This is as easy in pN as it is in JPl 1 , so we look at the general case. Let
p = [xo, X1, ... , XN] E pN (K)
be a point defined over K. We cannot immediately reduce the coordinates of P modulo p, since some of the coordinates might not be in R. However, since the coordinates of P are homogeneous, we can replace them with
p = [cxo, CX1, ... , CXN]
for any c E K*. Choosing c to be highly divisible by p, we can ensure that every cxi
is in R. However, if we overdo this process and end up with every cxi in the prime
ideal p, then when we reduce modulo p, we end up with [0, 0, ... , 0], which does not
represent a point of projective space.
The trick is to "clear the denominators" of the Xi's as efficiently as possible. To
do this, we choose an element a E K* satisfying
v(a) = min{v(xo), v(x1), ... , v(xN)}.
(2.2)
For example, a could be the Xj having minimal valuation. Then a- 1 xi E R for
every i, so we can reduce these quantities modulo p. We define the reduction of P
modulo p to be the point
2.3. Reduction of Points and Maps Modulo p
49
Note that P has at least one nonzero coordinate, since at least one of the numbers o:- 1 xi is a unit. Hence Pis a well-defined point in lP' 1 (k ).
We say that P = [x 0, ... , x N] has been written using normalized coordinates if
min{v(xo),v(xl), ... ,v(xN)} = 0,
in which case Pis simply [xo, ... , XN].
l i
J
2
1
5 , ~~, 2 5 ] E lP' ( Q). We can reduce P
modulo 11 without any modification, since every coordinate is an 11-adic integer and
not all coordinates vanish modulo 11. Thus P = [1, 10, 7, 9] (mod 11). However, if
we want to reduce P modulo 3, then we first need to divide all of the coordinates
by 3,
Example 2.6. Consider the point P = [ 4 ,
p - [ 3
-
9 24 27 ] - [ 1 3 8 9 ]
14' 35' 49' 245 - 14' 35' 49' 245 '
and then P = [2, 0, 2, 0] (mod 3). Similarly, in order to compute P modulo 7, we
first multiply the coordinates by 49,
p -
[2_14' ~35' 49'
24 :!!__] - [21 63 24 27]
245 2' 5' 1' 5 '
and then P = [0, 0, 3, 4] (mod 7).
It may appear that the reduction P depends on the choice of o:. We now check
that this is not the case.
Proposition 2.7. Let P = [x0 , •.. , XN] E lP'N (K). Then the reduction Pis independent of the choice of o: satisfYing (2.2).
Proof Suppose that o: and f3 both satisfy (2.2). Then o: and f3 have the same valuation, so o:/3- 1 E R*. This allows us to compute
[------Xo,O:-----0: -1
------ ] - [----------------- ,o:/3
------ ------XN ]
o:/3 -----Xo,o:/3
]
- [f3-----Xo,/3-----X1, ... ,/3-----XN.
-1 X1, ... ,0: -1 XN
-
-
-1 0: -1
-1
-1 0: -1 X1, ...
-1
-1 0: -1
-1
Hence the reduction of P modulo pis independent of the choice of o: satisfying (2.2).
D
We next prove an easy and useful lemma that relates reduction modulo p to v-adic
distance.
Lemma 2.8. Let P1 and P2 be points in lP' 1 ( K). Then
2. Dynamics over Local Fields: Good Reduction
50
Proof Write P1 = [X1, Y1] and P 2 = [X2, Y2] using normalized coordinates, so in
particular Pv(Pl, P2) = IX1Y2 :=- X2Y1lv· Supp_ose firs~ that P1 = P2. This means
that there is au E k* such that X1 = uX2 and Y1 = uY2. In other words, there is a
u E R* such that that X 1 = uX2 (mod p) and Y1 = uY2 (mod p). Hence
which completes the proof that Pv (P1, P2) < 1.
Next suppose that Pv(Pl, P2) < 1, which implies that X1Y2
If X1X2 =/:. 0 (mod p), then X1, X2 E k*, so we have
= X 2Y1 (mod p).
P2 = [X2, Y2] = [X1X2, X1Y2] = [X1X2, X2Y1] = [X1, Y1] = P1.
=
On the other hand, if X 1X 2 0 (mod p), then the fact that the coordinates are norX 2Y1 (mod p) imply that X 1 X 2 0 (mod p),
malized and the equality X 1Y2
D
so P 1 = P2 = [0, 1]. This completes the proof of the lemma.
= =
=
As a first application, we show that fractional linear transformations in PGL 2(R)
respect reduction modulo p.
Proposition2.9. Let P,Q E lP' 1(K) and f E PGL2(R). Then
P = Q ifandonlyif J(P) = f(Q).
Proof We combine Lemmas 2.5 and 2.8. Thus
p=Q
{=::?
Pv(P, Q) < 1
from Lemma 2.8,
{=::?
Pv (f(P), f( Q)) < 1
from Lemma 2.5,
{=::?
f(P) = f(Q)
from Lemma 2.8 again.
D
Example 2.10. Let f = (~§),so f ~ PGL2(Z3). Consider the points P = [7,5]
and Q = [4, 2] in lP' 1 (Ql 3). They satisfy P = Q= [1, 2] in lP' 1 (1F 3), but
f(P) = [45, 75] = [3, 5] = [0, 1]
f(Q) = [24, 36] = [2, 3] = [1, 0]
(mod 3),
(mod 3),
so f(P) =j:. f( Q). This shows the necessity of the condition f E PGL 2(R) in Proposition 2.9.
It is easy to see that if K is a field and P 1, P 2, P3 are distinct points in lP' 1( K),
then there is an element ofPGL 2(K) that moves them to the points 0, 1, oo. (See Exercise 1.4.) The next proposition gives a stronger result for points whose reductions
are distinct. It is especially useful because Lemma 2.5 says that the nonarchimedean
chordal metric is invariant for maps in PGL2(R), so changing coordinates via an
element ofPGL 2(R) does not change the underlying dynamics.
2.3. Reduction of Points and Maps Modulo p
51
Proposition 2.11. Let P1, P2 , P3 E IP'1(K) be points whose reductions P1, P2 , P3
are distinct. Then there is a linear fractional transformation f E PGL 2 ( R) such that
Proof Write Pi = [Xi, Yi] with normalized coordinates. If v(XI) > v(Y1 ), we begin by applying the map f = Y /X E PGL 2 ( R) to each of the three points, so we
may assume that v(X1 ) ~ v(YI). Since the coordinates are normalized, this implies
that v(Yl) = 0, so Y1 is a unit. We next apply the map
to the three points. Having done this, we see that P1 = [0, 1].
Next consider the point P3. Since F3 =1- P1 = [0, 1], we see that v(X3 ) = 0,
so we can apply the map X/(Y3 X- X 3 Y) E PGL 2 (R) to the three points. This
fixes P1 and sends P3 to [1, 0], i.e., P3 gets sent to oo. Finally, since P2 is equal to
neither P1 = [0, 1] nor F3 = [1, 0], we see that v(X2 ) = v(Y2 ) = 0. Applying the
map Y2 X/ X 2 Y E PGL 2 (R) to the three points then fixes P1 and P3 and sends P2
to [1, 1].
0
Having looked at the reduction of a point, we next turn to the problem of reducing
a rational map modulo p. Let ¢ : IP' 1 --> IP' 1 be a rational map of degree d defined
over K, so ¢ is given by a pair of homogeneous polynomials of degree d,
F(X, Y), G(X, Y) E K[X, Y].
Note that the map ¢(X, Y) = [F(X, Y), G(X, Y)] does not change ifF and G are
each multiplied by a nonzero constant c E K*, since the coordinates are homogeneous.
Definition. Let ¢ : IP'1 --> IP'1 be a rational map as above and write
¢ = [F(X, Y), G(X, Y)]
with homogeneous polynomials F, G E K[X, Y]. We say that the pair (F, G) is
normalized, or that¢ has been written in normalized form, ifF, G E R[X, Y] and at
least one coefficient ofF or G is in R*. Equivalently,¢= [F, G] is normalized if
F(X, Y) = aoXd + a1Xd-ly + · · · + ad-lxyd-l + adYd
and
satisfy
min{ v(ao), v(ai), ... , v(ad), v(bo), v(b1), ... , v(bd)} = 0.
(2.3)
Given any representation¢ = [F, G], it is clear that one can always find some c E K*
such that [cF, cG] is a normalized representation. Further, the element cis unique up
to multiplication by an element of R*.
52
2. Dynamics over Local Fields: Good Reduction
Writing¢ = [F, G] in normalized form, the reduction of¢ modulo pis defined
in the obvious way,
J>(X, Y) = [F(X, Y), G(X, Y)]
=
[iioXd + ii1Xd-Iy + · · · + iidYd, boXd + b1Xd-Iy + · · · + bdYd].
In other words,¢ is obtained by reducing the coefficients ofF and G modulo p. (If
the prime ideal is not clear from context, we write J>P or J> mod p.)
The fact that at least one coefficient of F or G is a unit ensures that at least
one of P and G is a nonzero polynomial, so the reduction J> gives a well-defined
map J> : IP' 1 ( k) ___, IP' 1 ( k). Further, the reduced map J> is independent of the choice
ofF and G, a fact whose proof we leave to the reader (Exercise 2.4) since it is quite
similar to the proof of Proposition 2.7.
The mere existence of the reduction J> of a rational map ¢ does not imply that J>
has good properties, as is shown by the following simple example.
Example 2.12. Let a E K* and consider the rational map
If a E R*, then ii -=J 0 and the reduced map J>a (X, Y) = liiXd, Yd] is again a rational
map of degree d. However, ifv(a) > 0, then ii = 0, so ¢a(X, Y) = [0, Yd] = [0, 1]
is a constant map! Similarly, if v( a) < 0, then a- 1 = 0, so
is again a constant map, but not to the same point!! To summarize, the reduction of
the map ¢a(X, Y) = [aXd, Yd] separates into three cases,
[iiXd, Yd]
{
¢a= [0, 1]
[1,0]
ifv(a) = 0,
ifv(a) > 0,
ifv(a) < 0.
Clearly it is only in the first case that the reduced map is interesting.
Keep in mind that our goal is to use the dynamics of J> to help us understand the
dynamics of¢. In the above example, if v( a) = 0, then it is easy to see that for any
P E IP' 1 (K),
¢(P) = J>(P),
and hence by induction,
q;n(P) = J>n(P).
Thus the J> orbit of P yields valuable information about the¢ orbit of P. However,
ifv(a) -=J 0, then J>(P) is constant, independent of P, so the¢ orbit of P contains no
information. We formalize this notion in Section 2.5 after a preliminary discussion
of the theory of resultants.
2.4. The Resultant of a Rational Map
2.4
53
The Resultant of a Rational Map
A rational map¢ : lP' 1 --> lP' 1 is given by a pair of homogeneous polynomials
¢ = [F(X, Y), G(X, Y)]
having no nontrivial common roots. However, if we reduce the coefficients of F
and G modulo some prime, they may acquire common roots in the residue field. In
order to understand this phenomenon, it is useful to have a tool that characterizes the
existence of common roots in terms of the coefficients ofF and G. This tool is called
the resultant. Resultants and their generalizations are widely used, both theoretically
and computationally, in number theory and algebraic geometry. We give in this section only a brief introduction to the theory of resultants. The reader desiring further
information might consult [105, Section 3.3], [112, Chapter 3], [259, Section V.10],
or [436, Sections 5.8, 5.9], while the reader interested in applications to dynamics
may wish to peruse only the statements of Proposition 2.13 and Theorem 2.14 and
return to the proofs at a later time.
Proposition 2.13. Let
A(X, Y) = aoXn + a1Xn- 1Y + · · · + an-1xyn- 1 + anYn,
B(X, Y) = boXm + b1Xm- 1Y + · · · + bm-1xym- 1 + bmYm
be homogeneous polynomials ofdegrees n and m with coefficients in a field K. There
exists a polynomial
Res(ao, ... , an, bo, ... , bm) E Z[ao, ... , an, bo, ... , bmJ,
in the coefficients of A and B, called the resultant of A and B, with the following
properties:
(a) Res( A, B) = 0 if and only if A and B have a common zero in lP' 1(K).
(b) If aobo -=f. 0 and if we factor A and B as
m
and
then
i=1
n
Res(A, B)= a~b0
m
j=1
II II (a;- /3j)·
i=1 j=1
(c) There exist polynomials
F1, G1, F2, G2 E Z[ao, ... , an, bo, ... , bm][X, Y],
homogeneous in X and Y of degrees m - 1 and n - 1, respectively, with the
property that
F1 (X, Y)A(X, Y) + G 1(X, Y)B(X, Y) = Res(A, B)xm+n- 1,
F2(X, Y)A(X, Y) + G2(X, Y)B(X, Y) = Res(A,B)Ym+n- 1.
Notice that in the first equation, the variable Y has been eliminated, and similarly X has been eliminated in the second equation.
2. Dynamics over Local Fields: Good Reduction
54
(d) The resultant is equal to the (m + n) x (m + n) determinant
ao a1 a2 ... an
ao a1 a2 ... an
ao a1 a2 ... an
Res( A, B) = det
ao a1 a2 ... an
bo b1 b2 ......... bm
bo b1 b2 ......... bm
bo b1 b2 ......... bm
m
n
In particular, Res( A, B) is homogeneous of degree m in the variables a0 , ... , an
and simultaneously homogeneous ofdegree n in the variables b0 , .. . , bm.
Proof We begin by showing that the following three conditions are equivalent.
(i) A(X, Y) and B(X, Y) have a common zero in JP> 1 (K).
(ii) A(X, Y) and B(X, Y) have a common (nonconstant) factor in the polynomial
ring K[X, Y].
(iii) There are nonzero homogeneous polynomials C, DE K[X, Y] satisfying
A(X, Y)C(X, Y) = B(X, Y)D(X, Y)
with deg(C) ::::; m- 1 and deg(D) ::::; n- 1.
(2.4)
The equivalence of(i) and (ii) follows immediately from the fact that the greatest
common divisor of A and Bin K[X, Y] vanishes at exactly the common zeros of A
and B in JP> 1 ( K). (What we are really using here, of course, is the fact that the ring
of homogeneous polynomials K[X, Y] is a principal ideal domain.) It is also clear
that (ii) implies (iii), since if A and B have a common factor F, then we simply
choose C and D using the formulas A = F D and B = FC. Finally, to prove
that (iii) implies (i), we suppose that (2.4) is true. If we factor both sides of (2.4) into
linear factors in K[X, Y], then A(X, Y) has n factors, while D(X, Y) has at most
n- 1 factors. Therefore A(X, Y) shares at least one linear factor with B(X, Y)
in K[X, Y], and hence they have a common zero in JP> 1 (K).
We next multiply out equation (2.4), treating the coefficients of C and D
as unknowns. This gives a system of m + n homogeneous linear equations in
the m + n variables co, ... , Cm-1, do, ... , dn- 1 , and the matrix of this system is
(up to changing the sign of some columns and transposing) equal to the matrix given
in (d). For example, if deg(A) = 3 and deg(B) = 2, then equating coefficients
in (2.4) gives the system oflinear equations
ss
2.4. The Resultant of a Rational Map
coao
= dobo,
+ c1ao = dob1 + d1bo,
coa2 + c1 a1 = dob2 + d1 b1 + d2bo,
CQa3 + C1a2 =
d1b2 + d2b1,
coa1
d2b2,
c1a3 =
whose associated matrix
ao
-bo
a 1 ao -b1 -bo
a2 a1 -b2 -b1 -bo
a3 a2
-b2 -b1
a3
-b2
becomes equal to the matrix in (d) if we change the sign of the last three columns
and transpose. The general case is exactly the same.
To recapitulate, we have shown that equation (2.4) has a nontrivial solution if and
only if the system of homogeneous linear equations described by the matrix in (d)
has a nontrivial solution, which is equivalent to the vanishing of the determinant of
the associated matrix. Hence if we take the determinant in (d) as the definition of
the resultant Res( A, B), then the equivalence of (i) and (iii) proven above shows
that Res( A, B) = 0 if and only if A and B have a common zero in JID1 (K), which
proves (a).
In order to prove (c), we write
xiym-l-i A
for 0 ~ i < m
and
xjyn-l-1 B
for 0 ~ j < n
xn+m-1
xm-lA
xm-2yA
as a system of homogeneous equations,
ao a1 a2 ... an
ao a1 a2 ... an
ao a1 a2 ... an
xn+m-2y
xn+m-3y2
xm-3y2A
ym-lA
ao a1 a2 ... an
bo b1 b2 ......... bm
xn-IB
xn-2yB
xn-3y2B
bobl~·········bm
bo b1 b2 ......... bm
xyn+m-2
bo b1 b2 ......... bm
yn+m-1
yn-lB
Notice that the matrix M appearing here is exactly the matrix in (d) whose determinant equals Res( A, B). We multiply on the left by the adjoint matrix Madj of M.
(Recall that the entries of Madj are the cofactors of the matrix M, and that the product MadjM is a diagonal matrix with the quantity det(M) as its diagonal entries.)
This yields the following matrix identity, where for convenience we write R(A, B)
for Res( A, B):
2. Dynamics over Local Fields: Good Reduction
56
R(A,B)
0
0
0
R(A,B)
0
0
0
R(A,B) · · ·
0
0
0
xn+m-1
xn+m-2y
xn+m-3y2
xm-1A
xm-2yA
xm-3y2A
ym-1A
xn-1B
xn-2yB
xn-3y2B
0
0
0
· · · R(A,B)
xyn+m-2
yn+m-1
yn-1B
Examining the top entry on each side, we find that Res( A, B)xn+m- 1 on the lefthand side is equal to an expression of the form
F1 (X, Y)A(X, Y) + G1 (X, Y)B(X, Y)
on the righthand side, where H and G 1 are homogeneous polynomials of degrees
m-1 and n-1, respectively, whose coefficients are (complicated) polynomials in the
coefficients of A and B. Similarly, the bottom entry shows that Res( A, B)Yn+m- 1
is equal to an expression of the form
F2(X, Y)A(X, Y) + G2(X, Y)B(X, Y).
This completes the proof of part (c) of the proposition.
Finally, we leave the proof of (b) as an exercise for the reader, or see [436, Section 5.9].
D
We define the resultant of a rational map in terms of its defining pair of polynomials.
Definition. Let ¢ : IP' 1 --t IP' 1 be a rational map defined over a field K with a
nonarchimedean absolute value I · lv· Write¢= [F, G] using a pair of normalized
homogeneous polynomials F, G E R[X, Y]. The resultant of¢ is the quantity
Res(¢)= Res(F, G).
Since the pair (F, G) is unique up to replacement by (uF, uG) for a unit u E R*,
we see that Res(¢) is well-defined up to multiplication by the 2lf't-power of a unit.
In particular, its valuation v(Res( ¢)) depends only on the map ¢.
The resultant of a rational map ¢ provides an upper bound to the extent that ¢
is expanding in the chordal metric. In particular, a rational map is always Lipschitz
with respect to the chordal metric, and if its resultant is a unit, then the map is nonexpanding. (See also Exercise 2.10.)
Theorem 2.14. Let¢ : IP' 1 --t IP' 1 be a rational map defined over afield K with a
nonarchimedean absolute value I · lv· Then
2.4. The Resultant of a Rational Map
57
Proof Write¢= [F(X, Y), G(X, Y)] in normalized form. Proposition2.13(c) says
that there are homogeneous polynomials F 1, G1, F 2, G2 E R[X, Y] satisfying
F1(X, Y)F(X, Y) + G1(X, Y)G(X, Y) = Res(¢)X 2d-l,
F2(X, Y)F(X, Y) + G2(X, Y)G(X, Y) = Res(¢)Y 2d-l.
Now let P = [x, y] E JP> 1(K) be a point, which we assume written in normalized form. We substitute [X, Y] = [x, y] into the first equation and use the nonarchimedean triangle inequality to compute
IRes(¢)x 2d-llv = IF1 (x, y)F(x, y) + G1 (x, y)G(x, Y)lv
~ max{IFl(x,y)F(x,y)lv, IGl(x,y)G(x,y)lv}
~ max{IF1(x, y)lv, IG1(x, y)lv} · max{IF(x, y)lv, IG(x, Y)lv}
~ max{IF(x,y)lv, IG(x,y)lv}·
A similar calculation using the second equation gives the analogous estimate
IRes(¢)y 2d-llv ~ max{IF(x, Y)lv, IG(x, Y)lv }.
Since Pis normalized, i.e., max{lxlv, IYiv} = 1, we find that
IRes(¢)1v ~ max{IF(x, y)lv, IG(x,y)lv }.
(2.5)
Notice that this estimate bounds the extent to which F(x, y) and G(x, y) can be
simultaneously divisible by high powers ofp.
Returning to the proofofthetheorem, we write P1 = [x1, Yl], P2 = [x2, Y2], and
¢ = [F(X, Y), G(X, Y)] in normalized form. Then the distance from P 1 to P2 is
while we can use the inequality (2.5) (applied to both P 1 and P 2) to estimate
IF(xl,Yl)G(x2,Y2)- F(x2,Y2)G(xt,Yl)l v
- max{ IF(xl, yt)lv, IG(xt, Yl)lv} ·max{ IF(x2, Y2)lv, IG(x2, Y2)lv}
-
< IF(x1,yt)G(x2,Y2)- F(x2,Y2)G(x1,Yt)lv.
-
I Res(¢)1~
To complete the proof, we observe that the polynomial
vanishes identically if X 1 Y2 = X 2Y1 . It follows that it is divisible by the polynomial
X 1 Y2 - X 2Y1 in the ring R[X1, Y1, X2, Y2], so we can write
2. Dynamics over Local Fields: Good Reduction
58
( .-~,(P) .-~,(P. )) < l(x1y2 -x2yl)H(xl,Yl,x2,Y2)iv
Pv '!-'
1 ''!-'
2
IRes(¢)1;
-
< lx1Y2 - X2Y1Iv
-
IRes(¢)1;
Pv(Pl, P2)
I Res(¢)1; ·
2.5
D
Rational Maps with Good Reduction
As we saw in Example 2.12, the reduction ¢ of a rational map ¢ may bear little
resemblance to the original map. Indeed, even the degree of the map may change. In
this section we characterize maps for which deg(¢) = deg(¢). These maps are the
dynamical analogue of varieties that have good reduction, and they share many of
the same properties. See [410, Chapter VII], for example, and compare the results of
this section with the properties of elliptic curves that have good reduction.
Theorem 2.15. Let ¢ : IP 1 ---. IP 1 be a rational map defined over K and write
¢ = [F, G] in normalized form. The following are equivalent:
(a) deg(¢) = deg(¢).
(b) The equations F(X, Y) = G(X, Y) = 0 have no solutions [a, ;3] E IP 1 (k).
(c) Res(¢) E R*.
(d) Res(.F, G) =f. 0.
Proof The equivalence of (b), (c), and (d) is immediate from the basic properties of
the resultant given in Proposition 2.13, once we observe that
Res(F, G)= Res(F, G).
This equality follows from the fact that the resultant is simply a polynomial in the
coefficients ofF and G.
To complete the proof, we observe that the degree of¢ is equal to the degree of¢
minus any cancellation that occurs in F(X, Y)/G(X, Y). In other words,
d
eg
¢ = d
eg
¢ _ ( Nu~ber of com~on roots )
of F(X, Y) = G(X, Y) = 0 '
where the roots are counted with appropriate multiplicities in IP 1 ( k). In particular,
deg ¢ = deg ¢ if and only if P and G have no common roots, which proves the
D
equivalence of(a) and (b).
Definition. A rational map ¢ : IP 1 ---. IP 1 defined over K is said to have good reduction (modulo p) if it satisfies any one (hence all) of the conditions of Theorem 2.15.
2.5. Rational Maps with Good Reduction
59
Remark 2.16. There is a fancier, but useful, characterization of good reduction
in the language of schemes. The rational map ¢ is a morphism ¢ : JP>k ---. JP>k
over Spec(K), so it induces a rational map JP>k ---. JP>k over Spec(R). Then ¢
has good reduction if and only if this rational map over Spec(R) extends to a
morphism. In other words, good reduction is equivalent to the existence of an
R-morphism ¢R : JP>k ---. JP>k whose restriction to the generic fiber is the original
map ¢ : JP>k ---. JP>k. In this setting, the reduction ~ is then simply the restriction
of ¢R to a morphism of the special fiber ~: JP>k ---. JP>k over Spec(k). See Exercise 2.15.
As a first application of the notion of good reduction, we use Theorem 2.14 to
prove the somewhat surprising result that maps with good reduction have empty Julia sets. Later, in Chapter 5, we will prove that rational maps always have nonempty
Fatou set. This is exactly opposite to the situation that holds over the complex numbers C, where the Julia set is nonempty, but the Fatou set may be empty.
Theorem 2.17. Let¢ : JID 1 ---. JID 1 be a rational map that has good reduction.
(a) The map¢ is everywhere nonexpanding,
(b) The map ¢ has empty Julia set.
Proof (a) This is immediate from Theorem 2.14 and the fact that good reduction is
equivalent to Res(¢) E R*.
(b) It is clear from the definition of equicontinuity that a nonexpanding map is equicontinuous. Indeed, the iterates of a nonexpanding map are uniformly continuous,
D
and indeed, even uniformly Lipschitz (cf. Section 5.4 and Exercise 5.9).
As their name suggests, rational maps with good reduction behave well when
they are reduced.
Theorem 2.18. Let¢ : JP> 1 ---. JP> 1 be a rational map that has good reduction.
(a) ~(F)=¢(?) for all P E JID 1 (K).
(b) Let 7/J : JP> 1 ---. JP> 1 be another rational map with good reduction. Then the composition ¢ o 7/J has good reduction, and
¢o'ljJ = ¢o'ljJ.
Proof (a) Write ¢ = [F(X, Y), G(X, Y)] in normalized form with homogeneous polynomials F, G E R[X, Y], and write P = [a, ,8] in normalized form
with a, ,8 E R. The good reduction assumption tells us that at least one of F(a, ,8)
and G(a, ,8) is in R*, so the point
¢(P) = [F(a, ,8), G(a, ,8)]
is already in normalized form. Hence
60
2. Dynamics over Local Fields: Good Reduction
where the second equality simply reflects the fact that the reduction map R ---+ k is a
homomorphism.
(b) Write¢ = [F(X, Y), G(X, Y)] and 1/J = [f(X, Y), g(X, Y)] in normalized
form with homogeneous polynomials F, G, j, g E R[X, Y]. Then the composition
is given by
(¢ o 1/J)(X, Y) =[A( X, Y), B(X, Y)]
= [F(f(X, Y),g(X, Y)), G(f(X, Y),g(X, Y))].
Clearly A( X, Y) and B(X, Y) have coefficients in R. Suppose that their reductions
A and B have a common root [a, ;3] E lP' 1 ( k). This means that
fr(](a,;J),?J(a,;J)) = o
and
c(](a,;J),?J(a,;J)) = o,
so F and Ghave the common root [](a, ;3), g(a, ;3)]. But¢ has good reduction, so F
and Ghave no common root in lP' 1 ( k), and hence we must have
](a, ;3) = g(a, ;3) = 0.
But this contradicts the assumption that 1/J has good reduction. This proves that the
polynomials A(X, Y) and B(X, Y) have no common root in lP' 1 (k), and therefore
¢ o 1/J = [A, B] has good reduction. We have also shown that the pair (A, B) is
normalized, so
~ = [A,iJ] = [F(}:;),G(7;)] = [F(],g),c(],g)] = ¢o;jJ,
D
which completes the proof of the theorem.
Remark 2.19. (a) The good reduction assumption in both parts of Theorem 2.18 is
essential. See Example 2.12 and Exercise 2.13.
(b) For an alternative proof of Theorem 2.18(b) that uses formal properties of resultants and provides additional information about the reduction of the composition
of two maps, see Exercise 2.12.
(c) It turns out that the converse of Theorem 2.18(b) is false. In other words, a
composition ¢ o 1/J may have good reduction, while both ¢ and 1/J have bad reduction.
For example, let ¢([x, y]) = [x 2 , py 2 ] and 1/J([x, y]) = [p 2 x 2 , y 2 ]. Then¢ = [1, 0]
and if;= [0, 1] are constant maps, so¢ and 1/J have bad reduction. However,
so 1/J o ¢ has good reduction.
One might object to this example by noting that there is a change of variables
such that ¢( z) = z 2 j p has good reduction, and similarly for 1/J( z) = p 2 z 2 . Thus if
f(z) = pz, then
¢ 1 (z) =
u-l 0¢0 f)(z) = u-l 0 ¢)(pz) = r (pz = z
1
2)
2
61
2.5. Rational Maps with Good Reduction
has good reduction. However, it is not difficult to modify this example so that ¢ and 'lj;
have bad reduction for all possible changes of variable; see Exercise 2.14.
(d) If we use the scheme-theoretic definition of good reduction as described in Remark 2.16, then both parts of Theorem 2.18 are clear. For example, if¢ and 'lj; are rational maps with good reduction, then they extend to maps¢ R and 'lj; R over Spec( R),
and the commutativity of the diagram
lP'lk
;r,
------+
lP'lk
J;
------+
1 WR 1
lP'lk
1
<l>n
lP'lR ------+ JP>lR ------+ JP>lR
r
lP'lK
1/J
------+
r
r
lP'lK ------+ JP>lK
</>
immediately gives ~ = ¢ o ,((;. Similarly, a point P E lP' 1 ( K) corresponds to
a unique morphism (i.e., a section) PR : Spec(R) ---+ lP'k, from which the equality i{P) = ¢(P) is immediate using the fact that the composition of R-morphisms
¢R o PR behaves well when restricted to the special fiber oflP'k. In other words, the
following diagram commutes:
Spec(k)
1
p
------+
lP'lk
1
J,
------+
lP'lk
1
Spec(R) ------+ lP'lR ------+ JP>lR
Pn
<l>n
An easy, but important, consequence of the theorem on good reduction is that
periodic points behave well under reduction.
Corollary 2.20. Let ¢ : lP' 1 ---+ lP' 1 be a rational map with good reduction. Then
the reduction map sends periodic points to periodic points and preperiodic points to
preperiodic points:
Per(¢)------+ Per(¢)
and
PreFer(¢) ------+ PreFer(¢).
Further, if P E Per(¢) has exact period n and if P E Per(¢) has exact period m,
then m divides n.
Proof Suppose first that Pis periodic of exact period n, soP = ¢n(P). Reducing
both sides modulo p and using Theorem 2.18 yields
which shows that P is periodic. Let m be the exact period of P and write n = mk + r
with 0 ::::; r < m. Then
62
2. Dynamics over Local Fields: Good Reduction
-------
p = ~n(F) = ~r o ~mo ... o c/Jm(F) = ~r(P).
k iterations
The minimality of m implies that r = 0, and hence m divides n. This proves the
assertion about periodic points.
Similarly, if P is preperiodic, say ¢i(P) = ¢i (P), then Theorem 2.18 gives
~i(P) = ¢3 (P). Hence Pis preperiodic.
0
2.6
Periodic Points and Good Reduction
Corollary 2.20 tells us that if ¢ has good reduction, then its periodic points reduce
to periodic points of ~. In this section we analyze the reduction map Per(¢) --+
Per(~) and use our results to study Per(¢). We start with the following theorem,
which is an amalgamation of results due to Li [266], Morton-Silverman [312, 313],
Narkiewicz [325], Pezda [355], and Zieve [454].
Theorem 2.21. Let ¢ : JP> 1 ( K) --+ JP>1 ( K) be a rational jUnction of degree d 2:: 2
defined over a local field with a nonarchimedean absolute value I · lv· Assume that¢
has good reduction, let P E JP> 1 ( K) be a periodic point of¢, and define the following
quantities:
n
The exact period of P for the map ¢.
m
The exact period of Pfor the map ~.
r
The order of >..4> (F) = (¢m )' (F) in k*. (Set r = oo if >..4> (F) is not a root of
unity.)
p
The characteristic of the residue field k of K.
Then n has one ofthefollowingforms:
n =m
or n = mr or n = mrpe.
Remark 2.22. Let E / K be an elliptic curve defined over a local field, and assume
that E has good reduction. Then one knows [410, VII.3.1] that the reduction map
E(K) --+ E(k) is injective except possibly on p-power torsion, where pis the characteristic of the residue field k. This is very similar to the statement ofTheorem 2.21.
In the case of elliptic curves, it is also possible to bound the power of pin terms of the
ramification index of pinK. We discuss below (Theorem 2.28) analogous bounds
in the dynamical setting.
Proof We make frequent use of Theorem 2.18, which tells us that
j{Q) = ~i(Q)
for all Q E JP> 1 (K) and all i 2:: 0.
Recall that we used this relation in Corollary 2.20 to prove that the ¢-period of P is
divisible by the ~-period of P, which, in our current notation, says that m divides n.
63
2.6. Periodic Points and Good Reduction
Replacing¢ by ¢m and m by 1, we are reduced to the case that Pis a fixed point
of¢. Having done this, we note that >.J,(P) is equal to i(P).
If ¢(P) = P, then n = m and we are done. We thus assume that ¢(P) f. P. To
simplify notation, we use Proposition 2.11 to find a transformation f E PGL 2 (R)
with f([O, 1]) = P. Replacing P and¢ with f- 1 (P) and¢!= f- 1 o ¢of, respectively, we may assume that P = [0, 1]. Dehomogenizing z = X/Y, we write¢ in
the form
¢(z) = aozd + a1zd- 1 + · · · + ad-1z +ad
bozd + b1zd- 1 + · · · + bd-1Z + bd
with coefficients aq_, . .. , bd E Rand at least one coefficient in R*. The fact that [0, 1]
is a fixed point of¢ says that
so ad E p and bd E R*. (We are also using the fact that ¢ has good reduction, of
course.)
Multiplying numerator and denominator by 1 , we may thus write ¢( z) in the
form
¢(z) = ad+ ad-1Z + · · · + a1zd- 1 + a0zd
b;;
1 + bd-1Z + · · · + b1zd- 1 + bozd
The first couple of terms of the Taylor expansion of¢ around z = 0, which in this
case may be obtained by simple long division, look like
(2.6)
with
A(z),B(z) E R[z],
,\ = ¢'(0),
and
J.L =ad E p.
A simple induction argument using (2.6) shows that
(2.7)
In particular, since ¢n (0) = 0 and J.L E p, we find that
1+..\+..\ 2 +···+..\n- 1 =:0
(modp).
(2.8)
The analysis now splits into two cases. First, suppose that r 2 2, or equivalently,
,\ ;f:. 1 (mod p). Then formula (2.8) implies that,\n = 1 (mod p), so we find that r
divides n. If n = r, the proof is complete. Otherwise, we replace ¢ by ¢r and n
by n/r. By an abuse of notation, we continue to write
¢(z) = J.L + ..\z +
A(z)
1 + zB(z)
z2
with the understanding that the values of J.L, ..\, A(z), and B(z) may have changed.
The principal effect of replacing ¢ by ¢r is that we are now in the situation that
2. Dynamics over Local Fields: Good Reduction
64
A= 1
(mod p),
i.e., the new value of r is 1, which brings us to the second case that we need to
consider.
To recapitulate, we have a rational function ¢ satisfying
¢n(o) = 0,
J-L = ¢(0)
= 0 (modp),
and
A= ¢'(0) = 1 (modp).
We are further assuming that ¢(0) #- 0 (otherwise, we are done), so (2.8) becomes
Thus n is divisible by p. We replace¢ by <P' and n by njp. If now ¢(0) = 0, we are
done. If not, the same argument shows that n is again divisible by p. Repeating, we
continue dividing n by p until finally we reach n = 1. This concludes the proof that
the original period n has one of the forms
n =m
or
n = mr
or
n = mrpe
for some e :2 1.
0
We have seen that maps with good reduction are n~nexpanding. This implies that
their periodic points are nonrepelling. If the reduction¢ is separable, we can say even
more. (Recall that ¢( z) E k (z) is separable if it is not in k (zP). See Exercise 1.10
for details.)
Corollary 2.23. Let ¢ : lP'1 ~ lP'1 be a rational map that has good reduction.
(a) Every periodic point of¢ is nonrepelling.
(b) If the reduction ¢ is separable, then ¢ has only finitely many attracting periodic
points.
Proof (a) Theorem 2.18 tells us that ¢n has good reduction. Let P be a periodic
point of¢ of exact period n. Using Lemma 2.5, we can make a change of coordinates
so that P = [0, 1]. Then we can write ¢n(z) in normalized form as
The fact that ¢n has good reduction implies that b0 E R*, since otherwise z = 0
would be a common root of P and G. Hence
so IAp ( ¢) Iv ~ 1, which shows that P is a nonrepelling point for ¢.
(b) Again let P be a periodic point for ¢ of exact period n, and let m be the period
of the reduced point P. Then we have equivalences
2.6. Periodic Points and Good Reduction
65
Pis attracting¢::::::> l-\p(¢)1v = i(¢n)'(P)Iv < 1
¢::::::> (¢n )' (P) = 0
¢::::::> (J>n)'(F) = 0
¢::::::>¢'(F). ¢'(¢F). ¢'(¢2 F) ... J>'(J>n- 1 F)= o
¢::::::> ( ¢' (F) . ¢' (;f>F) . ¢' (¢ 2 F) ... ¢' (J>m- 1 F)
¢::::::} (J>m)'(F)n/m =
tim o
=
0
¢::::::> 0 4> (F) contains a critical point.
The fact that J> is separable implies that a version of the Riemann-Hurwitz formula
(Theorem 1.1) is valid; see Exercise 1.1 0. Hence the map J> _has finitely many (precisely, at most 2d- 2) critical points, and a fortiori, the map ¢ has only finitely many
periodic orbits containing a critical point. In particular, there is a finite list of possible
periods for F. Further, we know that the multiplier ,\J,(F) = (J>m)'(F) ofF is 0,
so Theorem 2.21 tells us that n = m. There are thus only finitely many possibilities
for the period of P, and since ¢ has finitely many points of any given period, we
conclude that ¢ has only finitely many attracting periodic points.
D
Remark 2.24. The separability assumption in Proposition 2.23 is necessary, as is
shown by the example ¢( z) = zP, all of whose periodic points are attracting. See
also Exercise 2.16.
Example 2.25. Let ¢(z) E Z 2 [z] be a polynomial of degree d 2:: 2 whose leading
coefficient is a 2-adic unit. Then ¢ has good reduction. Let P E JID1 ( Q2 ) be a periodic
point of exact period n 2:: 2. In the notation of Theorem 2.21, n = mr2e, where m
is the period ofF in JID 1 (JF 2 ) and r is the order of AJ> (F) in lF2. But JID 1 (JF 2 ) has only
three points, and the fact that ¢ is a polynomial means that the point at infinity is not
in the orbit of P, so either m = 1 or m = 2. Similarly, we note that lF2 has only one
element, so r = 1. It follows that n = 28 for some s 2:: 0.
Similarly, if¢( z) E Z 3 [z] is a polynomial of degree d 2:: 2 with leading coefficient a 3-adic unit, and if P E JID 1 ( Q 3) is a periodic point of exact period n 2:: 2, then
we find that n = mr3e with 1 :-::; m :-::; 3 and 1 :-::; r :-::; 2. Thus n = 2t · 3u for some
0 :-::; t :-::; 2 and some u 2:: 0.
Finally, let ¢(z) E Z[z] be a polynomial of degree d 2:: 2 whose leading coefficient is relatively prime to 6. Then ¢ has good reduction at 2 and 3, so the period n
of a periodic point P E JID 1 (Q) satisfies both n = 28 and n = 2t · 3u with t :-::; 2.
This proves that n is either 1, 2, or 4. The examples ¢(z) = z 2 with z = 0 and
¢(z) = z 2 - 1 with z = 0 show that n = 1 and n = 2 are possible. Can you find an
example with n = 4? See Exercise 2.20 for a stronger version of this example.
The above example illustrates how the local result given in Theorem 2.21 can be
used to derive strong bounds for the periods of periodic points defined over number
fields by applying the theorem to two different primes. We now use the same argument to give a general result that, although not the strongest possible bound using
these methods, is sufficient for many applications.
66
2. Dynamics over Local Fields: Good Reduction
Corollary 2.26. Let K be a number field, let <P : JP> 1 ~ JP> 1 be a rational map defined
over K, and let p and q be primes of K such that <P has good reduction at both p
and q and such that the residue characteristics of p and q are distinct. Then the
period n ofany periodic point of <P in JP> 1 ( K) satisfies
n:::; (Np 2 -1)(Nq 2 - 1),
where Np and Nq denote the norms of p and q respectively.
In particular, the set Per(¢, K) of K -rational periodic points is finite. (For an
alternative proof of the finiteness of Per(¢, K) using the theory of height functions,
see Theorem 3.12.)
Proof Using the obvious notation, we have
mp =(period of <f;(P) mod p) :::; #JP> 1 (lFp) = Np + 1,
rp = (period of >.;p(F) mod p) :::; #lF; = Np- 1,
and similarly for mq and r q. Let p and q denote the residue characteristics of p and q,
respectively. Then Theorem 2.21 says that
Since p and q are distinct primes, it follows that
n:::; mp. rp . mq. rq :::; (Np + 1)(Np- 1)(Nq + 1)(Nq- 1),
which is the first part of the corollary.
The finiteness of Per(¢, K) then follows from the fact that <P has good reduction
at almost all primes of K and the fact that it has only finitely many periodic points
of any given period n.
0
Remark 2.27. The bound for rational periodic points in Corollary 2.26 depends only
weakly on <Pin the sense that the bound is solely in terms of the two smallest primes
of good reduction for ¢. There are many results in the literature using local and/or
global methods that describe bounds for rational periodic points that depend in various ways on the rational map. See for example [52, 87, 90, 91, 162, 171, 312,
325, 326, 328, 332, 353, 355, 358, 359, 361, 454]. However, none of these articles achieves the uniformity predicted by a conjecture that we discuss in Chapter 3
(Conjecture 3.15). This conjecture asserts that for a number field K of degree D and
a rational map <P E K(z) of degree d ~ 2, the number of K-rational preperiodic
points of <P should be bounded solely in terms of D and d.
If K is a discrete valuation ring of characteristic 0, then it is possible to bound
the exponent e appearing in the formula n = mrpe in Theorem 2.21.
Theorem 2.28. (Zieve [454], see also Li [266] and Pezda [355]) We continue with
the notation and assumptions from Theorem 2.21. We further assume that K has
characteristic 0 and we let v : K* - Z be the normalized valuation on K. If the
period n of P E JP> 1 ( K) has the form n = mrpe, then the exponent e satisfies
2.6. Periodic Points and Good Reduction
p
e-1
67
2v(p)
<--.
- p-1
(2.9)
Further, ifp = 2, then the upper bound may be replaced with v(p)f(p- 1). (Note
that v(p) is the ramification index ofp inK.)
Remark 2.29. Let R be the local ring of K and let F be a formal group defined
over R. (See, e.g., [410, chapter IV] for basic material on formal groups.) Theorem 2.28 is a close analogue of the fact [410, IV.6.1] that the torsion in the formal
group F(R) consists entirely of p-power torsion, and that if o: E :F(R) has exact
order pe, then pe-l ~ v(p)f(p- 1).
Example 2.30. Let q be a power of an odd prime p, let ( E Qp be a primitive q1h root
of unity, let K = Qp((), and let ¢(z) = 1 + (z- zq. The maximal ideal of K is
p = (1 - (), so ¢ = 1 + z - zq has good reduction. The point o: = 1 is a point
of exact period q for ¢, since ¢1 (1) = (J, while is clearly a fixed point of¢.
Further, ¢'(a) = 1. Hence in the notation of Theorem 2.21, we haven= q, m = 1,
r = 1, and e is determined by q = pe. On the other hand, the extension K/Qp is
totally ramified, so the normalized valuation v : K* ---» Z satisfies
a
v(p) = [K: Qp] = q(1- 1/p) = pe-l(p- 1).
Thus the inequality (2.9) in Theorem 2.28 becomes pe-l ~ 2pe-l, which shows that
the power of p cannot be improved.
We do not give the proof of Theorem 2.28, but are content to prove the following
special case, which serves to indicate some of the combinatorial issues that arise.
Theorem 2.31. Let p 2: 5 be a prime, let K = Qp, or more generally, an unramified
extension of QP, let ¢ : lP' 1 ____, lP' 1 be a rational map defined over K with good
reduction, and let P E lP' 1 ( K) be a periodic point with the property that
and
¢'(P) = 1.
Then ¢( P) = P. (In the notation of Theorem 2.21, this theorem asserts that if m =
r = 1, then e = 0 and n = 1.)
Proof As in the proof of Theorem 2.21, we begin by moving the periodic point to
the origin and dehomogenizing ¢.Recall that during the proof of Theorem 2.21, we
used a first-order Taylor expansion (2.6) for ¢(z) around z = 0. In order to bound
the exponent e, we need to use the second-order expansion
3
A(z)
2
¢(z)=JL+Az+vz + 1+z B( z )z,
(2.10)
where
A(z), B(z) E R[z],
JL E p,
), = ¢'(0) :== 1 (mod p),
and
v E R.
2. Dynamics over Local Fields: Good Reduction
68
(In general, to prove a sharp estimate for the exponent as in Theorem 2.28, one
needs to consider a longer Taylor expansion. But for the unramified case that we are
considering, the second-order expansion (2.10) suffices.)
Using the results already proven, it suffices to show that if p, =I 0, i.e., if ¢(0) =I 0,
then cf>P(O) =I 0. This will then imply by induction that cf>P" (0) =I 0 for all e 2:: 1, contradicting the assumption that 0 is a periodic point of¢.
During the proof of Theorem 2.21 we gave a simple formula (2.7) for cf>i(O)
modulo J.L 2. In a similar manner we use (2.1 0) to find a formula modulo J.L 3 . To
derive this formula, we write cf>k(O) = p,ak + p, 2 vbk, substitute into (2.10), and do
some algebra to obtain
J.Lak+l + J.L 2 vbk+l = cf>(J.Lak + J.L 2 vbk)
+ A(f.Lak + P, 2 Vbk) + vp, 2 az (mod J.L3 )
= J.L(1 + >.ak) + J.L 2 v(az + >.bk) (mod p,3 ).
=: f.L
This yields the recurrences
ak+l = 1 + >.ak (mod J.L 2 )
and
Starting from a 1 = 1 and b1 = 0, it is now a simple matter to find formulas for ak
and bk and check them by induction. The end result is
(2.11)
We are going to apply (2.11) with k = p. Consider the sum I: >.1. We know that
>.
1 (mod p ), and we are assuming that p is unramified in K, so >. = 1 + cp for
some c E R. Assuming that c =I 0, i.e., that >. #1, we compute
=
~ >.1 = ~ ~ 11 = (1 + c:)P- 1 =
j=O
p
t (~)
i=l
(cp)i-1 = p
(mod p2).
(2.12)
m
Note that the final congruence is true because every binomial coefficient
with
1 :S i < pis divisible by p (and we are assuming that p =I 2). We also observe
that the congruence is true even for>. = 1, although the intermediate calculation is
incorrect.
We perform a similar calculation for the more complicated sum in (2.11 ), but this
time we are interested only in the value modulo p, so we can replace >. by 1:
(2.13)
Note that for the last step we are using the assumption that p ;:::: 5.
Substituting (2.12) and (2.13) into the iteration formula (2.11) (with k = p)
yields
69
2.7. Periodic Points and Dynamical Units
¢?(0) = f.L(P + ap2 ) + {.L 2 vbp
(mod f.L 3 ),
where a and b are in R. The fact that p is unramified in K means in particular that p
divides f.L, so
<f!P(O) := f.LP (mod f.LP 2).
Now using our assumption that f.L -1- 0, we deduce that <f!P(O) -1- 0, which completes
D
the proof of the theorem.
2. 7
Periodic Points and Dynamical Units
Let ( be a primitive p 1h root of unity. Then the expression
is a unit in the cyclotomic field Q( (), a so-called "cyclotomic unit." Similarly, if ( 1
and ( 2 are roots of unity of relatively prime orders, then the difference ( 1 - ( 2 is an
algebraic unit. The crucial fact underlying these constructions is that distinct roots
of unity remain distinct when they are reduced modulo primes. It follows that their
differences are not divisible by any primes, and hence that they are units.
Theorem 2.21 can be used to deduce conditions under which distinct periodic
points remain distinct when reduced modulo primes, so it can be used to construct
units in a similar fashion. These constructions can be done either using different
points in a single periodic orbit or using points of different periods. The following
proposition provides us with the information needed to construct units of various
kinds. In fact, since Lemma 2.8 tells us that the v-adic chordal metric satisfies
Pv(P, Q) < 1
<==:;.
P = Q,
the proposition actually says something stronger than the simple assertion that certain
pairs of points have distinct reductions.
Proposition 2.32. Let ¢( z) E K ( z) be a rational function of degree d 2 2 with
good reduction.
(a) Let P E lP' 1 ( K) be a point ofperiod n for ¢. Then
Pv(¢iP,¢JP) = Pv(¢i+kp,¢J+kp) foralli,j,k E Z,
where for i < 0 we use the periodicity ¢n P = P to define ¢i P.
(b) Let P E lP' 1 ( K) be a point of exact period n for¢. Then
Pv(¢iP, ¢1 P) = Pv(¢P, P)
for all i,j E Z satisfYing gcd(i- j, n) = 1.
(c) Let P1, P2 E lP' 1(K) be periodic points for ¢ of exact periods n 1 and n 2, respectively. Assume that n1 f n2 and n2 f n1. Then
2. Dynamics over Local Fields: Good Reduction
70
Proof (a) Proposition 2.14 and the good-reduction assumption imply that
Pv(Q, R) ~ Pv(¢Q, ¢R)
for all Q, R E JP>1 (K).
Applying this repeatedly yields
Pv(Q,R) ~ Pv(¢Q,¢R) ~ Pv(¢2Q,¢ 2R)
~ Pv(¢3 Q,¢3 R) ~ ··· ~ Pv(¢nQ,¢nR).
If we now make the further assumption that Q and R are points of period n, then
Pv(¢nQ, ¢n R) = Pv(Q, R), so all of the inequalities must be equalities. This proves
that if ¢n(Q) = Q and ¢n(R) = R, then
Pv(Q, R) = Pv(¢kQ, ¢k R)
for all k E Z.
Substituting Q = ¢i P and R = q) P for the given point P of period n completes
the proof of (a).
(b) We know from (a) that the distance Pv(¢iP,q)P) depends only on the difference i - j. We use this fact and the nonarchimedean triangle inequality (Theorem 2.4(d)) to estimate
Pv(P, ¢k P) ::::; max{pv(P, ¢P), Pv(¢P, ¢ 2 P), ... , p(¢k-t P, ¢k P)}
= Pv(P, ¢P).
(2.14)
In a similar fashion we obtain the estimate
Pv(P, ¢P) ::::; max{pv(P, ¢k P), Pv(cPk P, ¢ 2kP), Pv(¢ 2kP, ¢ 3 kP),
... 'Pv(<f}m-l)k P, cf>mk P), Pv(cf>mkp, cf>P)}
= max{pv(P, cPk P), Pv(cPmk P, ¢P)}.
(2.15)
If we now make the further assumption that gcd(k, n) = 1, then we can find an
integer m satisfying mk 1 (mod n). Then ¢mk P = ¢P, so (2.15) becomes
=
Pv(P, cf>P) ::::; max{pv(P, ¢k P), Pv(cf>P, cf>P)} = Pv(P, ¢k P).
(2.16)
Combining (2.14) and (2.16) yields
Pv(P, ¢P) = Pv(P, ¢k P)
for all k satisfying gcd(k, n) = 1.
...
In particular, if gcd (i - j, n) = 1, we can use this formula and (a) to compute
Pv(¢iP, 4) P) = Pv(P, ¢i-i P) = Pv(P, ¢P).
(c) Suppose that Pv(Pt,P2) < 1, or equivalently from Lemma 2.8, suppose that
A = P2 • Let m be the exact period of P1 and let r be the order of its multiplier
>-.¢(P1 ). Then Theorem 2.21 tells us that both n 1 and n 2 are in the set
{ m, mr, mrp, mrp 2 , mrp3 , ... } .
It follows that either n 1 divides n 2, or that n 2 divides nt. contradicting the assump0
tion on n1 and n2. Therefore Pv(Pt, P2) = 1.
2.7. Periodic Points and Dynamical Units
71
It is a simple matter to use Proposition 2.32 to construct units from periodic
points. We begin with the easier case of a polynomial mapping.
Theorem 2.33. (Narkiewicz [325]) Let ¢(z) E R[z] be a polynomial of degree
d 2:: 2 whose leading coefficient is a unit in R. Let a E K be a periodic point of¢ of
exact period n (with n 2:: 2), and let i, j E Z be integers satisfYing gcd (i - j, n) = 1.
Then
¢ia- q)a
¢a-a
----E
R*
.
Proof. The assumption that ¢( z) is a polynomial with unit leading coefficient implies that ¢ has good reduction, since
We next observe that every periodic point of¢ is integral over R, since it is a root of
an equation q;n (z) - z = 0. Finally, we note that the distance between integral points
a, (3 E R is given by
Pv(a,(3) = Pv([a, 1], [(3, 1]) = Ia- f31v·
Using this formula and Proposition 2.32(b), we compute
¢)a I = l¢ia- ¢3alv = Pv(¢ia, ¢)a) = 1.
I¢ia¢a- a v
i¢a- aiv
Pv(¢a, a)
0
We create units from periodic points of arbitrary rational maps using a cross-ratio
construction.
Definition. Let P 1 , P 2 , P3 , P4 E JP> 1 ( K), and choose homogeneous coordinates
Pi = [xi, Yi] for each point. The cross-ratio of Pt, P2, P3, P4 is the quantity
Notice that "-(P1 , P2 , P3 , P4 ) is independent of the choice of homogeneous coordinates for the points.
Theorem 2.34. (Morton-Silverman [313, Theorem 6.4(a)]) Let¢ E K(z) be a
rational map of degree d 2:: 2 with good reduction. Let P E JP> 1 ( K) be a periodic
point for ¢ of exact period n, and let i and j be integers satisfYing
gcd(j,n) = gcd(i -1,n) = gcd(i- j,n) = 1.
Then
72
2. Dynamics over Local Fields: Good Reduction
Proof Comparing the definition of the cross-ratio to the definition of the chordal
metric, we see that
The assumptions on i and j and Proposition 2.32(b) tell us that
Hence
which proves that the cross-ratio is a unit.
0
We can also construct units using periodic points of different periods. This may
be compared with the two different types of cyclotomic units,
and
where (n indicates a primitive nth root of unity and the condition gcd( m, n)
ensures that (m - (n is a unit.
1
Theorem 2.35. Let ¢ E K ( z) be a rational map of degree d ~ 2 with good reduction. Let n1, n2 E Z be integers with n1 f n2 and n2 f n1, let P1. P2 E 1P' 1( K) be
periodic points of exact periods n 1 and n 2 respectively, and write Pi = [xi, Yi] in
normalized form. Then
Proof Since we have taken normalized homogeneous coordinates, the chordal metric is given by
Pv(Pl,P2) = lx1Y2 -x2Y1Iv·
The assumptions on n 1 and n2 and Proposition 2.32(c) tell us that Pv(Pl, P2) = 1,
0
and hence X1Y2 - X2Y1 is a unit.
Remark 2.36. For further information on the geometric and arithmetic properties of
periodic points and the fields that they generate, see Sections 3.9, 3.11, and 4.1-4.6.
In some sense, periodic and preperiodic points play a role in arithmetic dynamics
analogous to the role played by torsion points in the arithmetic of elliptic curves or
abelian varieties, albeit the dynamical setting has considerably less in the way of
helpful structure.
Example 2.37. We use the polynomial <P(z) = z 2 + 1 to construct units. First we
compute
2.7. Periodic Points and Dynamical Units
73
cf>(z) - z = z 2 - z + 1,
¢ 2 (z)- z = z4 + 2z 2 - z + 2 = (z 2 - z + 1)(z 2 + z + 2),
¢ 3 (z)- z = z 8 + 4z6 + 8z 4 + 8z 2 - z + 5
= (z 2 - z + 1)(z6 + z 5 + 4z 4 + 3z 3 + 7z 2 + 4z + 5).
It is not surprising that cf>(z) - z divides both ¢ 2 (z)- z and ¢ 3 (z)- z, since any
root of the fonner is clearly a root of the latter two; cf. Exercise 1.19(a). We have
Per~*(¢)= { 1 ± f3}
and
Per;*(¢)= { -1 ±2R},
where recall that Per~* (¢) denotes the set of points of exact period n for¢.
Further, the six roots of the polynomial
constitute a set of the fonn
since we know that the roots of <1>3 (z) are pennuted by ¢ into two periodic cycles of
period 3. In particular, the splitting field K (a, (3) of <1>3 is a Galois extension of K
of degree at most 18.
Applying Theorem 2.33 with i = 2 and j = 1 gives a unit
We can check that u 1 is a unit in R[a] by computing
Similarly, we can apply Theorem 2.33 with i = 2 and j = 0 to obtain a second unit
And replacing a with ¢(a) or with ¢ 2 (a) in u 1 and u 2 gives four more units,
u~ = ¢(a) 2 +¢(a)+ 1 = a 4 + 3a 2 + 3,
u~ = ¢ 2 (a) 2 + ¢ 2 (a) + 1 = a 4 + 2a2 +a+ 2,
u~ = ¢(a) 2 +¢(a)+ 2 = a 4 + 3a 2 + 4,
u~ = ¢ 2 (a) 2 + ¢ 2 (a) + 2 = a 4 + 2a2 +a+ 3.
(Note that in doing these computations, we make use of the fact that ¢3(a) = 0.)
74
Exercises
It turns out that these six units generate a group of rank (at most) 2, since an easy
computation yields
This is not surprising, since one can check that the polynomial IP3(x) is irreducible
over Q and that all of its roots are complex. Thus the global field Q( o:) is a totally
complex extension field of degree 6, so its unit group has rank 2.
We can use Theorem 2.35 to create more units. Taking n 1 = 2 and n 2 = 3 and
letting 'Y = ~( -1 + v'-7), we have
for 0 5: i 5: 1 and 0 5: j 5: 2.
The extension Q(o:, 'Y)/Q is totally complex of degree 12, so has five independent
units. We leave as an exercise for the reader to compute the number of independent
units in the set {ui,j}.
Exercises
Section 2.1. The Nonarchimedean Chordal Metric
2.1. Let K be a field that is complete with respect to an absolute value v and let ¢>(z) E K[z]
be a polynomial. The filled Julia set IC( ¢>) of¢> is the set
IC(¢>) = {a E K: l1>n(a)lv is bounded for n = 1, 2, 3, ... }.
(Note that this definition applies only to polynomials, not to more general rational maps.)
(a) Prove that the filled Julia set IC(¢>) is a closed and bounded subset of K. Note that if K
is locally compact, for example K = Qp or K = C, then this implies that 1C( ¢>) is
compact. However, there are cases of interest for which this is not true, e.g., the completion Cp of an algebraic closure ofQp.
(b) Prove that the complement of JC(¢>) is equal to the set of all P E JP> 1 satisfying
lim ¢>n(P) = oo,
n-oo
{2.17)
where the limit is in the v-adic topology. Equivalently, prove that it is the set of P satisfying limpv(¢>n(P),oo)-+ 0 or, writing P = [a,l], the set of points satisfying
lif>n(a)lv-+ oo. The set determined by (2.17), which from (a) is an open set, is called
the attracting basin of oo.
(c) Prove that the Julia set .J(¢>) of¢> is the boundary of the filled Julia set IC(¢>).
(d) Prove that the filled Julia set IC(¢>) is completely invariant, i.e., prove that it satisfies
rl(IC(¢>)) = JC(¢>) = if>(JC(¢>)).
Section 2.3. Reduction of Maps Modulo a Prime
2.2. Let f = ( ~ ~) E PGL2(K) with a, b, c, d E R, at least one of a, b, c, din R*, and
ad be (mod p). Prove that there exist points P, Q E JP> 1 ( K) satisfying
=
P = Q and i{P)-::/= f(Q).
Thus Proposition 2.9 is false in general iff f. PGL2(R).
Exercises
7S
2.3. Generalize Proposition 2.11 to IP'2 as follows. Let
Q1 = [1,0,0],
Q2 = [0, 1,0],
Q3 = [0,0, 1],
Q4 = [1, 1, 1].
Suppose that P1 , P2, P3 , P4 E IP'2 ( K) are points whose reductions P1 , ... , P4 are distinct
and have the property that no three of them lie on a line in IP' 2 (k). Prove that there is a transformation f E PGL 3 (R) such that f(Pi) = Qi for alii ~ i ~ 4.
Formulate and prove an analogous statement for IP'N.
2.4. Let ¢ = [F, G] = [F', G'] be two normalized representations of the rational map
¢ : JP' 1 --+ IP' 1 . Prove that there is a constant u E k* such that
uF=F'
uG = 6'.
and
Deduce that the reduced map ~ : IP' 1 ( k) --+ IP' 1 ( k) does not depend on the choice of normalized
representation.
Section 2.4. The Resultant of a Rational Map
2.5. (a) Let A(X, Y) = a0 X + a1 Y be a linear polynomial and let B(X, Y) be an arbitrary
homogeneous polynomial. Prove that Res(A, B) = B( -a 1 , a0 ).
(b) Let A(X, Y) = a0 X 2 + a1XY + a2Y 2 and B(X, Y) = b0 X 2 + b1XY + b2Y 2 be
quadratic polynomials. Prove that
4Res(A, B)= (2aob2- a1b1 + 2a2bo) 2 - (4aoa2- ai)(4bob2- bi).
(Of course, in characteristic 2 one needs to cancel 4 from both sides before using this
formula!)
2.6. With notation as in the statement of Proposition 2.13, prove that the resultant of A and B
is related to the roots of A and B by
n
Res( A, B)= a~b~
2.7. Let
n
m
i=l
j=l
II II(a;- f1i) =a~ II B(a;, 1) = (-1)mnb~ II A((1j, 1).
m
i=l j=l
A(X, Y) = aoXn + a1Xn- 1Y + · · · + an-1xyn- 1 + anYn,
B(X, Y) = boXm + b1Xm- 1 Y + · · · + bm-lxym- 1 + bmYm,
be homogeneous polynomials of degrees n and m and let a, (1, '"Y, 8 E K be arbitrary.
(a) Prove that
Res( A( aX+ (1Y,"fX + 8Y),B(aX + (1Y,7X + 8Y))
= (a8- f37)mn Res(A(X, Y), B(X, Y)).
(Hint. Use Proposition 2.13(b).)
(b) Suppose that m = n = d. Prove that
Res( aA(X, Y) + (1B(X, Y), 7A(X, Y) + 8B(X, Y))
= (a8- f1'"Y)dRes(A(X, Y),B(X, Y)).
(Hint. Use Proposition 2.13(d).)
76
Exercises
(c) Continuing with the assumption that m = n = d, define new polynomials by the formulas
A*(X, Y) = M(o:X + {3Y,-yX + oY)- {3B(o:X + {3Y,-yX + oY),
B*(X, Y) = --yA(aX + {3Y,-yX + oY) + o:B(o:X + {3Y, -yX + oY).
Prove that
Res(A *, B*) = (ao- {3-y)d 2 +d Res(A, B).
(d) Suppose that m = n = d and that Res( A, B) =I 0, so ¢ = [A, B] is a rational map
¢ : IP' 1 ---+ IP' 1 of degree d, and suppose further that ao - {3-y =I 0, so the map
f = [o:X + {3Y,-yX + oY]
is a linear fractional transformation. Let¢* = [A*, B*]. Prove that¢* = f- 1 o ¢of.
Use (c) to prove that¢* : IP' 1 ---+ IP' 1 is a rational map of degree d.
2.8. Let f E PGL 2 (K) be a linear fractional transformation, and write
f(z) = az + b
cz +d
in normalized form.
(a) Prove that Res(!) =ad- be.
(b) Prove that
p(f(P),J(Q))::::; 1Res(f)l- 1 p(P,Q)
for all P, Q E IP' 1 (K).
(Notice that this strengthens Theorem 2.14 for maps of degree 1.)
2.9. Let f E PGL2(K) be a linear fractional transformation, let ¢(z) E K(z) be a rational
1 0 ¢ 0 f.
map, and let ¢ 1 =
(a) IfRes(f) E R*,provethatRes(¢f) =Res(¢).
(b) If Res(!) is not a unit, find a formula or an inequality relating the valuations of the three
quantities Res(!), Res(¢), and Res( ¢f).
r
2.10. Let ¢ : IP' 1 ---+ IP' 1 be a rational map defined over a field K with a nonarchimedean
absolute value I · Iv, and consider the statement
Pv ( ¢(P1), ¢(P2))
sup
P1,P2Eil" 1 (K)
1
Pv(P1, P2)
(2.18)
PFIPz
(a) Prove that (2.18) is true for the map ¢(z) = azd, where a E R. This example shows
that Theorem 2.14 cannot be improved in general.
(b) Prove that (2.18) is not true for the map
¢ = [pX 2 + XY, XY + pY 2 ]
over the field K = Qp by computing both sides. (You may assume that p =I 2.)
2.11. Let¢ : IP' 1 (!C) ---+ IP' 1 (!C) be a rational map of degree d.
Exercises
77
(a) Prove that there is a constant C(d), depending only on d, such that
(b) Find an explicit value for the constant C(d).
Note that this exercise provides an archimedean counterpart to Theorem 2.14, whose proof
may, mutatis mutandis, be helpful in doing this exercise.
Section 2.5. Rational Maps with Good Reduction
2.12. Let F(X, Y) and G(X, Y) be homogeneous polynomials of degree D, let f(X, Y)
and g(X, Y) be homogeneous polynomials of degree d, and let
A(X, Y) = F(f(X, Y),g(X, Y))
and B(X, Y) = G(f(X, Y),g(X, Y))
be their compositions.
(a) Prove that the resultants satisfy
2
Res( A, B) = Res( F, G) d · Res(!, g) D .
(b) Use (a) to give an alternative proof of Theorem 2.18(b).
(c) For any rational map¢ : IP'1 -+ !P'l, define Ov(¢) = v(Res(¢))/ deg(¢), so Ov(¢) is a
kind of normalized resultant of¢. Prove that Ov satisfies the composition formula
Ov(</J o 1/1) = Ov(¢) + deg(¢)8v(1/J).
2.13. Show that the good-reduction assumptions in Theorem 2.18 are necessary by constructing the following counterexamples:
(a) Find a rational map¢ : IP' 1 -+ IP'1 , which will necessarily have bad reduction, and a point
P E IP'1 (K) such that ;;;(P} -:/: ¢(P).
(b) Find rational maps¢ : IP'1 -+ IP' 1 and 1/J : IP' 1 -+ IP' 1 such that¢ has good reduction and
~-:f.¢o~.
(c) Same as (b), except now 1/J is required to have good reduction and ¢> is allowed to have
bad reduction.
2.14. Let p ~ 5 be a prime and define rational maps
¢(z) = z2 + p3z
p3 z2 + p
and
1/l(z) = p2z2 + p4
p3 z + 1
(a) Prove that ¢ 1 has bad reduction modulo p for all f E PGL2(Qp). (N.B. We are allowing f to have coefficients in Qp and/or the determinant off to be divisible by p.)
(b) Prove that 1f;1 has bad reduction modulo p for all f E PGL 2 (Qp).
(c) Prove that the composition 1/J o ¢ has good reduction at p.
2.15. Let¢ : IP' 1 -+ IP' 1 be a rational map defined over K.
(a) Prove that the map¢ has good reduction if and only ifthere is an R-morphism
</JR: JP>k-+ JP>k
whose restriction to the generic fiber is equal to the original map ¢ : IP'k -+ IP'k.
(b) Assume that ¢ has good reduction. Prove that the reduction ¢ : IP'~ -+ IP'~ is the restriction of rPR to the special fiber ofiP'k.
78
Exercises
2.16. Let K /QP be a p-adic field.
(a) Prove that every periodic point of the map ljJ(z) = zP is attracting.
(b) More generally, suppose that 'ljJ(z) E K(z) has good reduction, and let ljJ(z) = 'ljJ(zP).
Prove that every periodic point of 1/J is attracting.
(c) Is (b) true without the assumption that 'ljJ has good reduction?
2.17. Let K/QP be ap-adic field and let ljJ(z) E K[z] be a polynomial with good reduction.
Let a E K be a critical point of 1/J, i.e., 1/J' (a) = 0, and suppose that a E Per(~). More precisely, suppose that the reduced point a satisfies ~m (a) = a. Prove that there is an attracting
periodic point f3 of 1/J of period m such that
lim 1/Jnm(a) = {3.
n~oo
Generalize to the case of a rational map 1/J( z) E K (z) of good reduction.
2.18. Let 1/J( z) E K (z) be a rational map of degree d ~ 2 and suppose that there is a point
P E lP' 1 ( K) and an integer m ~ 1 such that the following limit exists:
T = lim 1/Jmn(P).
(2.19)
n~oo
Prove that 1/Jm(T) = T, i.e., T E Perm(I/J). (This is true for any complete field K, so for
example, it holds for Qp, Cp, R, C, etc.)
Section 2.6. Periodic Points and Good Reduction
2.19. Let a, b, c E Z and let 1/J( z) = ( az 2+ bz +c)/ z 2 . Suppose that P E lP'1 ( Q) is a periodic
point for 1/J of exact period n.
(a) If gcd( c, 6) = 1, prove that n = 1, 2, or 3.
(b) Give examples to show that it is possible for n to take on each of the values 1, 2, and 3.
(c) Show that if 1/J has a rational periodic point of exact period 3, then it has no other rational
periodic points.
(d) Give a similar description of possible rational periodic points under the assumption that
gcd(c, 10) = 1.
(e) Same as (d), but under the assumption that gcd(c, 15) = 1.
2.20. Let 1/J(z) E Z[z] be a polynomial of degree d ~ 2 whose leading coefficient is odd
and let P E lP'1 (Q) be a periodic point of exact period n. Prove that n E {1, 2, 4}. This
strengthens Example 2.25.
2.21. Let 1/J(z) E Z[z] be a polynomial of degree d ~ 2 whose leading coefficient is relatively
prime to 15 and let P E 1P' 1 ( Q) be a periodic point for 1/J of exact period n. What can you say
about the possible values of n?
2.22. Let R = lFP [T] be the ring offormal power series in one variable with coefficients in the
finite field lF p· Prove that for every e > 0 there exists an element c E R with c 1 (mod T)
such that the polynomialljJ(z) = zP + cz has a periodic point a
0 (mod T) of exact order pe. Thus Theorem 2.28 is false in characteristic p; there is no upper bound on the exponent
of p for the period of a periodic point.
=
=
2.23. Let 1/J E K (z) be a rational map with good reduction and let P E Per( 1/J) be a periodic
point. Prove that P is attracting if and only if the orbit of its reduction 0 4> ( P) contains a
critical point of~.
Exercises
79
2.24. Let c E Rand consider the quadratic map <P(z) = z 2 +c. Suppose that there is an
integer m 2:: 1 such that the limit a = limn-oo <Pmn(o) exists. Prove that a is an attracting
periodic point if and only if 0 E Per(¢), i.e., if and only if 0 is a purely periodic point of¢.
Section 2.7. Periodic Points and Dynamical Units
2.25. Theorem 2.33 gives a construction of units from periodic points of polynomial maps.
Prove the following generalization.
Let <P E K ( z) be a rational map of degree d 2:: 2. Assume that <P has good reduction and
that oo is a critical fixed point of¢. (That is, assume that ¢( oo) = oo and that e00 ( <P) 2:: 2.)
Let [a, 1] E Per~* (¢, K) be a point of exact period n 2:: 2. Let i, j E Z be integers satisfying
gcd(i- j, n) = 1. Prove that
<Piacf}a E R* .
.:..__
_ __:___
<(Ja -a
2.26. Prove the following version of Theorem 2.35, in which one of the periods divides the
other period. Let R be a local ring with residue characteristic p and let q = #k be the order
of the residue field. Let <P E K (z) be a rational map of degree d 2:: 2 with good reduction. Let
s be a positive integer with p f s and s f q - 1. Finally, let P 1 , P2 E lP' 1 ( K) be periodic points
of exact periods nand sn respectively, and write Pi = [xi, Yi] in normalized form. Prove that
2.27. Let H, P2, P3, P4 E lP' 1 , and let 71' be a permutation of the set {1, 2, 3, 4}. Prove that
the cross-ratio satisfies
where E( 71') = (-l)•ign( 11') equals 1 (respectively -1) if 71' is an even (respectively odd) permutation.
2.28. Let ¢( z) = z 2 + 1, let 'Y be a root of z 2 + z + 2, and let a be a root of
z 6 + z 5 + 4z 4 + 3z 3 + 7z 2 + 4z + 5.
In Example 2.37 we explained that there are units
for 0 :S i :S 1 and 0 :S j :S 2,
and that the unit group R["f, a]* has rank at most 5. Find generators and relations for the group
of units generated by the six units Ui,i.
Chapter 3
Dynamics over Global Fields
Just as algebraic number theory and Diophantine geometry can be studied over local
fields and over global fields, so too does arithmetic dynamics have a local theory
and a global theory. In this chapter we study some of the fundamental questions in
arithmetic dynamics over global fields. Many of these constructions, theorems, and
conjectures have direct analogues in the theory of Diophantine equations, especially
the arithmetic theory of elliptic curves. Among the topics covered in this chapter are
dynamical analogues of the theory of canonical heights, finiteness of integral points,
uniformity of rational torsion points, and cyclotomic and elliptic units.
We have attempted to keep this chapter self-contained, but the reader desiring
further motivation and background might consult standard textbooks on arithmetic
and Diophantine geometry, such as the following:
•
•
•
•
•
•
3.1
E. Bombieri and W. Gubler [76], Heights in Diophantine Geometry.
M. Hindry and J. Silverman [205], Diophantine Geometry.
S. Lang [256], Fundamentals ofDiophantine Geometry.
J.-P. Serre [397], Lectures on the Mordell-Weil Theorem.
J. Silverman [410], The Arithmetic ofElliptic Curves.
J. Silverman [412], Advanced Topics in the Arithmetic of Elliptic Curves.
Height Functions
In order to study the arithmetic properties of points in projective space, it is important
to have a method of measuring the size of a point. This "size" should reflect the
complexity of the point in an arithmetic sense. In particular, it is good if there are
only finitely many points of bounded size.
For example, suppose that we naively define the size of a rational number a
to be its absolute value lal. This reflects the size of a as a real number, but there
are infinitely many rational numbers with bounded absolute value, so it does not
correctly measure a's size from an arithmetic perspective. If we write a = a/bas
a fraction in lowest terms, then the integers a and b should each contribute to the
81
82
3. Dynamics over Global Fields
complexity of a, so we might define the size of a to be the larger of Ia I and jbj. With
this definition, it is obvious that there are only finitely many rational numbers of
bounded size, since there are only finitely many integers of bounded absolute value.
We can easily generalize this idea to rational points in projective space.
Example 3.1. Let P E pN (Q) and write P using homogeneous coordinates as
P = [xo,xl, ... ,xN]·
Since the coordinates are homogeneous, we can multiply through by an integer to
clear the denominators, and we can also cancel any common factors, so we may
assume that the homogeneous coordinates have been chosen to satisfy
xo, ... ,xN E Z
and
gcd(xo, ... , XN) = 1.
Having done this, we define the height of P to be the quantity
H(P) = max{lxol, ... , lxNI}.
It is clear that for any constant B, the set
{ P E pN (Q) : H(P) ::; B}
(3.1)
is finite. Indeed, this set clearly has fewer than (2B + l)N+l elements, since each
coordinate Xi of Pis an integer satisfying lxil ::; B, so has at most 2B + 1 possible
values. (See Exercise 3.2 for an asymptotic estimate for the size of the set (3.1).)
When trying to generalize Example 3.1 to an arbitrary number field K, we run
into the problem that the ring of integers of K may not be a principal ideal domain, so there is no uniform way to normalize the homogeneous coordinates of a
point in JP>N (K). For this reason we take a different approach based on the theory of
absolute values. We now describe the absolute values on Q, and then move on to general number fields. For further information about absolute values and completions of
number fields, see any standard textbook on algebraic number theory or Diophantine
geometry, for example [258, Section 2.1], [256, Chapters I, II], or [205, Section B.l].
Definition. The set of standard absolute values on Q, denoted by MQ, consists of
the following absolute values:
• MQ contains one archimedean absolute value that is associated to its embedding Q c lR into the real numbers,
lxloo =usual absolute value on lR = max{ x, -x }.
• Let p be a prime, and for any (nonzero) integer a, let
ordp (a) = exponent of highest power of p dividing a.
In other words, pordv(a) is the highest power of p dividing a. The set MQ contains nonarchimedean (p-adic) absolute values, one for each prime p, defined
by
3.1. Height Functions
83
Now let K jQ be a number field. The set ofstandard absolute values on K is denoted
by MK and consists of all absolute values on K whose restriction to Q is one of the
absolute values in MQ. We write M'j( for the (archimedean) absolute values on K
lying above I · loo. and Mi for the (nonarchimedean) absolute values of K lying
above the p-adic absolute values ofQ.
The archimedean absolute values of K correspond to embeddings of K into lR
or C, while the nonarchimedean absolute values correspond to prime ideals of the
ring of integers of K. Indeed, the ring of integers of K, denoted by RK, may be
characterized as the set
RK ={a E K: lalv:::; 1 forallv E Mi}.
More generally, if S c MK is any (finite) set of absolute values containing M'j(,
then the ring of S-integers of K is the set
Rs ={a E K: lalv:::; 1 forallv ~ S}.
Definition. For any absolute value v E MK on K, let Kv denote the completion
of Kat v. The local degree ofv, denoted by nv, is the quantity
For example, if v is archimedean, then Qv = lR and nv is 1 or 2 depending on
whether v corresponds to a real or complex embedding of K, respectively. Similarly,
if v is nonarchimedean, then Qv = Qp is the p-adic rational numbers for some
prime p, the absolute value v corresponds to some prime ideal p of K lying over p,
and using standard notation, nv = e (p) f (p) is the product of the ramification index
ofp and the degree of the residue field modulo p.
The next two results, which we quote without proof, will be used to define and
study height functions.
Proposition 3.2. (Extension Formula) Let L / K jQ be a tower ofnumberfields, and
let v E MK be an absolute value on K. Then
Here the notation wlv means that the restriction ofw to K is equal to v, so the sum
is over all absolute values on L that extend the absolute value v on K.
Proposition 3.3. (Product Formula) Let K jQ be a number field. Then
for all a E K*.
3. Dynamics over Global Fields
84
For proofs of these two formulas, see for example [258, Section II. I and Section V.l]. The astute reader will have observed that if we take K = Q and v nonarchimedean, then the extension formula becomes the well-known formula
L e(p)f(p) = [L: Q].
PIP
We now have all of the tools needed to define the height of algebraic points in
projective space.
Definition. Let KIQ be a number field, and let P E lP'N (K) be a point with homogeneous coordinates
P= [xa, ... ,xN],
xa, ... ,xN E K.
The height of P (relative to K) is the quantity
HK(P) =
II max{lxolv, ... , lxNivrv.
vEMK
Notice how the height measures the size of the coordinates of P with respect to
all of the absolute values on K. We begin by checking that the height is well-defined
and proving two elementary properties.
Proposition 3.4. Let K I Q be a number field and P E lP'N (K) a point.
(a) The height HK(P) is independent of the choice of homogeneous coordinates
forP.
(b) HK(P) 2: 1.
(c) Let L I K be a finite extension. Then
HL(P) = HK(P)[L:KJ.
Proof (a) Any other choice of homogeneous coordinates for P = [xo, ... , x N]
has the form P = [o:x 0 , ... , o:xN] for some o: E K*. Then the product formula
(Proposition 3.3) yields
II max{lo:xilv}nv = II lo:l~v max{xilv}nv = II max{lxilv}nv.
vEMK
t
vEMK
t
vEMK
t
(b) Choose an index j such that x 1 i- 0 and divide the homogeneous coordinates
of P by x 1. Then one of the homogeneous coordinates is equal to 1, so every factor
in the product defining HK(P) is at least 1.
(c) We choose homogeneous coordinates for P that are in K and use the extension
formula (Proposition 3.2) to compute
HL(P) =
II max{lxilv}nw = II II max{lxilv}nw
wEML
t
vEMK wEML
wlv
This completes the proof of Proposition 3.4.
t
0
3.1. Height Functions
85
Remark 3.5. Now that we know that the height is well-defined, we should check
that it agrees with the naive definition of the height on jp>N (Q) given in Example 3.1.
Thus let P = [x 0, ... , XN] E J!DN (Q) with homogeneous coordinates satisfying
Xi E Z and gcd(xi) = 1. Then every nonarchimedean absolute value v E
gives lxilv :::; 1 for all indices i and lxilv = 1 for at least one index i. Hence in the
product defining HQ(P), only the term corresponding to the archimedean absolute
value contributes, so
M8
We note again that the set
is finite for any fixed bound B. Later in this section we prove an analogous result for
arbitrary number fields.
It is sometimes easier to work with a height function that does not depend on
choosing a particular number field.
Definition. Let P E J!DN (Q) be a point whose coordinates are algebraic numbers.
The (absolute) height of P, denoted by H(P), is defined by choosing any number
field K such that P E J!DN (K) and setting
H(P) = HK(P)lf[K:Q]_
The transformation formula in Proposition 3.4(c) tells us that H(P) is well-defined,
independent of the choice of the field K.
In the following, K denotes an algebraic closure of a number field K.
Theorem 3.6. Let KIQ be a number field, let P E J!DN (K), and let a- E Gal(K I K).
Then
H(a-(P)) = H(P).
In other words, the height is invariant under the action of the Galois group.
Proof Let L I K be a finite Galois extension such that P E pN ( L). For any absolute
value v E ML and any a- E Gal(LIK), we can define a new absolute value a-(v)
on L by the formula
lala(v) = la-(a)lv·
Using the fact that a- : L ---. L is a field automorphism, it is easy to verify that a-( v)
is an absolute value on L, and indeed the map v t-t a-( v) is simply a permutation of
the set of absolute values M£. Further, the automorphism a- : K ---. K induces an
isomorphism of the completions a- : Kv ---. Ka(v)• since the effect of a- on K is to
transform the absolute value v into the absolute value a-( v). In particular, the local
degrees
and
3. Dynamics over Global Fields
86
are equal. We now write P = [x0 , ... , xN] E pN (L) and compute
= HL(P).
Taking [L: Q]th roots completes the proof that H(a(P)) = H(P).
0
Definition. Let P = [x0 , ••• , XN] E pN (K) be a point with algebraic coordinates.
The smallest field over which P can be defined is called the field of definition of P
over K and is denoted by K(P). It can be constructed by choosing a nonzero coordinate Xi and setting
XO, X1
K(P)=K ( , ... ,XN)
- .
Xi Xi
Xi
It is easy to see that this field is independent ofthe choice of the index i.
Our purpose in defining the height is to have a method of measuring the arithmetic size or complexity of points in projective space, analogous to the way in which
the size of a rational number is measured by taking the larger of its numerator and
its denominator. The following finiteness theorem is of fundamental importance.
Theorem 3.7. Let K/Q be a number field, and let B be any constant. Then the set
ofpoints
{P E pN (K) : HK(P) S B}
is finite. More generally, for any constants B and D, the set ofpoints
{P E pN (Q) : H(P) S B
and [Q(P) : Q] S D}
is finite. In other words, there are only finitely many points in pN (Q) of bounded
height and bounded degree.
Proof. It clearly suffices to prove the second statement, i.e., that there are only
finitely many points
P = [x0 , ••• , xN] E pN (Q)
satisfying H(P) ::; B
and [Q(P) : Q] =D.
Dividing the homogeneous coordinates of P by some nonzero coordinate, we may
assume that some coordinate equals 1. Then
Q(P) = Q(xo, ... , XN) :::) Q(xi)
foreachO::; i::; N,
87
3.1. Height Functions
s
D. Further, if we write K = Q(P) and d = [K : Q], then we can
so [Q(xi) : Q]
estimate the heights of the individual coordinates by
= ( H(xo)H(xl) · · · H(xN) )
for every 0
1/N
s i S N.
Thus each coordinate of P lies in a field of degree at most D and has height at most
BN. Replacing B by BN and taking numbers of exact degree d for each 1 d D,
it suffices to prove that the set
s s
{a E Q: H(a) S B and [Q(a): Q] = d}
(3.2)
is finite.
Let a be in the set (3.2) and write the minimal polynomial of a as
Fa(X) = Xd + a1Xd- 1 +···+ad E Q[X].
Also factor Fa(X) over Cas
The number a 1 , ... , ad are the conjugates of a, so Theorem 3.6 tells us that
Further, the coefficients a 1 , ... , ad of Fa (X) are the elementary symmetric polynomials of the roots (up to ±1). For example,
a1=-(a1+a2+···+ad)
More generally, for any 1
and
ad=(-l)da1a2···ad.
s k s d we have
ak = (-1 )k
L
ai 1 ai 2 • • • aik.
(3.3)
1::;il <i2<··-<ik::;d
Note that there are (~) terms in the sum (3.3).
We can use (3.3) and the triangle inequality to estimate the absolute values of the
coefficients a 1 , ... , ad in terms of the absolute values of the roots a 1 , ... , ad. Thus
for any v E MQ(a) we compute
3. Dynamics over Global Fields
88
laklv
ai1ai2 · · · aik I
L
=I l:'Oil<i2<···<ik:'Od
v
: :; (d)
max
k l:'Oil<i2<···<ik:'Od
::::;
lai1 ai2 · · · aik lv
(~) max{lallv, 1} max{la2lv, 1} · · · max{ladlv, 1}.
Further, if v is nonarchimedean, then we may discard the factor of (~). Taking the
maximum over k and using the fact that (~) ::::; 2d for all k, we find that
max{1, la1lv, la2lv, · · ·, ladlv}
::::; 2d max{lallv, 1} max{la2lv, 1} · · · max{ladlv, 1},
where again the 2d is needed only if v is archimedean. Now raise to the nv power,
multiply over all v, and take the rfh root to obtain
H([1, a1, a2, ... , ad]) ::::; 2d H(al)H(a2) · · · H(ad)·
Theorem 3.6 tells us that H(ai) = H(a) for every i, and we have H(a) ::::; B by
assumption, so we conclude that
H([1,al,a2, ... ,ad])::::; (2B)d.
In other words, the height of [1, a 1, a2, ... , ad] E lP'd(Q) is bounded by (2B)d. However, we already know that there are only finitely many Q rational points in projective
space (see Example 3.1 and Remark 3.5). Hence there are only finitely many possibilities for the minimal polynomial Fa(X) of a, and since each Fa(X) has only
d roots, there are only finitely many possibilities for a. This proves that the set (3.2)
D
is finite, which completes the proof of the theorem.
We can use Theorem 3. 7 to give a very brief proof of a famous theorem of Kronecker. It says that an algebraic integer whose conjugates all lie on the unit circle
must be a root of unity.
Theorem 3.8. Let a E Q be a nonzero algebraic number. Then
H (a) = 1
if and only if a is a root of unity.
Proof If a is a root of unity, then lalv = 1 for every absolute value v, so we clearly
have H(a) = 1. Now suppose that H(a) = 1. Directly from the definition of the
height we see that
for all (3 E Qand all n 2: 1.
Thus H(an) = H(a)n = 1 for all n 2: 1, which implies in particular that the set
{a,a 2 ,a 3 , ..• }
is a set of bounded height. This set is also clearly contained in the number field Q( a),
so Theorem 3. 7 tells us that it is a finite set. Hence there are integers i > j > 0 such
D
that ai = aj, which shows that a is a root of unity.
89
3.2. Height Functions and Geometry
3.2
Height Functions and Geometry
The height of a point P measures the arithmetic complexity of P. We now investigate
how the height of P changes when we map it to some other projective space. This
will allow us to relate geometric properties of maps to the arithmetic information
encapsulated by the height function.
Remark 3.9. For ease of exposition we restrict attention to heights on projective
space, but the reader should be aware that there is a general theory of height functions on algebraic varieties due to Weil. Height functions provide a powerful tool
for converting algebra-geometric relationships into number-theoretic relationships.
Theorem 3.11 below provides an example; it converts the geometric information that
a map ¢ has degree d into the arithmetic information that ¢( P) has height that is
more-or-less the rfh power of the height of P. A summary of the general theory of
heights on varieties is given in Section 7.3; see [205, Part B] or [256, Chapters 3-5]
for further details.
Definition. A rational map of degree d between projective spaces is a map
cP : JP>N - - t ]p>M
¢(P) = [fo(P), ... , fM(P)],
where fo, ... , fM E K[Xo, ... , XN] are homogeneous polynomials of degree d
with no common factors. (The polynomial ring K[X0 , ... ,XN] is a unique factorization domain, so it makes sense to talk about common factors.) The rational map¢
is defined at P if at least one of the values fo(P), ... , !M(P) is nonzero. The rational
map¢ is called a morphism if it is defined at every point of!P'N (K), or equivalently,
if the only solution to the simultaneous equations
fo(Xo, ... ,XN) = · · · = fM(Xo, ... ,XN) = 0
is the trivial solution Xo = · · · = XN = 0. If the polynomials j 0 , ... , fN have
coefficients in K, we say that ¢ is defined over K.
Our goal is to relate the height of a point P to the height of its image ¢( P). To
do this in general, we use an important theorem from algebraic geometry called the
Nullstellensatz. Here
Null= zero,
stellen =places,
satz = theorem,
so the Nullstellensatz is a theorem that relates a function to the points at which it
vanishes. We give a brief overview of some basic concepts from algebra and algebraic geometry that are needed to understand the statement of the Nullstellensatz.
However, we note that for rational maps ¢(z) E K(z) in one variable (i.e., maps
¢ : IP'1 --t IP' 1), the Nullstellensatz may be replaced by a simple argument based on
the fact that the ring K[z] is a principal ideal domain; see Exercise 3.8.
An ideal I in K[Xo, ... , X N] is homogeneous if it is generated by homogeneous
polynomials. The radical ofan ideal I is the ideal
3. Dynamics over Global Fields
90
..Ji = {f E R[Xo' ... 'X N l :
r
r E I for some n ~ 1}.
is the n 1h power of J, not its nth iterate.) The
(N.B. In this definition, the quantity
ideal I is the set
homogeneous
a
to
algebraic set attached
V(I) = {P E JIDN (K) : f(P) = 0 for all f E I}.
The Hilbert basis theorem [259, Chapter VI, Theorem 2.1] implies that I is finitely
generated, so V (I) is the set of simultaneous zeros of a finite collection of polynomials. If V C JIDN (K) is an algebraic set, the ideal attached to V is the ideal
ideal generated by_all homogeneous)
I(V) = ( polynomials f E K[Xo, ... , XN]
such that f(P) = 0 for all P E V
.
It is clear that if V = W, then I(V) = I(W). The converse is not quite true, since
the algebraic set attached to the radical /I is the same as the algebraic set attached
to I.
Theorem 3.10. (Hilbert's Nullstellensatz) Let I and J be homogeneous ideals properly contained in K[Xo, ... , XN]. Then
V(I)=V(J)
ifandonlyif ..Ji=viJ.
r
E I for some
Proof Suppose that /I= VJ. Let P E V(I) and f E J. Then
E V(J).
soP
J,
E
f
every
for
true
is
This
0.
=
f(P)
so
0,
n ~ 1, so r(P) =
interchangby
follows
inclusion
opposite
the
and
V(J),
c
V(I)
that
proves
This
ing the roles and I and J. Hence V(I) = V(J). This proves the trivial direction
of the theorem. For a proof of the nontrivial converse, see [198, 1.1.3A] or [259,
D
Section X.2].
We are now ready to prove an important result that says that up to a scalar factor,
a morphism of degree d causes the height to be raised to the rfh power.
Theorem 3.11. Let cjJ : JIDN (K) ---+ JIDM (K) be a morphism of degree d. Then there
are constants 1' c2 > 0, depending on c/J, such that
c
for all P E JIDN (K).
(In fact, the upper bound H (cjJ(P)) ~ C2H(P)d is validfor rational maps provided
we restrict attention to points P at which cjJ is defined.)
Proof We begin with some notation. For any point P = [x 0 , ..• , x N] with coordinates in K and any absolute value v E M K, we write
(This assumes that we have fixed particular homogeneous coordinates for P.) Similarly, we define the absolute value of a polynomial
91
3.2. Height Functions and Geometry
' aio ... iN xio
f(x o, · · ·' X N ) = "
N
0 . . . XiN
~
to be
iaio ... iN lv,
ifiv = . max
... ,'l.N
to,
and if¢ = [fo, ... , f M] is a collection of polynomials, we let
Notice that the height of a point P E JP'N (K) may now be written in the compact
form
H(P) =
II IPI~v
(
)
1/[K:<Q!]
•
vEMK
We define the height of a polynomial f or a collection of polynomials ¢ similarly,
II lfl~v
H(f) = (
)
1/[K:<Q!]
and
H(¢) =
(
II l¢1~v
) 1/[K:<Q!]
(3.4)
vEMK
vEMK
We set one final piece of notation that will make our computations easier. For
any absolute value v E Mx and any number m, we set
0
(m) = {m ifv M'j{ (i.e., ifv is archimedean),
v
E
1
if v E M~ (i.e., if v is nonarchimedean).
With this notation, we can write a uniform version of the triangle inequality as
Now letP = [x 0 , ... , XN] E lP'N (K) be a rational point and f E K[X 0 , ... , XN]
a homogeneous polynomial of degree d. Then for any v E M K we can estimate
if(P)Iv
=I L
ai 0 ... iNxbo
···X~ I
v
io, ... ,iN~o
io+··+iN=d
::::; Ov(# of terms). max !ai 0 ... iNXb0 • • • xiJ lv.
to, ... ,tN
The number of terms in the sum is equal to at most the number of monomials of
degree d in N + 1 variables, which is given by the combinatorial symbol (Ntd). For
our purposes it is enough to know that it depends only on N and d. Continuing with
the computation, we find that
3. Dynamics over Global Fields
92
We apply this inequality with f = fk for k = 0, ... , N and take the maximum
over k to obtain
(3.6)
Finally, we raise (3.6) to the nv power, multiply over all v E MK, and take the
[K : Q] root to obtain
Note that in deriving this formula, we have used the fact that
and
L nv [K: Q],
=
vEM~
where the latter is a special case of Theorem 3.2.
This completes the proof of the upper bound with the explicit constant
And as indicated in the statement of the theorem, we never assumed that ¢ is a
morphism, so the upper bound holds for rational maps.
In order to prove the opposite inequality, it is necessary to use the assumption
that¢ is a morphism, or equivalently, that¢= [fa, ... , !M] for homogeneous polynomials fa, ... , fM having no common zero in IP'N (.k). This condition tells us that
the ideals (fa, ... , fM) and (X a, ... , XN) in k[Xa, ... , XN] define the same algebraic set in IP'N (K), namely the empty set. The Nullstellensatz (Theorem 3.10) then
implies that these two ideals have the same radical. In particular,
Hence we can find an integer e 2 0 such that
(The Nullstellensatz says that there is an exponent ei for each Xi, and then we take e
to be the largest of the ei-) Writing this out explicitly, we find homogeneous polynomials 9iJ E k[Xa, ... , XN] such that
Xf = 9iafa + 9ilh + · · · + 9iM !M
for each 1 ::=; i ::::; N.
We observe that each gij is homogeneous of degree e - d, since each J1 has degree d.
We evaluate these polynomial identities at P = [xa, ... , x N] E IP'N (K) and
estimate v-adic absolute values,
93
3.2. Height Functions and Geometry
!PI~ = max !xi!~
O~i~N
=
max !9iO(P)fo(P) + 9il(P)!l(P) + · · · + 9iM(P)fM(P)!
O~i~N
::; 8v(M + 1) max !9ij(P)fi(P)!
O<i<N
v
o?;J~M
::; Dv(M + 1) m.ax Dv
O<•<N
o?;J~M
(
::; 8v (M + 1)
v
+ e d- d)) 19iJ!v!P!ve-d !fJ(P)!v
((Ne-
(Ne+_e d- d)) (0~~N 19iJ!v ) !P!ve-d !¢(P)!v·
O~j~M
So ifwe let !9!v = maxi,j !9iJ!v and aM,N,d,e = (M + 1)(N:~~d), then dividing
both sides by !P!~-d yields the estimate
(3.7)
Now raise this inequality to the nv power, multiply over all v E MK, and take the
[K: Q] root to obtain
H(P)d::; aM,N,d,eH(g)H(¢(P)),
where H (g) is the height determined by the coefficients of all of the 9iJ polynomials.
Its precise value is not important for our purposes; it is enough to note that it does
not depend on the point P. We have thus proven that H(P)d ::; CH(¢(P)) for a
constant C that does not depend on P, which completes the proof of Theorem 3.11.
D
Theorem 3.11 says that the height of ¢(P) is approximately equal to the cf!h
power of P. This means that His a multiplicative kind of function, similar in some
ways to an absolute value. It is often convenient notationally to work instead with an
additive function, which prompts the following definition.
Definition. The logarithmic height (relative to K) is the function
The absolute logarithmic height is the function
h(P) = logH(P).
Using this logarithmic notation, Theorem 3.11 says that h(¢(P)) and dh(P)
differ by a bounded amount. The notion of functions that differ by a bounded amount
appears so frequently in mathematics that it has its own notation.
Definition. LetS be a set and let J, g : S ---+ JR. be real-valued functions on S. The
formula
94
3. Dynamics over Global Fields
f=g+O(l)
means that the quantity
IJ(P)- g(P)I
is bounded as P ranges over S. More generally, if h : S -+ JR. is another function,
the formula
f=g+O(h)
means that there exists a constant C such that
IJ(P)- g(P)I :::; Clh(P)I
for all PES.
Using the theory of heights as developed in Theorems 3.7 and 3.11, it is easy to
prove that a morphism ¢ : JP>N -+ JP>N of degree at least 2 has only finitely many
preperiodic points defined over a given number field. This result was first discovered
by Northcott [343] and then independently rediscovered many times in varying degrees of generality. It is instructive to compare this global proof with the local proof
of finiteness of periodic points given in Corollary 2.26.
Theorem 3.12. (Northcott [343]) Let ¢ : JP>N -+ JP>N be a morphism of degree d ;:::: 2 defined over a number field K. Then the set of preperiodic points
PrePer( ¢) C JP>N (K) is a set of bounded height. In particular,
PrePer( ¢, JP>N (K)) = PrePer( ¢) n JP>N (K)
is a finite set, and more generally, for any D ;:::: 1 the set
U PrePer(¢, IP'N (L))
[L:K]~D
is finite.
Proof Theorem 3.11 tells us that there is a constant C = C(¢) such that
h(¢(Q)):::::: dh(Q)-
c
for all Q E JP>N (K).
Applying this with Q = R, ¢(R), ¢ 2 (R), ... , q;n- 1 (R) yields
h(¢n(R));:::: dnh(R)- C(l + d + d2 + · · · + dn- 1 );:::: dn(h(R)- C).
(3.8)
Now suppose that Pis preperiodic, say q;m+n(P) = q;m(P) with n ;:::: 1 and m ;:::: 0.
Setting R = q;m(P) and using the preperiodicity yields
and hence
(3.9)
3.3. The Uniform Roundedness Conjecture
95
For the last inequality we are using the fact that d ~ 2 and n ~ 1. Next we use
inequality (3.8) with n = m and R = P,
d"'(h(P)- C) ::; h(qr(P)).
(3.10)
Finally, combining (3.9) and (3.10) yields
1
1
h(P) ::; dm h(qr(P)) + C::; dm 2C + C::; 3C,
which completes the proof that the height of P is bounded by a constant depending
only on the map¢.
The remaining assertions of the theorem are immediate consequences of Theorem 3. 7, which says that sets of bounded height have only finitely many points of
bounded degree.
D
Remark 3.13. One may also consider preperiodic points defined over infinite extensions. For example, we might fix an infinite extension L/Q and ask which morphisms ¢ have infinitely many periodic or preperiodic points in pN ( L). Very little is
known about this question in general.
We consider a special case. Let K /Q be a number field and let Kcyci denote
the maximal cyclotomic extension of K, i.e., the field obtained by adjoining to K
all roots of unity. Now let ¢(z) E K[z] be a polynomial of degree d ~ 2 and suppose that the set PrePer (¢, JP> 1 ( Kcycl)) has infinitely many points. Then Dvomicich
and Zannier [146] prove that there is a Mobius transformation f E PGL 2 (K) such
that ¢! ( z) has the form ±zd or Td (±z), where Td (z) is a Chebyshev polynomial
(see Section 1.6.2).
Notice that an alternative dynamical definition of Kcyci is the field obtained by
adjoining to K all 'of the points in PrePer(zd) for some d ~ 2. This suggests a
natural question.
Question 3.14. Let¢, '1/J : pN ---+ pN be morphisms of degree at least 2 defined over a
number field K. Suppose that ¢ has infinitely many preperiodic points defined over
the field K (PrePer( '1/J)). What can one deduce about the relationship between ¢
and¢?
3.3
The Uniform Boundedness Conjecture
Northcott's Theorem 3.12 says that a morphism ¢ : pN ---+ JP>N has only finitely
many K -rational preperiodic points. It is even effective in the sense that we can, in
principle, find an explicit constant C (¢) in terms of the coefficients of ¢ such that
every point P E PrePer(¢) satisfies h(P) ::; C(¢). This also allows us to compute
an upper bound for #PrePer(¢,JP>N(K)), but the bound grows extremely rapidly
as the coefficients of ¢ become large. A better bound, at least for periodic points,
may be derived from the local estimates in Chapter 2 as described in Corollary 2.26.
However, even that estimate depends on the coefficients of¢, since it is in terms of
the two smallest primes for which ¢ has good reduction. The following uniformity
96
3. Dynamics over Global Fields
conjecture says that there should be a bound for the size of PrePer( ¢, lP'N (K)) that
depends in only a minimal fashion on¢ and K.
Conjecture 3.15. (Morton-Silverman [312]) Fix integers d ~ 2, N ~ 1, and
D ~ 1. There is a constantC(d, N, D) such that for all numberfields K/IQ ofdegree
at most D and all finite morphisms ¢ : lP'N ----+ lP'N of degree d defined over K,
# PrePer( ¢, lP'N (K)) ::::; C(d, N, D).
Remark 3.16. There are many results in the literature giving explicit bounds for the
size of the sets PrePer (¢, lP'N ( K)) or Per ( ¢, lP'N (K)) in terms of¢, especially in
the case N = 1. Some of these results use global methods, while others use a small
prime of good (or at least not too bad) reduction for¢. For example, we used local
methods in Corollary 2.26 to give a weak bound for #Per(¢,lP' 1 (K)). For further
results, see [52, 87, 90, 91, 92, 100, 101, 137, 162, 171, 191, 192, 193, 194, 195,
227,331,265,312,325,326,328,329,330,332,353,355,358,3 59,361,454].
Remark 3.17. Very little is known about Conjecture 3.15. Indeed, it is not known
even in the simplest case (d, N, D) = (2, 1, 1), that is, for IQ-rational points and
degree-2 maps on lP' 1 . Specializing further, if we let c/Jc : lP' 1 ----+ lP' 1 denote the
quadratic map c/Jc (z) = z 2 + c, then the conjecture implies that
sup# Per( c/Jc, lP' 1 (IQ)) < oo,
cEIQI
but the best known upper bounds for # Per ( c/Jc, lP' 1 ( IQ)) depend on c.
There are one-parameter families of c-values for which c/Jc(z) has a IQ-rational
periodic point of exact period 1, 2, or 3; see Exercise 3.9 and Example 4.9. And
one can show that c/Jc cannot have IQ-rational periodic points of exact period 4 or 5;
see [171, 309]. Poonen has conjectured that c/Jc cannot have any IQ-rational periodic
points of period greater than 3. Assuming this conjecture, he gives a complete description of all possible rational preperiodic structures for c/Jc; see [361].
Remark 3.18. Another interesting collection of rational maps is the family
b
c/Ja ,b(z) = az + -.
z
These maps have the symmetry property c/Ja,b( -z) = -c/Ja,b(z), i.e., conjugation by
the map f (z) = - z leaves them invariant. It is known that there are one-parameter
families of these maps with a IQ-rational periodic point of exact period 1 (in addition
to the obvious fixed point at oo), 2, or 4, and that none of the maps c/Ja,b(z) has a IQrational periodic point of exact period 3. See [286] for details, and Examples 4.69
and 4.71 and Exercises 4.1, 4.40, and 4.41 for additional properties of these maps.
Remark 3.19. Conjecture 3.15 is an extremely strong uniformity conjecture. For
example, if we consider only maps ¢ : lP' 1 ----+ lP' 1 of degree 4 defined over IQ, then
the assertion that # PrePer ( ¢, lP' 1 ( IQ)) ::::; C for an absolute constant C immediately
implies Mazur's theorem [292] that the torsion subgroup of an elliptic curve E j!Q
3.4. Canonical Heights and Dynamical Systems
97
is bounded independently of E. To see this, we observe that Proposition 0.3 tells us
that
Etors = PrePer([2], E),
and hence the associated Lattes map tPE, 2 described in Section 1.6.3 satisfies
Note that tPE, 2 has degree 4.
In a similar manner, Conjecture 3.15 for maps of degree 4 on JID 1 over number
fields implies Merel's theorem [297] that the size of the torsion subgroup of an elliptic curve over a number field is bounded solely in terms of the degree of the number
field. Turning this argument around, Merel's theorem implies the uniform boundedness conjecture for Lattes maps, i.e., for rational maps associated to elliptic curves;
see Theorem 6.65. Lattes maps are the only nontrivial family of rational maps for
which the uniform boundedness conjecture is currently known.
In higher dimension, Fakhruddin [162] has shown that Conjecture 3.15 implies
that there is a constant C(N, D) such that if K is a number field of degree at most D
and if A/ K is an abelian variety of dimension N, then
#A(K)tors :S C(N, D).
He also shows that if Conjecture 3.15 is true over Q, then it is true for all number
fields.
3.4
Canonical Heights and Dynamical Systems
It is obvious from the definition of the height that
for all a E Q.
(3 .11)
Notice that Theorem 3.11 applied to the particular map ¢(z) = zd gives the lessprecise statement
h(¢(P)) = dh(P) + 0(1).
(3.12)
Clearly the exact formula (3.11) is more attractive than the approximation (3.12). It
would be nice if we could modify the height h in some way so that the general formula (3.12) from Theorem 3.11 is true without the 0(1). It turns out that this can be
done for each morphism ¢. To create these special heights, we follow a construction
due originally to Tate.
Theorem 3.20. Let S be a set, let d > 1 be a real number, and let
¢:8-+S
and
h:S-+JR
be functions satisfYing
h(t/J(P)) = dh(P) + 0(1)
for all PES.
3. Dynamics over Global Fields
98
Then the limit
(3.13)
exists and satisfies:
(a) h(P) = h(P) + 0(1).
(b) h(¢(P)) = dh(P).
The function h: S-----; JR. is uniquely determined by the properties (a) and (b).
Proof We prove that the limit (3.13) exists by proving that the sequence is Cauchy.
Let n > m ;::: 0 be integers. We are given that there is a constant C such that
ih(¢(Q))- dh(Q)I::; c
for all Q E S.
(3.14)
We apply inequality (3.14) with Q = cpi- 1 (P) to the telescoping sum
t ~i
: ; t ~i
Idln h(cpn(P))- d~ h(cpm(P))I = .
( h(cpi(P))- dh(cpi-l(P)))
t=m+l
ih(¢i(P))- dh(¢i-l(P))i
i=m+l
C
C
C
::; 2:: di ::; 2:: di = (d- l)dm. (3.15)
n
i=m+l
00
i=m+l
The inequality (3 .15) clearly implies that
asm,n-----; oo,
so the sequence d-nh(cpn(P)) is Cauchy and the limit (3.13) exists.
In order to prove (a), we take m = 0 in (3 .15), which yields
Next we let n -----; oo to obtain
which is (a) with an explicit value for the 0(1) constant.
The proof of (b) is a simple computation using the definition of h,
Finally, to prove uniqueness, suppose that h' : S -----; JR. also has properties (a)
and (b). Then the difference g = h- h' satisfies
99
3.4. Canonical Heights and Dynamical Systems
g(P) = 0(1)
g(¢(P)) = dg(P).
and
These formulas hold for all elements P E S, so
for all n 2: 0.
In other words, the quantity dng(P) is bounded as n -+ oo, which can happen only
D
if g(P) = 0. This proves that h(P) = h'(P), soh is unique.
Definition. Let¢ : pN -+ pN be a morphism of degree d 2: 2. The canonical height
function (associated to¢) is the unique function
satisfying
h¢(P) = h(P) + 0(1)
and
The existence and uniqueness of h¢ follow from Theorem 3.20 applied to the maps
and
since Theorem 3.11 tells us that¢ and h satisfy
h(¢(P)) = dh(P) + 0(1)
for all P E JP>N (Q).
Remark 3.21. The definition h¢(P) = limn---.oo d-nh(¢n(P)) is not practical for
accurate numerical calculations. Thus even for P E JP> 1 ( Q), one would need to compute the exact value of ¢n(P), whose coordinates have O(dn) digits. A practical
method for the numerical computation of h¢(P) to high accuracy uses the decom-
position of hq, as a sum of local heights or Green functions. This decomposition
is described in Sections 3.5 and 5.9. See in particular Exercise 5.29 for a detailed
description of the algorithm.
The canonical height provides a useful arithmetic characterization of the preperiodic points of¢.
Theorem 3.22. Let ¢ : pN -+ pN be a morphism of degree d 2: 2 defined over Q
and let P E IP'N (Q). Then
P E PrePer( ¢)
if and only if h¢ (P) = 0.
Proof If P is preperiodic, then the quantity h( ¢n (P)) takes on only finitely many
values, so it is clear that d-n h( ¢n (P)) -+ 0 as n -+ oo.
Now suppose that h¢(P) = 0. Let K be a number field containing the coordinates of P and the coefficients of¢, i.e., P E pN (K) and¢ is defined over K.
Theorem 3.20 and the assumption h¢(P) = 0 imply that
100
3. Dynamics over Global Fields
Thus the orbit
is a set of bounded height, so it is finite from Theorem 3.7. Therefore Pis a preperiodic point for¢.
D
Remark 3.23. Further material on canonical heights in dynamics may be found in
Sections 3.5, 5.9, and 7.4, as well as [16, 20, 23, 36, 38, 39, 87, 88, 89, 147, 159,
231,228,230,232,234,406,409,446,453].
Theorem 3.22 is a generalization of Kronecker's theorem (Theorem 3.8), which
says that h( a) = 0 if and only if a is a root of unity. Kronecker's theorem follows
by applying Theorem 3.22 to the £ilh-power map ¢(z) = zd, whose canonical height
is the ordinary height h.
The fact that only roots of unity have height 0 leads naturally to the question
of how small a nonzero height can be. If we take the relation h(ad) = dh(a) and
substitute in a= 21/d, we find that
h(21/d) = 1h(2) = lo~ 2'
so the height can become arbitrarily small. However, this is possible only by taking
numbers lying in fields of higher and higher degree. For any algebraic number a, let
D(a) = [Q(a): Q]
denote the degree of its minimal polynomial over Q.
Conjecture 3.24. (Lehmer's Conjecture [264]) There is an absolute constant r;, > 0
such that
h(a):::: r;,fD(a)
for every nonzero algebraic number a that is not a root of unity.
There has been a great deal of work on Lehmer's conjecture by many mathematicians; see for example [7, 8, 75, 94, 264, 366,421, 422, 424, 445]. (The survey [422]
contains an extensive bibliography.) The best result currently known, which is due to
Dobrowolski [138], says that
h(a) > -""- (loglogD(a)) 3
- D(a)
log D(a)
The smallest known nonzero value of D (a) h( a) is
D(f3o)h(f3o) = 0.1623576 ... ,
where (30 = 1.17628 ... is a real root of
x 10 + x 9 - x 7 - x 6 - x 5 - x 4 - x 3 + x + 1.
Theorem 3.22 tells us that hq,(P) = 0 if and only if P is a preperiodic point
for¢. This suggests a natural generalization of Lehmer's conjecture to the dynamical
setting. (See [317] for an early version of this conjecture in a special case.)
3.4. Canonical Heights and Dynamical Systems
101
Conjecture 3.25. (Dynamical Lehmer Conjecture) Let ¢ : IP'N ---t IP'N be a
morphism defined over a number field K, and for any point P E IP'N (K), let
D(P) = [K(P) : K]. Then there is a constant"'= K-(K, ¢) > 0 such that
for all P E IP'N (K) with P ~ PrePer(¢).
There has been considerable work on this conjecture for maps ¢ : IP'1 ---t IP'1 that
are associated to groups as described in Section 1.6. For example, in the case that¢
is attached to an elliptic curve E, it is known that
D(P)3log 2 D(P)
in general [291],
if j(E) is nonintegral [203],
D(P) 2
_"-_ (loglogD(P)) 3 if E has complex multiplication [263].
D(P)
log D(P)
Aside from maps associated to groups, there does not appear to be a single example for which it is known that h¢(P) is always greater than a constant over a fixed
power of D(P). Using trivial estimates based on the number of points of bounded
height in projective space, it is easy to prove a lower bound that decreases faster than
exponentially in D(P); see Exercise 3.17.
Remark 3.26. The Lehmer conjecture involves a single map¢ and points from number fields of increasing size. Another natural question to ask about lower bounds for
the canonical height involves fixing the field K and letting the map¢ vary. For example, consider quadratic polynomials ¢c(z) = z 2 +cas c varies over Q. Is it true
that h<Pc (a) is uniformly bounded away from 0 for all c E Q and all nonpreperiodic a E Q? In other words, does there exist a constant "' > 0 such that
h¢c(a) 2: K- for all c, a E Q with a~ PrePer(¢c)?
We might even ask that the lower bound grow as c becomes larger (in an arithmetic
sense). Thus is there a constant "' > 0 such that
h<Pc (a) 2: K-h( c)
for all c, a E Q with a ~ PrePer( ¢c)?
This is a dynamical analogue of a conjecture for elliptic curves that is due to Serge
Lang; see [202], [254, page 92], or [410, VIII.9.9].
For the quadratic map z 2 + c, the height of the parameter c provides a natural
measure of its size, but the situation for general rational maps ¢(z) E K(z) is more
complicated. We cannot simply use the height of the coefficients of ¢, because the
canonical height is invariant under conjugation (see Exercise 3.11), while the height
of the coefficients is not conjugation-invariant. We return to this question in Section 4.11 after we have developed a way to measure the size of the conjugacy class
of a rational map.
102
3. Dynamics over Global Fields
3.5 Local Canonical Heights
The canonical height h,p attached to a rational map ¢ is a useful tool in studying
the arithmetic dynamics of¢. For more refined analyses, it is helpful to decompose
the canonical height as a sum of local canonical heights, one for each absolute value
on K. In this section we briefly summarize the basic properties of local canonical
heights, but we defer the proofs until Section 5.9. The reader wishing to proceed
more rapidly to the main arithmetic results of this chapter may safely omit this section on first reading, since the material covered is not used elsewhere in this book.
The construction of the canonical height relies on the fact that the ordinary height
satisfies h(¢(P)) = dh(P) + 0(1), so it is "almost canonical." The ordinary height
of a point P = [a, I] is equal to the sum
h(P)=h(a)=
L nvlogmax{lalv,l},
vEMK
so for each v E M K it is natural to define a local height function
>.v(a) = logmax{lalv, 1}.
We can understand >.v geometrically by observing that for v E Mi,
>.v(a) = -logpv(a, oo),
where Pv is the nonarchimedean chordal metric defined in Section 2.1. One says
that >.v(a) is the logarithmic distance from a to oo.
Unfortunately, the function >.v does not transform canonically, since >.v (¢(a)) is
not equal to d>.v(a) + 0(1). To see why, note that >.v(¢(a)) is large if a is close to
a pole of¢, while >.v (a) is large if a is close to the point oo E IP'1 . (Here the word
"close" means in terms of the v-adic chordal metric Pv·) Thus we can hope to find
a canonical local height only if we allow an appropriate correction term, as in the
following theorem.
Theorem 3.27. Let¢ : IP' 1 --+ IP' 1 be a rational function of degree d 2:: 2 defined
over K, write ¢(z) = F(z)/G(z) using polynomials F, G E K[z] having no common factors, and let v be an absolute value on K. Then there is a unique function
with the following two properties:
(a) For all a E IP'1 (Kv) with a -=f oo and ¢(a) -=f oo, the function .X,p,v satisfies
(3.16)
(b) The function
a f----+ .X,p,v(a) -logmax{lalv, 1}
extends to a bounded continuous function on all of IP' 1 ( kv ).
103
3.5. Local Canonical Heights
The function ~</>,vis called a local canonical height (associated) to¢.
Proof We defer the proof until Section 5.9; see Theorem 5.60.
D
Remark 3.28. The local canonical height function ~</>,v constructed in Theorem 3.27
depends on the choice of a decomposition of¢ as ¢ = FIG. If we use instead cFI cG
for some c E K*, so G is replaced by cG, then it is easy to see that the new function
differs from the old one by a constant,
1
Aq,,v,cc(o:) = Aq,,v,c(o:) + d _ 1 log iclv
A
A
In the sequel, when we refer to 5.q,,v without further specification, we assume that
some particular G has been fixed. However, we note that there are situations in which
it may be convenient to normalize ~</>,v differently for different v; see Exercise 3.29.
The utility of local canonical heights is that on the one hand, they are defined
on JP>1 ( K v) and have various nice metrical and analytic properties, while on the other
hand, they fit together to give the global canonical height, as described in the next
theorem.
Theorem 3.29. Let K be a number field, andfor each v E MK, let ~</>,v be the local
canonical height function constructed in Theorem 3.27. Then
hq,(o:) =
L nv~q,,v(o:)
for all a E JP> 1 (K) " {oo }.
(3.17)
vEMK
Proof We defer the proof until Section 5.9; see Theorem 5.61.
D
Remark 3.30. If¢( z) E K[ z] is a polynomial, then the local height may be computed
as the limit (Exercise 3.24)
1
Aq,,v(o:) = n~oo
lim -d
logmax{i¢n(o:)iv, 1}.
n
A
If v is a nonarchimedean absolute value and if¢ has good reduction at v, or more
precisely, if¢= FIG with IRes(F, G)lv = 1, then the local height is given by the
simple formula (Exercise 3.30)
~q,,v(o:) = logmax{io:lv, 1}.
In general there is no simple limit formula to compute ~</>,v (a). However, there is
an algorithm that leads to a rapidly convergent series for ~q,,v(o:). Roughly, the Nth
partial sum of the series approximates ~q,,v(o:) to within O(d-N). For details see
Exercise 5.29, which describes an efficient algorithm to compute the Green function gq,, and then use Theorem 5.60, which says that tlle local canonical height and
the Green function are related by tlle formula
5.q,,v ([x, yl) = gq,,v(x, y) -log IY!v-
3. Dynamics over Global Fields
104
Remark 3.31. Baker and Rumely [26, Chapter 7] give an interesting interpretation of the local canonical height function. They construct an invariant measure on
Berkovich space and prove that the measure is the negative Laplacian of a suitable
extension of the local canonical height function 5-q,, v. See Section 5.10 for a brief
introduction to the geometry/topology of Berkovich space. For the more elaborate
machinery required to do harmonic and functional analysis on Berkovich space, including the construction of the invariant measure, we refer the reader to [26, 29] and
to the other references listed in Remark 5. 77.
Remark 3.32. The theory oflocal canonical heights began with Neron's construction
on abelian varieties [333], or see [256, Chapter 11]. The general theory of global
and local canonical heights associated to morphisms on varieties with eigendivisor
classes is described in [88].
3.6 Diophantine Approximation
The theory of Diophantine approximation seeks to answer the question of how
closely one can approximate irrational numbers by rational numbers. The subject
now includes a large and well-developed body of knowledge, while at the same time
there is considerable ongoing research on many deep questions. In this section we
state a famous theorem on Diophantine approximation and show how it is applied to
deduce finiteness results for certain Diophantine equations. In Sections 3.7 and 3.8
we apply the theory of Diophantine approximation in a similar fashion to deduce
integrality properties of points in orbits 0 1> (a).
It is clear that any irrational number a E lR can be approximated arbitrarily
closely by rational numbers, since Q is dense in R The following elementary result
quantifies this observation by relating the closeness of y and to the arithmetic
complexity of the rational number xly. We leave the proof as an exercise (Exercise 3.31), or see any elementary number theory text that discusses Diophantine approximation.
xI
a
Proposition 3.33. (Dirichlet) Let a E lR "- Q be an irrational number. Then there
are infinitely many rational numbers xI y satisfYing
~~ -al ~ :
2 •
Dirichlet's theorem says that every irrational number can be fairly well approximated by rational numbers. Some irrational numbers can be much better approximated, but it turns out that this is not true for algebraic numbers. A succession of
mathematicians (Liouville, Thue, Siegel, Gel'fond, Dyson) derived ever better estimates for the approximability of algebraic numbers by rational numbers, culminating in the following deep theorem of Roth, for which he received the Fields Medal
in 1958.
Theorem 3.34. (Roth) Fix E > 0 and let a E Q be an algebraic number with
a~ Q. Then there exists a constant /'i, = /'i,(E, a) > 0 such that
3.6. Diophantine Approximation
105
X
for all- E Q.
y
Proof The proof of Roth's theorem is beyond the scope of this book. A nice exposition may be found in [393]. For more general versions, see [205, Part D] and [256,
Chapter 7].
0
We apply Roth's theorem to prove an important result of Thue on the representability of integers by binary forms.
Theorem 3.35. (Thue) Let G(X, Y) E Z[X, Y] be a homogeneous polynomial of
degree d and let B E Z. Assume that G(X, Y) has at least three distinct roots
in lP' 1 (C). Then the equation
G(X, Y) = B
(3.18)
has only finitely many integer solutions (X, Y) E 1:: 2 .
Proof We begin by factoring G(X, Y) into irreducible factors in Q[X, Y], say
Replacing G(X, Y) by cG(X, Y) and B by cB for a sufficiently large integer c, we
may assume without loss of generality that G 1 , ... , Gr have integer coefficients. 1
If r 2: 2, that is, if G(X, Y) has more than one irreducible factor, then any
solution (a, b) to (3.18) has the property that G1 (a, b) and G2(a, b) divide B. Since B
has only finitely many distinct factors, we are reduced in this case to showing that
there are only finitely many solutions to the pair of simultaneous equations
and
This is clear, since the plane curves G 1 = B 1 and G 2 = B 2 have no common
components, so they intersect in only finitely many points.
Next suppose that r = 1, so G(X, Y) = G 1 (X, Y)e 1 • If B is not equal to B~ 1
for some integer B 1 , then G(X, Y) = B has no solutions. Otherwise, we can take
roots of both sides and reduce to the case that G(X, Y) is irreducible in Q[X, Y].
Then G(X, Y) has only simple roots, and by assumption there are at least three such
roots.
We factor G(X, Y) in qx, Y], say
with distinct algebraic numbers a 1 , ... , ad. Now divide the equation G(X, Y) = b
by yd to obtain
1Or we can invoke Gauss's lemma, which implies that the factorization of G(X, Y) in IQ[X, Y]
can
already be achieved in Z[X, Y].
106
3. Dynamics over Global Fields
We observe that if (x, y) E '!} is a large solution to G(X, Y) = b, then the
righthand side of(3.19) is small, so at least one ofthe factors on the lefthand side
must also be small. However, since a 1 , ... , ad are distinct, the rational number xI y
can be close to at most one of a 1 , ... , ad. Hence we can find a constant C = C (G, b)
such that
I JyJd
. IX--a·<c
mm
l:Si:'Sd y
t
-
for all (x, y) E '!}satisfying G(x, y) =b.
(3.20)
In the other direction, Roth's Theorem 3.34 applied to each of a 1 , ... , ad tells
us that for any E > 0 there is a constant"' = "'( E, a 1 , •.• , ad) > 0 such that
I JyJ2+e
"'
mm
. IX--a·>--
l:Si:Sd Y
t
-
X
for all - E Q.
y
(3.21)
Combining (3.20) and (3.21), we find that
IYid- 2
-€
~
c I"'·
By assumption, d ~ 3, so this shows that there are only finitely many possible values
for y. Finally, we observe that for each y, the equation G(x, y) = b implies that there
are at most d possible values for x. Therefore the equation G(x, y) = b has only
finitely many integer solutions
0
Siegel observed that Thue's theorem can be reformulated in terms of integral
values of rational functions.
Theorem 3.36. (Siegel) Let ¢(z) E Q(z) be a rational function with at least three
distinct poles in IP'1 (C). Then
{a E Q: ¢(a) E Z}
is a finite set.
Remark 3.37. Note that Theorem 3.36 need not be true if the rational function¢ has
fewer than three poles. A simple example with two poles is the function
F(z)
¢(z) = (z2- D)d'
where D > 1 is a squarefree integer and F(z) E Z[zj is a polynomial of degree 2d.
If we now take any solution (u, v) E Z2 to the Pell equation u 2 - Dv 2 = 1, then
¢( ulv) = v 2d F( ulv) E Z. The Pell equation has infinitely many solutions, so there
are infinitely many rational numbers ulv E Q with ¢(ulv) E Z.
Proof Write
¢ = [F(X, Y), G(X, Y)]
using homogeneous polynomials F(X, Y), G(X, Y) E Z[X, Y] of degree d having
no common factors. For any fraction a = alb E Q written in lowest terms, we have
107
3.6. Diophantine Approximation
-+-( ) = F(a,b)
'l'a
G(a,b)'
so ¢(a) E Z if and only ifG(a, b) divides F(a, b).
Let R = Res(F, G) be the resultant ofF and G, which is a nonzero integer
since F and G have no common factors. Proposition 2.13 says that there are homogeneous polynomials JI, g 1 , h, g2 E Z[X, Y] satisfying
!I (X, Y)F(X, Y) + g1 (X, Y)G(X, Y) = RX 2d-1,
h(X, Y)F(X, Y) + g2(X, Y)G(X, Y) = RY 2d- 1.
Substituting (X, Y) = (a, b) into these equations, we see that if G(a, b) divides
F(a, b), then G(a, b) also divides both Ra 2 d-l and Rb 2d-l_ However, a and bare
relatively prime, so we have proven that
¢(a/b) E Z implies that G(a, b) divides R.
It is important to emphasize that the resultant R depends only on F and G and that
it is nonzero. Hence
{ ~ E Q: ¢ (~) E Z} C
U {~ E Q: G(a, b)= D}.
(3.22)
DIR
Thue's Theorem 3.35 tells us that each set on the righthand side of (3.22) is finite,
which completes the proof of Theorem 3.36.
0
Remark 3.38. Roth's Theorem 3.34 is not effective, in the sense that it does not provide a method for computing an allowable constant~( t, a) in terms off and a. Thus
our proofs ofThue's and Siegel's theorems (Theorems 3.35 and 3.36) are also ineffective. Baker's theorem on linear forms in logarithms gives an effective, although
weak, version of Roth's theorem, from which one can derive effective versions of
Theorems 3.35 and 3.36. This means that our finiteness result on integral points in orbits (Theorem 3.43) proven in the next section can be made effective, but the stronger
integrality statement (Theorem 3.48) cannot, since it relies on the full strength of
Roth's theorem.
Let C C IP'N be a smooth projective curve defined over Q and let ¢ : C -+ !P'1
be a nonconstant rational function on C. If C has genus g ;:=: 1, Siegel proved that
Cq,(Z) = {P E C(Q) : ¢(P) E Z} is a finite set. For g ;:=: 2, this is superseded by
Faltings' theorem that C(Q) is finite, but for elliptic curves (g = 1), the set C(Q)
is often infinite. In this case, Siegel greatly strengthened the qualitative statement
that Cq,(Z) is finite by showing that the coordinates of the points in C(Q) have
numerators and denominators of approximately the same size. The precise theorem,
which we generalize to the dynamical setting in Section 3.8, is as follows.
Theorem 3.39. (Siegel [403, 404]) Let E/Q be an elliptic curve, let¢ E Q(E) be
a nonconstant rational function onE, and for each rational point P E E(Q), write
3. Dynamics over Global Fields
108
¢(P) = [a(P), b(P)] E JID 1 (Q)
with a(P), b(P) E Z and gcd(a(P), b(P)) = 1.
Then
logJa(P) J = 1.
lim
logJb(P)J
PEE(Q)
h(<P(P))-+oo
Proof See [410, IX.3.3]. For a general version on curves of arbitrary genus g 2: 1,
see for example [205, D.9.4], although as noted above, if g 2: 2, then Faltings [164,
0
165] proves that C(Q) is finite.
For the convenience of the reader, we conclude this section by stating general
versions of Theorems 3.34, 3.35, and 3.36 for number fields. Proofs of these, and
even more general, results may be found in [205, Part D] or [256, Chapter 7]. We set
the following notation:
a number field;
K
a finite set of absolute values on K;
S
Rs the ring of S-integers of K.
Theorem 3.40. (Roth) Let E > 0. For each v E S, extend v to K in some fashion,
and choose an algebraic number av E k. Then there is a constant li > 0, depending
on K, S, E, and {av }vES· such that
II min{lz- aviv, 1} 2: HK(:)2+'
nv
for all z E K.
vES
Theorem 3.41. (Thue-Mahler) Let G(X, Y) E K[X, Y] be homogeneous of degreed with at least three distinct roots in lP' 1 (C), and let B E K. Then there are only
finitely many (X, Y) E R~ satisfying G(X, Y) =B.
Theorem 3.42. (Siegel) Let ¢(z) E K(z) be a rational function with at least three
distinct poles ink. Then there are only finitely many a E K satisfying¢( a) E Rs.
3. 7 Integral Points in Orbits
In this section we prove that the orbit of a rational point by a rational map contains
only finitely many integers, except in those cases in which that statement is clearly
false. For ease of exposition, we work with Q and Z, but the result holds quite generally for rings of S-integers in number fields; see [411] or Exercise 3.38. We begin
with a definition.
Definition. A wandering point for a rational map ¢ : JID 1 - t JID 1 is a point P E JID 1
whose forward orbit Oq,(P) is an infinite set. Thus every point is either a wandering
point or a preperiodic point. 2
2The reader should be aware that in topological dynamics, especially with respect to invertible maps,
the standard definition says that a point is wandering if it has a wandering neighborhood. In other
words, a point P is topologically wandering if there is a neighborhood U of P and an integer no such
that¢; (U) n ¢) (U) = 0 for all i > j 2: no. Our definition coincides with the topological definition if
we use the discrete topology.
109
3.7. Integral Points in Orbits
Theorem 3.43. Let¢( z) E IQ( z) be a rational map ofdegree d 2 2 with the property
that ¢ 2 (z) ~ IQ[z]. Let a E 1Q be a wandering point for¢. Then the orbit
contains only finitely many integer points.
Theorem 3.43 is an immediate consequence of the following elementary geometric result combined with Siegel's Theorem 3.36 concerning integral values of rational
functions.
Proposition 3.44. Let ¢ E C( z) be a rational function of degree d 2 2 satisfying
¢ 2 ( z) ~ C [z ], so Theorem I. 7 implies that no iterate of¢ is a polynomial map. Then
Ifd 2 3, then #¢- 3 (oo) 2 3.
Proof We give a pictorial proof using the Riemann-Hurwitz formula. Suppose
that¢ is a rational map with #¢- 3 (oo) s; 2. There are four possible pictures for
the backward orbit of oo, as illustrated in Figure 3.1.
• ---+- Q ---+- P ---+- oo
• --.......
•
•
•
__.,. Q---+- p---+-
(A)
00
(B)
p---+- 00
(C)
---+- Q--.._
---+- Q'__.,.
• ---+- Q--+
• ---+- Q'--+
P--.._
P'__.,-
00
(D)
Figure 3.1: Backward orbits containing few points
The weak form of the Riemann-Hurwitz formula (Corollary 1.3) says that
2d- 2 =
L (d- #¢- (P)).
1
(3.23)
PEIP' 1
We apply (3.23) to each of the four pictures in Figure 3.1. Before computing, we
need to determine to what extent the points in the four pictures are distinct. For (A)
110
3. Dynamics over Global Fields
and (B), the three points P, Q, and oo must be distinct, since P = oo would mean
that ¢ is a polynomial and Q = oo i- P would mean that ¢ 2 is a polynomial.
Similarly, in (C) we cannot have P = oo, so relabeling the points in (C) and (D) if
necessary, we may assume that P, Q, and oo are distinct in all four cases. We now
use the Riemann-Hurwitz formula to estimate
2d- 2 2: (d- #¢- 1 (oo)) + (d- #¢- 1 (P)) + (d- #¢- 1 (Q))
(d- 1) + (d - 1) + (d- 1) = 3d- 3
_ { (d-1)+(d-1)+(d-2)=3 d-4
(d- 1) + (d- 2) + (d- 1) =3d- 4
(d- 2) + (d- 1) + (d- 1) =3d- 4
in case (A),
incase(B),
in case (C),
in case (D).
Thus case (A) yields d ::::; 1, a contradiction, while the other three cases give d ::::; 2.
This proves that if d 2: 3, then #¢- 3 ( oo) 2: 3. Finally, if d = 2, then the RiemannHurwitz formula tells us that ¢ has only two ramified points, and by inspection we
see that those points already appear in the pictures for (B), (C), and (D). Since
#¢- 3 (oo) = 2 in each case, it follows that #¢- 4 (oo) 2: 3. (See Exercise 3.37
0
for an explicit lower bound for #¢ -n ( oo) in terms of d and n.)
Proof of Theorem 3.43. Proposition 3.44 tells us that #¢- 4 ( oo) 2: 3. (If d 2: 3, we
could even take ¢- 3 .) The condition #¢ - 4 ( oo) 2: 3 can be rephrased as saying that
the rational function ¢ 4 has at least three distinct poles, so we may apply Siegel's
Theorem 3.36 to conclude that the set
(3.24)
is finite.
Consider the set of integers n such that the nth iterate of¢ applied to a: is integral,
It clearly suffices to prove that N a is finite. If n E N a with n 2: 4, then
so we see that ¢n- 4 (o:) is in the finite set (3.24). However, for any fixed {3 in the
set (3.24), there is at most one iterate of a: that is mapped to {3 (remember that a: is a
0
wandering point). This completes the proof that Oq,(o:) n Z is finite.
Theorem 3.43 says that if ¢ 2 (z) is not a polynomial, then the orbit Oq,(o:) contains only finitely many integral points. It is natural to ask how large 0 q, (a:) n Z
can be. The answer is that it may be arbitrarily large, as shown by the following
construction.
Example 3.45. In order to create a rational function whose orbit contains a large
number of integral points, we simply take a finite (reasonably random) sequence of
integers z0 , z2 , ... , Zk-l and treat the equations
111
3.7. Integral Points in Orbits
¢n+ 1 (zn) = Zn+I
for n = 0, 1, ... , k- 1
as a system of homogeneous linear equations for the 2d + 2 coefficients of¢. If the
degree of¢ satisfies 2d + 1 ~ k, then there should be a nontrivial rational solution.
We carry out this procedure with d = 2 to find the rational function
¢(z) = 899x 2 - 2002x + 275
33x 2 - 584x + 275
such that the orbit of 0 contains quite a few integer points:
0 __!!:__. 1 __!!:__. 3 __!!:__. -2 __!!:__. 5 __!!:__. -7
Of course, as Theorem 3.43 predicts, the list of integer points must end, and indeed
the next few iterates make it seem likely that there are no further integral points in
the orbit ofO. Thus
-
¢
7 - -¢- + 2917
299 - - - +
296398306
10198813
17011876043966969359
938619769242091763
Example 3.45 (see also Exercise 3.41) shows how to construct orbits with many
integer points by using rational maps of large degree. The following trick, adapted
from work ofChowla [103] and Mahler [285] on elliptic curves (see also [405]), says
that 0¢ (a) n Z may be arbitrarily large even for rational maps of a fixed degree.
Proposition 3.46. For all integers N ~ 0 and d ~ 2 there exists a rational map
¢( z) E Q( z) with the following properties:
• ¢ 2 (z) ~ C[z].
• 0 is a wandering point for ¢.
• 0, ¢(0), ¢ 2 (0), ¢ 3 (0), ... , </>N (0) E Z.
Proof Let 'lj;(z) E Q(z) be any rational map of degree d for which 0 is not a preperiodic point. For each 0 :<::; n :<::; N, write
as a fraction in lowest terms, and let B = b0 b1 · · · bN. Consider the rational function
¢(z) = B'lj;(z/ B). Clearly ¢n(z) = B'lj;n(z/ B), so for 0 :<::; n :<::; N we have
Hence ¢n(o) E Z for all 0 :<::; n :<::; N.
0
Although Proposition 3.46 shows how to make O¢(a)nZ arbitrarily large, it is in
some sense a cheat. What is really happening is that we are clearing the denominators
of rational points in an orbit. We can use the following notion of minimal resultant
to rule out this behavior.
3. Dynamics over Global Fields
112
Definition. For a rational map </>(z) E Q(z), write </>(z) as </>(z) = F¢(z)/G¢(z),
where F¢, G<P E Z[z] have integer coefficients and the greatest common divisor of
all of their coefficients is 1. Then F¢ and G¢ are uniquely determined by</> up to
multiplication by ±1, and we define the resultant of</> to be the quantity
Res(</>)= IRes(F</J, G¢)1.
(See Section 2.4 for the definition and basic properties of the resultant of two polynomials. See also Section 4.11 for a general discussion of minimal resultants of rational
maps over number fields.)
We can use the resultant to rule out the denominator-clearing trick of Proposition 3.46, and having done so, we conjecture that the number of integer points in an
orbit is bounded solely in terms of the degree of the map. (See Theorem 6. 70 for a
special case.) This is a dynamical analogue of a conjecture ofLang [254, page 140];
see also [202, 407].
Conjecture 3.47. Let </>( z) E Q( z) be a rational map ofdegree d ~ 2 with 4> 2 ( z) ~
Q[ z ], and let a E JP> 1 ( Q) be a wandering point for </>. Assume further that </> is affine
minimal in the sense that
Res(</>)=
min
/EPGL2(1Q)
f(z)=az+b
Res(<j>f).
(In other words, we cannot make the resultant smaller by conjugating by an affine
linear transformation az + b.) Then there is a constant C = C (d) depending only
on the degree of</> such that
#(O<P(o:) n Z) :::; c.
3.8
Integrality Estimates for Points in Orbits
Theorem 3.43 says that, except for the obvious counterexamples, an orbit 0 <P (a) contains only finitely many integer points. In this section we prove a dynamical analogue
of Siegel's Theorem 3.39, which asserts that the numerators and denominators of certain rational numbers have approximately the same number of digits. Siegel's proof
of Theorem 3.39 uses the existence of the multiplication-by-m maps [m] : E --t E
on an elliptic curve. These finite unramified maps have the effect of significantly increasing the height of points while leaving distances relatively unchanged. We adapt
Siegel's argument to the dynamical setting, but since our maps are on JP> 1 , they are
always ramified. This causes some additional complications, since distances shrink
significantly near ramification points.
Theorem 3.48. (Silverman [411]) Let </>(z) E Q(z) be a rational map with the
property that <j> 2 (z) ~ Q[z] and 1/4>2 (1/ z) ~ Q[z]. Let a E Q be a wandering point
for </>, and write
3.8. Integrality Estimates for Points in Orbits
113
as a fraction in lowest terms. Then
lim log lanl = 1.
(3.25)
n->oo log Ibn I
Remark 3.49. It is clear why we must assume that ¢ 2 (z) rf. Q[z] in the statement of
Theorem 3.48, but it may be less clear why we also require that 1/¢2 (11 z) rf. Q[z].
Letting f (z) = 1I z, we observe that
1/¢k(11 z) =
u- 1 0 ¢k 0 f)(z) = u- 1 0 ¢0 f)k(z) = (¢f)k(z).
So if 1/¢2 (1lz) E Z[z], then (¢f) 2n(z) E Z[z] for all n ~ 1. Now consider the
orbit of a= 1lb for an integer b. We have
The quantity (¢f) 2n(b) is an integer, so a2n = 1 and b2n = (¢f) 2n(b) E Z. Hence
the limit in (3.25) for even values of n is 0. Thus the assertion of Theorem 3.48 is
not true for rational maps ¢( z) such that 1I ¢ 2 ( 1I z) is a polynomial.
Example 3.50. To illustrate Theorem 3.48, we take ¢(z) = z + 1lz and a= 1 and
list the first few values of ¢n(l) = anlbn in Table 3.1.
n
an
bn
log( an)
log(bn)
1
1
0
1
2
1 0.69315
2
5
2 1.60943
3
29
10 1.46239
4
941
290 1.20759
5
969581
272890 1.10128
1014556267661
264588959090 1.05110
6
7 1099331737522548368039021 268440386798659418988490 1.02613
Table 3.1: Orbit 01(¢) for ¢(z) = z + 1lz, writing ¢n(1) = anlbn.
Notice how both the numerator and denominator of ¢n(1) grow extremely
rapidly. Of course, the elementary height estimate (Theorem 3 .11) tells us that the
maximum of Ian I and Ibn I grows this rapidly, but the fact that they both grow at approximately an equal rate lies much deeper and is the content of Theorem 3.48. And
to illustrate the speed with which the fractions grow, even for a very simple map of
degree 2 such as ¢(z) = z + 11 z, here is the exact value of ¢ 9 (1), which is the last
value that fits on one line using very small (5 point) type:
114
3. Dynamics over Global Fields
172699903806694372485 75086385863865042815392793760910340864851121501213389899778415 73308941492781
3 77908709746050392481071609609580527 436122569261424131112048023467330784 739529329885668846964890 •
This ninth iterate has logarithmic ratio
log( an) = log(17269990380 ... 492781) ~ 1. 00690 .
log(bn)
log(37790870974 ... 964890)
Proof The idea underlying the proof of Theorem 3.48 is fairly simple. Choose some
E > 0, and suppose that
(3.26)
This means that cf>n(a) = anfbn is very large, so cf>n(a) is close to oo. It follows
that a is quite close to one of the points in the inverse image cf>-n( oo ), say a is
close to (3 E ¢-n( oo ). But a is in Q, so one can hope that if n is sufficiently large,
then a and (3 are so close to one another that they contradict Roth's Theorem 3.34.
Unfortunately, this naive approach does not work, because the point (3 depends on n,
so the constant in Roth's theorem changes with each new value of n.
A more sophisticated idea is to use a fixed (large) integer m and apply Roth's
theorem to the rational point cf>n-m (a) and a nearby point in ¢-m (oo ). Note that
so the height of cf>n-m (a) is much smaller than the height of cf>n (a).
We fix an integer m satisfying dm > 6/E (we will see later why this is a good
choice form) and we let (3 be the point in cf> -m (oo) that is closest to cf>n-m (a). How
close is (3 to a? It turns out that this depends on the ramification index of cf> at various
points. If we make the (incorrect) assumption that cf> is everywhere unramified, then cf>
preserves distances up to a scaling factor, which allows us to make the following
estimates, where the constants Ct, C2 , •.• may depend on¢, a, and m, but they do
not depend on n:
~
~ Ibn
lanl
an
I=
lc/>n(a)l-1
since lanl ~ Ibn IHe from {3.26),
= Clp(cf>n(a),c/>m((3)),
definition of p, since lc/>n(a)l > 1,
since cf>m ((3) = oo,
~ C2p(cf>n-m(a), (3),
assuming cf> is unramified,
~ C1p(¢n(a), oo),
~ C3lc/>n-m(a)-
>
c4
c5
- H(cf>n-m(a))3'
f31,
definition of p,
Roth's Theorem 3.34 (with exponent 3),
property of heights (Theorem 3.11 ),
3.8. Integrality Estimates for Points in Orbits
115
since m satisfies dm > 6I f..
Taking logarithms, we have proven that
2
log lanl S -log(C5)
f.
for all n satisfying (3.26), i.e., lanl ~ lbnll+e.
We reiterate that the constant C5 depends only on ¢, a, and m; it does not depend
on n. Hence if n satisfies (3.26), then an, and also bn, are bounded. It follows that
the ¢-orbit of a contains only finitely many points satisfying (3.26), and therefore
.
log lanl
hmsup 1 lb I S 1 +f..
n-+oo
og n
(3.27)
Repeating the argument with the rational map 1I¢( 1I z) and the initial point 1I a
yields
(3.28)
Since (3.27) and (3.28) hold for all f. > 0, we conclude that the limit in (3.25) is 1.
This would complete the proof of Theorem 3.48 except for the unfortunate fact
that rational maps JP>1 ---+ JP>1 are always ramified. Further, our proof sketch made no
mention of the assumption that ¢ 2 ( z) is not a polynomial, and the theorem is false
for polynomials! In order to fix the proof, we begin by studying how ramification
affects the distance between points.
Lemma 3.51. Let¢ : JP> 1 (C) ---+ JP>1 (C) be a rational map of degree d ~ 2, let
p : JP> 1 (C) x JP> 1 (C) ---+ lR be the chordal metric as defined in Section 1.1, and
for Q E JP> 1 (C), let
(3.29)
be the maximum of the ramification indices of the points in the inverse image ofQ.
Then there is a constant C = C (¢, Q ), depending on ¢ and Q, such that
min
Q'Ecf>-l(Q)
p(P, Q')eQ(</>) S Cp(¢(P), Q)
for all P E JP> 1 (C).
(3.30)
In other words, if ¢(P) is close to Q, then there is a point in the inverse image ofQ
that is close to P, but ramification affects how close.
Proof. We dehomogenize using a parameter z such that Q -=J oo and oo rf. ¢- 1 ( Q).
This means that we can write Q = (3 and P = a and that we are looking for the
(3' E ¢- 1 ((3) that is closest to a.
Writing ¢(z) = F(z)IG(z) as a ratio of polynomials, the set ¢- 1 ((3) is precisely
the set of roots of the polynomial F(z) - (3G(z). If we factor this polynomial over C
as
3. Dynamics over Global Fields
116
with distinct /3~, ... , f3r, then /3i E ¢- 1(/3) has ramification index ei. For notational
convenience, we write
e = eQ(¢) = maxei.
We may assume that ¢(P) is quite close to Q, i.e., that p(¢(P), Q) is small,
since otherwise, the fact that the chordal metric satisfies p ~ 1 lets us choose a C
for which the inequality (3.30) is true. In particular, P =1- oo, and writing a= z(P),
we may assume that a is quite close to at least one of the points f3k in ¢- 1(/3). For
example, we may require that a satisfy
and
This implies in particular that
for all i =1- k.
Hence
iF( a)- f3G(a)i = lblla- f31le 1 1a- f32le 2 • • ·Ia- /3rler
= lblla- f3klek
II Ia- /3ilei
~ lblla- f3klek II ~lf3k- /3ile;
i'fk
= C1la- f3klek,
where the constant C1 is positive and depends only on¢ and /3. Further, the exponent
satisfies ek ~ e, so we obtain the estimate
The factthat ¢(a) = F(a)/G(a) is close to /3 and the assumption that G(/3) =1- 0
implies that G(a) is bounded away from 0, so dividing by G(a) yields
This in turn implies the same estimate for the chordal metric, since we can estimate
l¢(a)l using l/31 and we can estimate lal using l/3kl· Therefore
D
Next we show that if we stay away from totally ramified periodic points, then
iteration of¢ tends to spread out the ramification.
3.8. Integrality Estimates for Points in Orbits
117
Lemma 3.52. Let ¢ : lP' 1 -> lP' 1 be a rational map ofdegree at least 2 and let Q E lP' 1
be a point such that Q is not a totally ramified fixed point of ¢ 2 . Then
(See Lemma 3.51for the definition of eQ( ¢ ).)
Proof The proof of this geometric result uses the Riemann-Hurwitz Theorem 1.1
in a manner similar to the way that we have used it in the past. Let d = deg( ¢ ), fix
an integer m, and let P E ¢-m(Q). We use the fact that the ramification index is
multiplicative,
To ease notation, we write
We consider several cases.
Case I. The points P, ¢(P), ¢ 2 (P), ... , ¢rn- 1 (P) are distinct.
Note that this covers the case that Q is a wandering point. The Riemann-Hurwitz
formula allows us to estimate
ep(cjym) = eoe1e2 · · · em-1
<(
_
=
eo+ e1 +m·· · + em-1)m
ceo-
arithmetic-geometric inequality
1) + (e1- 1~+ ... + (em-1- 1) + 1) m
:::; ( 2dr: 2 + 1) m
Riemann-Hurwitz formula.
(3.31)
This estimate is clearly much stronger than the stated result, since the righthand side
of (3.31) has a finite limit as m -> oo. Indeed, it is an elementary exercise to verify
that (1 + tjm)m:::; et is valid for all t 2:: 0, so
ep(¢m):::; e 2d- 2 for all m 2:: 1 provided P, cjy(P), ... , ¢m- 1(P) are distinct.
(N.B. ep is a ramification index, while thee in e 2d- 2 is the number 2.71828 ... !)
Case II. P is purely periodic of period k 2: 3, and m 2: k.
The assumption that ¢k(P) = P implies that ei = eq,i(P)(¢) depends only on i
modulo k. Writing m = qk + r with 0 :::; r < k, we use the multiplicativity of
ramification, applied to cjyk, to estimate
118
3. Dynamics over Global Fields
Note that P,¢(P), ... ,¢k- 1 (P) are distinct, so we can apply Case I to each ofthe
ramification indices on the righthand side of(3.32). This yields
2d - 2
) kq ( 2d - 2
)r
ep(</Jm)::::; ( - k - + 1
-r- + 1
2d- 2
(
<
- -k +
1)m-r 2d-2
applying (3.31) to <Pk and ¢r,
using (1 + t/rr::::; et,
e
2d- 2
)m 2d-2
::::; ( -k-+1
e
.
Finally, we observe that
~ ( 2d; 2 +1) =1- (1-~) (1-~)::::; ~
forall d ~ 2 and k ~ 3.
This proves that ep( <Pm) ::=:; e2d- 2 ( id) m.
Case III. P is purely periodic of period k ~ 2, and m 2:: k.
We use the assumption that Q is not a totally ramified fixed point of ¢ 2 to deduce
that at least one of ep (¢) and ecf>(P) (¢) is strictly smaller than d - 1, so
ep(¢2) = ep(<fJ)ecf>(P)(f) = eoe1 ::::; d2 - d.
(3.33)
Now using the factthat ¢ 2(P) = P, we see that em= ecf>m(P)(¢) is equal to eo ifm
is even and equal to e1 if m is odd, so
ep(if>m) = e0e 1e2 · · · em-l
=
::::;
multiplicativity of ramification index,
{
(eoe1)mf 2
(eoe1)(m- 1)1 2eo
{
(d2 - d)m/ 2
ifm is even,
(d2 - d)(m- 1)1 2 d ifm is odd,
= ( 1-
ifm is even,
ifm is odd,
~rm/2J dm.
Case IV. P is preperiodic.
If P is periodic, we are already done from earlier cases. Suppose that P is not periodic and let j ~ 1 be the smallest integer such that ¢i (P) is periodic. Let R = ¢i (P)
and let k be the exact period of R. Then P, ¢(P), ... , <f)- 1 (P) are distinct and
R = ¢i (P) is periodic, so from Case I we have
ep(if>m) == ep(¢}). ecf>i(P)(¢m-j) ::::; e2d-2eR(¢m-j) ::=:; e2d-2eR(¢m).
We also note that R is not a totally ramified fixed point of ¢ 2 , since ¢- 1 ( R) contains
at least the two distinct points ¢k- 1 ( R) and ¢i - 1 ( P). (This is where we use the assumption that Pis not itself periodic.) Hence we can apply Case II or III to eR(¢m),
which gives the desired estimate for ep ( ¢m).
3.8. Integrality Estimates for Points in Orbits
119
In all four cases we have proven estimates for ep ( ¢m) that imply that
This completes the proof of Lemma 3.52. More precisely, we proved that there are
constants c1 and c2 , depending only on d and with c2 < 1, so that
for all m ~ 1.
And if P is a wandering point, we have shown that ep ( ¢m) is bounded by a constant
depending only on the degree of¢.
0
We next prove an elementary lemma that relates the chordal metric p( x, y) to the
Euclidean distance lx- Yl·
Lemma3.S3. LetpbethechordalmetriconJIP1 (!C). Thenforallx,y E C c JIP1 (!C),
1
p(x,y)::; 2p(y,oo)
p(x, y) ~ lx- Yl·
p(y oo)2
'2 .
Proof Note that p(z, oo) = 1/ Jlzl 2+ 1, so directly from the definition of p we
have
lx-yl
p(x,y)= vlxi2+1Jiyl2+1 =lx-ylp(x,oo)p(y,oo).
Then the triangle inequality for pin the form p(x, oo) + p(x, y) ~ p(y, oo) (see (1.2)
in Section 1.1) yields
p(x, y) ~ lx- Yl {p(y, oo) - p(x, y) }p(y, oo ).
Making the further assumption that p(x, y) ::; ~p(y, oo), we obtain the desired inequality.
0
Proof of Theorem 3.48. We now have the tools needed to complete the proof of Theorem 3.48. Writing tjJn(a) = anfbn as usual, our initial goal is to prove that the set
is finite. We observe that for n EN(¢, a, E), the point ¢n(a) is close to oo, since
By assumption, ¢ 2(z) ¢ C[z), so oo is not a totally ramified fixed point of ¢ 2.
This allows us to apply Lemma 3.52, which gives us an integer m0 = m 0(d, E) such
that
(3.35)
for all m ~ mo.
Having fixed an m ~ m 0 , we apply Lemma 3.51 to the map ¢m and the points
P = tjJn-m (a) and Q = oo to obtain a point .Bn E ¢ -m (oo) satisfying
120
3. Dynamics over Global Fields
(3.36)
where C1 = C1 (¢m, oo) depends only on m and¢. Hence if we define
thenN(¢, a,~:) is the union of N(¢, a, ~:,(3) for (3 E ¢-m(oo), so it suffices to prove
that the set N(¢,a,~:,(3) is finite for each (3 E ¢-m(oo). Note that if N(¢,a,~:,(3)
is infinite, then (3.34) and (3.36) imply that
lim
nEN(<f>,o,€,{3)
p(¢n-m(a),f3) =0.
(3.37)
n-+oo
In other words, ¢n-m(a) approaches (3 in JID 1 (C) as n--+ oo.
We consider first the case that (3 I= oo. The limit (3.37) tells us that
p(¢n-m(a),f3):::; ~p((3,oo)
forallbutfinitelymanyn E N(¢,a,~:,(3). (3.38)
Note that the inequality (3.38) allows us to apply Lemma 3.53 withx = ¢n-m(a)
andy= (3, which yields
where as usual C2 > 0 depends only on ¢ and m.
We are now ready to repeat the calculation from page 114, but this time done
rigorously to account for the fact that ¢ has ramification. As usual, all constants may
depend on ¢ and m, but are independent of n:
Ia: I€ 2: p(¢n(a), oo)
from (3 .34),
2: C3p(c/Jn-m(a),f3t=(<l>"')
from (3.36),
2: C41¢n-m(a)- f31eoo(</>"')
from (3.39),
>
c5
>
c6
- H(c/Jn-m(a))3eoo(</>m)
- H( c/Jn(a) )3e (</>m)jdm
00
c6
>
- H(¢n(a)f/2
=
Roth's Theorem 3.34 with exponent 3,
from Theorem 3.11, which says that
h(P) = d-mh(c!Jm(P)) + 0(1),
from (3.35),
c6
Janl€/ 2
Hence Janl :'S Ci 1 is bounded for n E N(¢,a,~:,(3), and the same is true
of Jbnl since JbnJH€ :::; lanl. so ¢n(a) takes on only finitely many values for
€
3.8. Integrality Estimates for Points in Orbits
121
n EN(¢, a,~:, {3). But by assumption, a is a wandering point for¢, so we conclude
that N (¢,a,~:, {3) is finite in the case that {3 :f. oo.
Suppose now that {3 = oo. Then a similar, but elementary, argument not requiring
Roth's theorem yields
from (3.34),
from (3.36),
definition of p,
Cs
- H(¢n-m(a)too(4>m)
definition of height, where note that
bn-m :f. 0, since a is wandering and
oo is periodic,
>
from Theorem 3.11,
>
from (3.35),
>
Cg
- H(¢n(a)roo(4>m)/dm
Cg
- H(¢n(a))'/6
Cg
\an\e/6
As above, we conclude that an and bn are bounded, and hence that N(¢, a, E, oo) is
finite.
We have now proven that N (¢, a, E, {3) is finite for all {3 E ¢ -m ( oo), and hence
that N(¢,a,E) is finite. The finiteness of N(¢,a,E) is equivalent to the statement
that
for all but finitely many n 2: 0.
(3.40)
We now apply the same argument to the rational map '1/J ( z) = 1I¢( 1I z) and the
point 1la. It is easily verified that 'lj;n(11a) = bnlan, so we obtain the complementary inequality
log \bn \ < 1 + E
log\an\ -
for all but finitely many n 2: 0.
(3.41)
Since (3.40) and (3.41) hold for all E > 0, it follows that
lim log \an\ = 1
n--->oo log \bn \
'
which completes the proof of Theorem 3.48.
D
122
3.9
3. Dynamics over Global Fields
Periodic Points and Galois Groups
In this section we study the Galois groups of the field extensions generated by periodic points of a rational map. Much of the theory is valid for rational maps defined
over an arbitrary perfect field and even for nonperfect fields, as long as <P is separable and one replaces the algebraic closure k of K with the separable closure Ksep.
For further material on this topic and the more general Galois theory of iterates,
see[3, 179,220,222,310,311,345,346,347,425].
Let ¢( z) E K (z) be a rational function of degree d 2: 2 with coefficients in a
perfect field K. The periodic points of <P have coordinates in the algebraic closure k
of K, since they are solutions to equations of the form
More precisely, write <P = [F, G] using homogeneous polynomials F, G E K[X, Y].
Then
</Jn = [Fn, Gn]
for homogeneous polynomials Fn, Gn E K[X, Y] of degree dn, and the periodic
points of <P of period n are the solutions in IP 1 ( K) to the equation
YFn(X, Y)- XGn(X, Y) = 0.
Thus counted with multiplicity, there are exactly dn + 1 such points.
In this section we study the field extensions of K generated by the coordinates
of the periodic points. Recall that if P = [a 0, ... , aN] E pN (K), then the field of
definition of P over K is the field K(P) obtained by choosing a nonzero ai and
setting
It is easy to see that this field is independent of the choice of the index i.
The Galois group Gal(k I K) acts on the points of lPN (K) in the natural way,
a(P) = [a(ao), a(al), ... , a( aN)],
and it is not hard to verify (Exercise 3 .4 7) that
K(P) =Fixed field of {a E Gal(K I K) : a(P) = P}.
Recall that the set of n-periodic points of <P is the set
Some of the points in Pern( <P) may have period that is smaller than n. For example,
Per n ( <P) contains all of the fixed points of¢. This suggests that we look at a smaller
set consisting of the primitive n-periodic points,
Per~*(¢)= {P E IP 1 (K): <Pn(P) = P and <Pi(P)-=/= P for alll::; i < n}.
We begin by verifying that both Pern (<P) and Per~* (<P) are Galois-invariant.
3.9. Periodic Points and Galois Groups
123
Proposition 3.54. Let ¢( z) E K (z) be a rational function of degree d. The set of
n-periodic points Pern(¢) and the set of primitive n-periodic points Per~*(¢) are
Galois-invariant sets.
Proof Let a E Gal( k / K). Then
¢(a(P)) = a(¢(P))
for all points P E JP> 1 (K),
since¢ is a rational function with coefficients inK. Let P E Pern(¢). Then
so a(P) E Pern(¢). This proves that Pern(¢) is Galois-invariant.
Similarly, we see that
which proves that P is a primitive n-periodic point if and only if a( P) is a primitive
D
n-periodic point. Hence Per~* (¢) is also Galois-invariant.
Proposition 3.54 tells us that the set of primitive n-periodic points generates a
Galois extension of K. We denote this extension by
Kn,<f> = K(P: PEPer~*(¢)).
These fields are analogues of the fields generated by roots of unity (cyclotomic
fields), so we call Kn,</> the n 1h dynatomicfieldfor ¢.We denote the Galois group of
the nth dynatomic field by
Gn,</> = Gal(Kn,<t>/ K).
Example 3.55. Consider the quadratic polynomial ¢(z) = z 2 + 1. The sets of primitive 2nd, 3rd, 4th, and 6th periodic points are given, respectively, by the roots of the
polynomials
*() ¢2(z)-z
2
c/J2 z = ¢(z) _ z = z + z + 2,
¢*(z)
=
3
¢ 3 (z)- z
= z6 + z 5 + 4z 4 + 3z3 + 7z 2 + 4z + 5,
¢(z)-z
¢*(z) = ¢ 4(z)- z = z 12 + 6z 10 + z 9 + 18z8 + 4z 7 + 33z 6 + 8z 5
4
¢2(z)-z
+ 40z4 + 9z3 + 30z 2 + 6z + 13,
¢*(z) = (¢ 6 (z)- z)(¢(z)- z)
6
(¢3(z)- z)(¢2(z)- z)
= z 54 - z 53 + 27z 52 - 25z 51 + · · ·- 13750z + 45833.
As this simple example clearly shows, it is infeasible to study dynatomic fields via
explicit equations except for very small values of n.
124
3. Dynamics over Global Fields
The Galois group Gn,</> acts on the set of primitive n-periodic points Per~* (¢ ),
and it is clearly determined by this action, but except in trivial cases Gn,</> cannot
be the full group of permutations ofPer~*(¢). The reason why it cannot be the full
group of permutations is due to the relation
a(¢i(P)) = ¢i(a(P))
for all a E Gn,</> and all PEPer~*(¢).
Thus the action of a on points in the orbit of Pis determined by its action on P. This
suggests that we decompose the action of Gn,</> on Per~* (¢) into its action on the set
of orbits and its action within each orbit.
A nice way to visualize the action of Gn,</> on the points of Per~* (¢) is to consider
the following picture:
Pr-1
1/J(Pr-d
¢2 (Pr-1)
An element of Gn,</> permutes the orbits, i.e., it permutes the columns, and then
within each column it performs a cyclical shift.
We can write this algebraically by observing that applying a E Gn,</> to a point P1
gives a point ¢i(Pk) for a uniquely determined 0 :S i < n and 1 ::; k ::; r. We
indicate the dependence of i and k on a and j by adopting the notation
Notice that 1rcr is a permutation of the set { 1, 2, ... , r}, so we can think of it as an
element of the permutation group Sr. This gives a map 1r : Gn,</> --+Sr.
Similarly, if we set icr = (icr(1), icr(2), ... , ia(r)), then we obtain a map
We perform a computation to see how i and 1r interact with one another:
(aT)(Pj) = a(T(Pj)) = a(¢ir(j)(Prrr(j)))
= ¢ir(j) (a(P1rr(j)))
= 1/Jir(j) (I/Jia(1rr(j)) p1ra(1rr(j)))
= I/Jir(j)+ia(1rr(j)) (P(1ra1rr)(j))·
From this equation we obtain the two formulas
and
Thus the map 1r: Gn,</>--+ Sr is a homomorphism, but the map
(3.42)
3.9. Periodic Points and Galois Groups
125
is not a homomorphism, since it is twisted by the permutation action of Sr on
(Z/nZt. This is an example ofthe following general construction.
Definition. Let S and H be groups, and let A be an index set on which S acts.
For simplicity, we assume that H is an abelian group and we write its group law
additively. The group structure on H makes the collection of maps Map( A, H) into
a group. Thus ifi 1 , i 2 E Map(A, H), then
There is a natural action of Son the group Map( A, H) via its action on A,
7r: Map( A, H)----+ Map( A, H),
7r(i)(a) = i(7r(a)).
The wreath product of H and S (relative to A) is the twisted product of the
groups Map(A, H) and S for this action. In other words, as a set the wreath product
consists of the collection of ordered pairs
Wreath(H, S) =Map( A, H) x S,
and its group law is defined by twisting the natural group law on the product,
(i1, 71'1) * (i2, 7!'2) = (7r2(il) + i2, 71'171'2)·
Theorem 3.56. Let ¢ E K ( z) be a rational fimction ofdegree d. Let C:h, ... , Or be
the distinct¢ orbits in Per~* (¢) and choose a point Pi E Oi in each orbit. For each
a E Gn,¢ and each 1 :::; j :::; r, define integers 0 :::; i,.(j) < nand 1 :::; 7l',.(j) :::; r
by the formula
a(Pj) = ¢iuUl(P1ru(j))·
Let Wreath(Z/nZ, Sr) be the wreath product of the symmetric group Sr and Z/nZ
relative to the natural action of Sr on the set {1, 2, ... , r }. Then the map
W : Gn,¢ ----+ Wreath(Z/nZ, Sr ),
a f-----> ( i,., 7l',.),
is an injective homomorphism.
Proof In order to show that W is a homomorphism, we need to verify that
W(ar) = W(a)W(r).
Writing this out in terms of the (twisted) definition of the group law on the wreath
product, we need to prove that
This is precisely the formula (3.42) proven above, which shows that W is a homomorphism.
126
3. Dynamics over Global Fields
Now suppose that W(a) = 1. This means that iu(j) = 0 and 7ru(j) = j for all
1 ~ j ~ r, so a(Pj) = Pj. Hence
so a fixes every point in Per~* (4>). Therefore a fixes Kn,cf>• so a = 1, which proves
that W is injective.
0
The above discussion shows that the Galois group Gn,cf> of the dynatomic extension Kn,cf>l K roughly splits up into two pieces, a permutation piece determined by a
permutation action on the orbits, and a cyclic piece determined by the action within
each orbit. The permutation piece can be quite complicated, but one might hope that
the dynamics of 4> can be used to study the cyclic piece. We now try to make these
vague remarks more precise.
Let
G~,cf> = Ker(7r: Gn,cf> - - Sr) ={a E Gn,cf> : a(Oj) = Oj for all1 ~ j ~ r },
and let
K~,cf> = fixed field of G~,cf>.
Then
More precisely, if we fix a particular primitive n-periodic point P E Per~* (4>), then
the extension K~,cf> (P) / K~,</> is a cyclic extension of order dividing n whose Galois
group is determined by the homomorphism
i : Gal(K~,cf>(P)/ K~.cf>) <-----? ZfnZ,
satisfying a(P) = ¢i" (P).
(3.43)
This formula gives us some control over the Galois group of the relative abelian
extension K~,cf>(P)/ K~,cf>·
Remark 3.57. If L = K(a) is any Galois extension and if a E Gal(L/ K) is any
automorphism, then a( a) can always be expressed as a polynomial in a. Thus there
is nothing special, a priori, in the fact that the action of Galois in (3.43) is given by a
polynomial. However, if we fix¢( z) of degree d and taken very large compared to d,
then the extension K~.cf>(P)/ K~.cf> is interesting because there is an automorphism a
described by a polynomial (or rational function) 4> of small degree compared to the
degree of the extension.
3.10
Equidistribution and Preperiodic Points
There are many theorems and conjectures concerning the distribution of torsion
points and points of small height on elliptic curves and abelian varieties. In this
section we describe, without proof, some dynamical analogues. For further details,
see Zhang's survey article [453] and the papers listed in its references.
3.10. Equidistribution and Preperiodic Points
127
We begin with the Manin-Mumford conjecture, which asserts that if X is an irreducible subvariety of an abelian variety A such that X n A 10rs is Zariski dense in X,
then X is a translate by a torsion point of an abelian subvariety of A. The original
Manin-Mumford conjecture was proven by Raynaud [367, 368]. Replacing torsion
points by preperiodic points leads to a dynamical conjecture, where the dynamical
analogue of a translate of an abelian subvariety of A is a preperiodic subvariety of!P'N
as in the following definition.
Definition. Let ¢ : IP'N ----+ JP>N be a morphism. A subvariety X c JP>N is a periodic
variety for¢ if there is an integer n ;::: 1 such that cpn(X) = X. The subvariety is a
preperiodic variety for¢ if ¢m(X) is periodic for¢ for some m ;::: 0.
Conjecture 3.58. (Dynamical Manin-Mumford Conjecture) Let ¢ : IP'N ----+ IP'N be
a morphism defined over C and let X C IP'N be a subvariety. Then
X n PrePer( ¢)
is Zariski dense in X if and only if the subvariety X is preperiodic for ¢.
The set of torsion points on an elliptic curve E, or more generally an abelian
variety, is equidistributed with respect to the natural (Haar) measure on E(C). More
precisely, we identify E(C) = C/ Las described in Section 1.6.3, and then for any
open set U c C lying in a fundamental domain for E(C) we have
. #(E[n] nu)
2~~
#E[n]
= Area(U).
A deeper equidistribution result of an arithmetic nature says that the Galois conjugates of torsion points are equidistributed. In order to state a dynamical analogue for
morphisms </> : lP'N ----+ lP'N, we need a ¢-invariant measure on lP'N (C) as described in
the following proposition.
Proposition 3.59. Let ¢ : JP>N ----+ JP>N be a morphism of degree d defined over C.
There is a unique probability measure 1-L<P on JP>N (C) satisfying
and
We call 1-L<P the canonical ¢-invariant probability measure on JP>N (C).
Proof For a general construction that covers both archimedean and nonarchimedean
base fields, see [453]. We also mention a standard result in dynamics (the KrylovBogolubov theorem [226, Theorem 4.1.1]), which says that any continuous map
¢ : X ----+ X on a metrizable compact topological space X admits a Borel probability
measure J.L,p satisfying ¢* (J.L,p) = J.L,p, i.e.,
J.L,p(¢- 1 (A)) = J.L,p(A)
foreveryBorel-measurablesubsetAofX.
D
128
3. Dynamics over Global Fields
For the remainder of this section we fix an algebraic closure Q of Q and an embedding Q <......+ C. So when we speak of a number field K and its algebraic closure k,
we assume that they come with compatible embeddings into C.
Definition. Let K / Q be a number field. For any algebraic point P E JP>N (k),
let C(P/ K) denote the set of Galois conjugates of P, i.e.,
C(P/K) = {a(P) E JP>N(K): a E Gal(K/K)},
and let 8p denote the Dirac measure on JP>N (C) supported at P,
8 (U) = { 1 if P E U,
p
0 if p ~ u.
We associate toPE JP>N (K) the discrete probability measure
1
J.Lp = #C(P/K)
L
QEC(P/K)
8Q
supported on the Galois conjugates of P.
Definition. Let K/Q be a number field and let Pt, P2, P 3 , ... E JP>N (K) be a sequence of points with algebraic coordinates. Fix a probability measure J.L on JP>N (C).
We say that the sequence {Pi h> 1 is Galois equidistributed with respect to J.L if the
sequence of measures J.LP; converges weakly3 to J.L.
We are now ready to state a dynamical equidistribution conjecture for Galois
orbits of preperiodic points on JP>N, and more generally for points of small height.
Conjecture 3.60. (Dynamical Galois Equidistribution Conjecture) Let K/Q be a
number field, let ¢ : jp>N ---+ jp>N be a morphism ofdegree d 2: 2 defined over K, and
let P 1, P2, P3, ... E JP>N (K) be a sequence of distinct points such that no infinite
subsequence lies entirely within a preperiodic subvariety of JP>N.
(a) If Pt, P2, P3, ... E PrePer( ¢), then the sequence {Pi}i~ 1 is Galois equidistributed in jp>N (C) with respect to the canonical ¢-invariant probability measure J.Lc/J·
(b) /flimi--+oo h¢(Pi) = 0, then the sequence {Pih~1 is Galois equidistributed
in jp>N (C) with respect to the canonical ¢-invariant probability measure J.Lc/J·
It is clear that Conjecture 3.60(b) implies Conjecture 3.60(a), since preperiodic
points have canonical height equal to 0. A version of the conjecture is known provided that the sequence of points satisfies a somewhat stronger Zariski density condition as in the following theorem.
3 Recall that a sequence of measures /-Li on a compact space X converges wealdy to p, if for every
Borel-measurable set U, the sequence of values /-Li (U) converges to p,(U) as i --> oo.
3.11. Ramification and Units in Dynatomic Fields
129
Theorem 3.61. (Yuan [450]) Let¢ : lP'N -+ lP'N be a morphism of degree d 2 2
defined over K and let H, P2, P3, . . . E lP'N (K) be a sequence ofpoints satisfying
the following two conditions:
(a) Every irifinite subsequence of { Pi}i> 1 is Zariski dense in lP'N.
(b) hq,(Pi) -+ 0 as i -+ oo.
(In the terminology of [453], sequences with property (a) are called generic and
sequences with property (b) are called small.) Then the sequence {Pi}i>l is Galois equidistributed with respect to the canonical ¢-invariant probability measure 1-L<i>
onlP'N(C).
Proof The proof is beyond the scope of this book. See Yuan [450] for a general version over archimedean and nonarchimedean base fields and algebraic dynamical systems on arbitrary projective varieties. Earlier results and generalizations
are given by Autissier [15, 16], Baker-Ih [24], Baker-Rumely [28], ChambertLoir [98], Chambert-Loir-Thuillier [99], Favre-Rivera-Letelier [169] and Szpiro0
Ullmo-Zhang [432].
The classical Bogomolov conjecture, which says that sets of points of small
height on abelian varieties lie on translates of abelian subvarieties, was proven by
Ullmo [435] and Zhang [452]. We state a dynamical analogue.
Conjecture 3.62. (Dynamical Bogomolov Conjecture) Let¢ : lP'N -+ lP'N be a
morphism of degree d 2 2 defined over a number field K and let X C lP'N be an
irreducible subvariety that is not preperiodic. Then there is an E > 0 such that the
set
{P E X(K): hq,(P) < c}
is not Zariski dense in X.
Notice that Conjecture 3.62 implies Conjecture 3.58, since the set of points
with hq,(P) < E includes all of the preperiodic points.
Finally, in closing this section, we mention that canonical invariant measures
have been constructed on Berkovich spaces; see Remark 5.77 and the references
listed there.
3.11
Ramification and Units in Dynatomic Fields
In Section 3.9 we used periodic points to construct field extensions Kq,,n and studied
their Galois groups. We now take up the question of the arithmetic properties of these
algebraic number fields. In general, the three basic questions that one would like to
answer about a given number field are these: Where is it ramified? What is its ideal
class group? What are its units? In addition, one wants to know how the Galois group
acts on ideal classes and on units. In this section we provide partial answers to the
question of ramification and units for dynatomic fields.
We first recall the classical case of cyclotomic extensions, which provide a model
for the dynatomic theory. Let (is a primitive nth root of unity and CJ E Gal(Ql/Ql).
Then
3. Dynamics over Global Fields
130
IT(() = (j(a)
for a unique j( IT) E (Z/nZ)*.
This defines an isomorphism
j: Gal(Q(()/Q)--+ (Z/nZ)*,
IT~ j(IT),
expressing the action of IT on Q( () as a polynomial action ¢( z) = z1. Now let p be a
prime not dividing n, let p be a prime ofQ(() lying above p, and let ITp E Gal(Q/Q)
be the corresponding Frobenius element. The definition of Frobenius says that
(3.44)
However, the nth roots of unity remain distinct when reduced modulo p, so the congruence (3.44) implies an equality in Q((),
1Tp(() = (P.
This exact determination of the action of Frobenius as a polynomial map on certain
generating elements ofQ(() is of fundamental importance in the study of cyclotomic
fields.
To some extent, we can carry over the analysis of Q( () to dynatomic fields, although the final results are not as complete as in the classical case. Let p be a prime
ideal of the ring of integers of K~,n' let p be the residue characteristic of p, and
let q be the norm of p. We assume throughout that p f n. Choose a prime ideal s,p
in K~,n (P) lying above p. Assuming that p does not ramify in K~,n (P), the associated Frobenius element IT p is determined by the condition
ITp(a)
=aq (mod s,p)
for all a in the ring of integers of K~,n(P). On the other hand, by construction the
action of IT p on P E Per~* (¢) is given by by formula
ITp(P) = c/Ji"P (P).
The next proposition allows us to characterize the primes at which the extension
K~,n(P)/ K~,n may be ramified.
Proposition 3.63. Let K be a number field, let ¢ E K ( z) be a rational map of
degree d 2: 2, let P E Per~* (¢, K) be a point in IP' 1 ( K) of exact period n, and let p
be a prime of K satisfying the following three conditions:
¢ has good reduction at p.
p does not divide n.
(3.45)
If A.q:,(P) -:/= 1, then p does not divide A.q:,(P) - 1.
(3.47)
Then P mod p has exact period n. In particular, the set
{p : P mod p has period strictly less than n}
is a finite set ofprimes of K.
(3.46)
3.11. Ramification and Units in Dynatomic Fields
131
Proof Let p be a prime of K satisfying (3 .45), (3 .46), and (3 .4 7). Let m be the exact
period of P and let r be the order of >..J, ( P) in IF~. Theorem 2.21 tells us that either
n =morn= mr, since (3.46) rules out powers of p appearing inn. If >..q,(P) = 1,
then also >..J,(F) = 1, so r = 1 and n = m. On the other hand, if >..q,(P) :f= 1,
then (3.47) tells us that
But >..J,(Pr = i by definition of r, so njm cannot equal r. Hence n = m m
all cases, which shows that P mod p has exact period n for all primes p satisfying (3.45), (3.46), and (3.47). This proves the first part of the proposition, and since
each of the three conditions is satisfied for all but finitely many primes, the second
D
part is clear.
Corollary 3.64. Let K be a number field, let ¢(z) E K(z) be a rational map of
degree d :2: 2, and let Kn,</> be the n 1h dynatomic field for ¢. Let S be the set of
primes p of K such that either ¢ has bad reduction at p or p divides n or p divides
the quantity
IT
(>..q,(P)- 1).
(3.48)
PEPer;;(¢)
,\q,(P)#l
Then Kn,q,/ K is unramified outside of S.
Proof The field extension Kn,q,/ K is generated by the points of Per~* ( ¢ ). Proposition 3.63 tells us that if p ~ S, then those points remain distinct when reduced
D
modulo primes lying over p. Hence the extension Kn,q,/ K is unramified at p.
Example 3.65. We continue studying the quadratic polynomial ¢(z) = z 2 + 1 from
Example 3.55 (see also Example 2.37). The polynomial ¢(z) has everywhere good
reduction, so if we assume for the moment that none of its periodic points have
multiplier equal to 1, then the quantity (3.48) in Corollary 3.64 is equal to
II (>..q,(a) -1) = II ((¢n)'(a)- 1) = Res(¢~(z), (¢n)'(z) -1).
</>~(a)=O
</>~(a)=O
Denoting this resultant by L\n (¢), it is not hard to compute the values of L\n (¢) for
small values of n. We obtain
L\2(¢) = 72,
L\3(¢) = (33. 11)3,
L\4(¢) = (3 2 . 11. 13. 41) 4,
L\s(¢) = (3 3 · 7 · 83 · 331·140869) 5 ,
L\6(¢) = (3 4 . 5. 7. 23. 73.223. 2251. 347495839) 6.
132
3. Dynamics over Global Fields
The field K¢, 2 is generated by the roots of ¢2(z) = z2 + z + 2, so we have
explicitly K¢, 2 = K( A). For higher values of n, we can check the primes
dividing il.n(¢) by computing directly the discriminant of the polynomial ¢~(z),
whose roots generate K¢,n/ K. Thus for example, Disc( ¢3) = -3 6 · 11 3 and
Disc(¢4) = 34 · 11 4 · 133 · 41 4 .
The local theory of units described in Section 2.7 allows us to construct units (or
at least S-units) in dynatomic fields Kn,</> and their composita. For convenience, we
make two definitions.
Definition. Let S be a finite set of places of K and let L / K be a finite extension.
We say that u E L is an S-unil if it is an S L -unit, where S L is the set of places of L
lying overS.
Definition. Let ¢(z) E K(z) be a rational map. We write 84> for the set
S¢ = M'j( U {primes at which ¢ has bad reduction}.
Thus ¢ has good reduction at all primes in the localized ring Rs. In particular, if
¢(z) = aozd +···+ad
is a polynomial, then the finite primes in S¢ are the primes where some ai is nonintegral together with any primes for which a0 is not a unit.
Having set this notation, we now state globalized versions of the dynamical unit
theorems from Section 2.7.
Theorem 3.66 (Global version of Theorem 2.33). Let ¢(z) E K[z] be a polynomial of degree d 2: 2, let a: E Per~* (¢) with n 2: 2, and fix integers i and j
satisfying gcd(i- j, n) = 1. Then
¢i(a:)- ¢-i(a:)
¢(a:)- a:
is an 84>-unit in K¢,n·
For the next result, we recall that the cross-ratio of four points P1. P2 , P 3 , P4
in JP> 1 is the quantity
"'
(p p. p.
1,
2,
n ) __ (x1Y3- X3Yl)(X2Y4 - X4Y2)
3,£4
(
)(
)'
X1Y2 - X2Yl X3Y4 - X4Y3
Theorem 3.67 (Global version of Theorem 2.34). Let ¢(z) E K(z) be a rational
function ofdegree d 2: 2, let P E Per~* (¢ ), andfix integers i and j satisfying
gcd(j, n) = gcd(i- 1, n) = gcd(i- j, n) = 1.
Then
is an 84>-unit in K</>,n·
K-(P, ¢P, ¢i P, ¢) P)
3.11. Ramification and Units in Dynatomic Fields
133
Theorem 3.68 (Global version of Theorem 2.35). Let ¢( z) E K (z) be a rational
function ofdegree d :2: 2, let m and n be integers with m f n and n f m, and let
P = [x, y] E Per;,:'(¢)
and
Q = [x', y'] E Per~*(¢).
Denote by S p the set ofplaces
Sp = {v EM~: v(x) > Oandv(y) > 0} U {v EM~: v(x) < Oorv(y) < 0},
and similarly for Sq. Then
xy'- x'y
is an (S¢ U Sp U Sq)-unit in the compositum K¢,mK¢,n·
Theorems 3.66, 3.67, and 3.68 allow us to construct many S-units in dynatomic
fields Kcp,n and their composita K¢,mK</J,n· Further, since Gal(K¢,n/ K) frequently
contains an element O" <P whose action on points P E Per~* (¢) is characterized by the
equation O"¢(P) = ¢(P), we obtain a partial description of the action of the Galois
group on the dynamical units. For example, the units in Theorem 3.67 transform via
We compare this to the construction of cyclotomic units in cyclotomic fields.
Let q be a prime power and let (be a primitive qth root of unity. Then the cyclotomic
field Q( () contains the units
for 2 :::; i:::; q; 1 with gcd(i, q).
The Galois group Gal(Q(()/Q) is the set of elements O"t characterized by
O"t(() = (t withO < t < qandpft.
The action of the Galois group on the cyclotomic units is given by the explicit formula
(}" t ( (i - 1 ) = (it - 1 .
(t- 1
(- 1
Further, the cyclotomic units generate a subgroup of finite index in the full group of
units Z[(] *, and the index of this subgroup is related to the class number of Q( ().
The situation for dynatomic fields is not nearly as complete. One problem is that
the dynatomic fields K</J,n tend to have very large degree over K, so the dynamical
unit theorems cannot produce enough units to give a subgroup of finite index in the
full unit group. Further, the Galois group Gal(K¢,n/ K) is usually huge, and we have
an explicit description only of the subgroup generated by the element O"¢. However,
since for general number fields there is no known way to systematically produce any
units with any explicit Galois action, the dynatomic construction might be said to fall
under the heading of "half a loaf is better than none."
134
3. Dynamics over Global Fields
Example 3.69. Consider the rational map
¢>(z) = z2 - 4.
After some algebra, we find that
¢>(z) - z = z 2 - z- 4,
¢2(z)- z
2
¢>(z) _ z = z + z- 3,
¢ 3 (z)- z
¢>(z)- z
-.'--:-'--- = (z 3 - z 2 - 6z + 7)(z 3 + 2z 2 - 3z- 5).
It is not hard to check that ¢( z) - z divides q;n (z) - z for all n 2:: 1 (Exercise 1.19(a)),
but the further factorization of ¢ 3 ( z) - z into a product of cubics is less common;
see Exercise 3.49. If we let a, (3 E C satisfy
and
(3 3 + 2(32 - 3(3 - 5 :::;:: 0,
then we have
..ff7},
Peri* (¢>) = { 1 ± 2
Per;* (¢>) = { - 1 ~
y'I3},
Per;* (¢>) = {a,¢>( a), ¢ 2(a), (3, ¢>((3), ¢2((3)} ,
where we recall that Per~* (4>) denotes the set of points of exact period n for¢.
Let K = Q(a). The polynomial z 3 - z 2 - 6z + 7 is irreducible over Q, but it
factors completely inK since its roots are a, if>( a), and ¢ 2(a). Hence K is Galois
over Q with Galois group generated by the map a determined by a( a) = ¢(a).
Notice that the discriminant of z 3 - z 2 - 6z + 7 is 192, again confirming that its
roots generate a cyclic cubic Galois extension. Further, Z[a] must be the full ring of
integers RK of K, since
192 = Disc(Z[al) = [R: Z[al] 2 Disc(K/Q) > [RK: Z[aJt
(Note that K =f. Q, so Disc(K/Q) > 1.) Then the fact that 19 is prime forces
RK =Z[a].
We apply Theorem 3.66 with i = 2 and j = 1 to obtain the unit
u =
1
¢ 2(a)- ¢(a)
= a 2 + a - 4 E R*
¢(a)- a
K·
It is easy to check that NK;Q(u 1 ) = 1, confirming that u 1 is a unit. Similarly, taking
i = 2 and j = 0 gives the unit
135
Exercises
In this case, NK;Q(u2) = -1.
The field K is totally real of degree 3, so its unit group R'K has rank 2. It is not
hard to see that the two units u 1 and u 2 are independent, so they at least generate a
subgroup offinite index. (In fact, { -1, u 1 , u 2 } generates the full unit group R'K, but
we leave the verification of this fact to the reader.) We can compute the action of the
Galois group on u1 and u2,
a(u1) = a(a 2 +a- 4) = ¢(a) 2 +¢(a)- 4 = a 4 - 7a 2 + 8 =-a+ 1,
a(u2) = a(a 2 +a- 3) = ¢(a) 2 +¢(a)- 3 = a 4 - 7a 2 + 9 =-a+ 2.
These new units are related to the original units by
and
Using points of period 2 and 3 for¢, we can create units in larger fields. Thus let
L = K( v'I3). Then L contains both a and the points in Per;*(¢), so Theorem 3.68
with n1 = 2 and n2 = 3 says that
u 3 = 'Y- a
with
-1 + v'I3
'Y = - - - 2
is a unit in the ring of integers of L. If we take the norm from L down to K, we
find that NL;K(u3) = u2 is one of the units that we already discovered. Similarly,
NL;Kb- ¢(a))= -a+ 2.
·
If instead we compute the norm of u 3 from L down to Q("f), we find using
"( 2 + 'Y - 3 = 0 that
3
2
NL/Q{-y)('Y-a)="f -"'( -6"(+7=-"(+1=
3-v'I3 ·
2
This unit and -1 generate the unit group of the ring of integers ofQ( v'f3).
For additional information about cyclic cubic extensions generated by periodic
points of polynomials, see [306], and for a general analysis of units generated by
3-periodic points of quadratic polynomials, see [313, Section 8].
Exercises
Section 3.1. Height Functions
3.1. Show that the constant C(d, N, D) in Conjecture 3.15 must depend on each of the quantities d, N, and D by giving a counterexample if any one of them is dropped.
3.2. Let
v(B) = #{P E lP'N (Q): H(P):::; B}.
(a) Find positive constants c1 and c2 such that
c1BN+l :::; v(B) :::; c2BN+l
for all B 2: 1.
Exercises
136
(b) For N = 1, prove that
lim v(B) = 12.
B--+oo
B2
1r 2
(c) More generally, prove that the limit limB--+oo v( B)/ BN +1 exists and express it in terms
of a value of the Riemann (-function.
3.3. Prove that
#{P E lP'N (Q): H(P) :S Band D(P) :S D} :S (12D)N2ND 2 BND(D+l).
Aside from the constant, to what extent can you improve this estimate? In particular, can the
exponent of B be improved?
3.4. Let F(X) = a0 Xd + a 1xd-l + · · · +ad E Q[X] be a polynomial of degree d, and
factor F(X) as
F(X) = ao(X- a1)(X- a2) ···(X- ad)
over the complex numbers. Prove that
2-d H(a1) · · · H(ad) ::; H([ao, a1, ... , ad]) ::; 2d H(a1) · · · H(ad)·
(Hint. Mimic the proof of Theorem 3. 7 for the upper bound. To prove the lower bound, for
each v pull out the root with largest lailv and use induction on the degree of F.) Can you
increase the 2-d and/or decrease the 2d?
3.5. Prove that the number of (N + 1)-tuples (io, ... , iN) E zN+l of integers satisfying
io, ... , iN ~ 0
and
io +···+iN= d
is given by the combinatorial symbol ( N 1d). Note that this is equal to the number of monomials of degree din theN+ 1 variables xo, ... , XN.
Section 3.2. Height Functions and Geometry
3.6. Let¢ : lP'2 -+ lP' 2 be the rational map ¢(X, Y, Z) = [X 2, Y 2, X Z]. Although¢ is not
defined at [0, 0, 1], we can define Per(¢, lP'2) to be the set of points satisfying ¢n(P) = P for
some n ~ 0 and ¢i(P) f= [0,0, 1] for all 0::; i < n. Prove that Theorem 3.12 is false by
showing that Per(¢, lP'2(Q)) is infinite. What goes wrong with the proof? Try to find a "large"
subsetS oflP'2 such that Per(¢, S(K)) is finite for every number field K.
3.7. LetK/Qbeanumberfield,letP = [x 0, ... ,xN] E lP'N(K),andletbbethefractional
ideal generated by xo, ... , x N. Prove that
3.8. Let KjQ be a number field and let ¢(z) E K(z) be a rational map of degree d ~ 2.
Recall that the height H ( ¢) is defined by writing
¢ = [F(X, Y), G(X, Y)] = [aoXd + a1Xd-l + · · · + adYd, boXd + · · · + bdYd]
and setting H(¢) = H([ao, ... ,ad,bo, ... ,bdl). (See (3.4) on page 91.) Prove that there are
positive constants c1 (d) and c2 (d) such that
for all P E lP' 1(J?).
Find expressions for c1 and c2 in terms of d. This gives an explicit version of Theorem 3.11
for lP' 1.
Exercises
137
3.9. Let ¢c(z) = z 2 +c.
(a) Prove that there are infinitely many c E Q such that ¢c has a Q-rational fixed point.
(b) Prove that there are infinitely many c E Q such that ¢c has a Q-rational point of exact
period 2.
(c) Prove that there are infinitely many c E Q such that ¢c has a Q-rational point of exact
period 3.
(d) Prove that there are no c E Q such that ¢c has a Q-rational point of exact period 4.
3.10. Let d ~ 2 and let </Jd(z) = zd. Prove that there is an absolute constant c such that for
all number fields K jQ of degree n we have
# PrePer( </Jd, lP' 1 (K)) ::; c[K: Q]log log([K: QJ).
Prove that aside from the constant, this upper bound cannot be improved. (Hint. You will need
the fact that the Euler totient function 'P satisfies <p( m) ::; em/ log log m; see for example [ 11,
Theorem 13.14].)
In particular, the uniform boundedness conjecture (Conjecture 3.15) is true for </Jd. This
is one of the few maps for which the conjecture is known. The others are the Chebyshev
polynomials and Lattf:s maps; see Theorem 6.65.
Section 3.4. Canonical Heights and Dynamical Systems
3.11. Let¢ : lP'N ___, lP'N be a morphism defined over K, let f E PGLN+ 1 (K) be an
automorphism oflP'N, and let ¢f = f- 1 o ¢of. Prove that
N
-
for all P E lP' (K).
Thus the canonical height is conjugation-invariant, i.e., it is independent of change of coordinates.
3.12. Let ¢, 1/J : lP'N ___, lP'N be morphisms of degree at least two that commute with one
another, i.e.,¢(1/J(P)) = 1/J (¢(P)) for all points P E lP'N (Q). Prove that
for all P E lP'N (Q).
3.13. Let ¢ : lP'N ___, lP'N be a morphism of degree at least two defined over a number field K
and let P E lP'N (k) have the property that
Prove that either P is preperiodic for ¢ or else there is some point in the orbit of P satisfying ¢n(P) E lP'N (K). Is this result true if instead K jQp is a p-adic field?
3.14. In the setting of Theorem 3.20, suppose that the setS is a topological space and that the
maps ¢ : S ___, S and h : S ___, lR are continuous maps. Prove that iL : S ___, lR is also a continuous map. (Hint. Show that the functio~s hm ( P) = d-m h ( ¢m ( P)) form = 1, 2, 3, ... are
continuous and converge uniformly to h.)
3.15. Let hq, be the canonical height for ¢(z) = z 2 +c. For any given c E Q, there is a
minimum nonzero value for hq,(z) as z ranges over nonpreperiodic points in Q. Find that
minimum value for:
(a) c = 0.
138
Exercises
(b) c = -2.
(c) c= -1.
(d) **arbitrary c E Z.
(e) **arbitrary c E Q.
(Hint. Try to do (a), (b), and (c) directly, but we note that it is easier to do them using the
theory oflocal canonical heights for polynomials; see Exercises 3.24 and 3.28. For (d) and (e)
the goal is to describe the minimum value of hq, in terms of c.)
3.16. Let ¢>( z) = z 2 - 1 . Analyze the canonical height for points in the following sets.
(a) Bq,( oo ), the attracting basin of oo, where we recall (Exercise 2.1) thatthe attracting basin
of oo for a polynomial ¢> is the set
Bq,(oo) ={a E C: lim if>n(a) = oo} U {oo}.
n-->oo
(b) :Fq, '- Bq,( oo ). (Note that all of the points in this set are eventually attracted to the attract-
ing 2-cycle 0 ~ -1.)
(c) Jq,.
(See also Exercise 3.27.)
3.17. Let¢> : pN ---> pN be a morphism defined over a number field K, and for any point
P E pN (K), let D(P) = [K(P) : Q]. Prove that there is a constant C = C(¢>) > 0 such
that
N for all P E lP (K) with P ¢ PrePer(¢>).
(Hint. Use the estimate for the number of points of bounded degree and height given in Exer-
cise 3.3.) This is a very weak version of the dynamical Lehmer Conjecture 3.25.
3.18. **Let c E Q with c "1- 0, -2 and let ¢>(z) = z2+c. Prove thatthere exist K = K(c) > 0
and N = N(c) > 0 such that
for all a E Q with a¢ PrePer(¢>).
(This is not currently known for any value of c other than c = 0 and c = -2.)
3.19. Let a E Q* and let
f(x) = aoxd + a1xd-l +···+ad E Z[x]
be the minimal polynomial of a, normalized so that ao > 0 and gcd(ao, ... , ad)
Factor f(x) over Cas f(x) = ao Il(x- a;). Prove that
1.
The quantity H (a) d, or equivalently H Q( a) (a), is also known as the Mahler measure ofa and
denoted by M(a). Lehmer's conjecture (Conjecture 3.24) can be stated in terms of Mahler
measure: There is an absolute constant K > 1 such that if a E Q* is not a root of unity, then
M(a) > K.
Exercises
139
3.20. Let ¢(z) E Q(z). Write a program to estimate hq,(a) directly from the definition.
Use your program to compute the following heights to a few decimal places. (See also Exercises 3.30 and 5.31.)
(a) ¢(z) = z 2 - 1 and a= ~·
(b) ¢(z) = z 2 + 1 and a=~·
(c) ¢(z) = 3z 2 - 4 and a= 1.
1
(d) ¢(z) = z + -and a= 1.
z
3z 2 - 1
(e) ¢(z) = z 2 _ 1 and a= 1.
3.21. Let ¢(z) = z 2 - z + 1. The ¢-orbit of the point 2 is called Sylvester's sequence [428],
CJq,(2) = {2,3,7,43,1807,3263443,10650056950807,
113423713055421844361000443, ... }.
(a) Prove that Sylvester's sequence satisfies
¢n+1(2) = 1 + ¢0(2)¢1(2)¢2(2)¢3(2) ... ¢n(2).
(b) A rough approximation gives hq,(2) ~ 0.468696, so ¢n(2) is approximately equal to
e0.4 68696 n2 • Prove a more accurate statement by showing that
n = 0, 1, 2, ... ,
is positive, strictly increasing, and converges to ~·(Hint. First conjugate ¢(z) to put it
into the form z 2 + c.)
(c) Deduce that there is a real number H such that ¢n(2) is the closest integer to Hn 2 for
all n 2 0. Note that this is far stronger than the general height estimate
3.22. Let 6 > 0, let d 2 2 be an integer, and let (xn)n;:::o be a sequence of positive real
numbers with the property that
xo > 1 + 6
and
I
lxn+1 - x~ ::S: 6x~- 1
for all n 2 0.
(a) Prove that the sequence Xn is strictly increasing and that Xn -+ oo as n-+ oo.
(b) Prove that the limit
n---+oo
exists. (Hint. Take logs and use a telescoping sum to show that the sequence is Cauchy.)
(c) Prove that
IH n2 - Xn I ::S: d _6 1 for all n 2 0.
3.23. Let d 2 2 and let ¢(z) E Z[z] be a monic polynomial of degree d, say
¢(z) = zd + azd- 1 + ··· E Z[z].
Prove that for every E > 0 there is a constant C = C ( ¢, E) such that for all a E Z satisfying Ia! 2 C we have
for all n 2 0.
Exercises
140
Section 3.5. Local Canonical Heights
The general theory oflocal canonical heights is developed in Section 5.9. However, the theory becomes much simpler if ¢ is a polynomial, because the local canonical height then has
a simple limit definition similar to the limit used to define the global canonical height. Exercises 3.24-3.30 ask you to develop some of the theory oflocal canonical heights for polynomials.
3.24. Let ¢( z) E K [z] be a polynomial. Prove that the limit
~cf>,v(a) = n--+oo
lim d1 logmax{l¢n(a)lv, 1}
n
(3.49)
exists and that the resulting function has the following two properties:
(a) For all a E Kv,
(b) The function
a~----+ ~cf>,v(a) -logmax{lalv, 1}
is continuous on Kv and has a finite limit as lalv --+ oo.
Hence the function defined by (3.49) is a local canonical height as described in Theorem 3.27.
Prove that the (global) canonical height is equal to the sum of the local heights,
hcp(a) =
L nv~cf>,v(a).
vEMK
3.25. Let ¢(z) E K[z] be a polynomial and let v be an absolute value on K. Prove that the
local height ~</>,v as defined by (3.49) has the following properties.
(a) ~cf>,v(a) 2:: 0 for all a E Kv.
(b) Acp,v(a) = 0 if and only if I<Pn(a)lv is bounded, equivalently, if and only if a is in the
v-adic filled Julia set Kv(¢) of¢ (see Exercise 2.1).
3.26. Let ¢(z) E C(z] be a polynomial with complex coefficients and let Bct>(oo) be the
attracting basin of oo for¢. (See Exercises 2.1 and 3.16 for the definition of Bct>(oo).) Prove
that the local height
)..</> : C --+IR
as defined by (3.49) has the following properties:
(a) >.</>is a real analytic function on Bet>( oo) -.... { oo }.
(b) The function)..</> is harmonic on Bet>( oo)-.... { oo }. In other words, writing z = x + iy, the
function )..</> is a solution to the differential equation
on the open set Bet> (oo) -.... { oo}. It satisfies the boundary conditions that >.</> vanishes on
the boundary of Bet> (oo) and has a logarithmic singularity at z = oo, i.e.,
Act>(z) -log lzl
is bounded as z --+ oo.
(c) )..</> is the unique function that has the properties described in (b).
In classical terminology, the function ~</> is the Green function for the filled Julia set K (¢) =
lP' 1 (C)-.... Bcp(oo).
141
Exercises
3.27. Let ¢>(z) = z 2 - 1. Analyze the local canonical height ~</>,v for points in:
(a) theattractingbasinB.p(oo)ofoo.
(b) F.p"-B.p(oo).
(c) J.p.
(See also Exercise 3.16.)
3.28. Let cf>(z) E Z[z] be a monic polynomial of degree d :::: 2 and let o: = a/b E Q be a
rational number written in lowest terms. Prove that
h.p(a):::: log lbl,
with equality if and only if o: is in the filled Julia set K( ¢>) of¢>.
3.29. Let K/Q be a number field and let ¢>(z) E K(z) be a rational map of degree d:::: 2.
For each finite place v
write ¢>(z) = Fv(z)/Gv(z) as a ratio of polynomials that are
normalized for v, i.e., the coefficients of Fv and Gv are v-adic integers and at least one coefficient is a v-adic unit. Let~¢,':;;' be the local canonical height, as described in Theorem 3.27,
normalized using Gv in (3.16).
(a) Prove that~¢~':;;' is well-defined, independent of the decomposition cf>(z) as a ratio of vnormalized polynomials. For nonarchimedean places v this serves to pin down a specific
function~¢~':;;' that depends only on¢> and v (cf. Remark 3.28).
(b) Give an example to show that it may not be possible to write cf>(z) = F(z)/G(z) such
that F and G are simultaneously normalized for all finite places v E M~.
(c) More generally, prove that every¢> E K(z) can be written as F(z)/G(z) with F and G
simultaneously normalized for all finite places v E M~ if and only if K has class
number 1.
EMiL
3.30. Suppose that ¢>( z) E K (z) has good reduction at a finite place v E M K. Prove that the
function
~.p,v(a) = logmax{lo:lv, 1}
fora E Kv
is a local canonical height by showing that it has the required properties.
Section 3.6. Diophantine Approximation
3.31. Prove Dirichlet's Theorem 3.33, which says that for every o: E lR" Q there are infinitely
many x / y E Q satisfying
I~ -o:l ~ :
2 •
(Hint. Look at the numbers yo:- Lyo:J for 0 ~ y ~ A. They all lie in the interval [0, 1].
Divide the interval into A equal pieces and use the pigeonhole principle.)
3.32. Let B be a nonzero integer.
(a) Prove that every solution (x, y) in integers to the equation
(3.50)
satisfiesmax{lxl, IYI} ~ ...j4ii73. (Hint. ThepolynomialX 3 +Y 3 factorsinQ[X, Y].)
(b) Try to find an explicit upper bound for max{lxl, IYI} for the similar-looking equation
X 3 + 2Y 3 = B. The difficulty you face illustrates the fact, seen during the proof of
Theorem 3.35, that the equation G(X, Y) = B is relatively easy to solve if G(X, Y)
has distinct factors in Q[X, Y], but very difficult if it does not.
Exercises
142
(c) Find all solutions in integers x 2: y to the equation (3.50) for the following values of B:
(i) B = 2. (ii) B = 91. (iii) B = 728. (iv) B = 1729.
Section 3.7. Integral Points in Orbits
3.33. Let ¢(z) = z + 1/ z and write q;n(1) = an/bn in lowest terms as in Example 3.50.
(a) Prove by a direct computation that there are constants c, c' > 0 such that
logan
- logbn -
c < -- < c
,
for all n > 2.
-
(b) Try to prove directly that log an /log bn ---+ 1 as n ---+ oo.
(c) Prove that q;n(1) ---+ oo as n---+ oo. (Notice that oo is a rationally indifferent fixed point,
since ¢' ( oo) = 1.)
3.34. Let c E !Z be a squarefree integer and let 4>c(z) = z + cjz. Further, let a E Q be a
wandering point for ¢c·
(a) Ifl- 4c is not a perfect square, prove that#( O¢(a) n z) :S: 2.
(b) Ifl-4c = d2 is a perfect square, prove that# ( 0¢( a)nz) :::; 2 unless a = (-l±d)/2,
in which case there are three integer points in the orbit.
3.35. Let ¢( z) E Q( z) be a rational function of degree d 2: 2 such that ¢ 2 is not a polynomial. Suppose that ¢ has everywhere good reduction and that ¢( oo) = oo. Let a E Q be a
wandering point for ¢. Prove that
# (0 <t> (a) n !Z) :::; 2d - 1.
Show that this is sharp for d = 2. Try to improve the bound for larger values of d.
3.36. Let ¢(z) E Q(z) be a rational function of degree d 2: 2, let a E Q be a wandering
point for¢, and write q;n(a) = an/bn E Q as a fraction in lowest terms.
(a) Suppose that there is a nonzero integer B such that bniB for infinitely many n. Prove
that ¢ 2 (z) E Q[z]. (Hint. Make a change of variables and apply Theorem 3.43.)
(b) Suppose that there are infinitely many n such that bn Ibn+ 1· Prove that either oo is a fixed
point of¢ or else ¢ 2 (z) E Q[z].
3.37. Let ¢(z) E C(z) be a rational function of degree d. This exercise gives a quantitative
strengthening of Proposition 3.44.
(a) Let P be a nonperiodic point for¢. Prove that
for all n 2: 0.
(b) Let P be a fixed point of¢ that is not totally ramified. Prove that
for all n 2: 2.
(c) Generalize (b) to the case that Pis a periodic point for¢ of exact period m under the
assumption that q;rn is not totally ramified at P.
Use (a), (b), and (c) to deduce that if¢ is not a polynomial map, then
#¢ -n (P) 2: 3
if either
{
n 2: 4
and
d = 2,
n 2: 3
and
d 2: 3.
Exercises
143
3.38. Let K/Q be a number field, letS C MK be a finite set of absolute values on K, and
let Rs be the ring of S-integers of K. Let cf>(z) E K(z) be a rational function of degree d :2: 2
with ¢>2 (z) ~ K[z], and let a E K be a wandering point for¢>. Prove that
CJ<t>(a) n Rs
is a finite set. (This exercise generalizes Theorem 3.43.)
3.39. Let K be a number field, letS C MK be a finite set of absolute values on K that
includes all the archimedean absolute values, and let Rs be the ring of S-integers of K. Let
cf>(z) E K(z) be a rational function of degree d :2: 2 satisfying ¢>2 (z) ~ K[z].
(a) Let n = 4 if d = 2 and let n = 3 if d :2: 3. Prove that the set
{a E K: cf>n(a) E Rs}
is finite.
(b) Give an example with (n, d)= (3, 2) and oo nonperiodic for¢> such that the set in (a) is
infinite.
(c) Same question as in (b) with n = 2 and d arbitrary.
(d) Repeat (b) and (c) with oo a fixed point of¢>.
3.40. Let c/>1, ... , cf>r E Q( z) be rational functions of degree at least 2, and let <I> be the collection of rational functions obtained by composing an arbitrary finite number of ¢> 1, ... , cf>r.
Note that each c/>i may be used many times. For example, if r = 1, then <I> is simply the collection of iterates {¢>1}. Also note that in general, composition of functions is not commutative,
so if r :2: 2, then <I> is likely to be a very large set.
(a) A rational map ¢> E K(z) is said to be of polynomial type if it has a totally ramified fixed point. Prove that for such a map, there is a linear fractional transformation f E PGL 2 (K) such that ¢>f E K[z]. Further, if¢> is not conjugate to zd, prove
that one can take f to be in PGL2(K).
(b) Assume that <I> contains no maps of polynomial type. Prove that there are finite subsets <I>1 and <I>2 of <I> satisfying
and
(Here ¢><I> denotes the composition of¢> with every map in <I>.)
(c) Let a E Q. The <I>-orbit of a is the set CJof1 (a) = {¢>(a) : ¢> E <I>}. Continuing with
the assumption that <I> contains no maps of polynomial type, prove that the <I>-orbit of a
contains only finitely many integers, i.e., prove that
C).p (a) n Z is a finite set.
3.41. Define the (logarithmic) height of a rational map cf>(z) E Q(z) by writing
cf>(z) = F(z) = ao + a1z + · · · + adzd
G(z)
bo + b1z + · · · + bdzd
with F(z), G(z) E Q[z] and setting
h(¢>) = h([ao, a1, ... , ad, bo, b1, ... , bd]).
Prove the following quantitative version of Proposition 3.46.
For all d :2: 2 there is a constant C = C( d) such that for all integers N 2: 0 there exists a
rational map ¢>( z) E Q( z) of degree d with the following properties:
Exercises
144
• ¢?(z) rt C[z].
• 0 is a wandering point for ¢>.
• 0, ¢(0), ¢ 2 (0), ¢ 3 (0), ... , if>N (0) E Z.
e h(if>) ~ C · dN.
Try to do quantitatively better than this, for example, replace the height estimate with one of
the form h(¢>) ~ C ·ON for some o <d.
3.42. Recall that a rational map ¢( z) E Q( z) is affine minimal if its resultant Res(¢>) cannot
be made smaller via conjugation by an affine linear map f (z) = az +b. If¢> is affine minimal
and ¢ 2 is not a polynomial, we have conjectured (Conjecture 3.47) that the size of 0¢ (a) n Z
is bounded solely in terms of the degree of¢>. For each integer d ~ 2, let
¢>(z) E Q(z), ¢> 2 (z) rt Q[z], }
{
.
C(d)=sup #(O¢(a)nz): ¢isaffineminimal,and
a E IP'N (Q) 'PrePer(¢)
Thus Conjecture 3.47 says that C(d) is finite.
(a) Prove that for every integer N ~ 0 there exists an affine minimal rational map ¢(z) E
Q(z) such that ¢ 2 (z) rt Q[z], such that 0 is not a preperiodic point for¢, and such that
are all integers. In particular, C(d) --+ oo as d--+ oo.
(b) Prove that the function ¢>(z) in (a) can be chosen to have degree l(N- 1)/2J. Hence
C(d) ~ 2d+2.
(c) Let ¢( z) be the function
¢>(z) = 481z 5 - 2565z 4 + 9385z 3 - 14955z 2 + 12094z- 3720.
z 5 - 465z 4 + 2185z 3 - 6975z 2 + 8254z- 3720
Trace the orbit of 0. How many integers do you find?
(d) Find a rational function as in (a) of degree 2 and with the property that ¢i(O) is an integer
for all 0 ~ i ~ 6. Hence C(2) ~ 7, which improves the bound from (b). Can you find
infinitely many such functions? Can you prove that C(2) ~ 8?
(e) Find a rational function as in (a) of degree 3 and with the property that ¢;(0) is an integer
for all 0 ~ i ~ 8. Hence C(3) ~ 9, which improves the bound from (b). Can you find
infinitely many such functions? Can you prove that C(3) ~ 10?
Section 3.8. Integrality Estimates for Points in Orbits
3.43. Let Kv be a field complete with respect to the absolute value v, let ¢> : IP' 1 --+ IP' 1 be
a rational map of degree d ~ 2 defined over Kv, and let Pv : IP' 1 (Kv) x IP' 1 (Kv) --+ lR
be the associated chordal metric. (See Sections 1.1 and 2.1 for the definition of the chordal
metric when vis archimedean and nonarchimedean, respectively.) Generalize Lemma 3.51 by
showing that for every Q E IP' 1 ( K v) there is a constant Cv = Cv (¢, Q) such that
for all P E IP' 1 (Kv ).
Here eq (¢>) is as defined in Lemma 3 .51.
145
Exercises
3.44. Let K be a number field and let ¢> : lP' 1 -+ lP' 1 be a rational map of degree d ~ 2 defined
over K. Prove that there is a finite set of absolute values S C MK such that for all v i Sand
all Q E lP' 1 (K), the constant Cv(¢>,Q) in Exercise 3.43 may betaken equal to 1.
3.45. Let ¢>(z) E Q(z) be a rational map of degree d ~ 2 with ¢ 2 (z) i Q[z], let o: E Q, and
write ¢n(o:) = an/bn E Q as a fraction in lowest terms as usual. Prove that
( )
_ h, q,o:.
Ibn Ilog . _ 1lm
lanllog . lIm
dn
n~oo
dn
n~oo
3.46. Let K/Q be a number field, let v E MK be an absolute value on K, and let
¢>(z) E K(z) be a rational function of degree d ~ 2. Suppose that o: E lP' 1 (K) is a wandering point for ¢> and that 1 E IP' 1 ( K) is any point that is not a totally ramified fixed point
of ¢ 2 . Prove that
-log Pv ( ¢>n (0:), I)
.
=0.
lim
h( ¢Jn(o:))
n~oo
Taking first 1 = oo and then 1 = 0, explain how to use this result to generalize Theorem 3.48
to number fields.
Section 3.9. Periodic Points and Galois Groups
3.47. Let P E lP'N (K) be a point in projective space and let K(P) be its field of definition.
Prove that
K(P) =fixed field of {a E Gal(K / K) : a(P) = P}.
In mathematical terminology, this says that the field of moduli of P is a field of definition
for P. We discuss fields of moduli and fields of definition for (equivalence classes of) rational
maps in Chapter 4.
Section 3.11. Ramification and Units in Dynatomic Fields
3.48. Let ¢( z) = z 2 + c and let
be the polynomial whose roots are periodic points of period 3.
(a) Prove that <I>3(z) E Z[c, x] is a polynomial in the variables c and z and that it has integer
coefficients.
(b) Let o: be a root of <I>3 ( z) and assume that the field Q( c, o:) is an extension of Q( c) of
degree 6 (i.e., <I>3(z) is irreducible in Q(c)[z]). Theorem 2.33 implies that
and
are units. Compute these units explicitly as elements of Q[c, o:].
(c) Prove that there is a field automorphism a : Q(c, o:) -+ Q(c, o:) characterized by the
fact that it fixes Q( c) and satisfies a( o:) = ¢>( o:) = o: 2 +c. (Note that in general, Q( c, o:)
will not be a Galois extension ofQ(c).) Prove that a 3 is the identity map.
(d) Compute the units a(u1) and a(u2).
(e) Compute the units a 2(u1) and a 2 (u 2 ).
(f) Express a(u1), a(u2), a 2(u!), and a 2(u2) in the form ±uiu~.
Exercises
146
3.49. Let ¢(z) = z 2 + c and let <P3(z) be as in the previous exercise.
(a) Prove that <P3(z) factors into a product of two cubic polynomials in the ring Q(c)[z] if
and only if c has the form c = -(e 2 + 7)/4 for some e E Q(c).
(b) Suppose that c = -(e 2 + 7)/4. Show that there is a polynomial 9e(z) E Q[e, z]
such that the factorization <P3(z) has the form 9e(z)9-e(z). Compute the discriminant
of 9e(z) and verify that it is a perfect square in Q[e]. Conclude that if 9e(z) is irreducible
over Q( e), then its roots generate a cyclic Galois extension Ke of Q( e).
(c) Let a be a root of 9e(z). Use the results of the previous exercise to construct units
in Q( e) (a). Analyze these units for some small values of e, say e = 1, 5, 7, 9. (Note that
we investigated the case e = 3 in Example 3.69.)
3.50. **Let ¢(z) be a generic monic polynomial of degree d 2: 2, i.e., ¢(z) is a polynomial
of the form zd + a 1 zd-I + · · · + ad, where a 1 , ... , ad are indeterminates. Let p be a prime
and let a E Per;• (¢).Theorem 3.66 says that the elements
u;,j =
¢;(a)- ¢1(a)
'
¢(a)- a
0:::; j < i < p,
are units. What is the rank of the group that they generate? Is the answer the same for the
polynomial ¢(z) = z 2 + c?
3.51. **Let ¢(z) be a generic rational function of degree d 2: 2, let p be a prime, and let
P E Per;• (¢ ). Theorem 3.67 describes how to construct units K(P, ¢P, ¢; P, ¢1 P). What
is the rank of the group that they generate? Is the answer the same for the rational function ¢(z) = (z 2 + b)/(bz 2 + 1)?
Chapter 4
Families of Dynamical Systems
Most of our work in previous chapters has focused on a single rational map ¢(z)
and the effect its iterates have on different initial values. We now shift focus and
consider the effect of varying the rational map ¢( z). In order to do this, we study the
set of all rational maps. This set turns out to have a natural structure as an algebraic
variety, as does the set of rational maps modulo the equivalence relation defined
by PGL 2 conjugation. There are many threads to this story. The specific topics that
we touch upon in this chapter include:
1. Dynatomic polynomials and fields generated by periodic points.
2. The space of quadratic polynomials (the simplest nontrivial case).
3. Rational maps that are PGL 2 -equivalent over k, but not over K (twists).
4. Field of defintion versus field of moduli for rational maps.
5. Minimal models for rational maps.
6. Moduli spaces of rational maps (with marked periodic points).
All of these topics have close analogues in the geometric and arithmetic theory
of elliptic curves and abelian varieties. For purposes of comparison, we list the correspondences:
1. Division polynomials and fields generated by torsion points.
2. The space of elliptic curves (the simplest abelian varieties).
3. Abelian varieties isomorphic over k, but not over K (twists).
4. Field of definition versus field of moduli for an abelian variety.
5. Minimal Weierstrass equations and Neron models.
6. Elliptic modular curves and moduli spaces of abelian varieties.
In this chapter, we let K be a perfect field and fix an algebraic closure k of K,
although much of what we do is also valid for nonperfect fields if we use instead a
separable closure of K. We write Gal(k I K) for the absolute Galois group of K. It
is convenient to use the profinite group Gal(K I K), but we observe that it generally
suffices to work with finite extensions and finite Galois groups, since the coefficients
of any particular rational map ¢( z) E k (z) lie in a finite extension of K.
147
148
4.1
4. Families of Dynamical Systems
Dynatomic Polynomials
Let ¢(z) E K[z] be a polynomial. Then the fixed points of¢ are the roots of ¢(z) - z
(plus the point at oo), and more generally, the points of period n for ¢ are the roots
of ¢n(z)- z. However, the polynomial ¢n(z)- z may have roots of period smaller
than n, since if ¢k(a) =a and kin, then also ¢n(a) =a. It is natural to try to
eliminate these points of strictly smaller period and focus on the points of exact
period n.
Example 4.1. We recall an analogous situation. The roots of the polynomial zn - 1
include all nth roots of unity, not only the primitive ones. The nth cyclotomic polynomial is defined using an inclusion-exclusion product,
nth cyclotomic polynomial=
IT(zk- 1)JL(n/k).
(4.1)
kin
It is the polynomial whose roots are the primitive nth roots of unity. Here J.L is the
Mobius function J.L defined by J.L(1) = 1 and
( e1 • • • er) _ { (-1 Y if e1 = · · · = er = 1,
Pr .f
>
0
1 any ei _ 2.
J.L Pt
(4.2)
See [216, Section 2.2] or [11, Chapter 2] for basic properties of the Mobius function
and the Mobius inversion formula. It is easy to check that the product (4.1) is a
polynomial, using the fact that the (complex) roots of zn- 1 are distinct and the
following basic property of the J.L function (Exercise 4.2):
LJ.L
kin
U~) =
k
{1 ~fn =
0
1,
tfn>l.
(4.3)
Taking our cue from the example provided by the cyclotomic polynomials, we
might define the nth dynatomic polynomial by the formula
<I>n(z) = IT(<t>k(z)- zt(n/k).
kin
However, it is not clear that <I>n(z) is a polynomial, since ¢n(z) may have multiple
roots, as shown by the following example.
Example 4.2. Let ¢( z) be the polynomial
3
¢(z) = z2 - -.
4
Then
4.1. Dynatomic Polynomials
149
¢( z) - z = z2 - z 2
4
~ = ( z - ~) ( z + ~) ,
32
3
3
¢ (z)- z = z - 2z - z- 16 = ( z- 2)
(
1
z + 2)
3
,
z (z + ~)2
¢2(z)- =
¢(z)-z
2
Thus ¢ 2 ( z) - z vanishes with multiplicity 3 at the point z = - ~, and although it is
true that the ratio ~(~t:zz is a polynomial, it is somewhat distressing to observe that
its root is a fixed point of¢, not a point of primitive period 2.
For simplicity, the preceding discussion dealt with polynomial maps ¢(z). We
now turn to general rational maps ¢(z) E K(z) and develop tools that are useful for
studying their periodic points. Let ¢( z) be a rational function of degree d and write
¢ = [F(X, Y), G(X, Y)] using homogeneous polynomials F and G. Then the roots
of the polynomial
YF(X, Y)- XG(X, Y)
in lP' 1 are precisely the fixed points of¢. If we count each fixed point according to
the multiplicity of the root, then ¢ has exactly d + 1 fixed points. More generally,
we can apply the same reasoning to an iterate ¢n of ¢ and assign multiplicities to
the n-periodic points.
Definition. Let ¢( z) E K (z) be a rational function of degree d, and for any n ~ 0,
write
with homogeneous polynomials Fn, Gn E K[X, Y] of degree dn. (See Exercise 4.9
for a formal inductive definition of Fn and Gn.) Then-period polynomial of¢ is the
polynomial
ll>q,,n(X, Y) = Y Fn(X, Y)- XGn(X, Y).
Notice that ll>q,,n(P) = 0 if and only if ¢n(P) = P, which justifies the name assigned to the polynomial ll>q,,n·
The n 1h dynatomic polynomial of¢ is the polynomial'
q,,n (X ' Y)
I!>*
=IJ(Y Fk(X ' Y)- XG k(X ' Y))J-<(n/k) =II I!> </>,k (X ' Y)J-<(n/k) '
where J.l is the Mobius function. If¢ is fixed, we write ll>n and~!>~. If ¢(z) E K[z]
is a polynomial, then we generally dehomogenize [X, Y] = [z, 1] and write ll>n(z)
and ll>~(z).
All ofthe roots P ofll>¢,n(X, Y) satisfy ¢n(P) = P, but we saw inExample4.2
that ll>¢,n (X, Y) may have roots whose periods are strictly smaller than n. Following
Milnor [302], we make the following definitions.
1See Theorem 4.5 for the proof that <l>¢,n is indeed a polynomial.
4. Families of Dynamical Systems
150
Definition. Let ¢( z) E K (z) be a rational map and let P E lP' 1 be a periodic point
for¢.
• Phasperiodnif~n(P) =0.
• P has primitive (or exact) period n if ~n(P) = 0 and ~m(P) :/:- 0 for all
m<n.
• P has formal period n if~~ (P) = 0.
We set the notation
Pern(¢) = {P E lP' 1 : ~n(P) = 0},
Per~(¢)= { P E lP' 1 : ~~(P) = 0},
Per~*(¢) = { P E lP' 1 : ~n(P) = 0 and ~m(P) :/:- 0 for all1 :::; m < n }.
Thus Pern(¢) is the set of points of period n, Per~(¢) is the set of points of
formal period n, and Per~* (¢) is the set of points of primitive (or exact) period n. Sometimes we treat these as sets of points with assigned multiplicities, e.g.,
if P E Per~(¢), the multiplicity of Pis the order of vanishing of~~ at P.
It is clear that
primitive period n
=*
formal period n
=*
period n,
but neither of the reverse implications is true in general.
Remark 4.3. The polynomial <I></>,n is homogeneous of degree dn + 1, so counted
with multiplicity, the map¢ has exactly dn + 1 points of period n. And if we let
vd(n) = deg(~¢,n(X, Y)) =
LJ-L (~) (dk + 1),
(4.4)
kin
then counted with multiplicity, the map ¢has exactly vd(n) points of formal period n. The number vd(n) grows very rapidly as d or n increases. See Exercise 4.3.
The first few period and dynatomic polynomials for ¢( z) = z 2 + c are listed
in Table 4.1 on page 156. Notice how complicated ~n and~~ are, even for small
values of n. For example, ~6 has degree 54 as a polynomial in z and degree 27 as a
polynomial in c.
Remark 4.4. Rather than using the homogeneous polynomials ~nand~~, it is sometimes more natural and convenient to consider instead the associated divisors in lP' 1 ,
especially in generalizing the theory to higher-dimensional situations. This is the
approach taken in [313], where for any (nondegenerate) morphism¢: V---. V of
smooth algebraic varieties, the 0-cycle ~</>,n is defined as the pullback of the graph
of ¢n by the diagonal map A: V---. V x V. See Exercise 4.8.
We now prove the important fact that the dynatomic polynomial ~¢,n (X, Y) is
indeed a polynomial.
151
4.1. Dynatomic Polynomials
Theorem 4.5. Let¢( z) E K (z) be a rational function ofdegree d 2: 2. For each P E
JID 1 ( K), let
ap(n) = ordp(<I>q,,n(X, Y))
and
or, in terms ofdivisors in Div(JID 1 ),
div(<I>q,,n) = L
and
ap(n)(P)
div(<I>:i,,n) = L
aj,(n)(P).
p
p
(a) <I>:i,,n E K[X, Y], or equivalently,
aj,(n) 2:0 foralln 2: 1 and all P E JID 1 .
(b) Let P be a point ofprimitive period m and let A(P) = ( ¢m )' ( P) be the multiplier ofF. ThenPhasformalperiodn, i.e., aj,(n) > 0, ifandonlyifoneof
the following three conditions is true:
(i) n = m.
(ii) n = mr and A(P) is a primitive r 1h root of unity.
(iii) n = mrpe, A(P) is a primitive r 1h root of unity, K has characteristic p,
and e 2: 1.
In particular, if K has characteristic 0, then a j, (n) is nonzero for at most two values
ofn.
Proof By the definition of <I>~, we have the relation
aj,(n) = LJL(n/m)ap(m).
min
We begin with a lemma that describes the value of a p ( n) for fixed points.
Lemma 4.6. Let '¢( z) E K (z) be a rational function of degree d 2: 2, let P E
JID 1 (K) be a fixed point of'¢, let A= Ap('lj;) = 'lj;'(P) be the multiplier of'¢ at P,
and let
ap(N) = ordp(<I>,;;,N(X, Y))
be as in Theorem 4.5. Then for any N 2: 2,
AN# 1
===> ap(N) = ap(1) = 1,
(4.5)
A # 1 and AN = 1
===> ap(N) > ap(1) = 1,
===> ap(N) = ap(1) 2: 2,
===> ap(N) > ap(1) 2: 2.
(4.6)
A = 1 and N # 0 in K
A = 1 and N = 0 in K
(4.7)
(4.8)
Proof Making a change of variables, we may assume that P = 0, and then the
assumption that P is a fixed point of 'lj; means that 'lj; ( z) has the form
(4.9)
4. Families of Dynamical Systems
152
where a i- 0, e 2: 2, and O(ze+l) is shorthand for a function that vanishes to order
at least e + 1 at 0. Note that
if>. i- 1,
1
.
ap(1) = ord('ij!(z)- z) = {
e 2: 2 If A = 1.
z=O
We consider first the case that A i- 1. Using the weaker form 'ij!(z) = AZ + O(z 2 )
of (4.9) and iterating gives
so
ap(N) = ord('ij!N (z)- z)
z=O
= ord((AN- 1)z + O(z 2 ))
z=O
-{ - 1
-
2: 2
if AN f- 1,
.f ,N __ 1_
1
A
This completes the proof of the lemma in this case.
Next we consider the case A = 1. Then ap(1) = ordz=o('ij!(z)- z) = e, and
iteration of (4.9) yields
(4.10)
Hence
ap(N) = ord('l/JN (z)- z)
-{ = e = ap(1)
z=O
= ord(Naze + O(ze+l))
> e = ap(1)
z=O
if N =f. 0 inK,
if N = 0 inK.
D
This completes the proof of Lemma 4.6
Resuming the proof of Theorem 4.5, we observe that if the primitive period m
does not divide n, then ap(n) = 0, and hence also af,(n) = 0. We may thus assume
that min. Since we will deal with several different maps, we write ap(¢, n) and
af,( ¢, n) to indicate the dependence on the map. Let
and
N = !!___
m
Using the fact that ap( ¢, k) = 0 unless mlk, we find that
af,(¢, n) =
L M (~) ap(¢, k) = L M ( ~=) ap(¢, km)
kin
kiN
= LM ( ~) ap(</Jm,k) = af,('ij!,N).
kiN
153
4.1. Dynatomic Polynomials
We further observe that Pis a fixed point of 'lj;, so Lemma 4.6 applies to 'ljJ and P.
We consider several cases, the first two of which may done simultaneously.
Case I(a) . .X(P)N =f. 1.
Case l(b) . .X(P) = 1 and N =f. 0 inK.
In Case l(a) we have )..(P)k "I 1 for all kiN, so we can apply (4.5) of Lemma 4.6
to conclude that a p ('ljJ, k) = a p ('ljJ, 1) for all kIN. Similarly, in Case l(b) we can
apply (4.7) of Lemma 4.6 to conclude that ap('lj;, k) = ap('lj;, 1) for all kiN. Hence
in both cases we find that
aj,('l/J,N) = l:tL
kiN
(~) ap('lj;,k) = (2:/L (~) )ap('lj;,1)
kiN
=
{ap('lj;, 1) 2 1 if N = 1,
if N > 1.
0
Using the equality ap('lj;, 1) = ap(qr, 1) = ap(<fy,m), this shows in Case l(a,b)
that
ifn = m,
ifn > m.
Case II• .X(P) = 1 and N = 0 inK.
Let p be the characteristic of K and write N = pe M with p f M. We observe that
if kiM and 0:::; i:::; e, then k "I 0 inK and Ap('lj;Pi) = Ap('ifJ)Pi = 1, so (4.7) of
Lemma 4.6 applied to 'lj;Pi tells us that
ap('lj;,pik) = ap('lj;Pi, k) = ap('lj;Pi, 1) = ap('lj;,pi).
This allows us to compute
=
(2:/L (~)) (ttL(Pe-i)ap('ljJ,pi))
kiM
t=O
= {ap('lj;,pe)- ap('lj;,pe-l) 2 1
0
if M = 1,
if M > 1.
The fact that the value is positive when M = 1 follows from applying condition (4.8)
of Lemma 4.6 to the difference ap('lj;P"- 1 ,p)- ap('lj;Pe- 1 , 1). Hence in Case II, we
have shown that aj,( ¢, n) 2 0, and further, aj,( ¢, n) 2 1 if and only if n = mpe.
Since Case II includes the assumption that )..(P) = 1, this proves Theorem 4.5 in
this case.
Case III. .X(P) =f. 1 and .X(P)N = 1.
Let r be the exact order of )..(P) inK*, so riN and r > 1. Then (4.5) of Lemma 4.6
4. Families of Dynamical Systems
154
tells us that if r f k, then ap('ljJ, k) = ap('ljJ, 1). This allows us to write the sum
defining aj,( '1/J, N) as
aj,('ljJ,N) =
(2:: + L )JL (~)
kiN
rfk
kiN
rlk
ap('ljJ,k)
=
(LJL (~) ap('ljJ,1)) + (LJL (~) ap('ljJ,k))
=
(LJL (~) ap('ljJ, 1)) + (LJL (~) (ap('ljJ, k)- ap('ljJ, 1)))
=
kiN
rfk
kiN
rlk
kiN
kiN
rlk
L JL (Nt) (ap('I/J, kr)- ap('ljJ, 1))
kllY_
r
=aj,('ljJr,N/r)-{a p('ljJ,1)
0
ifN=r,
if N =f r.
(4.11)
If N = r, son= mr, then we have
aj,(¢, n) = aj,('ljJ, N) = aj,('ljJr, 1)- ap('ljJ, 1) = ap('ljJ, r)- ap('ljJ, 1) 2: 1
from (4.6) of Lemma 4.6.
We may thus assume that N > r, so (4.11) says that
aj,(¢,n) = aj,('ljJ,N) = aj,('I/Jr,N/r).
We now observe that Ap('ljJr) = Ap('I/JY = 1. Hence ifNi=- 0 inK, then we can
apply Case l(b) to aj,('I/Jr,N/r), and if N = 0 inK, then we can apply Case II
0
toaj,('I/Jr,N/r). ThiscompletestheproofofTheo rem4.5.
As an easy application of Theorem 4.5, we now prove that a rational map
¢(z) E K(z) possesses periodic points of infinitely many distinct periods, i.e., the
set of primitive-n periodic points Per~* (¢) is nonempty for infinitely many n.
Corollary 4. 7. Let ¢( z) E K ( z) be a rational map of degree d 2: 2. Then for
all prime numbers £ except possibly for d + 2 exceptions, the map ¢ has a point of
primitive period£.
Proof We begin by discarding the finitely many primes £ satisfying either of the
following conditions:
• K has characteristic£.
• There is some Q E Fix(¢) with >.(Q) i=- 1 and >.(Q)c = 1.
4.2. Quadratic Polynomials and Dynatomic Modular Curves
155
The set Fix(¢) contains at most d + 1 points, so these conditions eliminate at
most d + 2 primes.
For any of the remaining primes £, take a point P E Pere( ¢ ). If P does not
have primitive period £, then it must be a fixed point of¢, since its period certainly
divides£. Then our assumptions on£ imply that either :\(P) = 1 or :\(P)e =1- 1, and
further that£ =f- 0 inK. It follows from (4.5) and (4.7) of Lemma 4.6 that in both
cases we have ap(£) = ap(1). Hence
ap(£) =
PEPere(¢)nFix(¢)
ap(1):::;
PEPere(¢)nFix(¢)
L ap(1) d + 1.
=
PEFix(¢)
Thus the total multiplicity of all of the fixed points of¢ in the £-period polynomial
<Pq,,e(X, Y) = YFe(X, Y)- XGe(X, Y)
is at most d + 1. However, the degree of <Pq,,£ is d£ + 1, so we conclude that <Pq,,e
has at least one root that is not fixed by ¢. Hence there exists at least one primitive£periodic point.
D
For many applications, Corollary 4.7 is sufficient, but a more detailed analysis
yields a much stronger result. We state the full theorem in characteristic 0 and refer
the reader to Pezda's two papers [354, 357] for the more complicated description
required in characteristic p. See also [161, 172, 173, 174] for higher-dimensional
results of a similar nature.
Theorem4.8. (I.N.Baker[l9]) Let ¢(z) E K(z)bearationalmapofdegreed 2:2
defined over afield K ofcharacteristic 0. Suppose that¢ has no primitive n-periodic
points. Then (n, d) is one of the pairs
(2,2),(2,3), (3,2), (4,2).
If¢ is a polynomial, then only (2, 2) is possible.
Proof For the proof, which is function-theoretic in nature, see [19] or [43, § 6.8].
D
4.2
Quadratic Polynomials and Dynatomic
Modular Curves
In this section we expand on the material from the previous section in the special
case of the quadratic polynomial
¢(z) = c/Yc(z) = z 2 +c.
For further material on the iteration of quadratic polynomials, see for example [17,
18,113,115, 171,220,221,222,305,309,350,361,388].
4. Families of Dynamical Systems
156
<I>1(c,z)=z 2 -z+c
<I>2 (c, z) = z 4 + 2cz 2 - z + (c2 + c)
<I>3(c, z) = z 8 + 4cz 6 + (6c 2 + 2c)z 4 + (4c3 + 4c2)z 2 - z + (c4 + 2c3 + c2 +c)
<I>4(c, z) = z 16 + 8cz 14 + (28c 2 + 4c)z 12 + (56c3 + 24c2)z 10
+ (70c4 + 60c3 + 6c2 + 2c)z 8 + (56c5 + 80c4 + 24c3 + 8c2)z 6
+ (28c6 + 60c5 + 36c4 + 16c3 + 4c2)z 4
+ (8c7 + 24c6 + 24c5 + 16c4 + 8c3)z 2
- z + (c8 + 4c7 + 6c6 + 6c5 + 5c4 + 2c3 + c2 +c)
<I>r(c,z)=z 2 -z+c
<I>; (c, z) = z 2 + z + (c + 1)
<I>~(c, z) = z 6 + z 5 + (3c + l)z 4 + (2c + 1)z3 + (3c2 + 3c + 1)z 2
+ (c 2 + 2c + 1)z + (c 3 + 2c 2 + c + 1)
<I>:(c, z) = z 12 + 6cz 10 + z 9 + (15c2 + 3c)z 8 + 4cz 7 + (20c3 + 12c2 + 1)z 6
+ (6c2 + 2c)z 5 + (15c4 + 18c3 + 3c2 + 4c)z 4 + (4c 3 + 4c 2 + 1)z3
+ (6c5 + 12c4 + 6c3 + 5c2 + c)z 2 + (c4 + 2c3 + c2 + 2c)z
+ (c6 + 3c5 + 3c4 + 3c3 + 2c 2 + 1)
<I>~(c, z) = z 54 - z 53 + 27cz 52 + ( -26c + 1)z 51 + (351c2 + 13c- 1)z 50
+ ( -325c 2 + 12c)z 49 + (2925c 3 + 325c2 - 24c + 1)z 48
+ ( -2600c3 - 12c2 + 12c)z 47 + · ·.
+ ( -c26 - 12c25 - 66c 24 - 226c23 - · · · + 2c3 + c2 + 2c- 1)z
+ (c 27 + 13c26 + 78c 25 + 293c 24 + 792c 23 + · · · + 3c3 + c2 - c + 1)
Table 4.1: Period and dynatomic polynomials for¢( z) = z 2 + c.
4.2.1
Dynatomic Curves for Quadratic Polynomials
Note that as long as the field K does not have characteristic 2, then any quadratic
polynomial
can be put into the form z 2 + c by a simple change of variables working entirely
within the field K. More precisely, ifwe let f(z) = (2z- B)/(2A), then
¢1(z) = (f- 1 ocpo f)(z) = z 2 + ( AC- ~B 2 +~B).
(4.12)
The roots of the period polynomials <I>n(z) = ¢~(z) - z and associated dynatomic polynomials <I>~ ( z) are periodic points of the map c/Jc (z) = z 2 + c. In order to investigate how the periodic points of c/Jc(z) vary as a function of c, we observe that <I>~ (z) is a polynomial in the two variables z and c. Thus in studying
4.2. Quadratic Polynomials and Dynatomic Modular Curves
157
quadratic polynomial maps, it is natural to write <I>~ (c, z) and treat <I>~ as a polynomial in Z[c, z]. (Table 4.1 lists the first few period and dynatomic polynomials
for <Pc(z).) Treating <J>~(c, z) as a polynomial in two variables leads to a natural
association
{
polynomial ¢(z) E K[z] of }
degree 2 with a point
P E K of formal period n
---->
{ ( c, a) E K
x K : <I>~ (c, a) = 0}.
Thus given a polynomial ¢(z) E K[z] of degree 2 and a point P of formal period n, we make a change of variables as in (4.12) so that ¢! ( z) = z 2 + c and
set a = f- 1 (P). Note that this entire procedure takes place within the field K
(which we assume has characteristic different from 2). Thus the solutions to the
equation <I>~(y, z) = 0 parameterize pairs (¢, P), where¢ is a conjugacy class of
quadratic polynomials and P is a point offormal period n for¢. Further, the solution
is K -rational if and only if¢ and P are K -rational.
Definition. The dynatomic modular curve Y1 ( n) c A2 is the affine curve defined
by the equation
<I>~(y, z) = 0.
The normalization of the projective closure of Y1 ( n) is denoted by X 1 ( n).
Example 4.9. It is easy to see that X 1 (1) and X 1 (2) are rational curves. Indeed, the
projective closures of Y1 ( 1) and Y1 ( 2) are smooth conics,
and
It turns out that X 1 ( 3) is also rational, but this is less clear from the equation in
Table 4.1. In order to parameterize X 1 (3), suppose that ¢(z) = Az 2 + Bz +Cis
any quadratic polynomial with a periodic point of primitive period 3. Conjugating
by a linear map z f---4 az + (3, we may assume that the given 3-cycle has the form
0 ___, 1 ___, t ___, 0 for some t. This gives the equations
¢(0) =
c = 1,
¢(1)=A+B+C=t,
¢(t) = At 2 + Bt + C = 0.
Solving for A, B, C in terms oft yields
"'( ) - t 2 - t + 1 2 - t 3 - t 2 + 1
'+' z
-
t-t 2
z
t-t 2
z+l.
Now we apply the linear change of variables (4.12) to put¢ into the form z 2 +c.
Thus letting f(z) = (2z- B)/(2A), we find that
Our computation shows that for every value oft tfc {0, 1}, the point
4. Families of Dynamical Systems
158
( t - 4t + 9t - 8t + 4t - 2t + 1 -t + t - 1)
6
4
5
3
-4t 4 + 8t 3 -
2
4t 2
2
2
' -2t + 2t
3
is a solution to the equation
ll>j(y, z) = z6 + z 5 + (3y + l)z 4 + (2y + l)z 3 + (3y 2 + 3y + 1)z2
+ (y 2 + 2y + l)z + (y 3 + 2y2 + y + 1) = 0.
(You may check this directly using a computer algebra system.) We have thus constructed a nonconstant rational map
tl---+ (
t 6 - 4t 5 + 9t 4 - 8t 3 + 4t 2 - 2t + 1 t 3 - t 2 + 1)
.
' 2t(t-1 )
-4t2(t-1) 2
(4.13)
General principles (Liiroth's theorem [198, IV.2.5.5]) tell us that X 1 (3) is birational
to IP' 1 .
More concretely, we can prove that the map (4.13) has degree 1 by constructing
+ c- b)z + b, so g
its inverse. Thus let (c, b) be a root of 11>3. We set g(z) =
2
sends 0 to band 1 to ¢(b). Thus the 3-cycle b----+ ¢(b) ----+ ¢ (b) ----+ 0 becomes the
following 3-cycle for ¢9:
W
0 ~ 1 ~ b2 +b+1+c ~ 0.
This gives the map
(c, b) !----+ b2 + b + 1 + c,
which is inverse to (4.13), a fact that can also be checked directly with a computer
algebra system.
4.2.2
Dynatomic Curves as Modular Curves
The curve Y1 ( n) and its completion X 1 ( n) are modular curves in the sense that their
points are solutions to the moduli problem of describing the isomorphism classes of
pairs (¢,a), where ¢ is a polynomial of degree 2 and a E A 1 is a point of formal
period n for¢. Here two pairs (¢ 1 , a 1 ) and (¢2, a2) are PGL2-isomorphic ifthere
is a linear fractional transformation f E PGL 2 satisfying
and
In order to state this more carefully, we define
{
Formal(n) =
. ¢ E k[z], deg(¢) = 2, a E k}
(¢,a) · a has formal period n for¢
.
h"tsm
.
PGL -tsomorp
2
We have demonstrated that the elements ofFormal(n) are in one-to-one correspondence with the points of Y1 ( n). But much more is true: the correspondence is algebraic in an appropriate sense. Before stating this important result, we must define
what it means for a family of maps and points to be algebraic.
4.2. Quadratic Polynomials and Dynatomic Modular Curves
159
Definition. Let V be an algebraic variety. An algebraic family of quadratic polynomials over V with a marked point offormal period n consists of a quadratic polynomial
A, B, C E K[V],
7/J(z) = Az 2 + Bz + C,
whose coefficients A, B, C are regular functions on V and such that A does not
vanish on V(K), and a morphism A : V ---+ A 1 such that for all P E V(K), the
point A(P) is a point of formal period n for the quadratic polynonomial
7/Jp(z) = A(P)z 2 + B(P)z + C(P) E k[z].
The family is defined over Kif the variety V and morphism A are defined over K
and the functions A, B, Care in K[V].
Example 4.1 0. The pair
and
A(t)=t-1
is an algebraic family of quadratic polynomials over IP' 1 ' {0, oo} with a marked
point of formal period 2.
Theorem 4.11. Let K be a field of characteristic different from 2.
(a) The map
Y1 (n) -----> Formal( n),
( C, CY) r---+ ( z 2 + c, CY),
(4.14)
is a bijection ofsets.
(b) Let V be a variety and suppose that the points ofV algebraically parameterize
a family of quadratic polynomials 7/J together with a marked point A offormal
period n. Then there is a unique morphism of varieties
with the property that
ry(P) = (7/Jp(z), A(P)) E Formal(n)
for all P E V(K),
(4.15)
where we use (4.14) to identifY Formal( n) with Y1 ( n ).
(c) If the family is defined over the field K, then the morphism ry is also defined
over K.
Proof (a) We have shown this earlier in this section.
(b) By definition 7/J has the form
7/J(z) = Az 2 + Bz + C,
A,B,C E K[V],
with A not vanishing on V and A a morphism A : V ---+ A 1 such that for all P E
V(K), the point A(P) is a point offormal period n for the quadratic polynonomial
7/Jp(z) = A(P)z 2 + B(P)z + C(P) E K[z].
160
4. Families of Dynamical Systems
For any point P E V(K), let fp(z) be the linear fractional transfonnation
fp(z) =
2z- B(P)
2A(P) .
Note that fp is well-defined for every P E V(K), since we have assumed that A is
nonvanishing on V. We define a map 'T/ from V to A? by the fonnula
ry(P)=(cp,ap) with
{
Cp = A(P)C(P)- ~B(P) 2 + !B(P),
ap = A(P)>.(P) + !B(P).
(4.16)
Note that 'T/ : V -+ A2 is a morphism, i.e., it is given by everywhere-defined algebraic
functions on V. We are now going to verify that the image of 'T/ is the curve Y1 ( n).
The computation that we perfonned in deriving fonnula (4.12) shows that
1/JV (z) = z 2 + Cp.
To ease notation, we let cjJp(z) = z 2 + cp. We also let <I>n,P, <I>~,P• Wn,P, and w~,P
be the period and dynatomic polynomials for c/Jp and 1/Jp, respectively. Note that the
period polynomials are related by
w;~p(z) = f? 1 o(1j;[,(z)-z)ofp = U? 1 o1/Jpfpt(z)-z = ¢[,(z)-z = <I>n,P(z).
Hence the dynatomic polynomials also satisfy
(w~,P )fp = <I>n,P·
(4.17)
We are given that >.(P) E Per~(1j;p), which is equivalent to saying that >.(P) is a
root of w~ ,P· It follows from (4.17) that
f? 1 ( >.(P)) = A(P)>.(P) + ! B(P)
is a root of <I>~ P• and thus is in Per~(c/Jp ). This proves that the image of the map 'T/
defined by (4.16) is contained in Y1(n), so 'T/ is amorphism from V to Y1 (n). Further,
this map 'T/ respects the identification of Y1 (n) with Formal(n) from (a), since it
takes P to a pair (cp,ap) that is isomorphic to the pair (1/Jp(z),>.(P)) via the
conjugation fp E PGL2(K).
Finally, it is clear from the construction that the map 'T/ is uniquely detennined as
a map (of sets) from V(K) to Formal(n), so it is the unique morphism V-+ Y1(n)
satisfying (4.15).
(c) The definition (4.16) of 'T/ shows immediately that 'T/ is defined over K, since all
of A, B, C, and ), are assumed to be defined over K.
D
Remark 4.12. In the language of algebraic geometry, Theorem 4.11 says that Y1 (n)
is a coarse moduli space. In fact, the curve Y1 ( n) is actually a fine moduli space for
all n ~ 1; see Exercise 4.18. The underlying reason is that there are no nontrivial
elements ofPGL2 that fix a quadratic polynomial and its points offonnal period n,
i.e., the moduli problem has no nontrivial automorphisms.
4.2. Quadratic Polynomials and Dynatomic Modular Curves
4.2.3
161
The Dynatomic Modular Curves X 1 (n) and X 0 (n)
The curve Y1 ( n), and by extension X 1 ( n), has the interesting property that the rational map ¢ acts on the points of Y1 ( n) via the map
¢: Yt(n)---+ Yt(n),
(y, z) f - t (y, z2 + y) = (y, c/>y(z)).
(4.18)
By abuse of notation, we will also use¢ to denote the map (4.18). This map is welldefined, since if a: is a point of formal period n for the polynomial c/>y (z), then c/>y (a:)
is also a point of formal period n for c/>y· Further, the nth iterate cf>n is the identity map
on Y1 (n) and X 1 (n), so Aut(Y1 (n)) and Aut(X1 (n)) contain subgroups of order n
generated by ¢.
In general if Vis any algebraic variety and if G c Aut(V) is any finite group of
automorphisms of V, then there exist a quotient variety W = VI G and a projection
map 1r: V---+ W with the property that 1r(y) = 1r(x) if and only if there is agE G
such that y = g(x). (See, for example, [321, II §7].) Further, ifV is defined over a
field K and ifG is K-invariant,2 then VIC is defined over K.
The construction of VI G when V is a nonsingular curve is particularly simple.
Each g E G induces an automorphism of the function field
g* : K(V)---+ K(V)
fixing k, so we may view Gas a subgroup of Aut(K(V)I K). Then one shows
that the fixed field K(V)G of G has transcendence degree 1 over k, so there exists
a unique nonsingular projective curve WI k with function field K(W) = K(V)G.
The inclusion K(W) c K(V) defines the projection 1r: V---+ W making W into
the quotient of V by G. Finally, if G is K -invariant, one shows that W has a model
defined over K.
Remark 4.13. We note that taking the quotient of a variety by an infinite group of
automorphisms, as we will need to do in Section 4.4, is considerably more difficult
than taking the quotient by a finite group of automorphisms. Indeed, in the infinite
case it often happens that the quotient does not exist at all in the category of varieties.
Definition. With notation as above, we let Yo (n) be the quotient of Yi (n) by the
finite subgroup of Aut(Y1 (n)) generated by¢. Similarly, Xo(n) is the quotient
of X 1 (n) by the finite subgroup of Aut(X1 (n)) generated by¢.
By construction, the points ofY0 ( n) classify isomorphism classes of pairs (¢, 0),
where ¢ is a quadratic polynomial and 0 is the orbit of a point of formal period n.
The moduli-theoretic interpretation of the projection map from Y1 ( n) to Yo (n) is
Yt(n)---+ Yo(n),
2 The elements in G are morphisms V --> V that are defined over some extension of K. The Galois
group Gal(K I K) acts on G by acting on the coefficients of the polynomials defining the maps in G. We
say that G is K -invariant if each element of Gal( K I K) maps G to itself.
4. Families of Dynamical Systems
162
Example 4.14. We have seen in Example 4.9 that X 1 (2) and X 1 (3) are rational
curves, i.e., they are isomorphic to lP'\ so their quotient curves X 0 (2) and X 0 (3)
must also be rational curves. However, it is still of interest to make the quotient maps
explicit. We do this for X 1 (2) and leave X 1 (3) as Exercise 4.19.
The affine curve Y1 (2) has equation
Y1 (2) : z 2 + z + y + 1 = 0,
where a point (c, o:) E Y1 (2) corresponds to the quadratic map ¢c(z) = z 2 + c and
point o: E Per~( ¢c)· The automorphism¢ ofY1 (2) is given by (y, z) f-+ (y, z 2 + y).
From general principles we know that ¢ maps Y1 (2) to itself, or we can see this
explicitly by the formula
(z 2 + y) 2 + (z 2 + y) + y + 1 = (z 2 + z + y + 1)(z2 - z + y + 1).
The affine coordinate ring ofY1 (2) is
""C[ ]
C[z, y]
(z 2 + z + y + 1) = z'
since we can express y in terms of z as y = - z 2 - z - 1. In terms of z alone, the
automorphism¢ ofY1 (2) corresponds to the map on C[z] given by
z ~----> z 2 + y = z 2 + (- z 2 - z - 1) = - z - 1.
So the affine coordinate ring of the quotient curve Yo (2) is the subring of C [z] that is
fixed by the automorphism z f-+ - z - 1. Since the map YI( 2) ___, Y0 ( 2) has degree 2,
we look for an invariant quadratic polynomial in z. Setting
Az 2 + Bz + C =A( -z- 1) 2 + B( -z- 1) + C
and equating coefficients, we find that A = B. Hence the invariant subring is
C[z 2 + z], which is the affine coordinate ring ofY0 (2).
Suppose now that ( ¢, { o:, ,8}) is a quadratic polynomial and an orbit consisting
of two points. How does this correspond to a point on Y0 (2) whose coordinate ring
is C[z 2 + z]? First, we use a PGL 2 conjugation as in (4.12) to put¢ into the standard
form ¢c(z) = z 2 +c. Then o: and ,Bare related by
,B = rPc(o:) = o: 2 + c and o: = rPc(,B) = ,8 2 +c.
Notice that this implies that
o:2 + o: = (,B _c)+ (,82 +c) = ,32 +,B.
Letting c = - o: 2 - o: - 1 and using the isomorphisms Y1(2) ~ A 1 and Y0 ( 2) ~ A 1
given earlier, we have the following commutative diagram:
4.2. Quadratic Polynomials and Dynatomic Modular Curves
163
Remark 4.15. Morton [307] describes an alternative method for finding an equation
for the curve Yo (n) and the map Y1(n) --+ Yo (n). The roots of <I>~ (c, z) = 0 can be
grouped into orbits, say o: 1, ... , O:r are representatives for the different orbits, so
The points in each orbit all have the same multiplier, and we define a polynomial
r
On (c, X) =
IT (X - \p (O:i)).
i=1
Then one can show that 8n(c, x) E Z[c, x] and that the equation
gives a (possibly singular) model for Y0 (n). Using this model, the natural map
from Y1(n) to Yo(n) is
(y, z) f--------7 (y, (¢~)'(z)).
See Morton's paper [307] and Exercise 4.13.
Remark 4.16. The dynatomic modular curves Y1(n) and Yo (n) defined in this section
are analogous to the modular curves that appear in the classical theory of elliptic
curves. Briefly, the elliptic modular curve Y1ell ( n) classifies isomorphism classes of
pairs (E, P), where E is an elliptic curve and P E E is a point of exact order n.
Similarly, the elliptic modular curve Y0e11 (n) classifies pairs (E, C), where E is an
elliptic curve and C c Eisa cyclic subgroup of order n. The group (Z/nZ)* acts
on Y1e11 (n) via
m * (E, P) = (E, [m]P)
for
mE (Z/nZ)*,
and the quotient ofY{11 (n) by this action is Y~ 11 (n).
The reader should be aware that standard terminology is to write
for elliptic modular curves. But since in this book we deal exclusively with dynatomic modular curves, there should be no cause for confusion. In situations in
which both kinds of modular curves appear, it might be advisable to use identifying
superscripts such as Y1dyn ( n) and Y1e11 ( n) to distinguish them.
We showed earlier that Xl(l ), X 1(2), and X 1(3) are all (irreducible) curves of
genus 0, and similar explicit computations show that X 1( 4) has genus 2 and X 1(5)
has genus 14. See [171, 305, 309] and Exercise 4.20. The geometry of X 1(n)
and Xo(n) for general n is described in the following theorem, which is an amalgamation of results due to Bausch and Morton.
164
4. Families of Dynamical Systems
Theorem 4.17. (a) The affine curve Y1 ( n) defined by the equation
Y1(n): if?~,q,(Y, z) = 0
is nonsingular.
(b) The dynatomic modular curves X 1 ( n) and Xo (n) are irreducible.
(c) The projection map
(z, c) ---+ c,
exhibits X 1 ( n) as a Galois cover of lP' 1 . The Galois group is maximal in the
sense that it is the appropriate wreath product, cf Section 3.9.
(d) Let cp denote the Euler totient function (not to be confused with the rational
map ¢ ), let f.L be the Mobius function, and let /'i, be the function
/'i,(n) = ~
LM (~) 2k.
kin
(This is essentially half the degree of if?¢,n; cf Remark 4.3.) Then the genera
of X 1 ( n) and Xo (n) are given by the formulas
genusX1(n) = 1 + n ~ 3 /'i,(n)- ~
L m/'i,(m)cp (:),
min, m<n
genusXo(n) = 1 + n- 3 /'i,(n)- ~
2
2n
L
4~
"'"'
/'i,(m)cp
~
mjn, m<n
(!!._)
m
f..l ( : ) 2m/2.
mjn, m odd
n/m even
(Ifn is odd, then the final sum in the formula for the genus of Xo(n) is empty.)
Proof The properties ofY1(n) and X 1(n) were originally proven by Bousch [83],
with subsequent proofs by Lau and Schleicher [261] using analytic methods and
Morton [307] via algebraic arguments. The latter two papers give various generalizations, including results for the dynatomic modular curves associated to maps of
the form zd +c. The formula for the genus of X 0(n) is due to Morton [307].
D
Remark 4.18. The genera of X 1(n) and X 0 ( n) grow rapidly; see Table 4.2.
n
genusX1 (n)
genusXo(n)
1
0
0
2
0
0
3
0
0
4
2
0
5
14
2
6
34
4
7
124
16
8
285
32
9
745
79
10
1690
162
Table 4.2: The genera of the dynatomic modular curves X 1(n) and Xo (n) for z 2 +c.
4.2. Quadratic Polynomials and Dynatomic Modular Curves
165
Res( <Pi, <P;) = 4c + 3
Res( <Pi, <Pj) = -16c2 - 4c- 7
Res( <Pi, <P~) = -16c2 + 8c- 5
Res( <Pi, <P5) = -256c4 - 64c3 -16c2 + 36c- 31
Res( <Pi, <P~) = -16c2 + 12c- 3
Res( <P;, <P~) = -( 4c + 5) 2
Res(<P;, <P~) = -(16c2 + 36c + 21) 2
Res(<Pj, <P~) = -(64c3 + 128c2 + 72c + 81) 3
Table 4.3: The first few bifurcation polynomials for z 2 +c.
4.2.4
Bifurcation, Misiurewicz Points, and the Mandelbrot Set
When does the polynomial ¢(z) = z 2 + c have a point of formal period n whose
exact period m is strictly less than n? This will occur if and only if<P~ (z) and <P:n(z)
have a common root, so if and only if cis a root of the resultant equation
Res(<P~(z), <P~(z)) = 0.
Note that this is a polynomial equation for the parameter c. We list the first few
examples in Table 4.3. Thus the polynomial ¢(z) = z 2 - ~ given in the last section
is the only example with a fixed point of formal period 2. Similarly, the polynomial
¢(z) = z 2 - ~has a point offormal period 4 whose exact period is 2.
One pattern that is apparent from even the small list in Table 4.3 is the fact that
ifmln, then Res(<P~(z), <P:n(z)) is the m1h power of a polynomial in Z[c]. See Exercises 4.7 and 4.12 for a description of the mth root ofRes(<P~(z), <P:n(z)).
The roots ofRes(<P~(z), <P:n(z)) are special points in the Mande/brot set, which
we recall is the subset of the c-plane given by
M = { c E C : <Pn(o) is bounded as n--+ oo }.
Alternatively, the Mandelbrot set M is the set of c for which the Julia set :T(<Pc) is
connected. See Remark 1.34 and the picture of the Mandelbrot set (Figure 1.2) on
page 27.
The solutions to the equation
Res(<I>~(z), <~>:n(z))(c) = 0
4. Families of Dynamical Systems
166
are called bifurcation points. 3 They connect the components of M's interior (the
bulbs of M). For example, the point c = - ~ connects the main cardioid of M to the
disk to its left. It is clear that the bifurcation points are algebraic numbers. Beyond
that, little is known about their arithmetic properties, although Morton and Vivaldi
conjecture that the bifurcation points of type (m, n) are all Galois conjugate to one
another. (See Exercise 4.12.)
An elementary property of the Mandelbrot set M, which we now prove, is that it
is contained in a disk of radius 2. Note that M is not contained in any smaller disk,
since
shows that -2 E M.
Proposition 4.19. The Mandel brat set is contained in the disk of radius 2,
M c {c E C: lei :S 2}.
Proof Suppose that lei > 2 and let Zn = ¢~(0). Then
lzn+ll2iz;i-lcl = lznl· {lznl-1) + {lznl-lcl).
(4.19)
We have lz1i = lei > 2 by assumption, so (4.19) and induction tell us first that lznl
is an increasing sequence, and indeed that
Hence lznl --> oo, soc is not in M.
0
The set of quadratic maps whose critical points are preperiodic, but not periodic,
defines an important subset of the Mandelbrot set.
Definition. A point c E C is called a Misiurewicz point if 0 is strictly preperiodic
for ¢c (z) = z 2 + c. We say that c E C is a Misiurewicz point of type ( m, n) if
• m 2 1 is the smallest integer such that ¢~(0) is periodic, and
• n is the primitive period of ¢~(0).
Example 4.20. Examples of Misiurewicz points include c = -2 and c = i. Thus
for c = -2 we have 0 --> -2 --> 2 --> 2, so c = -2 has type (2, 1). Similarly
for c = i we have 0-+ i --> i - 1 -+ -i --> i - 1, soc= i has type (2, 2).
Remark 4.21. If cis a Misiurewicz point, then the map ¢cis subhyperbolic without
being hyperbolic, ¢c has no neutral cycles, and the orbit Oci>JO) is repelling. The
Misiurewicz points are all contained in the boundary 8M of the Mandelbrot set, and
they are dense in 8M. For proofs of these analytic properties ofMisiurewicz points,
see for example [95, §VIII.1,6] or [143].
3 Bifurcation point is a general term for a point in moduli space whose polynomial <P satisfies Res(<I>~,q,(z), <l>;',,q,(z)) = 0. They are points (polynomials) at which periodic points of different
orders merge and break apart. For the quadratic polynomial ¢( z) = z 2 + c, whose moduli space is the cplane, the bifurcation points are also known as the roots of the hyperbolic components of the Mandelbrot
setM.
4.2. Quadratic Polynomials and Dynatomic Modular Curves
167
It is clear that every Misiurewicz point is contained in the Mandelbrot set M,
since preperiodic points certainly have bounded orbit. This suffices to prove an arithmetic result.
Proposition 4.22. The set ofMisiurewicz points is a set ofbounded (absolute) height
in Q. More precisely, the height of a Misiurewicz point 1 satisfies H (1) :::; 2. Hence
there are only finitely many Misiurewicz points defined over any given number field.
Proof A Misiurewicz point c = 1 is the root of a polynomial of the form
for some m 2 1 and n 2 1. These are monic polynomials with coefficients in Z,
so not only is 1 an algebraic number, it is an algebraic integer. Further, the minimal
polynomial of1 is a factor of Mm,n(c). On the other hand, every root of Mm,n(c) is
in the Mandelbrot set M, since if cis a root of Mm,n(c), then 0 is preperiodic for ¢c,
so it certainly has bounded orbit. It follows from Proposition 4.19 that every Galois
conjugate of 1 has absolute value at most 2. Let K = (()!( 1) and let 1 1 , ... , "Yr be the
full set of Galois conjugates of 1· Then
T
Taking the rth root yields H (1) :::; 2.
0
We now describe an analytic characterization of Misiurewicz points. It depends
on the following deep result giving an analytic uniformization of the complement of
the Mandelbrot set.
Theorem 4.23. (Douady-Hubbard) There is a conformal isomorphism from the exterior of the unit disk to the complement of the Mandelbrot set,
B: {wE <C: lwl > 1} ----+ <C "M.
Proof It is not hard to show that for all sufficiently large z (depending on c) there is
a consistent way to choose square roots so that the limit
1/Jc(z) = lim
n--+oo
2 V¢~(z)
converges and defines a holomorphic function 1/Jc on some region lzl > Rc. (See
Exercise 4.15.) If c ~ M, then one can prove that 7/Jc has an analytic continuation
to <C" M. The isomorphism in the theorem is the inverse of the map
e
<C "M ---+ {wE <C: lwl > 1},
See [142, 143, 141], [43, §9.10], or [95, VIII §§3,4] for details.
0
Remark 4.24. An immediate consequence of Theorem 4.23 is the connectivity of the
Mandelbrot set, since the theorem implies that lP' 1 (<C) "M is simply connected.
168
4. Families of Dynamical Systems
The unifonnization map f) from Theorem 4.23 can be used to give an analytic
description of the Misiurewicz points.
Theorem 4.25. Let f) be the isomorphism {wE C: lwl > 1} ---+ C "M described
in Theorem 4.23. Consider the doubling map x f-t 2x on QjZ. Let t E Q/Z be
preperiodic, but not periodic, for the doubling map. Let m and n be the smallest
positive integers for which we can write t in the form
(The fraction need not be in lowest terms. See Exercise 4.16.)
Then the limit
exists and is a Misiurewicz point of type (m, n) in M, although distinct values of
t E Q/Z may yield the same Misiurewicz point.
Remark 4.26. For a given t E Q/Z, the "spider algorithm" [213] can be used to compute Ct numerically. The spider algorithm is mainly topological and combinatorial in
nature, although the limiting process that yields Ct E C is analytic.
4.3
The Space Ratd of Rational Functions
The set of quadratic polynomials { Az 2 + Bz + C} has dimension three, since it may
be identified with the set of triples (A, B, C) with A =I 0. In fancier language, the
space of quadratic polynomials is equal to the algebraic variety
{(A,B,C) E A3 : A -=f. 0}.
We have seen in Section 4.2.1 that every quadratic polynomial can be conjugated
to a polynomial of the fonn z 2 + c, and that polynomials with different c values
are not conjugate to one another. Thus the space of conjugacy classes of quadratic
polynomials has dimension 1. It may be identified with the variety A 1 .
In the next few sections we study analogous parameter spaces for more general
rational maps and their conjugacy classes. We begin in this section by explaining
how the set Ratd of rational maps of degree d has a natural structure as an algebraic
variety and how the natural action of the algebraic group PGL2 on Ratd is an algebraic action. Then in Section 4.4 we discuss (mostly without proof) how to take the
quotient of Ratd to construct the moduli space Md of conjugacy classes of rational
maps of degree d. We continue in Section 4.5 by describing a natural collection of
algebraic functions on Md that are created using symmetric functions of multipliers
of periodic points. These functions can be used to map Md into affine space. Finally,
in Section 4.6 we use these functions to prove that M 2 is isomorphic to A 2 .
A rational map ¢ : IP'1 ---+ IP'1 of degree d is specified by two homogeneous
polynomials
4.3. The Space Ratd of Rational Functions
169
such that F and G have no common factors, or equivalently from Proposition 2.13,
such that the resultant Res(F, G) does not vanish. Thus a rational map of degree d
is determined by the 2d + 2 parameters ao, a1, ... , ad, bo, b1, ... , bd. However, if u
is any nonzero number, then [uF, uG] = [F, G], so the 2d + 2 parameters that determine ¢ are really well-defined only up to homogeneity. This allows us to identify in
a natural way the set of rational maps of degree d with a subset of projective space.
To ease notation, for any (d +I)-tuple a= (ao, ... , ad), let
Fa(X, Y) = aoXd + a1Xd-ly + · · · + adYd
be the associated homogeneous polynomial. Similarly, if a and b are (d + 1 )-tuples,
we write [a, b] E lP' 2d+l for the point in projective space whose homogeneous coordinates are [ao, ... , ad, bo, ... , bd].
Definition. The set of rational functions ¢ : lP' 1 ----+ lP' 1 of degree d is denoted
by Ratd. It is naturally identified with an open subset ofJP 2d+ 1 via the map
{[a, b] E JP 2 d+ 1 : Res( Fa, Fb) -:/:- 0} ~
Ratd,
[a, b] f---+ [Fa, Fb]·
The collection of rational maps Ratd, which a priori is merely a set, thus has the
structure of a quasiprojective variety. In fact, Ratd is an affine variety, since it is the
complement of the hypersurface Res( Fa, Fb) = 0 in the projective space lP' 2 d+l.
Proposition 4.27. The variety Ratd is an affine variety defined over Q. The ring of
regular functions Q[Ratd] ofRatd is given explicitly by
lfl'I[R
"J!
atd
]
aioail ... aidbiobi1 ... bjd
lfl'l [ o 1
d o 1
= "J!
Res( Fa, Fb)
d
l
: 'to+ ... + Zd + Jo + ... + ]d = 2d
.
.
.
.
.
Equivalently, Q[Ratd] is the ring of rational functions ofdegree 0 in the localization
ofQ[a 0 , a1, ... , ad, bo, b1, ... , bd] at the multiplicatively closed set consisting of the
nonnegative powers ofRes(Fa, Fb).
Proof We remind the reader that in general, ifF E K[Xo, ... , Xr] is a homogeneous polynomial of degree n, then the complement of the zero setofF,
is an affine variety of dimension r. (See [ 198, Exercise 1.3.5].) Explicitly, each rational function
with io + i1 + · · · + ir = n
is a regular (i.e., everywhere well-defined) function on V. There are (r~n) such
functions, and together they define an embedding
4. Families of Dynamical Systems
170
of V into affine space. The affine coordinate ring of V is the ring of polynomials in
these fi 0 ... ir'
K[V] = K[f•..a•.. 1 .. •"r
•]·to +.'tl +··· +.'l.r=n .
In the language of commutative algebra,
K[V] = K[Xo, ... , Xn, 1/F](o)
is the set of rational functions of degree 0 in the localization of K[X0 , ••• , Xn] at
the multiplicatively closed set (Fik:~o- Applying this general construction to
gives the results stated in the proposition.
D
Remark 4.28. The geometry ofRatd (q, especially near its boundary, presents many
interesting problems. See [121, 122, 369, 370].
Example 4.29. Let
be the resultant of a0 X 2 + a 1 XY + a 2Y2 and b0 X 2 + b1 XY + b2Y 2. Then the
collection of 84 functions
gives an embedding of Rat 2 into A84 . Of course, this is not the smallest affine
space into which Rat 2 can be embedded. Projecting onto appropriately chosen hyperplanes, there is certainly an affine embedding of the 5-dimensional space Rat 2
into A11 ; see [198, Exercise IV.3.11].
Example 4.30. The set of rational functions of degree 1 is exactly the set of linear
fractional transformations,
Rat1 = PGL2 ={[aX+ {3Y, 'YX + 8Y]: a8- f3'Y # 0} c IP3 .
We note that PGL2 is not merely a variety, it is a group variety, which means that
the maps
PGL2 x PGL2 - t PGL2,
and
(h,h) I - t hh,
defining the group structure are morphisms.
PGL2 - t PGL2,
fI-t J-1,
4.3. The Space Ratd of Rational Functions
171
Each point of Ratd determines a rational map JP> 1 --+ JP> 1 . Further, as we vary
the chosen point in Ratd, the rational maps "vary algebraically." We can make this
vague statement precise by saying that the natural map
¢ : JP> 1 x Ratd ----> JP> 1 x Ratd,
([X,Y],[a,bl) ~----+ ([Fa(X,Y),Fb(X,Y)j,[a,bl),
(4.20)
is a morphism of varieties. The following definition is useful for describing families
that vary algebraically.
Definition. Let V be an algebraic variety. The projective line over V is the product
JP>~ = JP> 1 XV.
A morphism 'ljJ : JP>~ --+ JP>~ over Vis a morphism that respects the projection to V,
i.e., the following diagram commutes, where the diagonal arrows are projection onto
the second factor:
"'" v ,/
Then 'ljJ can be written in the form
'ljJ = [F(X, Y), G(X, Y)],
where F, G E K(V)[X, Y] are homogeneous polynomials with coefficients that are
rational functions on V. The degree of'ljJ is the degree of the homogeneous polynomials F and G. 4
Any morphism A : V --+ W of varieties induces a natural morphism JP>t, --+ JP>h,r,
which, by abuse of notation, we also denote by A. Thus
A(P, t) = (P, A(t))
for (P, t) E JP>~.
With this notation, the map (4.20) says that there is a natural morphism
over Ratd. The next proposition says that this¢ is a universal family of rational maps
of degree d.
Proposition 4.31. Let V be an algebraic variety and let
'ljJ : jp>~ ----'> jp>~
be a morphism over V ofdegree d. Then there is a unique morphism
4 More generally, if'I/J: lP'~--> lP'~ is a morphism of S-schemes, then '1/J*OI'N (1) '="' oi'N (d) and d
8
8
is the degree of '1/J.
4. Families of Dynamical Systems
172
such that the induced map >. : lP'~ ----+ lP'katd fits into the commutative diagram
lP'lv
1/J
______.
lP'lv
lA
lA
<P
lP'katd ______.
(4.21)
lP'lRatd
Remark 4.32. The commutative diagram (4.21) in Proposition 4.31 says that any
algebraic family of degree d rational maps \fJ : lP'~ ----+ lP'~ factors through the Ratd
family. Thus ¢ : lP'katd ----+ lP'katd is a universal family of rational maps of degree d.
It is an example of a .fine moduli space. See [322, 323, 335] for further information
about moduli spaces.
Proof Let U C V be an affine open subset and write K[U] for its affine coordinate ring. The fact that 'ljJ is a morphism over V implies that it restricts to give a
morphism 'ljJ : lP'h ----+ lP'h over U. This restriction of 'ljJ to lP'h has the form
'ljJ = [o:oXd + o:lXd~l y + ... + o:dYd, f3oXd + f3IXd~l y + ... + f3dYd],
with o:o, ... , o:d, f3o, ... , f3d E K[U].
In other words, the coefficients of the polynomials defining 'ljJ are regular functions
on the affine open set U. In particular, if there is a map)..: V----+ Ratd making (4.21)
commute, then it must be given by A. = [o: 0 , ... , o:d, (30 , ... , ;3d] at any points of V
at which the o:i and f3i are defined and the homogeneous polynomials in (4.21) have
no common roots.
Now given any point in V, we can find a neighborhood U of that point and
o:i, f3i E K[U] as above such that 'ljJ : lP'h ----+ lP'h is given by
'1/J([x, y], t) = [o:o(t)xd + · · · + o:d(t)yd, f3o(t)xd + · · · + f3d(t)yd]
for ([x,y],t) E lP'h.
(4.22)
Using a natural notation, we abbreviate this by writing
\fl(P, t) = [Fa(t)(P), Fi3(t)(P)].
In order to make the diagram (4.21) commute on lP'h c lP'~, we are forced to define>.
onUby
>.(t) = [o:o(t), o:1 (t), ... , o:d(t), f3o(t), fJ1 (t), ... , f3d(t)] E lP' 2d+ 1.
(4.23)
Further, this ).. will have the desired properties provided that its image lies in Ratd.
However, the fact that 'ljJ is given by (4.22) for every point in lP'h implies that for
every t E U, the homogeneous polynomials Fa(t) and Fi3(t) have no common root
in lP' 1 . It follows from Proposition 2.13(a) that their resultant
4.3. The Space Ratd of Rational Functions
173
Res( Fa, Ff3) E K[U]
is nonzero at every point in U. Hence the image of the map A = [a, b] defined
by (4.23) is in RatJ.
We have now proven that every point in V has a neighborhood U for which
there is a map A : U ____, Ratd making (4.21) commute. Further, the maps on different U must agree on the intersection by the uniqueness discussed earlier. Fitting
them together, we find that there is a unique map A V ____, Ratd making (4.21)
commute.
0
An ongoing theme of this text has been the observation that a pair of rational
maps ¢ 1 , ¢ 2 E Ratd determine (arithmetically) equivalent dynamical systems if they
are PGL 2-conjugate, i.e., if there is a linear fractional transformation f E PGL 2
such that ¢ 2 = ¢{ = f- 1 ¢f. The conjugation action ofPGL 2 on Ratd is algebraic
in the following sense.
Proposition 4.33. The map
(4.24)
is an algebraic group action ofPGL 2 on Ratd and is defined over Q. This means
that the map (4.24) is both a morphism defined over Q and a group action.
Proof The proof is mostly a matter ofunsorting the definitions. Let
f = [aX+ (3Y, "(X+ 5Y] E PGL2 c lP'3
and
¢ = [Fa, Fb] E Ratd c lP'2d+l.
Then
¢1 ([X, Yl) = [5Fa(aX + (3Y, 'IX+ 8Y)- (3Fb(aX + (3Y, "(X+ 8Y),
-"(Fa(aX +(3Y,"(X +5Y) +aFb(aX +(3Y,"(X +8Y)].
(4.25)
The homogeneity of Fa and Fb shows that (4.24) at least gives a well-defined rational
map
Further, an elementary resultant calculation (Exercise 2.7) shows that the resultant
of the two polynomials appearing in the righthand side of (4.25) is equal to
It follows that iff E PGL 2 and¢ E Ratd, then ¢1 is a well-defined point in RatJ.
This proves that the map (4.24) is a morphism. The fact that it is a group action is
then a straightforward, albeit tedious, calculation.
0
174
4. Families of Dynamical Systems
Example 4.34. In principle it is possible to explicitly write down the action ofPGL 2
on Ratd, but in practice the expressions become hopelessly unwieldy for even moderate values of d. As illustration, we describe the action for d = 2. Let
f =[aX+ (3Y,1X + 8Y]
and
and set
Multiplying out both sides of (4.25) and equating the coefficients of X 2, XY, and Y 2
yields the following formulas for the coefficients of Fa' and Fb':
a~ = a 28a 0 + a18a1 + 1 28a2- a 2 f3bo - af31b1 - f3'·-?b2,
a~ = 2a(38ao + (a8 + f31)8a1 + 218 2a2- 2a(3 2bo- (a8 + f3!)f3b 1 - 2f318b2,
a~ = (3 28ao + (38 2a1 + 83 a2 - (3 3 bo- (3 28bt - (38 2b2,
b~ = -a21ao - a1 2a1 -13 a2 + a 3 bo + a 21b1 + a1 2b2,
b~ = -2af31ao- ((31 + a8)!a1- 21 28a2 + 2a 2f3bo + (a8 + f31)ab1 + 2a18b2,
b~ = -f3 21ao- f318a1 -182a2 + a(3 2bo + a(38b1 + a8 2b2.
4.4
The Moduli Space Md of Dynamical Systems
The intrinsic properties of the dynantical system associated to a rational map ¢ depend only on the PGL 2-conjugacy class of¢, so it is natural to take the quotient of
the space Ratd by the conjugation action ofPGL 2.
Definition. The moduli space of rational maps of degree d on JP> 1 is the quotient
space
Md = Ratd / PGL2,
where PGL 2 acts on Ratd via conjugation¢! = f- 1 ¢! as described in Proposition 4.33. For the moment, the quotient space Md is merely a set, in the sense that
for any algebraically closed field R,
We denote the natural map from Ratd to Md by
In order to endow Md with the structure of a variety, we observe that the action
ofPGL2 on Ratd induces in the usual way an action ofPGL 2 on the ring of regular
functions Q[Ratd] on Ratd. Thus an element R E Q[Ratd] is a function whose
domain is the set { ¢ : JP>1 --t JP> 1} of rational functions on JP> 1, and for f E PGL 2 we
define Rf E Q[Ratd] by the formula
Rf(¢) = R(¢f).
4.4. The Moduli Space Md of Dynamical Systems
175
In this way we obtain a map
so it makes sense to talk about functions in Q[Ratd] that are invariant under the
action ofPGL 2. We write
for the ring ofPGL 2-invariant functions on Ratd.
Remark 4.35. For technical reasons, it is sometimes advantageous to replace PGL 2
by a slightly different group. The special linear group is the subgroup ofGL2 defined
by
and the projective special linear group PSL 2 is the image of SL2 in PGL 2. The
groups SL 2 and PSL 2 act on Ratd via conjugation in the usual way. Geometrically,
the actions ofPSL 2 and PGL 2 on Ratd are identical, since it is easy to see that the
natural inclusion SL 2 c G 1 2 induces an isomorphism
However, if K is not an algebraically closed field, then the map
need not be an isomorphism; see Exercise 4.22.
The following theorem provides the abstract quotient Ratd / PSL2 with the
structure of an algebraic variety.
Theorem 4.36. There is an algebraic variety Md defined over Q and a morphism
(4.26)
defined over Q with the following properties:
(a) The map (4.26) is PSL2-invariant, i.e., the following diagram is commutative:
1 ---t
PSL2 x Ratd _('---f,_<l>)_-+_<1>_
l(ft
Ratd
(·)
In terms of elements, (eli) = (¢)for all¢ E Ratd and all f E PGL2.
4. Families of Dynamical Systems
176
(b) The map on complex points
is surjective and each fiber is the full PSL2 (C)-orbit of a single rational map.
Thus there is a bijection of sets
(c) The variety Md is a connected, integral (i.e., reduced and irreducible), affine
variety ofdimension 2d - 2 whose ring ofregular functions is the ring ofPSL2invariant functions on Ratd,
(d) Let VIC be a variety and let T : Ratd ----+ V be a morphism with the property
that T(qi) = T(cj;)for all¢ E Ratd(C) and all f E PGL2(C). Then there is
a unique morphism T: Md----+ V satisfYing f'( (¢)) = T(¢).
Proof Sketch. A full proof of Theorem 4.36 (see [416]) uses the machinery of geometric invariant theory [322] and is thus unfortunately beyond the scope of this book.
Geometric invariant theory tells us that there is a certain subset of JP2d+ 1 , called the
stable locus, on which the conjugation action of PSL2 is well behaved. The main
part of the proof is to use the PSL 2-invariance of the resultant to verify that
is a PSL 2-invariant subset of the stable locus of JP2d+l. Then the existence of the
quotient variety Md with affine coordinate ring equal to the ring of invariant functions in Q[Ratd] follows from general theorems of geometric invariant theory [322,
Chapter 1). Further, the fact that Md is connected, integral, and affine follows immediately from the corresponding property ofRatd [322, Section 2, Remark 2). The
dimension of Md is computed as
dimMd = dimRatd- dimPSL2 = (2d + 1)- 3 = 2d- 2.
This proves (a), (b), and (c). Finally, (d) follows directly from the description of Md
and Q[Md] in (c), since the morphism T induces a map T* : Ov ----+ C[Ratd], and
the assumption that T satisfies T(¢1) = T(¢) implies that the image ofT* lies in
0
the ring C[Ratd]PGL 2 (1C) = C[Md].
Remark 4.37. Theorem 4.36 says that the quotient Ratd I PSL 2 is an algebraic variety. Milnor [301, 302) shows that Ratd(C)I PGL 2 (C) has a natural structure as a
complex orbifold, which roughly means that locally it looks like the quotient of a
complex manifold by the action of a finite group. Thus its singularities are of a fairly
moderate type, although they can still be quite complicated. However, for rational
maps of degree 2, we will see in Section 4.6 that not only is M2 nonsingular, it has
a particularly simple structure.
4.4. The Moduli Space Md of Dynamical Systems
177
Remark 4.38. According to Theorem 4.36(c), the affine coordinate ring of Md is the
ring ofPSL 2 -invariant functions Q[RatdjP 8 L 2 • However, this is the same as the ring
of PGL 2 -invariant functions, since it suffices to check for invariance by the action
ofPGL 2 (C) = PSL 2 (C) on an element ofQ[Ratd].
Remark 4.39. The moduli space Md is the quotient of Ratd by the conjugation action of the group PSL 2. In particular, if k is any algebraically closed field, then the
set of points Md(K) is exactly the collection of cosets of Ratd(k) by the conjugation action ofPSL 2(K) = PGL 2(K). However, if K is not algebraically closed,
then the natural map
is generally neither injective nor surjective. The correct description of Md(K), at
least if K is a perfect field, is
M (K)-{(¢)EM (K)· foreveryTE_Gal(K/K)thereisan
d
-
d
.
/7
EPGL2(K)suchthatT(¢)=¢fT
}
.
The field of moduli of a rational map ¢ E Ratd (K) is the smallest field L such
that (¢) E Md(L ). Fields of moduli and related questions are studied in detail in
Section 4.10. In particular, see Example 4.85 for a map¢ whose field of moduli is Q,
yet¢ is not PGL 2 (C)-conjugate to any map in Ratd(Q).
The moduli space Md classifies rational maps up to conjugation equivalence,
just as we want, but it has the defect that it is an affine variety. It is well known that if
possible, it is generally preferable to work with projective varieties. How might we
naturally complete Md by filling in extra points "at infinity"? Note that we should
not do this in an arbitrary fashion. Instead, we would like these extra points to correspond naturally to degenerate maps of degree d.
One possibility is simply to start with all of JP' 2d+ 1 and take the quotient by the
conjugation action of PSL 2 . Unfortunately, there is no natural way to give the quotient lP'2d+l j PSL 2 any kind of reasonable structure. For example, it is not a variety.
So lP' 2 d+l is too large. Ideally, we would like to find a subset S c lP'2d+l with the
following properties:
1. S contains Ratd.
2. PSL 2 acts on S via conjugation.
3. There is a variety T and a morphism S ----t T that induces a bijection
4. The quotient variety T is projective.
Geometric invariant theory gives us two candidates for S. The smaller candidate
is the largest variety satisfying (3), but its quotient T may fail to be projective. The
larger candidate has a projective quotient, but the map S(C)/ PSL 2 (C) ----t T(C)
in (4) may fail to be injective. The following somewhat lengthy theorem describes the
application of geometric invariant theory to our situation, that is, to the conjugation
action of PSL 2 on lP' 2 d+ 1 .
4. Families of Dynamical Systems
178
Theorem 4.40. There are algebraic sets
Ratd C Ratd C Ratd8 c lP'2d+ 1
with the following properties:
(a) The conjugation action ojPSL2 on Ratd extends to an action ojPSL2 on Ratd
andRatds·
(b) There are varieties M 8 and M 88 and morphisms
and
( · ) : Ratd ---+ Md
(4.27)
that are invariant for the action ojPSL2 on Ratd andRatd8 • The varieties M 8
and Mss and morphisms (4.27) are defined over IQl.
(c) Two points [a, b] and [a', b'] in Ratd(C) have the same image in Md(C) if and
only if there is an f E PSL2 (C) satisfYing
[a', b'] =[a, b]f.
Thus as a set, Md(C) is equal to the quotient ofRatd(C) by PSL2(C).
(d) Two points [a, b] and [a', b'] in Ratd8 (C) have the same image in Md8 (C) if
and only ifthe Zariski closures oftheir PSL2 (C)-orbits have a point in common,
{[a, b]f: f E PSL2(C)} n {[a', b']f: f E PSL2(C)} -1-0.
Equivalently, they have the same image if and only if there is a holomorphic
map f: {t E C: 0 < itl < 1}--> SL2(C) such that
E~ [a, b]f' =[a', b'].
(e) Md is a quasiprojective variety and Mds is a projective variety.
(f) (Numerical Criterion) A point [a, b] E lP'2d+l (C) is not in Ratd8 if and only if
there is an f E PSL 2(C) such that [a', b'] =[a, b]f satisfies
d-1
1
ai = 0 for all i :S - 2 -
and
d+1
1
bi = 0 for all i :S - 2 -.
(4.28)
Similarly, [a, b] is not in Ratd if and only if there is an f E PSL2(C) such that
[a', b'] = [a, b]f satisfies
d-1
ai = 0 for all i < - 2 -
and
. d+1
bi = 0 for all z < - 2 -.
(4.29)
(g) Md is isomorphic to Mds if and only if d is even.
Proof The proof of this theorem is beyond the scope of this book. However, we note
that half of (g), namely Md ~ Mds for even d, follows directly from the numerical
criterion in (f), since for even d the criteria (4.28) and (4.29) are the same. See [416]
for a proof of a general version of Theorem 4.40 over Z. See also [301, 302] for a
D
similar construction over C.
4.5. Periodic Points, Multipliers, and Multiplier Spectra
179
Remark 4.41. In Theorem 4.40, the set denoted by Ratd is called the set of stable
rational maps, and the set denoted by Ratd8 is called the set of semistable rational
maps. Note that points in these sets need not represent actual rational maps of degree d on JP> 1 . The intuition is that points in Ratd and Ratd8 that are not in Ratd
correspond to maps that want to be of degree d but have degenerated in some reasonably nice way into maps oflower degree.
Remark 4.42. Theorem 4.40(c) says that the stable quotient Md(C) has the natural quotient property, since its points correspond exactly to the PSL 2 (C)-orbits of
points in Ratd (C). Quotient varieties with this agreeable property are called geometric quotients. The semistable quotient Mds (C) has a much subtler quotient property.
According to Theorem 4.40(d), points in Ratd8 (C) with distinct PSL 2 (C)-orbits
give the same point in Md 8 (C) if their orbits approach one another in the limit.
Quotients of this kind are called categorical quotients. As compensation for the lessintuitive notion of categorical quotient, Theorem 4.40(e) tells us that Mds is projective, so Mds (C) is compact. Finally, Theorem 4.40(g) says that if dis even, then Md
and Mds coincide, so in this case, the moduli space Md has a projective closure with
a natural (geometric) quotient structure.
Remark 4.43. Applying the full machinery of geometric invariant theory to the action
of PSL2 /71.. on JP>~d+l, it is possible to prove versions of Theorems 4.36 and 4.40
over 71... In other words, there is a filtration of schemes over 71..,
such that the group scheme PSL2 /71.. acts on each of these schemes and such that
the quotient schemes Md c Md c M 88 exist in a suitable sense. In particular,
Theorems 4.36 and 4.40 are true with Q replaced by the finite field lFp· The proof
is similar to the proof over Q, but requires Sheshadri's theorem that reductive group
schemes are geometricallly reductive. See [416] for details.
4.5
Periodic Points, Multipliers, and Multiplier
Spectra
The moduli space Md of rational functions modulo PSL2 -equivalence is an affine
variety whose ring of regular functions
consists of all regular functions on Ratd that are invariant under the action of PSL2 ,
or equivalently under the action of PGL 2 (C); see Remark 4.38. Abstr~ct invariant
theory, as described in the proof sketch of Theorem 4.36, says that there are many
such functions. In this section we use periodic points to explicitly construct a large
class of regular functions on Md.
4. Families of Dynamical Systems
180
Example 4.44. Let¢ E Ratd(C) be a rational function of degree d defined over
the complex numbers. Associated to each fixed point P E Fix(¢) is its multiplier >.p(¢) E C*. A simple calculation (Proposition 1.9) shows that Ap(¢)
is PGL 2 -invariant in the sense that iff E PGL 2 (C), then
and
AJ-l(P)(¢f) = >.p(¢)
for all P E Fix(¢).
Thus in some sense the function¢ r-+ >.<t>(P) is a function on Md, where Pis a fixed
point of¢. Unfortunately, this is not quite correct, since there is no way to pick out a
particular fixed point P for a given map ¢.
Recall that a rational map ¢ of degree d has d + 1 fixed points counted with
appropriate multiplicities, say
The points in Fix(¢) do not come in any particular order, so the set of multipliers for
the fixed points,
{ Apl (¢), >.P2 (¢), ... , Apd+l (¢) },
is an unordered set of numbers, but as a set, it depends only on (¢), the PGL2 equivalence class of¢. Hence if we take any symmetric function of the elements in
this set of multipliers, we get a number that depends only on(¢).
The elementary symmetric polynomials generate the ring of all symmetric functions, so we define numbers
by the formula
d+l
II (T + Ap(¢)) = I>·i(¢)Td+l-i.
PEFix(</>)
i=O
In other words, the quantity ai(¢) is the ith elementary symmetric polynomial of the
multipliers Ap1 (¢), ... , Apd+l (¢).
From this construction, it is clear that ai(¢f) = ai(¢) for all f E PGL2 • Further,
if we treat the coefficients of¢= [Fa, Fb] as indeterminates, then the fixed points Pi
and the multipliers >.pi(¢) of¢ are algebraic over Q(ao, ... , bd) and form a complete set of Galois conjugates, from which it follows that symmetric expressions in
the Ap; (¢),for example the functions ai(¢), are in the field Q(a0 , ... , bd)· With a
bit more work, which we describe in greater generality later in this section, one can
show that the multipliers Ap; (¢) are integral over the ring
and hence that ai(¢) E Q[Md]. Thus symmetric polynomials in the multipliers of
the fixed points of¢ are regular functions on the moduli space Md.
4.5. Periodic Points, Multipliers, and Multiplier Spectra
181
Example 4.45. We illustrate the construction of Example 4.44 for rational maps of
degree 2. As usual, we write
The map <P has three fixed points P 1 , P 2 , P2 , and after much algebraic manipulation
one finds that the first elementary symmetric function of the multipliers,
is given explicitly by the horrendous-looking formula
Notice that the denominator of a 1 (¢)is Res( Fa, Fb), so a1 (¢)is in Q[Ratd]. It is far
less obvious that this expression for a 1 (¢)is PGL 2 -invariant. One can verify directly
that a1 ( <P) = a1 (¢f) by checking that a 1 ( <P) does not change when a0 , .. . , b2 are
replaced by the expressions a~, ... , b~ described in Example 4.34. We leave this
task to the interested reader who has, we hope, access to a suitably robust computer
algebra system.
We have used the set of multipliers of the fixed points of a rational map ¢ to
create PGL2 -invariant functions on Rat d. More generally, we can use the multipliers
associated to periodic points of any order to create such functions.
Recall from Section 4.1 (page 149) that for any <P E Ratd we write
</Jn = [Fq,,n(X, Y), Gq,,n(X, Y)]
with homogeneous polynomials Fq,,n, Gq,,n E K[X, Y] of degree dn. (See also Exercise 4.9.) Then the set Pern(¢) ofn-periodic points of¢ are the roots of then-period
polynomial
<I>q,,n(X, Y) = YFq,,n(X, Y)- XGq,,n(X, Y),
and the set Per~ (<P) offormal n-periodic points of¢ are the roots of the n1hdynatomic
polynomial of¢,
<I>*</>,n (X ' Y) = Il(YF.</>,k (X ' Y)-XG </>,k (X ' Y))~-t(n/k) = Il<P </>,k (X ' Y)~-t(n/k) '
kin
kin
where we proved in Theorem 4.5 that <I>:i,,n is a polynomial.
The polynomial <I>q,,n is homogeneous of degree dn + 1. For the purposes of this
section, it is convenient to let Pern(¢) be a "set with multiplicity" in the sense that
a point appears in Pern (¢) according to its multiplicity as a root of <I> </>,n. Similarly,
we denote the degree of <I>:i,,n by vd(n) (see Remark 4.3) and we assume that points
appear in Per~ (¢) according to their multiplicity as roots of <l>:i,,n.
4. Families of Dynamical Systems
182
Definition. Let¢ E Ratd. Then-multiplier spectrum of¢ is the collection of values
An(¢)= { Ap(cp) : P E Pern(¢) }.
The formal n-multiplier spectrum of¢ is the analogous set of values
A~(¢)= { Ap(¢): PEPer~(¢)}.
In both sets, the multipliers are taken with the appropriate multiplicity.
Example 4.46. Let ¢(z) = zd with d ~ 2. Then Pern(¢) = {0, oo} U l£dn_ 1
consists of the points 0, oo, and the (dn- 1) 1h roots of unity. It is easy to check
that >..o( ¢) = A00 (¢) = 0, and for ( E l£dn_ 1 we have
dn-1 copies
,.---"-..
Hence
An(¢)= {O,O,dn,dn, ... ,dn}.
And if n ~ 2, then A~ (¢) consists of cp( dn - 1) copies of dn, where cp( m) is the
Euler totient function (not to be confused with the rational function ¢( z) ).
Example 4.47. Let ¢(z) = z 2 + bz. Then
Per1(¢) = {0, 1- b, oo}
and
A1(¢) = {b, 2- b, 0}.
Next we compute
*
¢2(z)- z
2
<I></>, 2 = ¢(z)-z =z +(b+1)z+b+l.
The two points of formal period 2 are the roots of <I>¢, 2,
Letting a and {3 denote these two values, we substitute them into
to compute their multipliers, which tum out to be identical,
Thus
A2 (¢) = {4 + 2b - b2 , 4 + 2b - b2 } .
The multiplier spectra An (¢) and A~ (¢) depend only on the PGL2-equivalence
class of¢, so we can use them to define functions on Md.
4.5. Periodic Points, Multipliers, and Multiplier Spectra
183
Definition. Let <P E Ratd and n 2: 1. Define quantities <Tin) (<P) for 0 :S i :S dn + 1
by the relation
dn+l
II (T + .\) = L (};n)(</J)Tdn+l-i.
>.EAn(¢)
i=O
Similarly, define quantities ;~n) ( <P) for 0 :S i :S vd( n) by
II (T + .\) = L ;~n)(</J)Tdn+l-i.
vd(n)
>.EA~(¢)
i=O
Example 4.48. Continuing with Example 4.46, let <P(z) = zd. Then
II (T + .\) = T (T + dn)dn-l and
II (T + .\) = (T + dn)'PWl.
2
>.EAn(¢)
>.EA~(¢)
Example 4.49. Continuing with Example 4.47, let <P(z) = z 2 + bz. We computed
AI(¢)= {b,2- b,O}, so
II (T + .\) = (T + b)(T + 2- b)T = T + 2T2 + (2b- b )T,
3
2
>.EA1(¢)
which gives
(}il) = 2,
(}~1) = 2b- b2'
(}~1) = 0.
Similarly, using the set A2 (<P) computed in (4.30), we find that
II (T+A)=(T+4+2b-b2)2,
>.EA;(¢)
so
and
Theorem 4.50. For <P E Ratd, n 2: 1, and i in the appropriate range, let <Jin) (<P)
and ;~n) (<P) be the symmetric polynomials of then-multiplier spectra of¢.
(a) The/unctions
and
(4.31)
are in Q[Ratd], i.e., they are rational jUnctions in the coefficients ao, ... , bd of
the map <P = [Fa, Fb] with denominators that are a power ofRes(Fa, Fb).
(b) The functions (4.31) are PGL 2 -invariant, and hence are in the ring of regular
functions Q[Md] of the dynamical moduli space Md.
Proof We sketch the proof for <Tin)(¢) and leave ;~n\¢) as an exercise for the
reader (Exercise 4.26).
(a) We write
4. Families of Dynamical Systems
184
¢n = [Fa,n(X, Y), Fb,n(X, Y)]
using homogeneous polynomials of degree dn. Formal properties ofthe resultant and
an easy induction imply that
(See Exercise 4.9.) Thus the rings
are equal. This allows us to replace¢ by ¢nand consider the quantities CJj 1l(¢)
associated to the fixed points of¢. To ease notation, for the remainder of this proof
we write CJi ( ¢) instead of CJjl) ( ¢ ).
Let L denote the field Q(ao, ... , bd), where we treat a0, ... , bd as indeterminates. The fixed points of¢ = [Fa, Fb] are the roots of a polynomial with coefficients in L, so they are defined over an algebraic extension of L. It follows that the
fixed points Fix(¢) and their set of multipliers A 1 ( ¢) are Gal( LI L )-invariant sets,
so the symmetric polynomials CJi( ¢)of the fixed points of¢ are Gal(LI L )-invariant
elements of L. This proves that each CJi ( ¢) is in L. Further, it is clear from the construction that CJi ( ¢) is homogeneous in a and b, in the sense that CJi ( ¢) gives the
same value for fFca, Feb] for any nonzero constant c. Thus the CJi ( ¢) are in L(o),
where the 0 denotes rational functions of a 0 , ... , bd whose numerator and denominator are homogeneous of the same degree.
Our next task is to prove that they are regular functions on Ratd, i.e., that they
are well-defined at every point of Rat d. In order to do this, we must show that their
only poles occur when Res( Fa, Fb) = 0. Let
be the ring of polynomials in the indeterminates a 0 , ... , bd, and to simplify notation,
let
r =Res( Fa, Fb) E A
denote the resultant of Fa and Fb. With this notation, Theorem 4.27 says that the
coordinate ring ofRatd is Q[Ratd], which equals A[r- 1 ]( 0 ), where again the 0 indicates that we take rational functions of degree 0. We also note that the polynomial r
is irreducible in A. 5
We are first going to prove that CJi( ¢) is in the ring A[b;J 1 , r- 1 ]. We dehomogenize by setting [X, Y] = [z, 1] and write ¢(z) = Fa (z) I Fb(z ). The fixed points of¢
are the roots of the polynomial ¢( z) - z, or equivalently, the roots of the polynomial
zFb(z)-Fa(z) = bdzd+l+(bd-1-ad)zd+(bd-2-ad_I)zd- 1+ · ·+(bo-al)z+ao.
(4.32)
5 Indeed, the resultant polynomial is geometrically irreducible, which means that it is irreducible
in K[ao, ... , bd] for any field K, see [436, §5.9].
4.5. Periodic Points, Multipliers, and Multiplier Spectra
185
The roots a 1 , ... , ad+ 1 of this polynomial in L are integral over the ring A[b:i 1 ]. For
any such root a, the corresponding multiplier is
¢'(a)= F~(a)Fa(a)- F~(a)Fb(a).
Fb(a)2
Let B be the integral closure of A [b;; 1 ] in the field L (a). It is clear that the numerator
of¢' (a) is in B. We claim that its denominator is a unit in B [r - 1 ]. Suppose not. Then
we can find a maximal ideall,p C B[r- 1 ] with Fb(a) E l,p. But a is a root of(4.32),
so
Thus z = a is a simultaneous root of
Fa(z) = Fb(z) = 0
(mod l,p),
so a standard property of resultants (Proposition 2.13(a)) implies that
But this is a contradiction, since r is a unit in B[r- 1 ], so it cannot be an element of a
maximal ideal. Hence Fb(a) is a unit in B[r- 1 ], and therefore the multiplier ¢'(a)
is in B[r- 1 ].
We have now shown that each of the multipliers ¢' (a 1 ), ... , ¢' (ad+ 1 ) E L is
integral over the ring A[b:i 1 , r- 1 ]. Hence the symmetric polynomials in these quantities, i.e., a 1 ( ¢), ... , ad+ 1 ( ¢ ), are in L(o) and are integral over A[b:i 1 , r- 1 ]. It follows that they are in A[b:i\ r- 1 ]( 0 ), since the ring A[b:i 1 , r- 1 ] is integrally closed
in its fraction field L.
A similar argument dehomogenizing [X, Y] = [1, w] shows that the ai are in the
ring A[ad 1 , r- 1 ]( 0 ). Therefore
(b) Our earlier calculation (Proposition 1.9) showed that for every complex point
[a, b] E JP' 2 d+ 1 (C) with Res(Fa, Fb) -=f. 0, the set of multipliers of¢ = [Fa, Fb]
is PGL 2 (C)-invariant. Hence the same is true for the quantities ai ( ¢) for all i. This
invariance says the following: Let f(z) = (az + (3)/(rz + 8) be a linear fractional
transformation with indeterminate coefficients a, (3, '/, 8 and consider the difference
(4.33)
n
Clearly (4.33) is a rational function in Q(a 0 , ... , bd, a, (3, '/, 8), and the PGL 2 (C)invariance of si(¢) says that (4.33) vanishes for all choices of ( ~
E GL 2 (C). It
follows that (4.33) is identically zero.
Finally, Theorem 4.36( c) says that Q[Md] is the subring ofPGL 2 -invariant functions in Q[Ratd] (see also Remark 4.38), so in particular the functions (4.31) are
inQ[Md]·
D
4. Families of Dynamical Systems
186
Remark 4.51. As noted earlier in Remark 4.43, the moduli space Md exists as a
scheme over Z. Theorem 4.50 is also valid over Z, so in particular the affine coordinate ring Z[Md] is the ring of PSL 2-invariant functions in Z[Ratd]. See [416,
Theorems 4.2 and 4.5].
The functions ain) and ~~n) constructed in Theorem 4.50 are regular functions
on Md, so they can be used to map Md to affine space. For example, in the next
section we show that
*(1)) ·M
( a*(1)
. 2---+~,~,2
1 ,a 2
is an isomorphism. In general, Md is not isomorphic to A 2d- 2, but we might ask
whether using a sufficient number of the a;n) or ~~n) gives an embedding of Md
into affine space.
In particular, do the values of all ain) or ~~n)(¢) determine the PGL2(C)conjugacy class of ¢? The answer is no. The Lattes maps that we studied in Section 1.6.3 provide nontrivial families of rational maps whose multiplier spectra, and
hence whose ain) and ~~n) values, are all the same.
Example 4.52. For each t E C* with t -=f. - 2 consider the rational map
J,
4>t(x) =
x 4 - 2tx 2 - 8tx + t 2
4x3 + 4tx + 4t
It is the Lattes map associated to multiplication-by-2 on the elliptic curve
Et : y 2 = x 3 + tx + t;
see Section 1.6.3. Following standard notation, we write Et [m] for the points of Et
of order m. Then the n-periodic points of¢ are given by
and it is not hard to compute the multipliers at these points,
ifa E x(Et[2n -1]) and a -=f. oo,
if a E x(Et[2n + 1]) and a -=f. oo,
if a= oo.
(For proofs of these statements, see Proposition 6.52 in Section 6.5.)
For any m, the set E[m] of m-torsion points has order m 2, and for odd m the
map x : E[m] ---+ JP> 1 is exactly 2-to-1 except at x- 1 (oo), so we can use the listed
values of Aa ( 4>t) to compute
11 (T +.X) = (T + 2n)22n-1_2n (T- 2n)22n-1+2n (T + 22n).
AEAn(t/.>t)
In particular, we see that every map 4>t has the same set of multipliers, so a; n) (4>t)
does not depend on t. (A similar statement holds for ~~n) ( 4>t).) Hence no matter how
4.5. Periodic Points, Multipliers, and Multiplier Spectra
187
many functions O"~n) we use, the resulting map M 4 --+ A_k always compresses all of
the maps 4>t down to a single point. On the other hand, we will later prove that the 4>t
are not PGL 2 -conjugate to one another (see Theorem 6.46). Hence the image ofthe
map
is a curve in M 4 that is compressed to a single point in Ak. An important theorem
says that aside from Lattes examples of this kind, the map on Md defined by the O"~n)
is finite-to-one.
Theorem 4.53. (McMullen [294, §2]) Fix d;::: 2, and for each N;::: 1let
ud,N: Md-----. Ak
(4.34)
be the map defined using all ofthe functions O"~n) with 1 ~ n ~ N. If N is sufficiently
large, then the map u d,N is finite-to-one on Md (<C) except for certain families of
Lattes maps that it compresses down to a single point. In particular, it is finite-to-one
if d is not a perfect square.
Further, the same statement is true for the map
*
U d,N
.· M d _____. f..l e
defined using all ofthe functions ;~n) with 1 ~ n ~ N. (The flexible Lattes maps for
which u d,N and u d,N• are not finite-to-one are discussed in detail in Section 6.5.)
McMullen's theorem says that aside from the flexible Lattes maps, the maps u d,N
and u d,N• are finite-to-one onto their image. One might hope that they are actually
injective if we avoid the flexible Lattes maps, but it turns out that this is far from true.
Theorem 4.54. Define the degree ofud,N to be the number ofpoints in u;:t,Jv(P)for
a generic point Pin the image u d,N(Md)· One can show that the degree of u d,N
stabilizes as N--+ oo. We write deg(u d)/or this value. Then for every E > 0 there is
a constant CE such that
for all d.
In particular, the multipliers of a rational function ¢ E Ratd determine the conjugacy class of¢ only up to OE(d~-E) possibilities.
Proof We will prove this in Chapter 6 using Lattes maps associated to elliptic curves
D
with complex multiplication; see Theorem 6.62.
Definition. Following Milnor [300], we define the multiplier spectrum of a rational
map ¢ to be the function A(¢) that to each positive integer n assigns the set An (¢).
In other words, A(¢) is the set-valued function
A(¢) : N -----. Sets with Multiplicities,
188
4. Families of Dynamical Systems
Two rational maps are said to be isospectral if they have the same multiplier spectrum. Then another way to state McMullen's Theorem 4.53 is to say that aside from
the flexible Lattes maps, the multiplier spectrum determines the rational map ¢ up
to finitely many possibilities. The fact that the flexible Lattes maps form nontrivial
isospectral families is proven in Section 6.5.
Remark 4.55. We show in the next section that the map
is an injection, and in fact it maps M 2 isomorphically to the plane z = x - 2,
so M 2 ~ A. 2. On the other hand, the existence of the Lattes maps described in
Example 4.52 shows that u 4,N cannot be finite-to-one on M4. The degree of the
map
does not appear to be known.
4.6
The Moduli Space M 2 of Dynamical Systems of
Degree 2
Theorem 4.50 tells us that symmetric combinations of the numbers in the multiplier
spectra give well-defined functions on the moduli space Md of degree d dynamical
systems on lP'1. In this section we describe Milnor's explicit identification of M 2
with A. 2 using two of these functions.
Theorem 4.56. (Milnor [301], see also [416]) Let a1, a2, a3 E Q[Rat2] be the three
functions constructedfrom the .fixed points ofa rational map ofdegree 2, i.e.,
11 (T + Ap(¢)) = Lai(¢)T -i.
3
3
PEFix(</>)
i=O
(In the notation from page 183, we have ai = a} 1 ) for 0 ~ i ~ 3.)
(a) a1 = a3 + 2.
(b) The morphism
u = (a1, a2) : Rat2 ---+ A. 2
has the following three properties:
(i) u(¢f) = u(¢)for all¢ E Rat2(C) and all f E PGL2(C).
(ii) Let ¢1,¢2 E Rat2(C). If u(¢1) = u(¢2), then there exists a Mobius
transformation f E PGL2(C) such that ¢ 2 = ¢{.
(iii) Let (s 1, s 2) E A. 2(C). Then there exists a rational map¢ E Rat2(C) such
thatu(¢) = (s1.s2).
Hence u induces a bijection u: M 2(C) ~ A. 2(C).
4.6. The Moduli Space M 2 of Dynamical Systems of Degree 2
189
(c) The functions a 1 and a2 are in Q[M2] and the map
u = (a1, a2) : M2 --+ A2
is an isomorphism of algebraic varieties defined over Q. Equivalently, the in-
duced map
is an isomorphism of rings.
(d) All ofthe functions a~n) and ~~n) can be expressed as polynomials in a1 and a2
with rational coefficients.
(e) For any field extension K jQ, the map u : M2 ---> A 2 in (c) induces a bijection
M 2 (K) +-+ A2(K). (Note that Md(K) is not the same as Ratd(K)j PSL2(K);
see Remark 4.39 and Section 4.1 0.)
Remark 4.57. For some applications it is useful to have an explicit description of the
map
u = (a1, a2) : Rat2 --+ A2
described in Theorem 4.56(b ). Let
p( a, b) = a~b6 - a 1 a2 b0 b1 + a0 a2 bi + ai b0 b2 - 2a0 a2 b0 b2 - a0 a 1 b1 b2 + a6b~
denote the resultant of Fa and Fb. Then, after some algebraic manipulation, one
finds that a 1 and a 2 are given by the expressions
p(a, b)a1(¢) = afb0 - 4a0 a1 a2 b0 - 6a~b6- a0 aib 1 + 4a6a2b1 + 4a 1 a 2 b0 b1
- 2a0 a 2 bi + a 2 bf - 2aib0 b2 + 4a0 a2 b0 b2 - 4a 2 b0 b1 b2
- a 1bib2 + 2a6b~ + 4a 1b0 b~,
p(a, b)a2(¢) = -a6ai + 4aga2 - 2afb0 + 10a0 a 1 a2 b0 + 12a~b6- 4a6a 2b1
- 7a 1a2 b0 b1 - aibi + 5a0 a2 bi - 2a 2 b~ + 2a5a 1b2 + 5aib0 b2
- 4a0 a2 b0 b2 - a0 a 1 b1b2 + 10a2b0 b1b2 - 4a 1b0 b~ + 2a0 b1b~
-bib~+ 4b0 b~.
Remark 4.58. According to Theorem 4.56(c), every function ~~n) E Q[M 2 ] is a
polynomial in a 1 and a 2 . In practice, it can be quite challenging to find explicit
expressions. Milnor [301] gives the examples
~~2 ) = 2a1 + a2,
~~3 ) = a1(2a1 + a2) + 3al + 3,
~~3 ) = (a1 + a2) 2(2a1 + a2)- a1(a1 + 2a2) + 12a1 + 28.
Notice that these expressions are in Z[a1, a 2], rather than merely in Q[ab a 2]. This
reflects the fact that the map u : M 2 ---> A 2 is actually an isomorphism of schemes
over Z as described in Remarks 4.43 and 4.51. The functions ~~n) are regular functions on the scheme M2 jZ, so they are in Z[a1, a2]. See [416] for details.
190
4. Families of Dynamical Systems
Proof of Theorem 4.56. (a) Let <P E Rat 2(C) and let At, >. 2, >.a be the multipliers of
the fixed points of¢. If we assume that none of >. 1, >. 2, >.a is equal to 1, then we can
apply Theorem 1.14 to deduce that
1
1
1
--+--+--=1.
1 - >.1 1 - >.2 1 - >.a
After some algebraic manipulation this becomes
>.1>.2>.a - >.1 - >.2 - >.a + 2 = 0,
which completes the proof that a 1(<P) = aa(<P) + 2 for all <P E Rat 2(q whose
multipliers are not equal to 1. It is not hard to see that such <Pare dense in Rat 2(C)
(see Exercise 4.21), from which it follows that the function
a1 - aa - 2 E C[Rat2] = C [ao, a1, a2, bo, bt, b2, Res( Fa, Fh)- 1]
is identically zero.
(b) The first property u(¢1) = u(¢) is a special case of Theorem 4.50, or more
directly it is an immediate consequence of our earlier calculation (Proposition 1.9)
showing that the multipliers of a rational map are PGL 2-invariant.
In order to prove the second property, it is convenient to show that every rational
map of degree 2 is PGL 2-equivalent to a map of a particular shape.
Lemma 4.59. (Normal Forms Lemma) Let <P E Rat 2(C) be a rational map of
degree 2 and let >.1, >.2, >.a be the multipliers of its .fixed points.
(a) /f>..1>..2 f. 1, then there is an f E PGL2(C) such that
¢1 (z) = z2 + >.1z.
A2Z
(4.35)
+1
Further, Res(z 2 + >.1z, >.2z + 1) = 1 - >.1>.2.
(b) If >.1>.2 = 1, then >.1 = >.2 = 1 and there is an f E PGL2(C) such that
¢1(z) = z+ ~+ !.
z
(4.36)
Proof. We recall that if a E Fix( <P) has multiplier >."" then the Taylor expansion
of <P around a looks like
</J(z) =¢(a)+ <P'(a)(z- a)+ O(z- a) 2 =a+ >.a(z- a)+ O(z- a) 2 •
Hence
<I>q,,1(z) = </J(z)- z =(>.a -1)(z- a)+ O(z- a) 2,
so <P has multiplicity 1 at the fixed point a if and only if >.a -=/:- 1.
We also note the formal identity
(X- 1) 2 - (XY- 1)(XZ- 1) = X(X + Y + Z- 2- XYZ)
4.6. The Moduli Space M 2 of Dynamical Systems of Degree 2
191
and apply it using the relation
AI + .A2 + A3 = ai = a3 + 2 = AIA2A3 + 2
from (a). This yields the useful formulas
(.AI- 1) 2 = (.AI.A2- 1)(.AIA3- 1),
(.A2- 1) 2 = (.A2.AI- 1)(.A2.A3- 1),
(4.37)
(.A3- 1) 2 = (.A3.AI - 1)(.A3.A2- 1).
We now start with a rational map
2
+ aiz + a2
boz 2 +biz+ b2
¢(z) = aoz
and change coordinates in order to put ¢ into the desired form.
(a) For this part we assume that .AI.A 2 f. 1, so (4.37) tells us that AI f. 1 and
.A 2 f. 1. Thus the fixed points associated to AI and .A 2 have multiplicity 1, so in
particular they are distinct and we can find an element ofPGL 2 (C) that moves them
to 0 and oo, respectively. After this change of variables has been made, the rational
map¢ satisfies ¢(0) = 0 and¢( oo) = oo, so it has the form
2
+ aiz
biz+ b2
¢(z) = aoz
with
Since ao f. 0, we can dehomogenize by setting ao = 1, and then a simple calculation
yields
and
Thus f has the form
¢(z) = z 2 + b2AIZ
A2Z + b2
with
Finally, replacing ¢(z) by b2I¢(b 2z) yields the desired form (4.35), and we calculate
Res(z 2
1 AI 0
+ AIZ, .A2z + 1) = det .A2 1 0 = 1- .AI.A2,
0 .A2 1
which completes the proof of (a).
(b) The proof of this part is similar. We begin by moving the fixed point associated
to AI to oo, so the map¢ has the form
with
The assumption that AIA2 = 1 combined with (4.37) tells us that .AI = .A 2 = 1, so
we have bi = a0 . Dehomogenizing a0 = 1 puts ¢ into the form
192
4. Families of Dynamical Systems
2
cf>(z)=z +a1z+a2.
z + b2
Next we replace cf>(z) by cf>(z- b2 ) + b2 , so now cf>(z) looks like
cf>(z) = z2 + a1z + a2
z
with
(Of course, the values of a 1 and a 2 have changed.) Finally, replacing cf>(z) by
cf> (y'a2 z) / y'a2 gives ¢( z) the desired form
..+..( ) _ z 2 + a1z + 1 _
'+' z
-
1
- z + a1 + -.
z
z
(4.38)
We can compute the value of a 1 by observing that cf> has a double fixed point
at oo and that its other fixed point is at -a1 1 . (If a 1 = 0, then there is a triple fixed
point at oo.) Thus the third multiplier is
so we find that a1 = v1 - .X3. (This is also correct if a1 = 0.) Substituting this value
of a 1 into (4.38) completes the proof of (b).
0
We resume the proof of Theorem 4.56(b,ii). Let ¢ 1 and ¢ 2 be rational maps of
degree 2 satisfying u (¢ 1 ) = u (¢ 2 ). Thus
and
and then the relation a3 = a1- 2 from (a) implies that also a3(¢1) = a3(¢2). It
follows that the set of multipliers of the fixed points of c/>1 and c/>2 are the same,
say {A1, .X2, .X3}, since they are the three roots (counted with multiplicity) of the
polynomial
T 3 - a 1 T 2 + a2T- a3.
We consider two cases. Suppose first that .X 1.X2 ":11. Then Lemma 4.59(a) says
that ¢ 1 and c/>2 are 2-equivalent to the function (z 2 + .X 1z)/(.X 2z + 1), so they are
PGL 2-equivalent to each other, and similarly if .X1.X 3 ":11 or if .X2.X3 ":11.
We are left to consider the case .X1.X2 = .X1.X3 = .X2.X3 = 1. Then Lemma 4.59(b)
tells us that .X1 = .X2 = .X 3 = 1 and that c/>1 and ¢2 are both PGL 2-equivalent to the
rational map z + z- 1 .
This completes the proof of (b,ii), so we turn to (b,iii), the surjectivity of u. Given
(8 1 , 8 2 ) E A 2 (C), we set 8 3 = 8 1 - 2 and factor the polynomial
Notice that the condition 8 3 = 8 1 - 2 gives the familiar relation
(4.39)
4.6. The Moduli Space M 2 of Dynamical Systems of Degree 2
which in turn implies the usual formal identities (4.37) for )q, ..\ 2 , ..\3 •
Suppose first that some Ai is not equal to 1, say ..\ 1 # 1. Then ..\ 1 ..\ 2
from (4.37), so we can set
¢(z)=z 2 +..\1z,
A2Z
193
# 1
+1
since Res(z 2 + ..\ 1z, ..\2z + 1) = 1 - ..\ 1..\2 from Lemma 4.59(a). The fixed points
of¢> are o:1 = 0, o:2 = oo, and o:3 = ~=~~. Its multipliers at o: 1 and o: 2 are easily
computed,
and
and the multiplier at 0:3 is
where we use (4.39) for the middle equality. Hence 0"1 (¢) = 81 and 0"2(¢) = 82.
We are left with the case ..\ 1 = ..\ 2 = ..\ 3 = 1, which corresponds to the values 81 = 82 = 3. It is easy to check that the rational map cp(z) = z + z- 1
has a triple-order fixed point at oo and that all three multipliers are equal to 1,
sou(¢) = (3, 3).
(c) We briefly sketch the proof, which uses basic methods from algebraic geometry.
We refer the reader to [416, §5] for further details.
The first step is to verify that the map u : M 2 ____, A 2 is proper. This follows
easily from the valuative criterion [198, 11.4.7] using the fact proven in (b) that every
fiber u- 1 ( t) consists of a single point. (Roughly speaking, a morphism of varieties
over C. is proper if its fibers are complete.) Next one checks that u is finite, which
can be proven using the fact that both M 2 and A 2 are affine varieties (cf. [416,
Lemma 5.6]). Alternatively, one can show in general that a proper quasifinite map
is finite ([299, I, Proposition 1.10]). Then one uses the fact that A 2 is nonsingular
and the bijectivity of u on complex points to prove that u is an isomorphism (cf.
[416, Lemma 5.7]). Finally, it is clear from the explicit formulas for 0" 1 and 0"2 in
Remark 4.57 that u is defined over Q.
(d) This is immediate from (c), which says that Q[M2] = Q[O"b 0"2], since we
already know from Theorem 4.50(b) that O"~n) and ;;.in) are in Q[M 2].
(e) This is a consequence of the fact that the isomorphism u : M 2 ____, A 2 in (c) is
defined over Q.
D
Remark 4.60. Regarding the proof sketch of Theorem 4.56(c), we observe that a
morphism of irreducible varieties V ____, W may be bijective on complex points, yet
not be an isomorphism. A simple example of this phenomenon is the cuspidal cubic
map
Thus in the proof that u : M 2 ____, A 2 is an isomorphism, it is crucial that u
maps M 2 onto the nonsingu/ar variety A 2 . It does not appear to be known in general
whether Md is nonsingular.
4. Families of Dynamical Systems
194
The content of Theorem 4.56 is that the moduli space M 2 of dynamical systems
of degree 2 is isomorphic to A 2 and an explicit isomorphism is provided by the pair
of functions (a1 , a 2 ) created from the multipliers of the fixed points of a rational
map. From our general theory (Theorem 4.40), the space M 2 sits naturally inside
two larger spaces M~ and M~s, but since d = 2 is even, these two spaces coincide
and will be denoted by M 2. We conclude this section with a description of M2.
Theorem 4.61. Let M 2 = M~ = M~ 8 be the completion ofM2 constructed using
geometric invariant theory in Theorem 4.40. Then the isomorphism
extends to an isomorphism ii : M2 ---. ]pl2 such that the following diagram commutes:
The points in M2(C) that are not in M2(C) correspond to degenerate maps of
degree 2 that may be informally described as maps of the form
c/>A,B(X, Y) = [AXY, XY + BY 2 ],
(4.40)
The point [A, B] is uniquely determined up to reversing A and B.
Proof See [302] and [416, Theorem 6.1 and Lemmas 6.2 and 6.3].
D
Remark 4.62. We expand briefly on what it means to say that the points in the
set (M 2 "M2)(C) correspond to the maps cf>A,B given by (4.40). Let U c A1
be a neighborhood of 0 and let
be a rational map that is a morphism away from 0. We denote the image oft E U
by cf>t to help remind the reader that cf>t is itself a map, i.e., cf>t : JP> 1 ---. JP> 1 .
Consider the composition
which by abuse of notation we again denote by ¢. It is a rational map, and since
U c A 1 and M 2 is complete, we see that it is a morphism from U to M 2 . In particular, the point ¢ 0 is a well-defined point in M2(C). If c/>o E M2(C), there is nothing
to say, so we suppose that c/>o rf_ M2(C).
Then the second part of Theorem 4.61 means that possibly after choosing a
smaller neighborhood of 0, there exists a morphism
195
4.7. Automorphisms and Twists
f : U ---+ PGL2
such that the conjugate map ¢! has the form
with ao, ... , b2 regular functions on U and satisfying
ao(O) = a2(0) = bo(O) = 0,
and
a1 (0), b2(0) not both 0.
In other words, the degeneration of¢! at t = 0 has the form [AXY, XY + BY 2 ].
Further, the map¢ determines the point [A, B] = [a 1 (0), b2 (0)] E lP' 1 (C) up to
switching the two coordinates. In this way we identify
[A, B] ,_____. [AXY, XY + BY 2 ],
where the symmetric group on two letters s2 acts on lP' 1 by interchanging the coordinates. (It is an exercise to show that lP' 1 / S 2 is isomorphic to lP' 1 .)
4. 7 Automorphisms and Twists
As we have repeatedly seen, from a dynamical perspective the geometric properties
of a rational map ¢( z) and its conjugates ¢! = f- 1 ¢ f are the same, since conjugation by f E PGL 2(C) is simply an invertible change of variables. However, matters
become more complicated if we restrict the coefficients off to lie in a field that is
not algebraically closed. This leads to a notion of conjugation equivalence relative to
a particular field, as in the following definitions.
Definition. Two rational maps ¢(z), '1/J(z) E K(z) are PGL 2 -equivalent if there is
a linear fractional transformation f E PGL 2( K) such that 'ljJ = ¢!. More generally,
we say that ¢ and 'ljJ are PGL 2 (K)-equivalent if there is a linear fractional transformation f E PGL 2(K) such that 'ljJ = cpf. When PGL 2 is clear from context,
we refer more simply to K -equivalence and K -equivalence. We leave as an exercise
(Exercise 4.31) the proof that these are equivalence relations.
We denote the set of rational maps that are K -equivalent to ¢ by
[¢] = {¢ 1 : f E PGL2(K)},
and similarly the set of rational maps that are K -equivalent by
[¢]x = {¢1 : f E PGL2(K)}.
Remark4.63. In some sense we already have a notation(¢) for the PGL2(K) equivalence class of¢. However, we generally view (·)as a map Ratd -+ Md and(¢) as
a point in the moduli space Md, while we think of[¢] as a set of rational maps. This
is the reason for the notational distinction.
196
4. Families of Dynamical Systems
Not all conjugates qi of cp need be distinct. For example, the rational map
b
cp(z) = az +z
satisfies
- ¢( -z) = cp(z),
so cpf = cp for the linear fractional transformation f(z) = -z. The set of transformations f E PGL 2 (K) that fix cp is an interesting group whose properties play an
important role both geometrically and arithmetically.
Definition. Let ¢( z) E K (z) be a rational map. The automorphism group ofcp is the
group
Aut(¢)= {f E PGL 2 (K): ¢1 (z) = cp(z)}.
(Another common name for Aut( cp) is the group of self similarities of cp.)
Remark 4.64. It is easy to check that Aut(¢) is a subgroup ofPGL 2 (K) and that
for any hE PGL 2(K) the map
f f-------7 h-l fh,
is an isomorphism (see Exercise 4.32). Thus as an abstract group, Aut( cp) depends only on the PGL 2 -conjugacy equivalence class of cp; more precisely, the Kequivalence class of cp determines Aut( cp) in PGL 2 up to conjugation.
Proposition 4.65. Let cp(z) E K(z) be a rational map ofdegree d 2': 2. Then Aut( cp)
is a finite subgroup ofPGL2(K), and its order is bounded by a function of d.
Proof Let f E Aut( cp ). Then for any point P E lP' 1 (K) and any n 2': 1 we have
¢n(P) = (¢f)n(P) = (f-1¢n f)(P),
and hence
j(cpn(P)) = </Jn(f(P)).
In particular, if P is a periodic point of (primitive) period n, then f(P) is also a
periodic point of(primitive) period n. Thus each f E Aut(¢) induces a permutation
of each ofthe sets Pern(¢) and Per~*(¢).
Choose three distinct integers n 1 , n 2 , n 3 such that¢ contains primitive n periodic
points for each value of n. Corollary 4. 7 says that we can find such values, and further
that their magnitude can be bounded solely in terms of d. More precisely, they may
be chosen from among the first d + 5 primes. Letting
fori = 1, 2, 3,
the action of cp on the sets of primitive periodic points yields a homomorphism
(4.41)
from Aut(¢) into a product of three symmetric groups.
We claim that the homomorphism (4.41) is injective. To see this, we observe that
any fin the kernel of(4.41) fixes each Per~*(¢); hence f fixes at least three points
in lP' 1 (K); hence f is the identity map. Thus'the homomorphism (4.41) is injective,
which clearly implies that Aut(¢) is finite.
D
197
4.7. Automorphisms and Twists
Remark 4.66. Proposition 4.65 tells us that the automorphism group Aut(¢) of a
rational map is a finite subgroup of PG1 2(K). For R = <C, or more generally for
any field of characteristic 0, the classical description of finite subgroups of PG1 2(q
says that every such subgroup is conjugate to either a cyclic group, a dihedral group,
or the symmetry group of a regular three-dimensional solid (i.e., the tetrahedral,
octahedral, and icosahedral groups). See, e.g., [414].
Example 4.67. Let (be an n 1h root of unity, let 7./J(z) E K(z) be any rational map,
and let ¢(z) = z?jJ(zn). Then Aut(¢) contains a cyclic subgroup of order n generated by the map f(z) = (z.
Example 4.68. The map ¢(z) = (z 2 - 2z)/( -2z + 1) has an automorphism group
Aut(¢) that is isomorphic to the symmetric group S 3 on three letters. More precisely
(see Exercise 4.36), the automorphism group of¢ consists of the following six linear
fractional transformations:
}
z
1
1 z-1
Aut(¢)= { z, -, - - , - - , - - , 1 - z ~ S 3 .
z
z
1-zz-1
Example 4.69. Consider the rational map
b
</Jb(z) = z + -,
z
whose automorphism group Aut(¢b) = {z, -z} has order 2. These maps are
all PG1 2-equivalent, since the linear fractional transformation f (z) = z /b(C gives
<Pc = ¢{.Thus the geometric dynamical properties of </Jb are the same for all b. However, the arithmetic properties of </Jb may change quite substantially depending on the
value ofb, since the change of variable involves a square root. So although </Jb and <Pc
are always PG1 2(K)-equivalent, they are not PG1 2(K)-equivalent unless b/c is a
perfect square in K. The underlying reason for the existence of these "twists" is the
fact that Aut( ¢1) is nontrivial.
Definition. Let ¢( z) E K ( z) be a rational map. The set of twists of¢ over K is the
set
Twist (¢/ K) = { K -equivalen~e ~lasse~ of maps 7./J such} .
that 7./J 1s K -eqmvalent to ¢
Remark 4.70. As noted earlier, the geometric dynamical properties of the maps
in Twist(¢/ K) are identical, but their arithmetic properties may be significantly
different. For example, if two rational maps are K-isomorphic, then the field extensions of K generated by their periodic points are the same. This is clear from the fact
that
Pern(¢f) = {f- 1 (P): P E Pern(¢) },
so iff E PG12(K), the fields generated by Pern(¢) and Pern(¢f) are identical.
However, if¢ and 7./J are only R -isomorphic, these fields may well be different. This
often provides a convenient method for proving that two maps are not K- isomorphic,
as in the following example.
198
4. Families of Dynamical Systems
Example 4.71. Continuing with Example 4.69, for each bE K* we let
b
¢b(z) = z + -.
z
We saw that these maps are all k -isomorphic, which gives a map of sets
K* -+ Twist(¢1/ K),
(4.42)
Note that if blc is a square inK, say blc = a2 , then ¢band ¢care K-isomorphic,
since ¢c = ¢{ for the map f(z) = az E PGL2(K). Thus (4.42) induces a welldefined map
K* IK* 2 - + Twist(¢1/K),
(4.43)
We can use Remark 4.70 to prove that the map (4.43) is injective. (We assume that K does not have characteristic 2.) A quick computation shows that
<I>¢b, 2(X, Y) = 2X 2 + bY 2 , so the primitive 2-periodic points of ¢bare±~.
FCfi)
Hence if ¢band ¢care K-isomorphic, then the fields K( Fb[2) and K(
are the same, so blc must be a square6 inK. Notice that we could not use Per 1 (¢b)
to prove this result, since the only fixed point of ¢b is oo.
These quadratic twists of ¢ 1 ( z) are analogous to quadratic twists of elliptic
curves (cf. [410, §X.5]). Thus fix A, B E K and, for each D E K*, let ED be
the elliptic curve ED : DY 2 = X 3 +AX+ B. Then all of the ED are isomorphic
over k, but E D 1 and E D 2 are isomorphic over K if and only if the ratio D 1 I D 2 is a
square inK. This gives a natural map K* I(K*) 2 ____, Twist(EI/ K).
Remark 4.72. How does the theory of twists fit in with the moduli spaces Md constructed in Section 4.4? The answer is that if¢ and '1/J are twists of one another, then
their corresponding points (¢) and ('1/J) in the moduli space Md are equal. This is
true because points in Md (k) classify rational maps of degree d modulo PG L2 (k)
conjugation, so points in Md(k) do not detect whether the conjugation is defined
over K. In other words, there is a natural map
(4.44)
but this map is not one-to-one. It fails to be injective precisely for those maps
in Ratd(K) that have nontrivial twists; cf. Remark 4.39. On the other hand, the
next proposition and Exercise 4.38 imply that the map (4.44) is injective on a Zariski
open subset ofRatd.
We now prove that a rational map with no automorphisms has no nontrivial
twists. Later, in Theorem 4. 79, we prove a much stronger result.
Proposition 4. 73. Let ¢( z) E K ( z) be a rational map of degree d 2 2 and assume
that its automorphism group Aut(¢) is trivial. Then ¢ has no nontrivial twists, i.e.,
Twist (¢I K) has only one element, the K -equivalence class of¢ itself
6 It is an easy exercise to prove that K ( vA) and K (vB) are isomorphic if and only if A/ B is a
square inK, assuming that K does not have characteristic 2.
4.8. General Theory of 1\vists
199
Proof Suppose that 'ljJ E K(z) is a twist of¢, so there is an f E PGL 2 (K) such
that 'ljJ = ¢!. We let an element a E Gal( R I K) act on f (z) and ¢( z) in the natural
way by applying a to each of the coefficients. Notice that a(¢) = ¢and a( '1/J) = '1/J,
since the coefficients of¢ and 'ljJ are inK. Hence
¢! = 'ljJ = a('ljJ) = a(cpf)
= a(fcpf- 1 ) = a(f)a(cp)a(f- 1 ) = a(f)cpa(f)- 1 =cpu(!).
Hence a(f)f- 1 E Aut(¢). But Aut(¢) is trivial by assumption, so a(f) =f. This
is true for all a E Gal(R I K), so we conclude7 that f E PGL 2 (K). Hence 'ljJ = ¢!
is K -isomorphic to ¢, so it represents the trivial twist.
D
Remark 4.74. As we have seen in Section 4.3, the set Ratd of all rational maps of
degree dis a Zariski open subset oflP' 2dH. It is the complement of the hypersurface
described by the vanishing of the resultant Res( Fa, Fb) = 0. One can show that the
set of rational maps ¢ E Ratd having nontrivial automorphism group forms a proper
Zariski closed subset of Ratd; see Exercise 4.38. Thus most rational maps have no
nontrivial automorphisms, and those that do, fall into finitely many irreducible algebraic families. Further, since for all f E PGL 2 the automorphism groups of¢ and¢!
are isomorphic as abstract groups, there is a proper Zariski closed subset of Md determined by the conjugacy classes of rational maps with nontrivial automorphisms.
It is an interesting geometric problem to describe the irreducible subvarieties making up this set and an interesting arithmetic question to describe their rational and
integral points. See Exercises 4.28 and 4.41.
4.8
General Theory of Twists
In this section we develop the basic theory of Galois twists in an abstract setting.
Although we apply this material only to rational maps, in Section 4.9, and to lP' 1 ,
in Section 4.10, we develop the theory in some generality in order to clarify the
underlying principles.
Let X andY be objects defined over the field K. For example, X andY might be
curves or algebraic varieties or rational maps. 8 We consider X and Y to be equivalent
if they are isomorphic over K. However, it may happen that they are isomorphic
over R, but not over K. This leads us to consider the following set.
Definition. Let X be an object defined over the field K. The set of twists of X I K is
the set
7 The proof that PGL2(K) is the subgroup ofPGL2(K) fixed by Gal(K I K) is not hard, although
it does use Hilbert's Theorem 90.
8 Formally, X and Y should be objects in a category on which the Galois group Gal( KI K) acts in
an appropriate way. We do not concern ourselves with such formalism and leave it to the reader either
to formulate the correct abstract concepts or to restrict attention to those situations, namely algebraic
varieties and rational maps, in which the Galois action is clear.
4. Families of Dynamical Systems
200
K -isomorphism classes of objects Y}
.
Twist (XI K) = { such that Y is defined over K ~nd
Y is isomorphic to X over K
In other words, an element ofTwist(XI K) is an object Y that is defined over K
such that there is an isomorphism i : X ---+ Y, but the isomorphism i might be
defined only over an extension of K. Two elements Y andY' in Twist(XI K) are
considered to be equivalent if there is an isomorphism j : Y ---+ Y' that is defined
over K. The following examples should help to clarify this definition.
Example 4.75. Our first example deals with twists of rational maps. Let ¢(z) E K(z)
be an odd rational map, i.e., a rational map satisfying¢( -z) = -¢(z). Then for
each b E K* we can define a new rational map ¢b by the formula
1
¢b(z) = y'b¢(Vbz).
The odd parity of ¢(z) implies that¢ has the form ¢(z) = z'lj;(z 2 ) for some rational
map 'lj;(z) E K(z), so ¢b(z) = z'lj;(bz 2 ) is in K(z). Further, ifb E K* 2 , say b = c2 ,
then ¢b is PGL 2 (K)-conjugate to¢ via ¢b = ¢!with f(z) = cz. In this way we
obtain a map
(4.45)
K*IK* 2 -----> Twist(¢1K),
If Aut(¢) = {z, -z }, then it is not hard to show that (4.45) is an isomorphism.
Notice how this example generalizes Example 4.71.
Example 4.76. Our second example deals with twists of the variety lP' 1 . For any
nonzero a E K*, let Ca be the plane curve
Ca : x 2 + y 2 = a.
All of these curves are isomorphic over K via the explicit isomorphism
i(x,y) = (x~,y~).
(4.46)
Further, if bI a is a square in K, say bI a = c2 , then Ca and Cb are isomorphic over K
via the isomorphism i(x, y) = (ex, cy). Thus exactly as in Example 4.75, we obtain
a natural map
K*IK* 2 --->Twist(CI/K),
where [Ca]K denotes the K-isomorphism class of Ca.
Notice that if two curves C and C' are K-isomorphic, then the K-isomorphism
i: C---+ C' identifies their K-rational points i: C(K)---+ C'(K). This suffices to
prove that the curves c1 and c_1 are not isomorphic over Q, since
and
c_1(Q) = 0.
Hence C _ 1 represents a nontrivial element of Twist (C1 I Q), and indeed a nontrivial
element of Twist (C 1 IIR). More generally, a famous theorem ofF ermat says that if p
is an odd prime number, then
4.8. General Theory of Twists
201
p =1
if and only if
(mod 4).
Hence if p = 3 (mod 4), then Cp represents a nontrivial element of Twist( Cl/Q).
It is not hard to show that different primes p = 3 (mod 4) yield distinct elements
of Twist( Cl/Q); see Exercise 4.44.
Returning to the general situation, let X be an object defined over K and let Y
represent an element of Twist( X/ K). This means that there is a K-isomorphism
i: Y ____,X.
We wish to determine whether X and Y are K -isomorphic. If i is itself a K -isomorphism, then we are done; but even if i is not a K -isomorphism, it may be possible to
modify i and tum it into a K -isomorphism.
In order to measure the extent to which i fails to beaK -isomorphism, it is natural
to make use of Galois theory, since i is defined over K if and only O"( i) = i for
every O" E Gal(K / K). Thus for each element O" E Gal(K / K), we consider the
composition of maps
a-(i- 1 )
i
~X.
ga-:X ~ Y
The map gu = iO"( i- 1 ) is a K -automorphism of X, i.e., it is an isomorphism from X
to itself defined over K. If i is already defined over K, then gu (x) = xis the identity
map, but in general go- will be a nontrivial automorphism.
Proposition 4.77. Let X be an object defined over K, let Y be a twist of X/ K,
choose a K -isomorphism i : Y ----+ X, and define a map
ga-(x) = (iO"(i- 1 ))(x).
g : Gal(K / K) ---->Aut( X),
(a) The map
Gal(K / K) ---->Aut( X),
satisfies
for all O", T E Gal(K / K).
Maps satisfYing guT = guO"(gT) are called 1-cocycles.
(b) Y is the trivial twist of X if and only if there is an f E Aut(X) satisfYing
for all O" E Gal(K / K).
Maps of the form fO"(f- 1) are called 1-coboundaries.
Proof (a) We have guT = i o (O"T) (i- 1 ) = i o O"( T(i) - 1 ). Consider the following
commutative diagram of maps:
X
a-( T( i) -l)
u(i)
y
i
~
1
X
ir
X ~y
4. Families of Dynamical Systems
202
The top row is 9an but if we travel around the diagram the long way, we get
i o a(i- 1 ) o a(i) o a(r(i)- 1 ) = 9a o a(9r)·
(b) Suppose first that 9 is a coboundary, say 9a = fa(f- 1 ) for some f E Aut(X).
We verify that the isomorphism f- 1i : Y __.. X is defined over K. For any
a E Gal(K I K) we have
a(f- 1 i) = a(f- 1 )a(i) = f- 19aa(i) = f- 1 ia(C 1 )a(i) =
r i.
1
This proves that f- 1 i is defined over K, soY is K-isomorphic to X.
Next suppose that Y is K-isomorphic to X, say j : Y __..X is a K-isomorphism.
Then a(j) = j for every a E Gal(K I K), so we have
9a = i o a(C 1 ) = i o F 1 o a(j) o a(i- 1 ) = (i o T 1 ) o a((i o T 1 )- 1 ).
ir
1 E Aut(X), then 9a = fa(f- 1 ), which proves that 9 is a
Thus if we let f =
D
coboundary and completes the proof of the proposition.
As the terminology suggests, there is a cohomology theory underlying Proposition 4.77.
Definition. Let r be a group that acts on another group A. A 1-cocycle (from r to A)
is a map
g : r ----> A
satisfying
9ar = 9ad9r)
for all a, T E r,
and a 1-coboundary (from r to A) is a map of the form
r----> A,
a f------+ fa(f- 1 )
for some f EA.
The cohomology set H 1 (f, A) is defined to be the collection of 1-cocycles r __.. A
modulo the equivalence relation that two 1-cocycles 91 and 92 are cohomologous if
the map 91 1 92 is a 1-coboundary.
Remark 4.78. If the group A is abelian, then H 1 (r, A) is itself an abelian group, and
in this situation it is possible to define cohomology groups Hn(r, A) for all n :2: 0.
For the general theory of group (and Galois) cohomology, with many important applications to class field theory, Diophantine equations, and arithmetic geometry, see
for example [97, 396, 410].
A slight elaboration of the proof of Proposition 4.77 shows that if Aut(X) is
abelian, then there is a well-defined one-to-one map
[Y]K f------+ (a f--+ iya(i¥ 1 )),
(4.47)
where iy : Y __.. X is a k -isomorphism; see Exercise 4.42. In some cases, for
example when X is an algebraic variety, the map (4.47) is an isomorphism.
Twist(XIK)----> H 1 (Gal(KIK),Aut(X)),
4.9. Twists of Rational Maps
4.9
203
Twists of Rational Maps
According to Proposition 4.77, every twist corresponds to a 1-cocycle, and a twist
is trivial if and only if its 1-cocycle is a 1-coboundary. The natural question that
arises is whether every 1-cocycle comes from a twist. It turns out that the answer
depends on the category from which the objects are being chosen. For example, in
the category of algebraic varieties, every 1-cocycle does come from a twist. We will
not prove this general result, but in the next section we treat the case of twists of lP' 1 .
In this section we describe what happens for rational maps.
Definition. We define an action of the Galois group Gal(K I K) on the group of
linear fractional transformations PGL 2 (K) in the natural way, thus
a(!) =a (az +b) = a(a)z + a(b)
cz + d
a(c)z +a( d)
az + b
for f = - - d E PGL2(K) and a E Gal(KIK).
cz +
The automorphism group Aut(¢) of a rational map ¢ is a subgroup of the full
group PGL 2 of linear fractional transformations. Thus ifwe have a 1-cocycle
g: Gal(KIK)-----+ Aut(¢)
with values in Aut(¢), we can compose it with the inclusion of Aut(¢) into PGL 2
to obtain a 1-cocycle
Gal(KIK) ~Aut(¢)'----+ PGL 2(K)
(4.48)
with values in PGL 2(K). This way of extending g provides the key to describing
which 1-cocycles correspond to actual twists of¢.
Theorem 4.79. Let ¢(z) E K(z) be a nonzero rational map and let
g: Gal(KIK)-+ Aut(¢)
be a 1-cocycle with values in Aut (¢ ). Then the following are equivalent:
(a) There is twist of ¢1 K whose 1-cocycle is g.
(b) The 1-cocycle g becomes a 1-coboundary when it is extended to a 1-cocycle
with values in PGL2(K).
Hence in cohomological terms, the set of twists of¢ I K is given by
Twist(¢1K) = {~ E H 1 (Gal(KIK),Aut(¢)):
~becomes trivial in H 1 (Gal(K I K), PGL2(K)) }·
Proof Suppose first that there is a twist¢! of ¢1 K whose 1-cocycle is g. Then by
the definition given in Proposition 4.77, the 1-cocycle g is given by gcr = fa(f- 1 ).
4. Families of Dynamical Systems
204
Hence g is the PGL 2 (R) 1-coboundary associated to the element f E PGL 2 (R).
This proves that (a) implies (b).
Next suppose that g is an Aut(¢) 1-cocycle that is a PGL 2 (K) 1-coboundary.
This means that there is an f E PGL 2 (K) with the property that 9u = fa(f- 1 ) for
all a E Gal(R I K). We claim that ¢f is a twist of 4>1 K with 1-cocycle g. What we
need to check is that ¢f is defined over K, i.e., that ¢f E K(z), since once we know
that, it is clear from the definitions that g is its associated 1-cocycle.
Let a E Gal(R I K). Then
<j>9a = 4>
since we are given that 9u E Aut (4>),
<J>gu = 9u4>
by definition of ¢9",
4>fa(r 1 ) = fa(r 1 )4>
r
1 4>f
= a(r 1 )¢a(f)
since 9u = fa(r 1 ) by assumption,
multiplying by r l and by a(!),
4>! = <j>u(f)
since a(r 1 ) = a(f)-I,
¢f = a( 4> )"(f) = a( ¢f)
since ¢(z) E K(z), so a(¢)=¢.
We have proven that ¢f = a( ¢f) for all a E Gal( R I K), which shows that ¢f (z) E
K(z) and thus completes the proof that (b) implies (a).
In order to prove the cohomological description of Twist (4> I K), we first use
Proposition 4.77, which says that there is a natural embedding of Twist( 4>1 K)
into the cohomology set H 1 ( Gal(R I K), Aut( 4>)) (cf. also Remark 4.78). Then the
equivalence of(a) and (b) tells us that an element of H 1 ( Gal(K I K), Aut( 4>)) comes
from an element of Twist( 4>1 K) if and only if it becomes trivial when mapped to
D
H 1 (Gal(KI K), PGL 2 (K)).
Remark 4.80. A formal argument with commutative diagrams shows that
where Br(K) is the Brauer group of K. The Brauer group plays an important role
in class field theory and many other areas of number theory and arithmetic geometry. For example, Br(Qp) = QIZ, so Br(Qp)[2] has only two elements, and the
same is true for finite extensions of Qp. The Brauer group of a number field is more
complicated.
Example 4.81. Let K be a field of characteristic not dividing n, and let ¢(z) E K(z)
be a rational map whose automorphism group is
Aut(¢)= { (z: ( E 1-Ln},
where we recall that 1-Ln c R denotes the set of nth roots of unity. Then there is an
isomorphism
K* IK*n ~ Twist(¢1K),
(4.49)
4.9. Twists of Rational Maps
205
Note that by assumption we have ¢(z) = (- 1 ¢((z) for all ( E #Ln· In particular, the function z- 1¢(z) is invariant under the substitution z --+ (z, so it has the
form ¢(z) = z'ljJ(zn) for some 7/J(z) E K(z) (Exercise 4.37). Thus the twist of ¢(z)
given in (4.49) is equal to z'ljJ(bzn), so it is indeed in K(z).
Since we are assuming that the automorphism group of¢ is isomorphic to ILn•
not merely as an abstract group, but also in terms of the way in which Gal( k I K)
acts on Aut(¢) and on ILn• a standard result in Galois cohomology9 says that there
is an isomorphism
b 1-------t
(0'
f-+
0' (y'b) ) .
y'b
(4.50)
This allows us to identify Twist (¢I K) with a subset of K* I K* n. In order to show
that they are isomorphic, Theorem 4. 79 says that we must show that every cocycle in (4.50) becomes a coboundary when we consider it as a cocycle with values
in PGL 2(K). But this is clear, since with our identification of #Ln with Aut(¢), the
cocycle in (4.50) is equal to
z
with f(z) = y'b E PGL2(K).
Note that this example generalizes Example 4.75, which dealt with the case n = 2.
Example 4.82. Let ¢(z) E K(z) be a rational map with automorphism group
Aut(¢)= {z,z- 1 }. (See Exercise 4.35.) The Galois group Gal(KIK) acts trivially on Aut(¢), so we have
The isomorphism is given explicitly by associating to any bE K* / K* 2 the cocycle
(4.51)
To ease notation, we let j3 = -/b and let g be the cocycle described by (4.51 ).
Then Theorem 4. 79 says that g comes from a twist of ¢ if and only if there is
somef E PGL 2(K) satisfyingg,. = jO"(f- 1 ). Usingthefactthatg,.(z) E {z,z- 1 },
we are looking for an f E PGL 2(K) satisfying
9 This statement is equivalent to H 1 (Gal( k / K), K*) = 0, which is a version of Hilbert's Theorem 90. Using this and taking Galois invariants of the Kummer sequence
1 ---+ 11-n ---+ K
yields the cohomology long exact sequence
* O!f---+.O!n
---+ K
*
---+ 1
4. Families of Dynamical Systems
206
a(f)(z) =
{
f(z)
11 f(z)
if a(j3) = j3,
if a(j3) = -j3.
It is not hard to construct such an f, for example
f(z)= j3z+1
-j3z + 1
Note that det( !f3 ~) = 2j3 =1- 0, so f is in PGL 2 (K). Hence every g gives a twist
of¢, and indeed we can write the twist associated to b = j3 2 E K* I K* 2 explicitly
using f,
f
¢ (z) =
j3¢ (
_A.. (
'1-'
!;::1) + j3 .
f3z+1 ) + 1
-f3z+1
For example, let Md (z) = zd be the rfh-power map. Then a judicious use of the
binomial theorem yields a formula for the b-twist Mdb) of Md,
Mdb)(z) =
2: (2dk)bkz2k
k
/I: (2k ~
1)bkz2k+l .
k
In particular, the first few b-twists of Md are
M(b)(z) = 1 + 3bz 2 ,
3
3z + bz 3
4.10
Fields of Definition and the Field of Moduli
If¢( z) is a rational map whose coefficients lie in a field K, then it is interesting to
study the orbits of points whose coordinates lie inK or in finite extensions of K.
The smallest field K containing the coefficients of ¢ can generally be determined
by inspection. For example, the map ¢(z) = z 3 + 1 is clearly in Q(z) and the
map ¢(z) = z 3 + i is just as clearly in Q(i)(z). However, ifwe make a change of
variables f(z) = iz in the latter map ¢(z) = z 3 + i, we find that
¢! (z) = f- 1 (¢(f(z))) = -i¢(iz) = -i( -iz 3 + i) = -z 3 + 1 E Q(z).
Thus the map ¢(z) = z 3 + i is really a Q(z) map that has been altered by an
injudicious change of variables.
Definition. Let K be a field of characteristic 0, let ¢( z) E K (z) be a rational map
with coefficients in an algebraic closure of K, and let K' I K be an extension field.
We say that K' is afield ofdefinition for¢ if there is a linear fractional transformation
f(z) E PGL2(K) such that
¢! = f- 1 ¢f E K'(z).
In other words, K' is a field of definition for ¢ (or an FOD for short) if, after a change
of variables, ¢ has coefficients in K'.
4.10. Fields of Definition and the Field of Moduli
207
As in Section 4.9, we can use Galois theory to investigate the possible fields of
definition of a given rational function ¢( z) E k (z). We let Gal( k I K) act on k (z)
in the natural way by applying <J E Gal (k I K) to the coefficients of¢ E k (z),
<J(¢) = <J (ao + a1z + · · · + adzd) = <J(ao) + <J(al)z + · ·. + <J(ad)zd
bo + b1z + · · · + bdzd
<J(bo) + <J(bl)z + ... + <J(bd)zd ·
Then Galois theory (and Hilbert's Theorem 90) tell us that
¢ E K' (z)
for the field
K' = (fixed field of{ <J E Gal(k I K) : <J( ¢) = ¢}).
Suppose instead that ¢ is merely equivalent to a map whose coefficients are
in some field K'. That is, suppose that there is a linear fractional transformation
f E PGL 2(K) such that¢! E K'(z). Then<J(¢f) = ¢! forall<J E Gal(KIK'),so
f-1¢! = ¢! = <J(¢f) = <J(f-1¢!) = <J(j-1)<J(¢)<J(f) = <J(j)-1<J(¢)<J(f).
Solving for <J( ¢) yields
<J(¢) = <J(f)f-1¢j<J(f-1) = (/<J(f-1))-1¢(/<J(f-1)) = ¢fa(f-1).
Thus <J(¢) is equal to¢ conjugated by the map j<J(f- 1) E PGL 2(K), so in particular <J( ¢) and ¢ are equivalent. We have proven that
( K' is a field of)
definition for ¢
(for every <J E q_al(k I K') there exists )
===> a 9a E PG 1 2( K) such that <J (¢) = ¢9" ·
(4.52)
Turning this around, we start with an arbitrary rational map ¢( z) E k (z) and
study the automorphisms <J E Gal( k I K) for which <J( ¢) is equivalent to ¢.
Definition. Let rjJ(z) E K(z) be a rational map. We associate to rjJ a subgroup G¢
ofGal(K I K) and a field K¢ defined by
Gq, = {<J E Gal(KIK): <J(¢) = ¢ 9 " forsomega E PGL2(K)},
Kq, =fixed field ofGq, = {a E R: <J(o:) =a for all <J E Gq,}.
The field Kq, is called the .field of moduli (FOM) of¢.
Remark 4.83. The group PGL 2(K) acts on the space of rational maps via the usual
conjugation action,¢! = f- 1¢f. If we consider as usual the equivalence class
consisting of all maps that are conjugate to a particular map ¢, then G q, is the subgroup of Gal(k I K) consisting of elements that map the set [¢] to itself. Equivalently, if¢ has degree d, let (¢) be the image of¢ in the moduli space Md =
Ratd I PSL 2 that we defined and studied in Section 4.4. The space Md is defined
over Q (Theorem 4.36) and the field of moduli of ¢ is exactly equal to the field
generated by the coordinates of the point(¢) E Md.
4. Families of Dynamical Systems
208
We begin by proving some elementary properties ofFOM and FOD, in particular,
the important fact that the field of moduli is contained in every field of definition.
Proposition 4.84. Let ¢(z) E K(z).
(a) The set G¢ is a subgroup ofGal(K I K).
(b) Let K' be a field of definition for ¢. Then K¢ ~ K'. Informally, we say that
FOM ~ FOD.
Proof The proof of this proposition is simply a matter ofunsorting definitions. Thus
let O", T E G¢. Then
Thus (O"T) ( ¢) is equivalent to ¢, so O"T E G¢. Similarly,
which shows that 0"- 1 ( ¢) is equivalent to¢, and hence that 0"- 1 E G¢. This proves
that G¢ is a subgroup of Gal( k I K).
Next let K' be a field of definition for ¢. Under this assumption, we proved
earlier (4.52) that for every O" E Gal(K I K') there is a 9a E PGL2(K) such that
O"( ¢) = rjJ9a. In other words, O"( ¢) is equivalent to¢, so O" E G¢ = Gal(K I K¢) by
definition. This proves that Gal(K I K') c Gal(K I K¢), and hence by Galois theory,
D
that K¢ c K'. (We are also using the fact that K¢ is a finite extension of K.)
It follows from Proposition 4.84 that the smallest possible field of definition for 4>
is the field of moduli of¢, but it is not clear whether the field of moduli is always a
field of definition. The following example shows that we can have FOM "/=- FOD.
Example 4.85. Let
¢(z) = i
3
(~)
z+1
Clearly Q( i) is a field of definition for ¢. Let O" be complex conjugation, so
Gal(Q(i)IQ) = {1, O"}, and let g(z) = -1lz. Then we obtain
1
i(-1lz-1)
-1lz+1
= i (-1lz +
1)3 = (~)3 =
-1lz-1
-i
z+1
3
O"(¢)(z).
This shows that O" E G¢, so G¢ = {1, O"} and K¢ = Q. In other words, Q is the field
of moduli of¢.
Now suppose that Q is actually a field of definition for ¢. This means that we
can find some f E PGL 2(Q) such that ¢1 E Q(z). In particular, letting O" denote
complex conjugation, we have
209
4.10. Fields of Definition and the Field of Moduli
It is not hard to verify (see Exercise 4.39) that Aut(¢) = 1, so we deduce that
f = ga(f). Using the fact that g(z) = -11 z, this equation says that
az + b
cz + d
cz + d
- az + b'
where we use an overscore to denote complex conjugation. The fact that these two
linear fractional transformations are equal means that there is a .\ E Q* such that
a = - .\c,
b = - .\d,
c = .\a,
d = .\b.
Multiplying the first and third equations gives
This is a contradiction, since.\ f. 0 and we cannot have a = c = 0. Hence Ql is not
a field of definition for ¢, so we have an example of a map with FOM f. FOD.
In order to investigate more closely the question of when the field of moduli of a
rational map ¢ is a field of definition, we make the simplifying assumption that
Aut(¢)= 1.
(4.53)
Replacing K by Kt>, we may assume that K is the field of moduli of¢. This means
that for every a E Gal( K I K) there exists a 9a E PG L2 (K) satisfying
Note that the assumption (4.53) that Aut(¢) = 1 implies that 9a is uniquely determined by a. The next proposition describes some of the properties of the map
a 1--+ 9a·
Proposition 4.86. Let ¢ E K (z) be a rational map of degree d ;::: 2 with field of
moduli K and satisfying Aut(¢)= 1, andfor each a E Gal(KIK) write a(¢)=
¢ 9 " as above.
(a) Themap
Gal(K I K) ----> PGL2(K),
is a 1-cocycle, i.e., it satisfies
for all a, T E Gal(K I K).
(b) K is afield ofdefinition for¢ if and only if g is a 1-coboundary, i.e., if and only
if there is an f E PGL2(K) such that
for all a E Gal(K I K).
4. Families of Dynamical Systems
210
Proof (a) Let a, T E Gal(k I K). Then
¢9ar = a(r(¢)) = a(¢9-r) = a(¢t(9r) = cjJ9a17(9r).
Since Aut(¢) = 1, we conclude that g177 = g17 a(g7 ).
(b) Suppose first that K is a field of definition for ¢. This means that there is an
f E PGL2(k) such that¢! E K(z). Hence for any a E Gal(K I K) we have
¢! = a(¢f) = a(cp)u(f) = ¢9au(f)_
Again using the fact that Aut(¢) = 1, we find that f = g17 a(f), and hence g17 =
fa(f- 1).
Conversely, suppose that there is an f E PGL2(K) such that g17 = fa(f- 1 ).
Then
a(cpf) = a(¢tU) = ¢9au(f) =¢f.
This is true for every a E Gal( k I K), so ¢! E K (z). Hence K is a field of definition
D
~~
The criterion for FOM = FOD given in Proposition 4.86 may seem complicated,
but it represents a tremendous simplification. If we try to use the definition of FOD
directly, we need to search for an f E PGL 2(K) that makes the coefficients of¢!
lie in K. The substitution
¢! (z) =
(az+b) _ b
'
cz+d
+a
-cA. (az+b)
cz+d
d¢
'f'
even for a rational function ¢( z) of small degree, has coefficients that are very complicated expressions in the quantities a, b, c, d. It is thus difficult to determine whether
there is some choice of a, b, c, d that makes the coefficients lie in K. On the other
hand, the cocycle-coboundary criterion
FOM = FOD
{:=::?
g17 has the form fa(f- 1)
in Proposition 4.86 involves only linear functions, i.e., elements ofPGL 2(K), so it
is often considerably easier to apply.
The 1-cocycles that arise in the FOD = FOM question have the form
g: Gal(KIK) ______, PGL 2(K).
The group PGL 2(K) is the automorphism group of the projective line IP' 1 , so these
cocycles should be associated to twists of!P' 1 . We saw in Section 4.8 that every twist
gives rise to a 1-cocycle, but in general it is a delicate question to determine whether
every 1-cocycle comes from a twist. It turns out that this is true for algebraic varieties,
but since we do not need the most general result, we are content to construct the
twists of IP' 1 that are needed to answer our question about FOD = FOM. (See also
Example 4.87.)
4.10. Fields of Definition and the Field of Moduli
211
Proposition 4.87. Let
g: Gal(KIK) ~ PGL2(K)
be a 1-cocycle, and assume that g has the property that there is a finite extension
L I K such that 9u = 1 for all a E Gal( k I L ). 10 Then there is an algebraic curve C
defined over K and an isomorphism i : C ---t JID 1 defined over k such that the 1cocycle
Gal(KIK) ~ PGL2(K),
a~ ia(i- 1 ),
is equal to the 1-cocycle g. Hence C is a twist ofJID 1 I K, and
C is the trivial twist ofJID 1I K
if and only if
g is a 1-coboundary.
Proof We construct the curve C by describing its field of rational functions. Note
that the field of rational functions for the curve JID1 is the field k (z) and that the
Galois group Gal(K I K) acts naturally on K(z) by acting on k and leaving z fixed.
Now consider another field of rational functions in one variable K(w), but this
time with a "twisted action" of Gal(kI K). We define this twisted action by letting Gal(K I K) act on k as usual, but setting
a(w) = g;; 1(w).
In other words, if 9u(z) = (az + b)l(cz +d), then
dw-b
a(w) = g;; 1(w) =
,
-cw+a
and for any rational function
F(w) = ao + a1w + a2w 2 + · · · + adwd E K(w),
bo + b1 w + b2w 2 + · · · + bdwd
we have
a(F)(w) = a(ao) + a(al)g; 1(w) + a(a2)g; 1(w) 2 + ... + a(ad)g;l(w)d.
a(bo) + a(b1)g,; 1(w) + a(b2)g,; 1 (w) 2 + · · · + a(bd)g,; 1(w)d
It is clear that a(F +G) =a( F)+ a( G) and that a(FG) = a(F)a(G). However,
to be an action, we must also have (ar) (F) = a( r( F)). We use the cocycle relation
to verify this condition,
a(r(w)) = a(g; 1w) = a(g; 1)a(w) = a(g; 1)g;; 1(w) = g;;}(w) = (ar)(w).
We now look at the subfield of k (w) that is fixed by the twisted action,
K, ={FE K(w): a(F) = F for all a E Gal(KIK)}.
We prove that the field K, has the following properties:
10 0ne says that g is a continuous 1-cocycle for the profinite topology on Gal(K / K) and the discrete
topology on PGL2(K).
4. Families of Dynamical Systems
212
(i) K n k = K.
(ii) K has transcendence degree 1 over K.
(iii) kK = K(z).
To verify (i), let a E K n k. Then a E k, so the action of Gal(K I K) on a is the
usual Galois action. On the other hand, the fact that a E K means that the Galois
action is trivial. Hence a E K. This shows that K n k c K, and the opposite
inclusion is obvious.
Next let Ll K be a finite Galois extension with the property that gT = 1 for every
T E Gal( k I L). This implies that the action of an element (} E Gal( k I K) on w
depends only on the image of(} in the quotient group
Gal(LIK) = Gal(KIK)IGal(KIL),
since ifT E Gal(k I L ), then
((JT)(w) = (J(gT(w)) = (J(w).
It follows that the polynomial
P(T) =
II
II
(T- A(w)) =
.\EGal(L/K)
(T- g): 1 (w)) E k(w)[T]
(4.54)
.\EGal(L/K)
is well-defined. Further, we claim that the coefficients of P(T) are in K. To see this,
for any(} E Gal(K I K) we note that
(J(P)(T) =
II
(T- (JA(w)) = P(T),
.\EGal(L/K)
since the effect of replacing A with (}A is simply to permute the order of the factors
in the product.
We have now constructed a polynomial P(T) E K[T], and we observe that w is a
root of P(T). Hence by definition, the extension K(w)IK is an algebraic extension.
Since the element w is transcendental over K, this proves that K has transcendence
degree at least 1 over K. On the other hand, K is contained in the field k (w), and
it is clear that k (w) I k has transcendence degree 1 over K. Therefore K has transcendence degree exactly 1 over K, which proves (ii).
Finally, consider the splitting field£ over K of the polynomial P(T). As already
noted, we have w E £.The definition of P(T) shows that there is a natural surjection
Gal(LI K) -+ Gal(£1K), which implies in particular that£= LK. Hence w E LK,
which proves that k (z) c k K. This gives (iii), since the other inclusion is true from
the definition of K.
We now have the tools needed to complete the proof of Proposition 4.87, but we
pause briefly for an example illustrating the general construction.
4.10. Fields of Definition and the Field of Moduli
213
Example 4.88. Define a map g by the rule
z
if a(i) = i,
ga(z)= { - 11 z 1"f a (")z - -z..
It is easy to check that g is a 1-cocycle, and indeed it is the 1-cocycle described in
Example 4.85. Fix an embedding ofQ into C and let p E Gal(QIQ) denote complex
conjugation. The twisted action on Q( w) is given by
p( a) = a for a E Q
and
p(w) = -1lw.
The field K is the subfield of Q( w) consisting of elements that are fixed by the
twisted action. The coefficients of the polynomial P(T) defined by equation (4.54)
give us elements in K,
P(T) = (T - w )(T - p( w)) = (T - w) ( T +
~) = T
2 -
(
w - ~) T - 1.
This yields one interesting element inK, namely u = w- w- 1 , and we observe that
the quantity v = i( w + w- 1 ) gives another element inK. It is not hard to show that u
and v generate K,
K = K (u, v) = K ( w -
~, i ( w + ~)) .
(See Exercise 4.45.) Of course, u and v are not independent, since
u 2 + v2 = ( w -
~) + ( i ( w + ~)) = -4.
2
2
The field K is the function field of the curve C : u 2 + v 2 = -4. Notice that C is
defined over Q, and Cis Q-isomorphic to IP' 1 , but Cis not Q-isomorphic to IP' 1 , since
C(Q) = 0. From our general theory, the fact that C is a nontrivial twist of IP' 1 IQ
is equivalent to the fact (proven by a direct calculation in Example 4.85) that the
1-cocycle g is not a 1-coboundary.
Resuming the proof of Proposition 4.87, we have constructed a field K that is the
function field of a curve C I K, and we have an isomorphism
kK = K(w) ~ K(z),
w~z,
that induces a k -isomorphism i : C ----> IP' 1 . In other words, the functions w on C
and z on IP' 1 are related by the formula w = z o i. The curve C is a twist of IP' 1 I K,
and its associated cocycle is given by a----> ia(i- 1 ). We compute
4. Families of Dynamical Systems
214
z o a( i) = a( z o i)
= a(w)
= g;;1(w)
dw-b
-cw+a
dz o i - b
-cz o i +a
= Z 0 ga-1 0 z..
since a(z) = z,
since w = z o i,
by definition of the twisted action on f< (w),
az + b
cz+
-
where ga(z) = - - d E PGL2(K),
since w = z o i,
Thus a(i) = g-; 1 o i, which proves that ga = ia(i- 1 ) is the 1-cocycle associated to
the f< -isomorphism i : C __... lP' 1 .
This completes the proof that the algebraic curve C is a twist of lP' 1 I K whose
associated 1-cocycle is g. Finally, Proposition 4.77 tells us that Cis the trivial twist
D
if and only if its associated 1-cocycle is a 1-coboundary.
Returning to the question ofFOD = FOM, let ¢(z) E f<(z) be a rational function with field of moduli K and trivial automorphism group. We have constructed a
1-cocycle
g: Gal(KIK)----+ PGL2(f<)
that is associated to ¢ (Proposition 4. 86) and a twist C,p of lP' 1 I K that is associated
to the 1-cocycle g (Proposition 4.87). We have also proven the following chain of
equivalences:
K is a field of definition for </>
~ g is a 1-coboundary
(Proposition 4.86),
~ Cq, is K-isomorphic to lP' 1
(Proposition 4.87).
It remains to find a way of determining whether a twist of lP' 1 is K -isomorphic
to lP' 1 . We will use the Riemann-Roch theorem to provide two sufficient conditions
for resolving this problem. For the convenience of the reader, we recall the general
statement of the Riemann-Roch theorem, although we will need it only for curves of
genus 0.
Theorem 4.89. (Riemann-Roch Theorem) Let C I K be a smooth projective curve
ofgenus g defined over K.
(a) There is a divisor on C of degree 2g- 2 that is defined over K.
(b) Let D be a divisor on C that is defined over K and assume that the degree of D
satisfies deg(D) 2 2g- 1. Then there is a/unction f E K(C) satisfying
div(f) + D 2 0.
Corollary 4.90. (Riemann-Roch in Genus 0) Let C I K be a smooth projective
curve ofgenus 0 defined over K.
(a) There is a K -rational divisor of degree 2 on C.
4.10. Fields of Definition and the Field of Moduli
21S
(b) Let DE Div(C) be a divisor defined over K and satisfying deg(D) ~ 1. Then
there is a.function f E K(C) with div(f) + D ~ 0.
Proof. The proof of the Riemann-Roch theorem over an algebraically closed field
is given in most introductory texts on algebraic geometry, such as [198, IV.l.3], or
see [255, Chapter I] for an elementary proof due to Weil and [410, II, §5] for an
overview. The divisor of degree 2g- 2 in (a) is a canonical divisor, i.e., the divisor of
any K -rational differential form such as df for any nonzero f E K (C). In particular,
for curves of genus 0 we get a K -rational divisor D of degree -2, so - D is a Krational divisor of degree 2.
D
Proposition 4.91. Let C be a twist ofJJ!' 1 I K. The following are equivalent:
(a) Cis the trivial twist of Jf!'1 I K, i.e., Cis K -isomorphic to Jf!'1 .
(b) C(K) is nonempty, i.e., C has a point with coordinates inK.
(c) There is a divisor D = L ni (Pi) on Jf!'1 ( K) ofodd degree such that D is defined
over K, i.e.,for all a E Gal(KIK) we have L:ni(a(Pi)) = L:ni(Pi) as a
formal sum ofpoints.
Proof. If C is the trivial twist of Jf!'1 I K, then there is a K -isomorphism j : C ---t Jf!'1 .
In particular, j : C (K) ---t Jf!'1 ( K) is a bijection, so C (K) is certainly nonempty.
This proves that (a) implies (b).
Next, it is clear that (b) implies (c), since if P E C(K), then the divisor D = (P)
has odd degree (one is an odd number!) and is clearly defined over K.
Finally, suppose that the degree n = deg(D) = L ni of D = L ni(Pi) is odd
and that Dis defined over K. We also use the fact that C and Jf!' 1 are k-isomorphic,
so in particular C is a curve of genus zero. It follows from the Riemann-Roch theorem (Corollary 4.90(a)) that there is a K-rational divisor on C having degree -2,
say
Consider the divisor
The divisor E is defined over K and has degree 1, so the Riemann-Roch theorem
(Corollary 4.90(b)) tells us that there is a rational function '1/J on C such that '1/J is
defined over K and '1/J has degree 1. In other words, '1/J is a map '1/J : C ---t Jf!' 1 of
degree 1 defined over K, and hence C is K- isomorphic to Jf!'1 . This shows that (c)
implies (a) and completes the proof of the theorem.
D
We have now assembled all of the tools that we need to prove the main theorem
of this section. We state the theorem in full generality, but give the proof only for
rational maps with trivial automorphism groups. The general case is proven similarly,
but there are many additional technical complications.
Theorem 4.92. Let cp( z) E K (z) be a rational map ofdegree d ~ 2. Then the field
ofmoduli of cp is a field ofdefinition for cp in the following two situations:
4. Families of Dynamical Systems
216
(a) ¢(z) has even degree.
(b) ¢( z) is a polynomial.
Proof We prove the theorem under the assumption that Aut(¢)= 1. See [414] for
a proof in the general case.
Without loss of generality, we may assume that K is the field of moduli of¢. Let
g : Gal( K I K) ----+ PGL 2 ( K) be the 1-cocycle associated to¢ (Proposition 4.86) and
let Cq, be the twist of lP' 1 I K associated to the 1-cocycle g (Proposition 4.87). This
means that there is a K -isomorphism
such that
ga = ia( C 1 )
for all(} E Gal(K I K).
Using many of the results proven in this chapter, we have the following chain of
equivalences:
K is a field of definition for ¢
{:::=:} g is a 1-coboundary
(Proposition 4.86),
{:::=:} Cq, is K -isomorphic to lP' 1
(Proposition 4.87),
Cq,(K) is not empty
(Proposition 4.91),
(There is a divisor on Cq,(K) of)
odd degree and defined over K
(Proposition 4.91 ).
{:::=:}
{:::=:}
We are going to produce divisors and points on the curve Cq, using the map
7/J : Cq, ----+ Cq, defined by the composition
We begin by checking that the map '1/J = i- 1 ¢i is defined over K. To verify this, we
let a E Gal(K I K) and compute
a('I/J) = a(C 1 ¢i) = a(i- 1 )a(¢)a(i) = a(i- 1 )¢9 "'a(i)
= a(i- 1 )g;; 1 ¢gaa(i) = a(i- 1 )(ia(C 1 ))- 1 ¢ia(i- 1 )a(i) = i- 1 ¢i = 7/J.
Hence 7/J : Cq, ----+ Cq, is defined over K. We also note that¢ and '1/J have the same
degree, since i is an isomorphism.
(a) Let D,p be the divisor of fixed points of '1/J, that is, the collection of fixed points
of 7/J counted with appropriate multiplicities. In the language of algebraic geometry,
D,p is the pullback by the diagonal map
Cq, ~ Cq, X Cq,,
P 1----t (P, P),
of the graph {(x, 7/J(x)): x E Cq,} of'ljJ.
A map of degree d has exactly d + 1 fixed points (counted with multiplicities),
and if the map is defined over K, then the divisor of fixed points is defined over K,
i.e., it is fixed by the Galois group. Thus D,p is a divisor of degree d + 1 on Cq,
4.10. Fields of Definition and the Field of Moduli
217
and D'I/J is defined over K. By assumption, dis even, so D.p has odd degree. Hence
by the chain of equivalences derived earlier, the field K is a field of definition for ¢.
(b) The map ¢ : lP' 1 -+ lP' 1 is a polynomial by assumption, so it has a totally ramified
fixed point P. In other words, ¢(P) = P and the ramification index of¢ at P satisfies
e p ( ¢) = d. The map i : C¢ -+ lP' 1 is an isomorphism, so the point
is a totally ramified fixed point of'¢. Suppose first that Pis the only fixed point of¢
in lP' 1 ( K) with ramification index d. Then Q is the only fixed point of 'lj; in C¢ ( K)
with ramification index d. However, for any a E Gal(K / K) and any point R E
C¢(K) we have
'lj;(a(R)) = a('lj;)(a(R)) = a('lj;(R))
and
eR('¢) = ecr(R)(a('¢)) = ecr(R)('¢),
since 'lj; is defined over K. In particular, taking R = Q to be the given fixed point of
ramification index d, we see that '¢(a( Q)) = a( Q) and d = eQ ('lj;) = ecr(Q) ('lj; ),
so a(Q) is also a fixed point of 'lj; of ramification index d. Hence a(Q) = Q,
and since this holds for every a E Gal(K / K), we conclude that Q E C¢(K).
Hence C¢(K) is not empty, so we again conclude that K is a field of definition
for¢.
We are left to consider the case that ¢ has a second fixed point of ramification
index d, say¢( P') = P' and ep' ( ¢) = d. (Note that the Riemann-Hurwitz formula,
Theorem 1.1, precludes more than two such points.) Choose some linear fractional
transformation h E PGL 2(K) satisfying h(oo) = P and h(O) = P'. Then ¢h
satisfies
so ¢h must have the form
¢h(z) = czd for some c E K*.
But any rational map of this form has a nontrivial automorphism group. Indeed, its
automorphism group is a dihedral group of order 2 (d - 1) generated by the maps
z ~------+ (z
and
z ~------+ a/ z,
where (d-l = 1 and ad-l = 1/c. In any case, we have ruled out this case by our
D
assumption that Aut(¢) = 1, which completes the proof of Theorem 4.92.
Remark 4.93. The distinction between the field of moduli and fields of definition is important in the study of abelian varieties; see for example the work of
Shimura [398]. More recently, the FOM = FOD question has been investigated for
the collection of of covering maps¢ : X -+ B up to automorphism of the base B. In
particular, this is much studied forB = lP' 1 (cf. Grothendieck's "dessins d'enfant").
If also X= B = lP'\ then one studies the set of rational maps ¢(z) E K(z) under the left composition equivalence relation¢ rv f¢ for f E PGL2(K). This bears
218
4. Families of Dynamical Systems
considerable resemblance to the material in this section, where we use instead the
equivalence relation¢ "' f- 1 ¢f, but there are significant differences. For example,
Couveignes [111] shows that using the relation equivalence¢ "' f ¢, there are polynomials in Q[z] with FOM -=f. FOD, in direct contrast to Theorem 4.92. For further
results, see for example work of Debes and Douai [ 117, 118, 119] and Debes and
Harbater [120].
4.11
Minimal Resultants and Minimal Models
Let P(X, Y) = 2.:: ai 1xiyi E K[X, Y] be a polynomial and let p be a prime of K.
We define the order of P at p to be
Notice that ordp (P) = 0 if and only if all of the coefficients of¢ are p-integral and
at least one coefficient is a p-unit.
In Section 2.4 we defined the resultant of a rational map ¢ (with respect to p)
by writing¢= [F, G] using homogeneous polynomials F, G E K[X, Y] satisfying
min { ordp (F), ordp (G)} = 0 and setting
Resp(¢) = Res(F, G).
The resultant Resp (¢) is well-defined up to multiplication by the 2tfh power of a
p-adic unit, so in particular ordp (Resp ( rjJ)) depends only on rjJ and p. We also recall
that ¢ has good reduction at p if and only if its resultant is a p-adic unit.
Example 4.94. Even if¢ has bad reduction at p, it may be possible to change coordinates and achieve good reduction. In other words, there may be some f E PGL2 ( K)
such that Res(¢!) is a p-unit. For example, the map ¢(z) = z + p 2 z- 1 has bad reduction at p, since
However, if we let f (z) = pz, then ¢! (z) = z + z- 1 , which has good reduction at p.
Our aim in this section is to study this phenomenon. In particular, we study the
extent to which we can eliminate, or at least ameliorate, bad reduction in ¢ by conjugating¢ with a linear fractional transformation in PGL 2(K).
Definition. Let¢ = [F, G] be a rational map given by homogeneous polynomials F, G E K[X, Y], let f E PGL 2 (K), and choose a matrix
representing f. We define polynomials FA, G A E K [X, Y] by the formulas
4.11. Minimal Resultants and Minimal Models
219
(4.55)
FA(X, Y) = oF(o:X +,BY, 1 X + oY)- ,BG(o:X +,BY, 1 X + oY),
GA(x, Y) = - 1 F(o:x +,BY, 1 x + oY) + o:G(o:x +,BY, "x + oY). (4.56)
It is often convenient to write this in matrix notation as
[
where Aadj = ( .!'"'~
~~] = A adj [~ : ~] ,
-!) is the adjoint matrix to A. Note that the conjugate cpf =
f- 1 o ¢of of¢ by f is equal to
cpf (X, Y) = [FA(X, Y), GA(X, Y)].
Proposition 4.95. Let¢= [F, G] be a rational map ofdegree d described by homogeneous polynomials F, G E K[X, Y] and let p be a prime ideal.
(a) The valuation of the minimal resultant of¢ is given by the formula
ordp (Resp (¢)) = ordp (Res( F, G)) - 2d min { ordp (F), ordp (G)}.
(4.57)
Note that there is no requirement that the coefficients ofF and G be p-integral
or that some coefficient be a p-adic unit.
(b) Let A E GL2(K). Then with FA and GA defined by (4.55) and (4.56),
ordp (Res( FA, GA)) = ordp (Res(F, G)) + (d 2 +d) ordp (det A),
min{ordp(FA),ordp(GA)} ~ min{ordp(F),ordp(G)} + (d+ l)ordp(A),
where ordp(A) denotes the minimum of the order of the coordinates of the matrix A.
(c) In particular, ifU E GL2(Rp), then
ordp (Res(Fu, Gu)) = ordp (Res(F, G)),
min{ ordp (Fu ), ordp( Gu)} =min{ ordp (F), ordp (G)}.
Proof (a) Choose a constant c E K* satisfying
ordp (c) =min{ ordp (F), ordp (G)}.
Then
where we have made use of the homogeneity property of the resultant (Proposition 2.13(d)). This gives the desired result (4.57).
(b) An elementary calculation (see Exercise 2.7(c)) shows that
Res(FA,GA) = (detA)d 2 +dRes(F,G),
so taking ordp gives the first part of (b). For the second part, we observe that every
coefficient ofFA and G A is a sum of terms of the form
220
4. Families of Dynamical Systems
( coefficient ) x ( homogeneous polynomial of )
ofF or G
degree d + 1 in Z[a, ,8, ')', 8]
·
Hence
min{ ordp (FA), ordp(G A)} 2: min{ ordp (F), ordp(G)} + (d + 1) ordp(A).
(c) The assumption that U E G 1 2( Rp) is equivalent to the two conditions
ordp(U) 2: 0
and
ordp (det U) = 0,
i.e., the coefficients of U are p-adic integers and the determinant is a p-adic unit.
Applying (b) with A = U gives
ordp(Res(Fu,Gu)) = ordp(Res(F,G)),
min{ ordp(Fu ), ordp(Gu)} 2: min{ ordp (F), ordp(G) }.
(4.58)
This almost completes (c). We next apply (b) to the polynomials Fu and Gu and to
the matrix A = u- 1 E GL2(Rp). Using the fact that (Fu )u-1 = F and similarly
for G, we find from (b) that
min{ ordp (F), ordp(G)} =min{ ordp ((Fu )u-1 ), ordp((Gu )u-1)}
2: min{ ordp (Fu ), ordp(Gu) }.
This gives the opposite inequality to (4.58), which completes the proof of (c).
D
Definition. Let K be a number field and let cp(z) E K(z) be a rational map. For
each prime p of K, define a nonnegative integer by
Ep(¢) =
min
/EPGL2(K)
ordp Resp(¢/).
In other words, Ep (¢) is the exponent of the power of p dividing the resultant of the
conjugate ¢! that is closest to having good reduction at p. Then the (global) minimal
resultant of¢ is the integral ideal
(4.59)
We say that (F, G) is a minimal mode/for¢ at p if¢= [F, G], the coefficients ofF
and G are p-integral and
ordp Res(F, G) = ordp 9l¢.
(See Section 6.3.5 for the analogous definition of mimimal models of an elliptic
curve.)
Remark 4.96. The product (4.59) defining 9l¢ makes sense, since Ep(¢) = 0 for
all but finitely primes p. To see this, write ¢ = [F, G] for any F and G having
coefficients in the ring of integers of K. Then there are only finitely many primes p
with ordp Res(F, G) > 0, and it is clear that Ep (¢) = 0 for all other primes.
4.11. Minimal Resultants and Minimal Models
221
The minimal resultant of a rational map is clearly invariant under PGL 2 (K)conjugation. It measures the extent to which the conjugates of¢ have bad reduction,
so provides a convenient measure of the arithmetic complexity of the conjugacy class
of¢. A coarser way to measure arithmetic complexity is simply to take the product
of the primes with bad reduction, which we denote by
(4.60)
It is clear that IJ1q, divides 9\q,, and it is tempting to conjecture an inequality in the
opposite direction that would be a dynamical analogue of Szpiro's conjecture [205,
F.3.2] for elliptic curves.
Conjecture 4.97. Let K/Q be a number field and d 2: 2. There is a constant c =
c( K, d) such that for all rational maps ¢( z) E K ( z) of degree d,
NK;IQ 9\q,::; (NK;Q snq,t
The minimal resultant gives one way to measure the arithmetic complexity of a
rational map, but note that there are infinitely many PGL 2 (K)-inequivalent rational
maps of a given degree whose minimal resultants are the same. For example, the
minimal resultants of the polynomials
are the same as u ranges over all units in the ring of integers of K.
The moduli space Md provides an alternative way to measure the arithmetic
complexity of the conjugacy class of a rational map. If we fix a projective embedding Md <.......t IP'N, then¢ E K(z) determines a point(¢) E Md(K) and we can
take the height of the corresponding point in JP>N ( K). However, this way of measuring arithmetic complexity is also not entirely satisfactory, since twists of a rational
map give the same point in Md, yet are arithmetically quite different. Note that the
same situation arises in the theory of elliptic curves, where curves with the same jinvariant need not be arithmetically identical.
This suggests combining the primes of bad reduction with the height coming
from moduli space. We do this and formulate a dynamical version of a conjecture of
Lang (cf. [202], [254, page 92], or [410, VIII.9.9]). Recall that the canonical height
of a point P satisfies hq, (P) = 0 if and only if Pis preperiodic for <P (Theorem 3.22).
The following conjecture says that the height of nonpreperiodic points grows as <P
becomes more arithmetically complicated.
Conjecture 4.98. Fix an embedding ofthe moduli space Md in projective space and
let hMd denote the associated height.function. Let K be a number field and d 2: 2 an
integer. Then there is a positive constant c = c(K, d) such that for all rational maps
¢ E K(z) ofdegree d and all wandering (i.e., non-preperiodic) points P E lP' 1 (K),
4. Families of Dynamical Systems
222
In the theory of elliptic curves, the notion of global miminal Weierstrass equation
is extremely useful; see the discussion in Section 6.3.5 and [410, VIII, §8] for further
details. We briefly discuss a dynamical analogue.
Definition. Let K be a number field and let ¢(z) E K(z) be a rational map.
Then ¢ has a global minimal model if there is a linear fractional transformation f E PGL2(K) and homogeneous polynomials F and G satisfying¢!= [F, G]
with the property that the coefficients ofF and G are in the ring of integers of K and
for every prime p.
In other words, the pair ( F, G) is simultaneously a minimal model for ¢ at every
prime p of K.
In the remainder of this section we develop some tools that help to determine
whether a given rational map has a global minimal model.
Proposition 4.99. Let K be a number field, let ¢(z) E K(z) be a rational map of
degree d, and write¢ = [F, G] with polynomials F and Gas usual.
(a) There is a (fractional) ideal ap,c of K satisfYing
9\</> = {
Res(F, G)a¥,0
if dis odd,
Res(F, G)a},c
if dis even.
(4.61)
(b) If d is odd, then the ideal class of aF,G depends only on ¢, independent of the
choice ofF and G.
(c) If d is even, then aF,G depends only on ¢ up to multiplication by the square of a
principal ideal.
Proof For any f E PGL 2(K) with corresponding A E GL 2 (K), we use Proposition 4.95(a,b) to compute
ordp (Resp (¢f)) = ordp (Res( FA, G A)) - 2dmin{ ordp (FA), ordp (G A)}
= ordp(Res(F, G))+ (d 2 +d) ordp( det A)
- 2d min{ ordp (FA), ordp (GA)}.
(4.62)
For each prime ideal p we choose a linear fractional transformation fp E PGL 2 (K)
and corresponding matrix Ap so as to minimize the resultant of ¢fP. In other words,
(4.63)
Combining (4.62) and (4.63) yields
ordp (9\¢) - ordp (Res(F, G))
= d[ (d + 1) ordp( det Ap) - 2 min{ ordp(FAp ), ordp( GAp)}].
(4.64)
223
4.11. Minimal Resultants and Minimal Models
Hence if we define an ideal OF,G by the rule
ordp(oF,c) = {
~ (d + 1) ordp( det Ap) -min{ ordp(FAP ), ordp (GAp)}, dodd,
(d + 1) ordp( det Ap) - 2 min{ ordp(FAp ), ordp( GAp)}, d even,
then (4.64) says that aF,G satisfies the desired formula (4.61). It remains to determine
the extent to which the ideal ap,c depends on the choice ofF and G.
Let¢= [F, G] = [F', G'] be two lifts of¢. Then there is a constant c E K* satisfying F' = cF and G' = cG. Suppose first that dis odd. We use (4.61) to compute
c2d Res(F, G)a~~,cG = Res(cF, cG)a~~,cG = 91¢ = Res(F, G)a~~G·
This is an equality of ideals, so by unique factorization of ideals we conclude
that COcF,cG = OF,G· Hence the ideals OcF,cG and OF,G differ by a principal ideal.
Next suppose that d is even. Then by a similar calculation we find that
c2d Res(F, G)a~F,cG = Res(cF, cG)a~F,cG = 91¢ = Res(F, G)a~,c·
Hence (c 2 )acF,cG = ap,c, so OcF,cG and OF,G differ by the square of a principal
D
~~
Definition. Let K be a number field, let ¢( z) E K (z) be a rational map of degree d,
and write ¢ = [F, G] with polynomials F and G as usual. If d is odd, we write a¢1 K
for the ideal class of aF,G in the ideal class group of K. If dis even, we write a¢; K
for the image of aF,G in the group of fractional ideals modulo squares of principal
ideals. In both cases, by analogy with the theory of minimal equations of elliptic
curves (cf. [410, VIII §8]), we call a¢; K the Weierstrass class of¢ over the field K.
By Proposition 4.99 the Weierstrass class a¢; K depends only on¢, independent of
the chosen lift [F, G].
The triviality of the Weierstrass class gives a necessary condition for the existence of a global minimal model.
Proposition 4.100. Let K be a number field and ¢(z) E K(z) a rational map
of degree d ~ 2. If¢ has a global minimal model over K, then its Weierstrass
class 0.¢/ K is trivial.
Proof Replacing ¢by¢! for an appropriate choice off E PGL2(K), we may
assume that¢= [F, G] with polynomials F and G having coefficients in the ring of
integers of K and satisfying
ordp (Res(F, G)) = ordp (91¢)
for every prime p.
It follows from the defining equation (4.61) of aF,G that
for every prime p.
Hence ap,c = (1), so its image in the ideal class group (if dis odd) or in the group
D
of ideals modulo squares of principal ideals (if d is even) is also trivial.
Exercises
224
Exercises
Section 4.1. Dynatomic Polynomials
4.1. Let ¢(z) = z + 1/z, or in homogeneous form, ¢([X, Y]) = [X 2 + Y 2 , XY].
(a) Compute the first few dynatomic polynomials 1>¢,n (X, Y) for¢, say for n = 1, 2, 3, 4.
(b) Prove that for all n 2:: 2, the dynatomic polynomial<I>¢,n satisfies <I>1,,n(±X, ±Y) =
<I>1,,n (X, Y). Deduce that <I>1,,n (X, Y) E Z[X 2 , Y 2 ].
(c) Prove that the field 1Ql2 ,q, generated by the points of exact period 2 is the field IQl (A).
(d) Prove that the field 1Ql3,¢ generated by the points of exact period 3 is an S3 extension of!Ql.
(Hint. Show that the roots of <I>q,,3 ( 1, fo,) E Z[w] generate a cyclic cubic extension
of!Ql.)
(e) Prove that 1>¢, 4 (X, Y) factors into a polynomial of degree 4 and a polynomial of degree 8. (See Exercise 4.40 for a more general result.) Describe the fields generated by
the roots of each factor.
4.2. Prove the following elementary properties of the Mobius function. (See (4.2) on page 148
for the definition of the Mobius function.)
(a) Let n 2:: 1 be an integer. Then
ifn = 1,
ifn 2::2.
(b) Let g(n) be a function whose domain is the positive integers, and define a new function f(n) by f(n) = l:din g(d). Prove that
g(n) = Lf(d)~Jo (~).
din
This is called the Mobius inversion formula.
(c) Prove that the n 1h cyclotomic polynomial
~(zk _ 1 )~(n/k)
kin
is indeed a polynomial. (Hint. Use (b) and the fact zk - 1 has distinct complex roots.)
4.3. Let lld(n) = deg(<I>1,,n) be the number of formal n-periodic points of a rational map of
degree d, counted with multiplicity (see Remark 4.3).
(a) Prove that vd(1) = d + 1 and
vd(n) =
LM (~) dk
forn 2::2.
kin
(b) Make a table of values of lid ( n) for some small values of n and d.
(c) Prove that
formally as power series. For what range of x > 0 do the series converge?
Exercises
J
225
Multiplier
Type of Point
Char K
aj,(n) values
aj,(n) = 0 for all n
Wandering
aj,(n) > 0 for exactly one n
Periodic
not root of unity
Periodic
root of unity
0
Periodic
root of unity
p>O
aj,(n) > 0 for exactly two n's
aj,(n) > 0 for n = tpk,
exactly two t's and all k 2 0
Table 4.4: Values of aj,(n).
4.4. Let ¢(z) E K(z) be a rational function of degree d 2 2 and suppose that z = 0 is a
fixed point of¢ with multiplier>.= 1. Write ¢(z) = z + ze'lj;(z) withe 2 2 and '1/J(O) =F 0,
where e = a0 (¢, 1) in the notation of Theorem 4.5.
(a) Prove that
<Pn(z) = z + nze'lj;(z) + O(z 2 e- 1 ).
(b) Assume further that K has characteristic p > 0. Prove that
for all k 2 1.
4.5. Verify that the description of the values of aj,(n) given in Table 4.4 is correct.
4.6. Let ¢( z) be a rational function, let n 2 1, let p be a prime with p f n, and let P E lP' 1 .
(a) Prove that
a'f,(n, ¢P) = a'P(n, ¢) + a'P(np, ¢).
(b) Deduce that the nth dynatomic polynomial for rjJP factors as
<P~,q,v = <P~,q,(z)<P~p,¢·
(c) Moregenerally,ifgcd(n,k) = l,provethat
<P*nk,¢- Il(<P*n,tjJJ. )p.(k/j) .
ilk
4.7. *Let
¢(z) = zd + ad-lZd-l + · · · + a2z 2 + a1z + ao
be a monic polynomial of degree d and let <P~,q,(z) be its n 1h dynatomic polynomial. Each
root a of<P~,q,(z) has an associated multiplier .Acp(a) = (¢n)'(a).
(a) Prove that there is a (unique) monic polynomial
whose nth power satisfies
Dn,cp(xt =
IT (x- .Aq,(a)) = Resz(<P~,q,(z),x- (¢n)'(z)),
<I>~,.p(<>)=O
where Resz means to take the resultant with respect to the z variable. (Hint. All n of the
points in the orbit Ocp(a) have the same multiplier.)
Exercises
226
(b) Let Ck(x) E Z[x] denote the kth cyclotomic polynomial, and for integers min with
m < n, define
~n,m = Res( Cn;m, Dm,¢) E Z[ao, a1, ... , ad-d·
Prove that the discriminant of <[> ~, ¢ ( x) is given by the formula
Disc <I>~,¢ = ±8n,¢(1t
II ~;;,';:,.
min
m<n
(c) Let min with m < n. Prove that the resultant of <I>~,¢ and <I>;',,¢ is given by the formula
Conclude that ¢ has a point of formal period n whose exact period m is strictly smaller
than n if and only if there is a point of formal period m whose multiplier is a primitive (n / m) th root of unity.
4.8. This exercise describes an analogue of Theorem 4.5 for automorphisms of projective
space. Let¢: lP'N _, lP'N be an automorphism defined over afield K, i.e.,¢ E PGLN+l (K).
We say that ¢ is nondegenerate if the equation ¢( P) = P has only finitely many solutions
in lP'N (K).
(a) Let A E GLN+ 1 (K) be an invertible matrix with coefficients inK representing the
map ¢. Prove that ¢ is nondegenerate if and only if every eigenspace of A has dimension 1.
(b) Assume that ¢n is nondegenerate, let r ¢n c lP'N X lP'N be the graph of ¢n' and denote
the diagonal map by~ : lP'N _, lP'N x lP'N. Following Remark 4.4, we define the 0-cycle
of n-periodic points of¢ to be the pullback
L ap(¢,n)(P)
<I>¢,n = ~*(f¢) =
PEI'N
and the 0-cycle of formal n-periodic points to be
<I>¢,n =
Lll (~) <I>¢,n = L af,(¢,n)(P).
kin
PEI'N
Let P E lP'N be a point of primitive period m for ¢. Prove that
ap (n ) = {
ap(m)
0
ifmln,
ifm fn,
* ( ) _ {ap(m)
ap n 0
and
ifm = n,
ifm f n.
In particular, <I>¢,n is an effective (i.e., positive) cycle.
4.9. Let F, G E K[X, Y] be homogeneous polynomials of degree d ~ 2 with no common
factors and let¢= [F, G] : lP' 1 _, lP' 1 be the associated rational map. Define two sequences
of polynomials inductively starting with F0 (X, Y) =X and Go(X, Y) = Y and then for
n ~ 0,
Fn+l = Fn (F(X, Y), G(X, Y))
and
Gn+l = Gn(F(X, Y),G(X, Y)).
(a) Prove that Fn and Gn have no common factors.
227
Exercises
(b) More precisely, prove that the resultant of Fn and Gn is given by
Res(Fn, Gn) = Res(F, G)
(2n-l)dn-l
.
(Hint. Use Exercise 2.12.)
(c) Prove that </>n = [Fn, Gn].
(d) For all n, m :2: 0, prove that
Fn+m(X, Y) = Fn(Fm(X, Y),Gm(X, Y)),
Gn+m(X, Y) = Gn(Fm(X, Y),Gm(X, Y)).
4.10. With notation as in Exercise 4.9, we define the (generalized) (m, n )-period polynomial
of 4> to be the polynomial
<I>m,n(X, Y) = <I>n(Fm(X, Y),Gm(X, Y)),
where <I>n(X, Y) = Y Fn- XGn is the usual n-period polynomial of¢.
(a) Prove that
<I>m,n(X, Y) = Gm(X, Y)Fn+m(X, Y)- Fm(X, Y)Gn+m(X, Y).
(b) Let P E JP1 ( K). Prove that
<I>m,n(P) = 0 if and only if 4>m+n(P) = <f>m(P).
Thus P is a root of <I>m,n if and only if P is a preperiodic point with "tail" of length at
most m and with period dividing n.
(c) Prove that for all m, n :2: 1, the quotient
<I>m,n
<I>m-l,n
is a polynomial.
4.11. We continue with the notation fromExercises4.9 and4.l0. Let <I>~(X, Y) be the nth dynatomic polynomial of¢. Then for m, n :2: 1 we define the (generalized) (m, n )-dynatomic
"polynomial" of 4> to be
•
<I>~(Fm(X,Y),Gm(X,Y))
<I>m n(X, Y) =
(
).
'
<I>;. Fm-l(X, Y),Gm-l(X, Y)
(a) Prove that <I>;,,n(P) = 0 if and only if <f>m(P) has formal period n. Points satisfying <I>;,,n(P) = 0 are calledpreperiodic points with formal preperiod (m, n).
(b) **Prove (or disprove) that <I>;,,n (X, Y) is a polynomial.
Section 4.2. Quadratic Polynomials and Dynatomic Modular Curves
4.12. We continue with the notation from Exercise 4. 7. Thus for any monic polynomial ¢( z),
we define a monic polynomial 8n,4> (x) by
<~>n,</>(a)=O
and for min with m < n we set
An,m = Res(Cn;m(x),8m,.p(x)),
where Ck(x) is the kth cyclotomic polynomial. We now specialize to </>c(z) = z 2 + c and
write 8n(c, x) = 8n,4>c (x) and An,m (c) to indicate the dependence on c.
228
Exercises
(a) Prove that Dn(c,x) E Z[c,x] and that ~n,m(c) E Z[c]. Then use Exercise 4.7(c) to
explain the powers that appear in Table 4.3.
(b) Prove that
bn(c,x)=Tdegz<I>;',8n(c,2nx) isin Z[c,x].
(This eliminates many powers of 2 from the coefficients of Dn (c, x).) Compute bn (c, x)
forn = 1,2,3,4.
(c) **Prove that there is a monic polynomialWn,m(x) E Z[x] such that
~n,m(c) = Wn,m(4c),
and deduce that the set
{cEQ: ~n,m(c) = 0 for some m < n with min}
is a set of bounded height. Conclude that the set of bifurcation points in the Mandelbrot
set M is a set of bounded height. (Hint. The formula ~n,m(c) = Wn,m(4c) bounds the
denominator of c, then mimic the proof of Proposition 4.22.)
(d) Prove that ~n,l (c) and ~ 2 n, 2 (c) are irreducible in Q[c] for all n ~ 1.
(e) **Prove the following conjecture of Morton-Vivaldi (314): The polynomial ~n,m (c) is
irreducible in Q[c] for all min with m < n.
4.13. Let cPc(z) = z 2 + c and write <I>~(c, z) for the n 1h dynatomic polynomial of cPc· Continuing with the notation from Exercise 4.12, we let Dn(c,x) E Z[c,x] be the polynomial
defined by (4.65) and bn(c, x) = T deg<t>;',8n(c, 2nx) the normalized polynomial in Exercise 4.12(b ).
(a) Recall that the dynatomic curve Y1 (n) CA.? attached to the polynomial ¢lc(z) is defined
by the equation <I>~(y, z) = 0. Let Z(n) c A2 be similarly defined by the equation
Z(n): bn(y,x) = 0.
Prove that there is a morphism
F: Y1(n)-----+ Z(n),
(b) Define an action of i E Z/nZ on Y1 (n) by
(y,z) = (y,¢~(z)).
(See Section 4.2.3.) Prove that the map Fin (b) is invariant for this action, i.e., prove
that
F(y,¢~(z)) = F(y,z) foralli E Z/nZ.
Deduce that there is a unique morphism Y0 (n) ____, Z(n) such that the composition
Y!(n) ____, Yo(n) ____, Z(n)
is the map F.
(c) Prove that the map Y0 (n) ____, Z(n) in (b) has degree 1, so the equation bn(y,x) = 0
gives a (possibly singular) model for Y0 (n).
(d) Prove that Z(1) and Z(2) are nonsingular and that Z(3) and Z(4) are singular. Resolve
the singularities of Z(3) and Z( 4) and check directly that Y0 (3) and Yo( 4) are curves of
genus 0.
Exercises
229
4.14. Let ¢c(z) = z 2 + c and let <I>:n,n be the generalized dynatomic polynomial defined in
Exercise 4.II.
(a) Use a computer algebra system to compute the first few generalized dynatomic polynomials <I>:n,n for ¢c, say for ( m, n) in the list
{(1, 1), (2, 1),(3, 1), (4, 1),(1,2), (2,2),(3,2), (4,2),(1,3), (2,3), (1,4)}.
(b) From your list, it should be apparent that many of the leading terms of <I>:n, 1 and <~>:n, 2
coincide. Prove that
for all m ~ 1.
(Note that this is a special property of the generalized dynatomic polynomials for ¢c.)
4.15. Let c E <C and let ¢c ( z) = z 2 + c. Prove that there is a number Rc > 0 and a
holomorphic function
1/Jc : { Z E <C : lzl > Rc} --+ <C
satisfying
and
Iogi'I/Jc(zl 1 ""log lzl
as lzl ___, oo.
(Hint. Show that there is a consistent way to choose square roots so that 1/Jc can be defined as
limn~oo 2 ~ .)
4.16. Consider the doubling map
D : Q_/71.. --+ Q/71..,
D(t) = 2t mod 71...
Fix t E Q/71.. and let m ~ 0 and n ~ 1 be the smallest integers such that the denominator oft
divides 2m(2n- 1).
(a) Prove that tis periodic forD if m = 0 and that tis strictly preperiodic forD if m ~ 1.
(b) Prove that m ~ 0 and n ~ 1 are the smallest positive integers such that t can be written
as a (not necessarily reduced) fraction of the form
(c) If t is preperiodic for D, prove that the preperiod oft is equal to m and that Dm (t) is
periodic with exact period n.
(d) Fort as above, we say that tis of type (m, n) for the doubling map D. For a given pair
of positive integers (m, n), how many t E Q/71.. are of type (m, n)?
The map Dis used in the analytic characterization ofMisiurewicz points; see Theorem 4.25.
However, note that (d) does not count the exact number ofMisiurewicz points of type (m, n),
because distinct t may give the same Misiurewicz point.
4.17. **Let ¢c(z) = z 2 +cas usual, and for integers m, n ~ 1, define a rational function
(of c) by
Fm n(c) = IJ (
'
¢;,rr+k(O)- ¢;,rr(o)
¢~-l+k(O)- ¢~-1(0)
kjn
) !'(n/k) = IJ(A,m-l+k(O) + A,rn-l(O))I'(n/k).
kin 'l'c
'l'c
Then set
Gm n(c) = {Fm,n(c)
'
Fm,n(c)/Fl,n(c)
ifm '/= 1 (modn) orm = 1,
ifm
= 1 (modn) andm #- 1.
230
Exercises
(a) Prove that Gm,n(c) is a polynomial in c. More precisely, prove that Gm,n(c) is in Z[c].
(b) Ifn ~ 2, prove that the roots ofGm,n are the Misiurewicz points of type (m, n).
(c) Prove that Gm,n(c) is irreducible in Q[c].
4.18. Let V be a variety and suppose that the points of V algebraically parameterize a family
of quadratic polynomials 'lj; together with a marked point >. of formal period n. Theorem 4.11
says that there is a unique morphism 'f/ : V ___, Y1 (n) satisfying
TJ(P) = (7j;p(z),>.(P)) E Formal(n).
Note that by construction, each point "f = (c, a:) E Y1 (n) is identified with a quadratic
polynomial ¢"1 and point 11-("f) of formal period n via
¢"/(z) = z 2 + c and
11-b) =a:.
Prove that there is a unique morphism of varieties
f: V---+ PGL2
such that the following identities are true for all P E V(K):
(a) (fp 1 0 'lj;p 0 fp )(z) = c/>TJ(P)(z).
(b) (TJ o
r
1o
>.)(P) = (11- o TJ)(P).
In other words, prove that the following two diagrams commute:
IP'~
,pf
----+
lT}
IP'~
v
lT}
lT}
IP'1
IP'~l(n) ----+ Y1(n)
q,
f-lo>.
----+
IP'~
lT}
IP'1Y1(n)
'"
Y1(n) ----+
(Hint. During the proof of Theorem 4.11 we defined a linear fractional transformation fp(z).
Prove that fp is uniquely determined by P. Deduce that the map P >-> fp is a morphism.)
4.19. In Example 4.14 we gave an explicit description of the quotient map Y1 (2) ---> Yo(2).
Perform a similar analysis and describe the map Y1 (3) ___, Yo (3).
4.20. We proved that the dynatomic modular curves X 1(1 ), X 1(2), and X 1(3) are isomorphic
to IP' 1. This exercise asks you to investigate X 1(n) for other small values of n.
(a) Prove that X 1(4) is a curve of genus 2 and X o(4) is a curve of genus 1.
(b) Prove that X 0 (4)(Q) is finite, and Y0 (4)(Q) is the empty set.
(c) Prove that X1 (5) has genus 14 and Xo(5) has genus 2.
(d) *Prove that X 0 (5)(Q) is finite, and Y0 (5)(Q) is the empty set.
(e) ComputethegeneraofX1(6) andXo(6).
(f) **Find all rational points in Y0 (6)(Q).
Section 4.3. The Space Ratd of Rational Functions
4.21. Let¢ = [Fa, Fb] E Ratd be a rational map of degree d. Prove that the following are
equivalent:
(a) At least one of the fixed points of¢ has multiplier equal to 1.
(b) The polynomial Y Fb - X Fa has a multiple root.
Exercises
231
Deduce that there is a nonzero homogeneous polynomial D( a, b) E Q[ao, ... , bd] such that
(
Hence
¢ has a fixed point whose)
. .
I
muIttp1ter equa s 1
¢:::::}
D(a, b)= 0.
{ ¢ E Ratd : ¢ has a fixed point whose multiplier equals 1}
is a proper Zariski closed subset ofRatd.
Section 4.4. The Moduli Space Md of Dynamical Systems
4.22. Let GLn be the group of n x n matrices with nonzero determinant, let SLn be its
subgroup of matrices with determinant 1, let PGLn be the quotient ofGLn by its subgroup of
diagonal matrices, and similarly let PSLn be the quotient of SLn by its subgroup of diagonal
matrices.
(a) Let K be an algebraically closed field. Prove that the natural map from PSLn(K)
to PGLn(K) is an isomorphism.
(b) More generally, prove that for any field K there is an exact sequence
1 - - - + PSLn(K) - - - + PGLn(K) ~ K* I K*n - - - + 1.
4.23. We say that a separable rational map ¢(z) E K(z) is very highly ramified if there is a
point P E IP' 1 such that the ramification index of¢ satisfies ep ( ¢) ~ 3. Let
V = {¢ E Ratd : ¢ is very highly ramified}.
(a) Prove that Vis a proper Zariski closed subset ofRatd. Hence "most" rational maps are
not very highly ramified. (One says that a generic map of degree dis not very highly
ramified.)
(b) Prove that Vis invariant under the conjugation action ofPGL2.
(c) Prove that the quotient VI PSL2 is a Zariski closed subset of Md = Ratd I PSL2.
4.24. Let ¢(z) E K(z) be a nonconstant rational function. The Schwarzian derivativeS¢
of¢ is the function
(S¢)(z) = <P"'(z)- ~ (¢"(z))2
<P'(z)
2
¢'(z)
It measures the difference between ¢ and the best approximation to ¢ by linear fractional
transformations.
(a) If f(z)
(az+b)l(cz+d) is a linear fractional transformation, prove that (S f)(z) 0.
(b) Let f be a linear fractional transformation. Prove that ¢ and f o ¢ have the same
Schwarzian derivative.
(c) Let ¢(z), 'lj;(z) E K(z) be nonconstant rational functions. Prove that
=
=
(S(¢ o '1/J))(z) = (S¢)('1/J(z)) · ('I/J'(z)) 2 + (S'Ij;)(z).
(d) In particular, iff is a linear fractional transformation, prove that
and deduce that the quadratic differential form
232
Exercises
W¢(z) = (S.P)(z) (dz) 2
is invariant under the substitution (¢, z) r-+ ( .pf, f- 1 z). Thus the map¢ r-+ W¢ induces
a natural map
Md ---> (quadratic differential forms on lP' 1 ).
(e) Suppose that ¢( z) has a multiple zero or pole at z = a:, say
¢(z) = a(z- o:)m + · · ·
with a=/= 0 and lml 2: 2.
Prove that S¢ has a double pole at a:. More precisely, prove that the Laurent series
expansion of s.p around 0: looks like
1-m2
(S.P)(z) = - 2 -(z- a:)
-2
+ .. · ·
(f) Prove that the map
(¢,P) f---> (S,P)(P),
is a morphism.
4.25. ** An algebraic variety V is called unirational if there is a rational map lP'N -+ V
whose image is Zariski dense in V, and the variety V is called rational if there is a rational
map lP'N -+ V that is an isomorphism from an open subset oflP'N to an open subset ofV. It is
clear that every moduli space Md is unirational, since Ratd is an open subset of lP' 2dH and
the map Ratd -+ Md is surjective. We also know that M 2 is rational, since Theorem 4.56
says that M 2 ~ A..Z. For which values of d is Md a rational variety? In particular, is M 3
rational?
Section 4.5. Periodic Points, Multipliers, and the Multiplier Spectrum
4.26. This exercise asks you to prove the part of Theorem 4.50 that was left undone in the
text. Prove that the map
Ratd---> A 1 ,
defines a function in Q[Ratd]. Prove that it is PGL2-invariant and deduce that it defines a
function in Q[Md].
4.27. **What is the degree of the map
0'3,N :
M3 ---> A k
for sufficiently large N? Similarly, what is the degree of a 4,N on M4 away from the Lattes
locus on which it is constant?
Section 4.6. The Moduli Space M2 of Dynamical Systems of Degree 2
4.28. Prove that ¢ E Rat2 is conjugate to a polynomial if and only 0'1 ( ¢) = 2. Thus the
polynomial maps in M 2 ~ A 2 trace out the line x = 2.
4.29. Let ¢ E Rat 2 ( q be a rational map of degree 2 and suppose that one of its fixed points
has a nonzero multiplier ,\,
Exercises
233
(a) Prove that there are an f E PGL2(1C) and acE C such that
¢/(z) = ~ (z+c+ ~).
(4.66)
This generalizes the normal form given in Lemma 4.59.
(b) Verify that the multiplier of¢/ at the fixed point z = oo is )...= ( ¢/) = >...
(c) Let )... 1, )... 2, )... 3 be the multipliers of the fixed points of¢ with >..1 = >... Prove that the
number c in (4.66) satisfies c 2 = 4- >..1 (2 + >..2 + >.. 3 ).
4.30. Fix d ~ 2 and consider the subset of Ratd defined by
BiCritd = { ¢ E Ratd : ¢has exactly two critical points}.
For d = 2 we have BiCrit 2 = Rat 2, but for larger d the set BiCritd is a proper subset
of Rat d.
(a) Prove that BiCritd is an algebraic variety of dimension 5.
(b) Prove that conjugation induces a natural action of PGL2 on BiCritd, i.e., there is a
morphism of varieties
(c) Suppose that¢ E BiCritd has critical points at 0 and oo. Prove that¢( z) has the form
(4.67)
(d) Let¢ E BiCritd and apply a conjugation to move the critical points of¢ to 0 and oo,
so¢ has the form (4.67) described in (c). Prove that the following quantities depend only
on the conjugacy class of¢:
(4.68)
(e) Let¢, 1/J E BiCritd and suppose that
w(¢) = w('ljJ),
T(cj;) = T('l/J).
Prove that¢ and 1/J are PGL 2 conjugate (working over an algebraically closed field).
(f) Prove that the three quantities (4.68) described in (d) satisfy the relation
aT= wd- 1(w + l)d+l
and no other relations. Conclude that M~icrit is isomorphic to A 2 . (This generalizes Theorem 4.56, since M~iCrit = M 2 .)
(g) ** Describe the stable and semistable completions of M~iCrit coming from geometric
invariant theory.
Section 4.7. Automorphisms and Twists
4.31. In Section 4.7 we defined two rational maps¢ and 1/J to be equivalent ifthere is an f E
PGL2(K) such that 1/J = rpf, and similarly to be K-equivalent ifthere is an f E PGL2(K)
such that 1/J = q;J.
Exercises
234
(a) Prove that these definitions do indeed define equivalence relations on the set of rational
maps.
(b) More generally, suppose that a group G acts on a set X. Define a relation on X by setting
y "' x if there exists a rJ E G such that y = rJ( x). Prove that this is an equivalence
relation.
4.32. Let ¢(z) E K(z) be a rational map.
(a) Prove that Aut(¢) is a subgroup ofPGL2(K).
(b) LethE PGL2(K). Prove that Aut(¢h) = h- 1 Aut(¢)h, so Aut(¢) and Aut(¢h) are
conjugate subgroups ofPGL2(K).
4.33. Let ¢(z) E K(z), let f E Aut(¢), and let a be a critical point of¢ (i.e., ¢'(a) = 0).
Prove that f(a) is also a critical point of¢. More generally, prove that a and f(a) have the
same ramification index.
4.34. Describe all polynomials ¢(z) E k[z] whose automorphism group Aut(¢) is nontrivial. (Hint. Iff E Aut(¢), what does f do to the totally ramified fixed point(s) of¢?)
4.35. Let ¢( z) E K (z) be a rational map of degree d and write ¢( z) as a quotient of polynomials
ao + a1z + · · · + adzd
¢(z)= bo+b1z+···+bdzd'
Prove that f(z) = z- 1 is in Aut(¢) if and only if bi = ad-i for all i.
4.36. Let ¢(z) = (z 2 - 2z)/( -2z + 1).
(a) Prove that Aut(¢) contains the maps
}
z
1
{ z ,1-z-1
, - - ,1-z ,
,--,1-z z- 1
z
z
and that they form a group isomorphic to 8 3. Prove that #Aut(¢) = 6, so in fact
Aut(¢) ~ 83. (Hint. Find the fixed points of¢.)
(b) Compute the values of rJ1 (¢ ), rJ2 (¢ ), and rJ3 (¢ ).
4.37. Let ( E K be a primitive n 1h root of unity and let ¢(z) E K(z) be a nonconstant
rational map.
(a) Prove that Aut(¢) contains the map f(z) = (z if and only if there is a rational
map 'ljJ(z) E K(z) such that ¢(z) = z'ljJ(zn).
(b) Prove that Aut(¢) contains both of the maps f(z) = (z and g(z) = 1/z if and only
if there is a polynomial F(z) E K[z] such that the rational map 'ljJ(z) E K(z) in (a)
has the form 'ljJ(z) = F(z)/(zdF(z- 1)), where d = deg(F). Verify that the group
generated by f and g is the dihedral group of order 2n.
(c) Let ¢(z) = zd. Prove that Aut(¢) is a dihedral group of order 2d- 2.
4.38. We identify the set of rational functions Ratd of degree d with an open subset of lP' 2d+ 1
as described in Section 4.3.
(a) Let f E PGL2. Prove that the set{¢ E Ratd: ¢I=¢} is a (possibly empty) Zariski
closed subset ofRatd. If f(z) =I= z, prove that it is a proper subset ofRatd.
(b) Let A C PGL2(K) be a nontrivial finite subgroup. Prove that the set
{ ¢ E Ratd : ¢I = ¢for all f E A}
is a proper Zariski closed subset ofRatd.
235
Exercises
(c) Prove that up to conjugation, PGL 2 (K) has only finitely many distinct finite subgroups
of any given order.
(d) Prove that the set {¢ E Ratd : Aut(¢) =/= 1} is a proper Zariski closed subset of Rat d.
(Hint. Note that the order of Aut(¢) is bounded by Proposition 4.65.)
(e) Prove that the set{¢ E Ratd : Aut(¢) =/= 1} is PGL2-invariant and defines a proper
Zariski closed subset of Md. (Hint. The groups Aut(¢) and Aut( ¢f) are conjugate
subgroups ofPGL 2; see Remark 4.64 and Exercise 4.32.)
4.39. Let a: E C* and d ~ 1 and consider the rational function
¢(z) =a: (~)d
z-1
Prove that Aut(¢)= {f E PGL 2(C): ¢!=¢}is trivial.
4.40. Let ¢( z) E K (z) be a rational map of degree d ~ 2 and assume that Aut (¢) contains
the element h(z) = -z of order 2.
(a) Suppose that d is even. Prove that at least one of the fixed points of ¢(z) is defined
over K.
(b) Suppose further that d = 2. Prove that there is an f E PGL2(K) such that h 1 =hand
such that ¢! has the form
b
¢ 1(z)=az+-.
z
{c) Write <Pa,b(z) = az + bz- 1 . Prove that for all c E K*, the maps <Pa,bc2 and <Pa,b
are PC L2 (K)-equivalent.
(d) Inhomogeneous form we have <Pa,b([X, Yl) = [aX 2 +bY 2 ,XY]. Let <P~(a,b; X, Y)
be the associated dynatomic polynomial. Prove that <P~(a, b; X, Y) E &::[a, b, X, Y].
(e) *If n ~ 4 is even, prove that <P~(a, b; X, Y) is reducible. More precisely, prove that
there are nonconstant polynomials Wn, An E Z[a, b, X, Y] such that
<P:(a, b; X, Y) = Wn(a, b; X, Y)An(a, b; X, Y).
(Hint. Divide the set of points P of formal period n into two subsets depending on
whether the involution h permutes the orbit 0 q, ( P) or sends it to a different orbit.) This
result is due to Manes [286].
4.41. Let¢= az + bz- 1 E Rah
(a) Prove that ( 0'1 ( ¢), 0'2 ( ¢)) depends only on a.
(b) We saw in Exercise 4.40 that Aut(¢) contains {±z}, so it has order at least 2. Find all
values of a for which Aut(¢) is strictly larger than { ±z }.
(c) As a varies, prove that the set of points
describes the curve
C : 2x 3 - 8xy + x 2 y- 4y 2 - x 2 + 12x + 12y- 36 = 0
inA?~ M2.
(d) Prove that the curve C in (c) is singular at the point ( -6, 12).
236
Exercises
(e) Move the singular point to the origin and perform a further change of variables to prove
that the curve C is isomorphic to the curve
y2 = 4x3 + x4.
Thus the singularity of C is a cubic cusp.
(f) The point ( -6, 12) is the unique singular point of the curve C, so the rational map ¢
satisfying ( 0" 1 ( ¢), 0"2 ( ¢)) = (- 6, 12) should be special in some way. How is it special?
(Hint. Compare with the answer to (a). See also Exercise 4.36.)
Section 4.8. General Theory of Twists
4.42. Let X be an object defined over K, and for each twist Y of X, fix a K-isomorphism iy : Y-> X. Assume that Aut( X) is abelian. Prove that
Twist(X/ K) ---+ H 1 ( Gal(K / K), Aut( X)),
is a well-defined one-to-one map of sets. (See Remark 4.78.)
Section 4.1 0. Fields of Definition and the Field of Moduli
4.43. Let¢ E K(z) be a rational function. We know from Proposition 4.84 that the field
of moduli K.p is contained in every field of definition of¢. Prove that K.p is equal to the
intersection over all fields of definition of¢.
4.44. Let Ca : x 2 + y 2 = a be the family of curves studied in Example 4. 76.
(a) Let p be a prime number satisfying p
1 (mod 4). Prove that Cp is Q-isomorphic
to C1.
(b) Let p be a prime number satisfying p
3 (mod 4). Prove that Cp(Q) = 0. Deduce
that Cp is not Q-isomorphic to C1, and hence that [Cp]K represents a nontrivial element
ofTwist(C1/Q).
(c) Let p and q be distinct prime numbers that are congruent to 3 modulo 4. Prove that Cp
and Cq are not Q-isomorphic. Deduce that Twist( Cl/Q) is an infinite set.
=
=
4.45. Let g be the cocycle
g: Gal(Q/Q)---+ PGL2(Q),
gu
(z) = {z
-1/z
if O"(i) = i,
ifO"(i) = -i,
described in Example 4.88, and let K be the associated fixed field in Q( w ).
(a) Prove that K = Q(u, v), where u = w- 1/w and v = i(w + 1/w). Deduce that
C : u 2 + v 2 = -4 is the twist of!P' 1/Q associated to the cocycle g.
(b) Let ¢(z) = i((z- 1)/(z + 1)) 3 be the rational map from Example 4.85 satisfying
O"(j) = ¢g". Our general theory says that K is a field of definition for ¢ if and only if
C(K) =f. 0, where Cis the curve in (a). For example, C(Q(H)) =f. 0, so Q(H)
must be a field of definition for ¢(z). Find an explicit linear fractional transformation
f E PGL2(Q) suchthat¢f(z) E Q(H)(z).
Section 4.11. Minimal Resultants and Minimal Models
4.46. ** This exercise raises some natural questions concerning global minimal models.
(a) Is it true that every rational map ¢(z) E Q(z) of (odd) degree d 2: 2 has a global
minimal model over Q?
237
Exercises
(b) Let K be a number field, let R be the ring of integers of K, let S be a finite set of
primes, and let ¢(z) E K(z) be a rational map of degree d 2: 2. We say that¢ has a
global S-minimal model ifthere is a linear fractional transformation f E PGL2(K) and
homogeneous polynomials F and G satisfying ¢' = [F, G] with the property that the
coefficients ofF and G are in the ring of S-integers Rs and
ordp(Res(F,G)) = ordp(mq,)
for every prime p rf. S.
Suppose that Rs is a principal ideal domain. Is it true that every ¢ of (odd) degree has a
global S-minimal model?
(c) Let K be a number field and¢( z) E K (z) a rational map of (odd) degree d 2: 2. Suppose
that the Weierstrass class aq,; K is trivial. Is it then necessarily true that ¢ has a global
minimal model? (This is true in an analogous situation for elliptic curves; see [410,
VIII.8.2].)
4.47. ** Let K be a number field and let ¢ E K (z) be a rational map of degree d 2: 2. Let S
be a finite set of primes of K. Prove that
{ 7P E Twist (¢/ K) : 7P has good reduction at all p rf. S}
is a finite set.
4.48. **Fix an embedding of the moduli space Md in projective space and let hMd denote
the associated height function. Let K/Q be a number field, let d 2: 2 be an integer, and
let B 2: 1 be a number. Prove that the set
contains only finitely many PG L 2 ( K) -conjugacy classes of rational maps.
4.49. * Let K be a number field, let R be the ring of integers of K, let S be a finite set
of primes of R, and let d 2: 2 be an integer. Prove that there is a finite set of rational
maps BK,S,d c Ratd(K) such that if¢ E Ratd(K) is a rational map satisfying
( 1) ¢ has three or more critical points,
(2) the critical points of¢ remain distinct modulo all primes not inS,
(3) the critical values of¢ remain distinct modulo all primes not inS, where we recall that
a critical value of¢ is the image of a critical point of¢,
then there are automorphisms f, g E PGL2(Rs) such that
f 0¢0 g E BK,S,d·
(Note that when we say that the critical points or values are distinct modulo a prime p, we
really mean that they are distinct modulo lfJ for all primes lfJ lying above p in a suitable finite
extension of K.)
Chapter 5
Dynamics over Local Fields:
Bad Reduction
In this chapter we return to the study of dynamical systems over complete local
fields such as Qlp. We saw in Chapter 2 that if a rational map ¢(z) E Qlp(z) has
good reduction, then its Julia set is empty, in which case considerable information
about the dynamics of¢ on IP' 1 ( Qlp) may be deduced by studying the dynamics of the
reduction ¢ on IP'1(JFP). But if¢( z) has bad reduction, then the situation is far more
complicated. Indeed, since "interesting" unpredictable dynamics occurs only in the
Julia set, we might say that the good reduction scenario studied in Chapter 2 is the
uninteresting situation. This chapter is devoted to the interesting case!
The field Qlp and its finite extensions have the agreeable property that they are
complete, but they are not algebraically closed, so they are more analogous to JR. than
they are to <C. This suggests that we work instead with an algebraic closure Qp ofQP,
but unfortunately we then lose the completeness property! Going one step further, we
take the completion ofQp, and it turns out that this field, denoted by
Cp =the completion of the algebraic closure ofQlp,
is both complete and algebraically closed. For proofs of the basic properties of Qp
and Cp, see for example [81, 175, 183, 249, 382].
The fields C and Cp share many common properties, but they also differ in crucial
ways. In particular, the field of p-adic complex numbers Cp is not locally compact!
However, it is often essential to work in Cp, rather than in a finite extension ofQlp, for
example if we want to guarantee the existence of large numbers of periodic points.
Table 5.1 compares some of the properties of the complete fields Qlp, Qp, Cp, JR.,
and C.
The subject of p-adic and more general nonarchimedean dynamics is relatively
new. After some early articles [50, 201, 278, 433, 438] in the 1980s, there was an
explosion of interest that put the subject on a firm footing with a body of significant
theorems and, just as importantly, an array of fascinating conjectures. In this brief
239
240
5. Dynamics over Local Fields: Bad Reduction
Nonarchimedean metric
.;
Algebraically closed
Complete
Locally compact
Totally disconnected
.;
.;
.;
.;
.;
.;
.;
.;
.;
.;
.;
.;
.;
.;
.;
Table 5.1: Comparison of complete fields.
chapter we can provide only a glimpse into this active area of current research, with
many important topics omitted entirely in order to keep the chapter at a manageable
length. For example, we do not touch on the important concept of local conjugacy,
nor do we describe Rivera-Letelier's classification ofFatou domains [373, 375]. For
the reader desiring further information, we mention the following articles on p-adic
dynamics that are listed in the references: [4, 5, 13, 14, 22, 23, 26, 30, 29, 50, 63, 53,
54, 56, 57, 58, 59, 60, 62, 70, 71, 72, 73, 82, 84, 104, 116, 170, 168, 169, 185, 188,
189,201,206,208,220,222,236,237,238,239,240,241,242,243,244,245,262,
266,267,268,271,272,273,274,278,280,282,283,320,318,319,334,337,338,
339,340,344,348,352,355,372,373,374,375,376,378,389,427,433,438,449].
5.1
Absolute Values and Completions
We recall from Section 2.1 that a valued field is a pair (K, I · I) consisting of a
field K and an absolute value I · I on K. In this section we briefly remind the reader
of the construction and basic properties of the completion of a valued field.
Definition. Let (K, I · I) be a valued field. A sequence a1, a2, a3, ... E K is
Cauchy if for every E > 0 there exists an N = N (E) such that
for all m,n 2 N.
In other words, the sequence {an} is Cauchy if Ian - am I --+ 0 as m, n --+ oo.
It is clear that every convergent sequence is Cauchy. A valued field K is said to
be complete if every Cauchy sequence in K converges.
Example 5.1. The real numbers JR. and the complex numbers Care complete with
respect to their usual absolute values.
Theorem 5.2. Let (K, I · IK) be a valued field. Then there exists a valued field
(k, I · Ik), unique up to isomorphism of valued fields, with the following properties:
(a) There is an inclusion K C k as valued.fields.
(b) k is complete.
5.1. Absolute Values and Completions
241
(c) K is a dense subset of k
Further, k is the smallest valued field satisfYing (a) and (b) in the following sense:
Let ( L, I · IL) be any complete valued field containing K. Then there is a unique
inclusion k '--+ L of valuedfields that respects the inclusions of K into k and L.
The field k is called the completion of K with respect to the absolute value I · Ix·
Proof (Sketch) The field k may be constructed as follows. Let C be the set of all
Cauchy sequences in K and make C into a ring by setting
and
We define an absolute value on C by setting
and we define an equivalence relation "' on C by
Then one checks that these definitions are consistent and that the quotient
is a complete field with the desired properties. Note that K is identified with the
subfield of k consisting of constant sequences. For further details on completions,
see for example [190, 4 §7] or [217, IV §1].
D
Theorem 5.3. Let ( K, I · IK) be a complete field and let L / K be a finite extension.
Then there is a unique absolute value I · IL on L extending the absolute value on K.
The field Lis complete with respect to I · 1£·
Proof (Sketch) One checks that I · IL is given by
laiL = INL;x(a)I~[L:KJ.
See [259, Proposition XII.2.6] or [382, II §4].
D
Example 5.4. The following table gives some examples of completions.
Field
Absolute Value
Completion
Q
laloo =max{ a, -a}
lR
Q(i)
Q
Ia + bil =
+
Ialp = p- ordp(a)
va 2
b2
c
Qp
242
5. Dynamics over Local Fields: Bad Reduction
Remark 5.5. Theorem 5.3 says that for a finite extension L/ K, there is a unique
absolute value on L extending the absolute value on K, and further that L is complete
with respect to this extended absolute value. If we go to the algebraic closure k
of K, then we still get a unique absolute value, since k is the compositum of the
finite extensions of K, but unfortunately k may not be complete. (See, e.g., [249,
III.4, Theorem 12] for a proof that Qp is not complete.) So we use Theorem 5.2 to
fonn the completion of the valued field k as in the next theorem.
Theorem 5.6. Let
Cp = QP =the completion ofthe algebraic closure ofQv.
Then Cp is both complete and algebraically closed. It is the smallest complete algebraically closedfield containing Qp.
Proof. More generally, the completion of an algebraically closed nonarchimedean
field is algebraically closed. We briefly sketch the proof. Let a be algebraic over Cv,
say a root of f(X) = 2: aiXi E Cv[X]. Choose bi E Qp that are very close to ai,
which can be done since Qp is dense in Cp. Then the roots of g(X) = 2: biXi
can be matched with the roots of f(X), so we can find a root f3 of g(X) that is
very close to a. Note that f3 E Qp, since Qp is algebraically closed. Thus we can
find elements in Qp (so a fortiori in Cp) that are arbitrarily close to a. This gives
a sequence of elements of Cp that converges to a, so the fact that Cv is complete
implies that a E Cp. For details of the proof, see [249, III.4, Theorem 13] or [382,
Lemma 11.4.2].
D
5.2
A Primer on Nonarchimedean Analysis
Throughout this section we take K to be a field that is complete with respect to an
absolute value I · I satisfying the nonarchimedean (ultrametric) triangle inequality
Ia + /31 :::; max{ Ia I, l/31}
for all a, /3 E K.
(5.1)
We recall from Lemma 2.3 that if Ia I =f. l/31, then (5.1) is an equality.
We set the following notation for the ring of integers of K, its unit group, and its
maximal ideal:
R = RK ={a E K: lal:::; 1},
R* = R"K ={a E K: lal = 1},
VJ1 = VJ1K ={a E K: lal < 1}.
We define "open disks," "closed disks," and "circles" in K by the usual fonnulas:
D(a, r) = {u E K: lu- al < r} =open disk of radius rat a,
D(a, r) = {u E K: lu- al :::; r} =closed disk of radius rat a,
S(a,r) = {u E K: lu- al = r} =circle of radius rat a.
5.2. A Primer on Nonarchimedean Analysis
243
However, it is important to note that in the nonarchimedean setting, the disks D( a, r)
and D(a,r) and the circle S(a,r) are simultaneously open and closed sets! To see
this, let x E S(a,r). Then for any s < r we have D(x,s) c S(a,r), since any
y E D(x, s) satisfies
iY- xi< s < r,
so
iY- ai = max{iY- xi, ix- ai} = r.
This shows that S(a, r) is open, and it is clearly closed by definition. It follows that
D(a, r) = D(a, r) U S(a, r) is open and that D(a, r) = D(a, r) "S(a, r) is closed.
Notice that D(O, r) and D(O, r) are groups under addition, i.e., they are closed
under addition and negation, with a nontopological meaning of the word "closed."
More precisely, the closed unit disk D(0, 1) is the ring of integers R of K and the
open unit disk D(O, 1) is the maximal ideal rot of R.
Recall that the chordal distance between two points P1 = [X1, Y1] and P2 =
[X2, Y2] in lP'1(K) is the quantity (cf. Section 2.1)
p
p.)-
(p
b
iX1Y2- X2Yli
2 - max{iX1i, IY1i} max{iX2i, iY2i}.
The chordal metric p satisfies 0 :::; p( P 1, P 2) :::; 1, so it is bounded. In particular,
if K is a finite extension of Qp, so K is locally compact, then lP'1(K) is compact,
since it is a locally compact bounded metric space. On the other hand, the field Cp is
not locally compact, so although lP'1(Cp) is a bounded space, it is not compact, nor
even locally compact.
We can define open and closed disks in lP'1(K) using the chordal metric. We will
use the same notation as above:
D(Q, r) = {P E lP'1(K) : p(P, Q) < r} =open disk of radius r in lP'1(K),
D(Q, r) = {P E IP' 1(K) : p(P, Q) ~ r} =closed disk of radius r in IP' 1(K).
It should be clear from context whether we are working in K or in lP'1(K). We
observe that lP'1(K) is equal to the union of the two unit disks
lP'1(K) = {[x,1]: ixi:::; 1} U {[1,y]: iYi:::; 1},
each ofwhich is metrically isomorphic to RK.
Remark 5.7. It is sometimes convenient to make the assumption that the radius r of a
disk is equal to the absolute value of some element of K*. In this case we say that the
disk has rational radius. We note that if K/Qp is a finite extension of ramification
degree e, then the value set iK*i has the form {pnfe: n E Z}. Similarly, the value
set iC;i is equal to {pr : r E Q}. Thus rational radius really means that the radius is
a rational power of p.
The nonarchimedean nature ofthe absolute value implies that a sequence {ai h~o
in K is Cauchy if and only if
)im lai+l - ai I = 0.
t->00
244
5. Dynamics over Local Fields: Bad Reduction
This follows immediately from the inequality
Jan- ami:::; max Jai+l- ail·
m::=;t<n
This observation and the completeness of K tell us that a power series
00
¢(z) = I>i(z- a)i E K[z]
i=O
converges if and only if
lim lai(z- a)i I = 0.
t--+00
In other words, the "nth term test" from elementary calculus becomes both necessary
and sufficient in the nonarchimedean setting.
Definition. A function¢: D(a, r) ~ K is holomorphic (or analytic) if it is represented by a power series
00
¢(z) = I>i(z- a)i E K[z- a]
(5.2)
i=O
that converges for all z E D(a, r). The order of¢ at a, denoted by orda(¢), is the
smallest index i such that ai i- 0.
A meromorphic function on jj (a, r) is a quotient ¢ = ¢1/¢2 of functions ¢ 1
and ¢ 2 i- 0 that are holomorphic functions on D(r, a). 1 A meromorphic function ¢ = ¢1 j ¢2 induces a well-defined map
¢: D(a, r) ---t JP> 1 (K),
The order of¢ at a is the difference
We say that¢ has a zero (respectively a pole) at z = a if orda (¢) > 0 (respectively
iforda(¢) < 0).
The next proposition describes some elementary properties of nonarchimedean
holomorphic and meromorphic functions.
Proposition 5.8. (a) Let ¢(z) be a holomorphicfunction on the closed disk D(a, r)
and let bE D(a, r). Then ¢(z) is a holomorphicfunction on D(b, r), i.e., ¢(z)
is given by a convergent power series in K[z- b].
(b) Let ¢(z) be a nonzero holomorphicfunction on D(a, r). Then the zeros of¢(z)
in D(a, r) are isolated. This means that if ¢(b) = 0, then there is a disk D(b, t:)
such that ¢(z) i- 0 for all z E D(b, t:) ' {b }.
1More precisely, a meromorphic function is an equivalence class of pairs (t/>I, t/>2) with t/>2 f- 0, with
two pairs (¢1,</>2) and (1/J1.1/J2) beingequivalentift/>11/J2 = t/>21/J1.
5.2. A Primer on Nonarchimedean Analysis
245
(c) Let ¢(z) be a meromorphicfunction on D(a, r) and suppose that the only pole
of ¢(z) in D(a, r), if any, is z = a. Then ¢(z) is represented by a convergent
Laurent series,
L ai(z- a)i K[z]
00
¢(z) =
E
for all z E D(a,r) "-{a}.
(5.3)
i=-m
(d) Let ¢(z) be a meromorphicfunction on D(a,r). Thenfor every bE D(a,r)
there is an s such that ¢(z) is represented by a convergent Laurent series
on D(b, s ).
Proof (a) Let ¢(z) = Eai(z- a)i, and fork 2': 0, define coefficients bk by the
formula
(These values are not mysterious. If K has characteristic 0 then they are the usual
Taylor coefficients bk = (1/k!)(dk¢/dzk)(b).) The series defining bk converges
since the convergence of ¢(z) on D(a, r) implies that
!ai(b- a)ii::; lailri ---t 0 as i---t oo.
Further, we have the estimate
so
Hence lbklrk __, 0 as k __, oo, so the power series E~=O bk(z- b)k converges
on D(b, r ). Finally, we check that the series represents¢ by computing
~bk(z-b)k = ~~ G)ai(b-a)i-k(z-b)k
= ~ ai ~G) (b- a)i-k(z- b)k
=
L ai(z- a)i.
i=O
(b) Let b E D( a, r) be a point with ¢(b) = 0. We will find a deleted neighborhood of bon which ¢(z) is nonvanishing. We use (a) to expand¢ as a power series ¢(z) = 2::: bi(zcentered at band converging on D(b, r). The assumption
that¢ =/= 0 means that some coefficient is nonzero; we let j be the smallest index such
that b1 =/= 0. The fact that¢ converges on D(b, r) implies that lbilri ---t 0 as i __, oo,
so there is a constant C such that lbilri < C for all i.
Let E = r1+ 1 lb1 1/2C. We claim that the only zero of ¢(z) on D(b, E) is z =b.
To see this, we take any z E D(b, E) and estimate
W
5. Dynamics over Local Fields: Bad Reduction
246
max lbil·lz- bli::::; max C ( lz- bl)i::::; C (lz- bi)J+l
•2:J+l
•2:J+l
r
r
·
11
·
CE I
.
b)1 1.
(z- WI=-2 b·(z·- bl ·l(z- WI<-.-·
= Clz
1
- rJ+l
rJ+ 1
Hence for all z E D(b, E) with z # b, the first term in the series
¢(z) = bj(z- b)1 +
L bi(z- b)i
00
i=j+l
has absolute value strictly larger than any of the other terms, so the nonarchimedean
triangle inequality (Lemma 2.3) implies that
for all z E D(b, E).
In particular, ¢(z) # 0 for all z E D(b, E) with z #b.
(c) Write ¢(z) = ¢1(z)/¢2(z) as a quotient of functions that are holomorphic
on D( a, r). We give the proof of (c) in the case that ¢ 2 ( z) is a polynomial, which is
the only case that we will need. For the general case, see [81, 175] or Exercise 5.7.
Let bE D(a, r) with b #a and ¢2(b) = 0. Taking the Taylor series of ¢1 and ¢2
around b, we can write
and
where 'lj;1 and 'lj;2 are nonvanishing holomorphic functions on D(b, r). Then the assumption that or db (¢) 2: 0 implies that we can write ¢( z) as a quotient
ofholomorphic functions on D( a, r) such that the denominator does not vanish at b.
Repeating this process for each of the zeros of ¢ 2 in D(a, r) other than z = a,
we find that ¢(z) is a quotient of holomorphic functions on D(a, r) such that the
denominator does not vanish except possibly at a. Further, this cancellation process
must stop, since we have assumed that ¢ 2 ( z) is a polynomial, so it has only finitely
many zeros.
By abuse of notation, we again write ¢(z) = ¢ 1(z)/¢ 2(z), where we may now
assume that the polynomial ¢ 2 (z) has no zeros in D(a, r) other than z = a. We
assume for the moment that K is algebraically closed and factor ¢ 2 ( z) as
n
¢2(z) = c(z- a)e
IT (z- ai)ei.
i=l
By assumption, the roots satisfy ai ~ D(a, r), or equivalently lai- al > r, for all
1::::; i::::; n.
The reciprocal 1/¢2 (z) has a partial fraction expansion
5.2. A Primer on Nonarchimedean Analysis
A.
- 1 ="""
~(ze
¢2(z)
247
n
ei
B ..
+ """"""
a)J
£:'t ~ (z- ai)J
J
2)
for certain coefficients Aj, Bij E K. (See [259, IV §5] or [436, §5.10].) The terms
with negative powers of z - a form part of the desired Laurent series. We claim that
all of the other terms are holomorphic on 15( a, r).
To verify this, we let a E K with Ia- al > r, and for each k 2 0 we consider
the function 1 j (z - a) k+ 1 . If k = 0 we get a geometric series
1
z-a
-a-a
(
) -1
1
(z-a)-(a-a)
1-(a-a)- 1 (z-a)
-1
oo
=a-a~
(
z-a
a-a
)n
(5.4)
This series converges for all z satisfying lz- al < Ia- al, so in particular for
all z E 15(a, r), since Ia- al > r. Hence 1/(z- a) is holomorphic on 15(a, r).
More generally, the same argument works using the identity2
1
(z- a)k+l
1
= (-1)k+ 1 ~(n+k)(~)n
((z-a)-(a-a))k+1
(a-a)k+ 1 ~
k
a-a
This completes the proof that 1/7/J2(z) is holomorphic on 15(r, a) in the case that K
is algebraically closed.
If K is not algebraically closed, we let L be the completion of the algebraic
closure of K and use the above argument to write ¢( z) as a power series in L [z - a]
that converges on 15( a, r) for L. We now observe that when we add up the series for
the different terms in the partial fraction expansion, the coefficients of each power
of z - a are symmetric expressions in the roots of 'lj; 2 ( z). Thus the resulting power
series actually lives in K[zWe have shown that 1/¢2 (z) is represented by a Laurent series on 15(a, r). Multiplying that Laurent series by the holomorphic function ¢ 1 ( z) gives a convergent
Laurent series (cf. Exercise 5.3) representing ¢(z) on 15(a, r).
(d) Write ¢(z) = ¢ 1 (z)/¢2(z). Then (b) says that there is a neighborhood 15(b, s)
of b such that ¢ 2 (z) =/=- 0 for z E 15(b, s) with z =/=-b. Hence ¢has no poles other
than bon 15(b, s ), so (c) says that it is represented by a Laurent series on 15(b, s ). D
4
Definition. Let ¢ : 15 (a, r) _., K be a holomorphic function represented by a power
series as in (5.2). The norm of¢, relative to the disk 15( a, r ), is the quantity
11¢11 = ll¢llr,a =sup lailri.
i2:0
Remark 5.9. The assumption that¢ converges on 15(a, r) implies that lailri _., 0, so
there will be at least one, and at most finitely many, indices i satisfying 11¢11 = lailri.
2 0ver a field a characteristic 0, this identity can be derived by taking the k 1h derivative of (5.4) and
dividing both sides by k!. This proof fails in characteristic p if k 2: p, but one can either prove the formula
directly by algebraic manipulations, or else one can clear denominators, observe that this yields an identity
in the two-variable polynomial ring Z[a, z], and then reduce modulo p.
248
5. Dynamics over Local Fields: Bad Reduction
Further, if D(a, r) has rational radius, then r E IK* I, so there is a b E K* satisfying 11¢11 = lbl.
The nonarchimedean nature of the absolute value gives immediately the inequality
l¢(z)l :::; sup lail·lz- ali:::; 11¢11
for all z E D(a, r).
i~O
We now show that 11¢11 is more or less a Lipschitz constant for a holomorphic map¢.
Proposition 5.10. Let ¢(z) E K[z] be a power series converging on D(a, r). Then
l¢(z)-¢(w)l:::; lltJllz-wl
r
forallz,wED(a,r).
Proof We take z, wE D(a, r) and compute
l¢(z)- ¢(w)l =
l~ai((z- a)i- (w- ani
~ lz- wll~a; ~(z- a)'-'-'(w- a)il
:::; iz- wl sup sup iail·lz- aii-l-jlw- alj
i~o
o-:o;j<i
:::; lz- wl sup lailri-1,
since lz- al, lw- al :::; r,
i~O
= lz-wlllt!l.
r
5.3
D
Newton Polygons and the Maximum Modulus
Principle
A powerful tool in complex analysis is the maximum modulus principle, 3 which asserts that a holomorphic function ¢( z) on an open set U c <C has no maximum
on U. Equivalently, if D C U is any closed disk in U, then i¢(z)i attains its maximum value on the boundary aD of D. In this section we prove a nonarchimedean
analogue of the maximum modulus principle that is of similar fundamental importance in the theory of nonarchimedean analysis. However, in the nonarchimedean
setting we cannot prove the maximum modulus principle using path integrals and
Cauchy's residue theorem. In their place we substitute the powerful method of the
Newton polygon.
3 Indeed, Ahlfors [I] says that "because of its simple and explicit formulation it is one of the most
useful general theorems in the theory of functions. As a rule, all proofs based on the maximum principle
are very straightforward, and preference is quite justly given to proofs of this kind."
5.3. Newton Polygons and the Maximum Modulus Principle
249
The Newton polygon of a nonarchimedean power series ¢(z) is very easy to
describe, and it can be used to give a precise description of the distribution of zeros
of the power series. To ease notation, we let
v(z) = -logP lzl
for z E Cp.
Notice that the valuation v is a sutjective homomorphism v :
have normalized I · I so that IPI = p- 1 .
c; _.. , Q, since we
Definition. Let ¢(z) = 2:: anzn E Cp[z] be a power series. The Newton polygon
of¢ is the convex hull of the set of points
{(n,v(an)) :n=0,1,2, ... },
where by convention we set v(O) = oo. Informally, the Newton polygon is created
as follows: take a vertical ray starting at the point ( 0, v( a0 )) and aiming down theyaxis. Then rotate the ray counterclockwise, keeping the point (0, v(a 0 )) fixed, until
it bends around all of the points (n, v(an)).
The Newton polygon consists of a set of line segments that connect the dots
required to create the convex hull. A typical Newton polygon is illustrated in Figure 5.l(a). It has a segment from (0, 5) to (2, 1), a segment from (2, 1) to (4, -1 ), a
segment from (4, -1) to (7, -1), etc. A Newton polygon may have infinitely many
line segments, or it may terminate with an infinite ray.
A fundamental theorem says that the Newton polygon of an analytic function
contains a tremendous amount of information about the roots of the function. It provides a very powerful tool for studying nonarchimedean power series.
Theorem5.11. Let¢(z) E Cp[z] beapowerseries. SupposethattheNewtonpolygon of¢ includes a line segment of slope m whose horizontal length is N, i.e., the
Newton polygon has a line segment running from
(n, v(an))
whose slope is
m=
to
(n + N, v(an+N ))
v( an+N) - v( an)
N
.
Suppose fort her that ¢ converges on the closed disk of radius pm. Then ¢( z) has
exactly N roots a, counted with multiplicity, satisfYing Ia I = pm.
Proof. See [249, IVA, Corollary to Theorem 14]. We observe that the proof of this
result for polynomials or rational functions is quite easy. For power series, one
first proves a p-adic version of the Weierstrass preparation theorem saying, roughly,
that ¢(z) factors into the product of a polynomial g(z) and a nonvanishing power series '1/J(z) such that the initial parts of the Newton polygons of ¢(z) andg(z) coincide.
Then the theorem for power series follows immediately by applying the elementary
result for polynomials to g(z).
D
250
5. Dynamics over Local Fields: Bad Reduction
•
•
¢has no roots in D(O, 1)
(a) Typical Newton polygon
(b) Positive-slope Newton polygon
Figure 5.1: Examples of Newton polygons.
Example 5.12. The Newton polygon of the power series
¢(z) = p5 +p4z+pz2 +pz3 +p-lz4 +p-1 z5 +p3z6 +p-lz7 +p2z8 +p3z9 + ...
is illustrated in Figure 5.1(a). The leftmost line segment has slope -2 and width 2,
so ¢(z) has exactly 2 roots a satisfying jaj = p- 2 (assuming that ¢(z) converges
on the appropriate disk). Similarly, ¢(z) has exactly 2 roots satisfying jaj = p-1,
exactly 3 roots satisfying jaj = 1, and exactly 2 roots satisfying jaj = p2 •
Our first application of the Newton polygon is a nonarchimedean version of the
classical maximum modulus principle from complex analysis.
Theorem5.13. (Maximum Modulus Principle) Let¢(z) E Cp[z] beapowerseries
that converges on a disk D( a, r) ofrational radius.
(a) There is a point {3 E D( a, r) satisfying
1¢({3)1 =
!1P lf/J(z)l = 11¢11.
8
zED(a,r)
(b) If¢ does not vanish on D( a, r ), then
l¢(z)l = 11¢11
for all z E D(a,r).
In other words, either¢ has a zero in D( a, r ), or else it has constant magnitude
on D(a,r).
Remark 5.14. For many applications, it suffices to know that the maximum modulus
principle is true for rational functions ¢(z) E Cp(z). In this case, both (a) and (b) are
quite easy to prove. We do (b) and leave (a) for the reader. Replacing z by (z- a)jc
for some c with jcj = r, we may assume that ¢(z) is well-defined and nonvanishing
on the disk D(O, 1). We factor ¢(z) as
¢(z) = zk (1- a1z)(1- a2z) · · · (1- arz)
(1- f31z)(1- fJ2z) · · · (1- f3sz)
251
5.3. Newton Polygons and the Maximum Modulus Principle
The assumption that ¢ has no zero or pole at z = 0 implies that k = 0. Further,
we have ia;l < 1 and I,B;I < 1 for all i, since otherwise ¢ would have a zero at
a; 1 E 15(0, 1) or a pole at ,8i 1 E 15(0, 1). It follows that
for all z E 15(0, 1),
11- a;zl = 11- ,B;zl = 1
andhence I<P(z)l = 1 forallz E 15(0,1).
Proof of Theorem 5.13. Write ¢(z) = L a;(z- a)i and choose constants b, c E K*
with lei =rand lbl = 11¢11 (cf. Remark 5.9). Consider the series
00
L
00
i
a~c zi.
1/J(z) = b- 1 </J(cz +a)= b- 1 I>i(cz)i =
i=O
i=O
The convergence of¢ on 15(a, r) clearly implies the convergence of 1jJ on 15(0, 1),
and we have
Replacing ¢ by 1/J, we are reduced to proving the theorem under the assumptions
that¢ converges on the unit disk 15(0, 1) and that 11¢11 = 1.
(a) The condition 11¢11 = 1 says that every coefficient of¢ lies in R, and the fact
that¢ converges on 15(0, 1) implies that all but finitely many coefficients lie in the
maximal ideal 9J1 of Cp. So when we reduce ¢(z) modulo 9J1, we get a nonzero
polynomial
Let a1, ... , ar E R be representatives for the distinct roots of¢( z) mod 9J1 in the
residue field R/9.n. The residue field is infinite, since K is algebraically closed, so
we can find a ,8 E R satisfying
,8 ='# ai
(mod 9J1)
for all 1 :::; i :::; r.
Then¢(,8) ='# 0 (mod 9J1), so l¢(,8)1 = 1 = 11¢11·
(b) Write ¢( z) = L aizi as usual and consider the Newton polygon of¢( z). Theorem 5.11 asserts that if some line segment of the Newton polygon of¢( z) were to
have slope m :::; 0, then ¢( z) would have at least one root a satisfying Ia I = pm :::; 1,
and hence ¢(z) would have a root in 15(0, 1). Thus our assumption that ¢(z) has no
roots in 15(0, 1) implies that every line segment of the Newton polygon has strictly
positive slope. (An illustrative Newton polygon is given in Figure 5.l(b).) In particular, directly from the definition of the Newton polygon, this implies that
v(ai) > v(ao) for all i 2: 1.
Equivalently, we have Iai I < Iao I for all i 2: 1. We first observe that this gives
252
5. Dynamics over Local Fields: Bad Reduction
Second, we note that it implies that the constant term a0 in the series
with z E D(O, 1)
has absolute value strictly larger than any of the other terms, so the ultrametric inequality (Lemma 2.3) says that
for all z E D(O, 1).
l¢(z)l = iaol = 11¢11
0
Remark 5.15. The maximum modulus principle (Theorem 5.13(a)), which we stated
over Cp, is true more generally as long as K has infinite residue field, but otherwise
it need not be true. For example, let
K = QP and ¢(z) = z- zP on D(O, 1).
Clearly 11¢11 = 1, but for every f3 E Zp = D(O, 1), Fermat's little theorem tells us
that f3 = f3P (mod p). Hence
l¢(!3)1 :Sp- 1 < 11¢11
forallf3ED(0,1).
We conclude this section with some useful consequences of the maximum modulus principle, including the fundamental fact that p-adic holomorphic and rational
functions send closed disks to closed disks.
Proposition 5.16. Let ¢(z) E Cp[z] be a nonconstant power series that converges
on a disk D( a, r) of rational radius.
(a) </>(D(a, r)) is a closed disk.
(b) Write ¢(D(a, r)) = D(¢(a), s). Then l<i>'(a)l :S sjr.
(c) If¢' (a) =f 0, then there exists a radius t > 0 such that
I<P(z)- ¢(w)l = l<i>'(a)l·iz- wl
for all z, wE D(a, t).
Proof Replacing ¢(z) with ¢(cz +a) -¢(a) for some c E Cp satisfying lei = r,
we may assume that ¢(z) E Cp[z] converges on D(O, 1) and that ¢(0) = 0.
Write ¢(z) = Ln~l anzn as usual (note that ao = 0 since ¢(0) = 0), and let
s=
s_up l¢(z)l.
zED(O,l)
(a) The maximum modulus principle (Theorem 5.13(a)) says that
s = 11¢11 =sup lanl·
n~O
Let j ~ 1 be the smallest index such that s = Iaj 1.
The definition of s clearly implies that ¢(D(O, 1)) ~ D(O, s), so we are reduced
to proving the opposite inclusion. Let f3 E D(O, s). Consider the Newton polygon of
the power series
5.3. Newton Polygons and the Maximum Modulus Principle
253
The fact that I!JI ::; s = la1 I and ian I ::; iaj I for all n ~ 1 implies that
v(!]) ~ v(a1)
and
v(an) ~ v(a1) for all n ~ 1,
so the Newton polygon of ¢(z) -!)includes one or more line segments connecting (0, v(!])) to (j, v( a1)). Further, since the point ( 0, v(!])) is no lower than the
point (j,v(a1 )), at least one ofthose line segments has slope rn::; 0. The fundamental theorem on Newton polygons (Theorem 5.11) then tells us that the power
series ¢( z) - !) has a least one root a satisfying Ia I = pm ::; 1. This proves that
there is a point a E D(O, 1) satisfying ¢(a) =!),and since!) E D(O, s) was arbitrary, this completes the proof that¢( D(O, 1)) = D(O, s ).
(b) As in (a), the maximum modulus principle (Theorem 5.13(a)) gives
s=
sup
zEi)(O,l)
l¢(z)l = 11¢11 =sup lanl ~ la1l·
n;;>O
This is the desired result, since r = 1 and a 1 = ¢' (0).
(c) Continuing with the notation from (b), we compute
(5.5)
For all n ~ 2, all 0 ::; i < n, all t ::; 1, and all z, w E D(O, t) we have
Hence ifwe choose a value oft satisfying 0 < t < 1¢'(0)1/11¢11, then the double sum
in the righthand side of(5.5) has absolute value strictly smaller than W(0) I, and (5.5)
reduces to the desired inequality
l¢(z)- ¢(w)l ::; lz- wl·l¢'(0)1.
This completes the proof of (c) and provides an explicit value fort.
0
Corollary 5.17. In each ofthe following situations, the indicated map¢ is both open
and continuous:
(a) ¢: D(a, r)----> Cp is a nonconstant analytic map.
(b) ¢ : D(a, r) ----> lP' 1 ( Cp) is a nonconstant meromorphic map.
(c) ¢ : lP' 1 ( Cp) ____, lP' 1 ( Cp) is a nonconstant rational function.
254
5. Dynamics over Local Fields: Bad Reduction
Proof (a) The continuity is a consequence of Proposition 5.10, which gives the
stronger assertion that ¢ is Lipschitz. The openness of¢ follows easily from Proposition 5 .16, since the collection of "closed" disks (which is also a collection of open
sets)
{D(b, t) : t > 0 and bE Cp}
forms a base for the topology ofCP.
(b) Let bE D(a, r). Proposition 5.8(d) says that we can write¢ as a Laurent series
in some neighborhood D(b, s) of b. If¢ does not have a pole at b, then (a) completes
the proof. If ¢ does have a pole at b, we consider instead the meromorphic fimction 1/¢(z). It too can be written as a Laurent series in some neighborhood D(b, s),
and it has no pole at b, so again we are done using (a).
(c) A rational function is clearly everywhere meromorphic, since it is the ratio of
two power series (i.e., polynomials) that converge on all ofCp. (For the point at oo,
D
change coordinates and use 1/¢(z).) Hence we are done from (b).
5.4
The Nonarchimedean Julia and Fatou Sets
In this section we recall some basic notions of convergence for collections of fimctions, define the Fatou and Julia sets of a rational map ¢( z) E K (z) over a field K
with an absolute value, and use a formula from Chapter 1 to show that in the nonarchimedean setting, the Fatou set is always nonempty.
We begin with three definitions.
Definition. Let U be an (open) subset of IP' 1 ( K) and let <I> be a collection of functions¢ : U--> JID 1 (K).
(a) <T> is equicontinuous on U if for every P E U and every e > 0 there exists a
t5 > 0 such that
¢(D(P, t5)) c D(¢(P), e)
for every ¢ E <T>.
(b) <T> is uniformly continuous on U if for every e > 0 there exists a t5 > 0 such that
¢(D(P, t5) n U) c D(¢(P), e)
for every ¢ E <T> and every P E U.
(c) <T> is uniformly Lipschitz on U if there is a constant C such that
p(¢(P),¢(Q)) 5o C·p(P,Q)
for every ¢ E <T> and every P, Q E U.
In the case that <T> = {¢n} is the collection of iterates of a single fimction, we say
simply that ¢ is equicontinuous, uniformly continuous, or uniformly Lipschitz.
It is important to understand that equicontinuity is weaker than uniform continuity, because equicontinuity is relative to a particular point, while uniform continuity
is uniform with respect to all points in U. In particular, uniform continuity is an open
condition, whereas equicontinuity is not. Similarly, the uniform Lipschitz property
5.4. The Nonarchimedean Julia and Fatou Sets
255
is an open condition, and indeed it is even stronger than uniform continuity. The
following implications are easy consequences of the definitions:
uniformly
uniformly
equicontinuous
Lipschitz ===} continuous ===} at every point
As we will discover throughout this chapter, in a nonarchimedean setting it is often
just as easy to prove that a family of maps is uniformly Lipschitz as it is to prove that
it is equicontinuous.
Definition. Assume first that K is algebraically closed. Then the Fatou set F( <P) is
the union of all open subsets ofJP> 1 (K) on which <Pis equicontinuous, i.e., F(¢) is the
largest open set on which <P is equicontinuous. The Julia set .J (<P) is the complement
of the Fatou set. In general, the Fatou set of <P over K, which we denote by F( ¢, K),
is the intersection of F( ¢, K) with JP>1 ( K). Similarly, the Julia set .J (¢, K) is the
complement ofF(¢, K) in JP> 1 (K).
Proposition 5.18. For every integer n ~ 1,
and
.J(</Jn) = .J(</J).
Proof We proved this over C in Proposition 1.25, and the same proof works for
nonarchimedean fields using the nonarchimedean Lipschitz property (Theorem 2.14)
in place of the archimedean version cited in Chapter 1.
D
We recall Theorem 1.14, which says that the multipliers of the fixed points of <P
satisfy
""'
L....t
PEFix(cf>)
1
1 - .Ap(.+.) -1
-
provided that Ap( <P) =/:. 1 for all P E Fix( <P ).
'f'
The following corollary of this formula has the useful consequence that nonarchimedean Fatou sets are never empty, a fact that is false in the archimedean setting
(cf. Theorem 1.30, Example 1.31, and Theorem 1.43).
Corollary 5.19. Let K be an algebraically closed field of characteristic 0 that is
complete with respect to a nonarchimedean absolute value. Let ¢( z) E K ( z) be a
rational function ofdegree d ~ 2. Then ¢ has a nonrepellingfixed point.
Proof If some fixed point P has multiplier .Ap( <P) = 1, then Pis nonrepelling and
we are done. Otherwise, we can use Theorem 1.14 to estimate
Hence there is at least one fixed point Q satisfying
256
5. Dynamics over Local Fields: Bad Reduction
It follows that
1-XQ(</>)1::; max{I-XQ(</>) -11, Ill} ::; 1,
so Q is nonrepelling.
D
Proposition 5.20. Let K be an algebraically closed field of characteristic 0 that is
complete with respect to a nonarchimedean absolute value. Let </>(z) E K(z) be a
rational function ofdegree d ~ 2.
(a) Let P E IP' 1 (K) be a nonrepellingperiodic point for</>. Then Pis in the Fatou
set :F(</>).
(b) Let P E IP' 1 ( K) be a repelling periodic point for </>. Then P is in the Julia
set J( </> ).
(c) The Fatou set :F( </>) of</> is nonempty.
Proof Making a change of variables, we can move P to 0, and then Proposition 5.18
lets us replace </> by </>n, so we may assume that 0 is a fixed point. This puts </> into
the form
z 2 F(z)
</>(z) = -\z + G(z)
with G(O) -=/= 0 and A= -\0 (</>) E K. Thus IG(z)l is bounded away from 0 if we stay
away from the roots ofG, and clearly F(z) is bounded on any disk around 0, so there
are a disk D(O, r) and a constant C such that IF(z)/G(z)l ::; C for all z E D(O, r).
Let 8 = min{r, 1/C}.
(a) The assumption that 0 is a nonrepelling fixed point means that I-XI ::; 1. Hence
for z E D(O, 8) we have
l</>(z)l = 1-\z+
z~;~)~::; lzlmax{I-XI,,z~~~)~}::; lzl.
This proves that</> is nonexpanding on the ball D(O, 8), so its iterates are uniformly
Lipschitz on that disk (with Lipschitz constant 1). Hence 0 is in the Fatou set.
(b) For this part the assumption that 0 is a repelling fixed point means that I-\ I > 1.
Hence for z E D(O, 8) with z-=/= 0 we have
Iz~;~) I ::; lzl < 1-Xzl.
The strict inequality allows us to conclude that
l<t>(z) I = 1-\z +
z~;~) I = 1-Xzl for all z E D(O, 8).
Suppose now that 0 E :F(</>). This implies that there is some E > 0 such that
if a E D(O, E), then every iterate satisfies </>n(a) E D(O, 8). But then
l</>n(a)l = 1-XInlal---+ oo
as n--> oo,
contradicting </>n(a) E D(O, 8). Hence 0 E J(</>).
(c) Corollary 5.19 says that</> has a nonrepelling fixed point and (a) tells us that
D
nonrepelling periodic points are in the Fatou set. Hence :F( </>) -=/= 0.
257
5.5. The Dynamics of (z 2 - z) / p
Remark 5.21. Corollary 5.19 is also true in characteristic p, since it follows directly
from the fixed point multiplier formula (Theorem 1.14), which can be proven in characteristic p either by reduction modulo p from the characteristic-a case or by using
the abstract theory of residues (see Exercise 5.10). It follows that Proposition 5.20 is
also true in characteristic p.
5.5
The Dynamics of ( z 2 - z) / p
In this section we illustrate the general theory by studying the p-adic dynamics of
the map
2
cp(z) = z - z
for a prime p 2 3.
p
Note that ¢ has bad reduction at p. An important observation is that the Julia set sits
inside two disjoint disks.
Proposition 5.22. Let cp(z) = (z 2 - z)jp.
(a)
(b)
I
I
lzl > - and lz- II > ====>
p
p
.J(¢) c D(O, I/p) u D(I, I/p).
Proof (a) We consider two cases. First, iflzl > 1, then lzl = lz-II, so we find that
In particular, I¢( z) I > I, so we can apply this inequality again. Repeating the process
shows that
Next suppose lzl ~ 1. Then max{lzl, lz -II} =I, so using the assumption that
lzl > I/p and lz- II > I/p, we compute
l¢(z)l =I z(z; I) I= p · max{lzl, lz- II}· min{lzl, lz- II}> 1.
Now we can apply the earlier result to cp(z) to conclude that l¢n(z) I ___. oo.
(b) From (a) we see that every point outside of the two disks D(O, I/p) U D(I, I/p)
is attracted to the superattracting fixed point at oo, so such points are in the Fatou
D
~
The proposition tells us that the Julia set is contained in D(O, I/p) U D(I, I/p),
but not every point in these disks is in the Julia set. For example,
cp(pa) = pa 2 - a= -a (modp),
5. Dynamics over Local Fields: Bad Reduction
258
so if -o: is not in D(O, 1/p) U D(1, 1/p), then ¢n(po:) is attracted to oo, hence it is
in the Fatou set. It is not easy to predict which points in the two disks are attracted
to oo. However, since only points with bounded orbit can be in .:J(¢),we let
A= A(¢)= { z E <Cp: i¢n(z)l is bounded for n::?: 0}.
It is clear that A is a completely invariant set, and Proposition 5.22 tells us that A
is contained in two disjoint disks, which for notational convenience we henceforth
denote by Io and h:
AcloUh,
where
10 = D(O, 1/p)
and
h = D(1, 1/p).
Definition. The orbit of a point z E A is contained within A, so each iterate ¢n (z)
is contained in one of the two disjoint disks 10 , h. The itinerary of z is the se-
quence [,80 ,81,82 ... ] of numbers ,Bn E {0, 1} determined by the condition
for n = 0, 1, 2, ....
In other words, the itinerary [,Bn] specifies how the points ¢n (z) jump back and forth
between the disks 10 and h.
Our goal is to show that the dynamics of the itineraries of the points in A accurately reflects the actual dynamics of ¢ on the set A. In particular, we will prove
that A = .:J (¢). In order to do this, we make a brief digression to discuss symbolic
dynamics.
5.5.1
Symbolic Dynamics
Symbolic dynamics is a tool for modeling seemingly more complicated dynamical
systems. It has a long history and is used in many areas of mathematics. See [276]
for a thorough introduction to the subject. In this section we briefly develop enough
of the theory to complete our analysis of the dynamics of (z 2 - z) j p.
Let S = { o- 1, o- 2, ... , o-s} be a finite set of symbols and let
SN = { sequence~ of}
elements m S
be the set of sequences [,Bo,81,82 ... ], where each ,Bn E S. (Here N = {0, 1, 2, ... }
denotes the set of natural numbers.) We put a metric on SN by fixing a number p > 1
and setting
p( 0:, ,B) = p -(smallest n with <>n # f3n).
Thus the more that the initial terms of o: and ,8 agree, the closer they are to one
another. It is easy to check that p is a metric, and indeed a nonarchimedean metric:
• 1::::: p(o:,,B)::::: 0.
• p( o:, ,8) = 0 if and only if o: = ,B.
5.5. The Dynamics of (z 2 - z) / p
259
• p(a,'Y) :S max{p(a,(J),p(fJ,'Y)}.
Symbolic dynamics is the study of the dynamics of continuous maps sN ___. sN.
An important map on sN is the left shift map L : sN ___. sN' defined by
In other words, L simply discards the first term in the sequence and shifts each
of the remaining terms to the left. More formally, the sequence L((J) is defined
by L(fJ)n = fJn+1· The next proposition describes some elementary properties of
the map L.
Proposition 5.23. Let sN be the space of S-sequences with associated metric as
above and let L : sN ___. sN be the left shift map.
(a) Ifp(a,(J) < 1, thenp(L(a),L(fJ)) =p·p(a,f3).
(b) L is continuous (indeed Lipschitz), and it is uniformly expanding on each of the
"disks"
{a E SN : ao = ui},
i = 1, 2, ... , s.
(c) The set SN contains exactly sn points satisfYing Ln(a) = a. In other words,
Pern(SN, L) has sn elements.
(d) The periodic points of L are dense in SN.
(e) There exists a point 'Y E SN whose orbit 0 L ( 'Y) = {Ln ('Y) : n 2:: 0} is dense
in SN. (Maps with this property are called topologically transitive.)
Proof (a) The condition p(a, (J) < 1 is equivalent to a 0 = (30 . It is then clear from
the definition that p( L( a), L((J)) = p · p( a, (J), since L( a )n =I L(fJ)n if and only if
an+ 1 =/= fJn+ 1 ·
(b) The metric p satisfies 0 ::; p ::;: l, so (a) implies that p( L( a), £(,6)) ::;: p· p( a, ,6)
for all a, (J. Hence Lis Lipschitz. Further, (a) says that L is expanding by a factor
of p on each of the disks.
(c) A sequence a E SN satisfies L n (a) = a if and only if its first n terms repeat,
i.e., it has the form
.._______...._______...._______..
a= [aoa1 · · · an-1 aoa1 ·. · an-1 aoa1 · · · an-1 · · .]
initial n terms
same n terms
same n terms
There are s choices for each of ao, a1, ... , an-1, so sn possible elements.
(d,e) We leave the proof of these elementary results as exercises for the reader (Exercise 5.12).
D
5.5.2
The Dynamics of (z 2 - z) / p
Recall that we are studying the p-adic dynamics ofthe map ¢(z) = (z 2 - z)jp for a
prime p 2:: 3, and that we have defined
260
5. Dynamics over Local Fields: Bad Reduction
10 = D(O, 1/p)
and
h = D(1, 1/p),
A= {z E Cp: cf;n(z) is bounded for all n ~ 0} C 10 U h,
{3(z) = [f3of3If32 ... ] =(Itinerary of z E A), where f3n is defined by cf;n(z) E I/3n·
The sequence f3(z) is an element of the space ofbinary sequences {0, 1 }N, so we
obtain a map (of sets)
{3(z) =itinerary of z.
Proposition 5.24. With notation as above, the itinerary map {3 : A - t {0, 1}l\1 has
the following properties:
(a) {3 is injective.
(b) {3(A n Qp) = {0, 1}N, i.e., {3 restricted to An QP is surjective.
(c) {3 respects the metrics on A and {0, 1}1\1, i.e.,
iz- wl = p(f3(z), {3(w))
for all z,w EA.
(d) Let L : {0, 1}l\1 - t {0, 1}l\1 be the left shift map on {0, 1}l\1. Then {3 o ¢ = L o {3.
In other words, the following diagram is commutative:
Proof (a) We begin with the following observation. Let u be 0 or 1.
If z, wE Iu, then [¢(z)- ¢(w)[ = p ·lz- wl.
(5.6)
To verify (5.6), we use the assumption that z, w E Iu to write z = u + px and
w = u + py with lxl :S 1 and IYI :S 1. Then
cf;(z)- cf;(w) = (x- y)((2u- 1) + p(x + y)).
The quantity 2u- 1 is ±1, so the final factor is a unit and we have
iz-wi
[¢(z)-¢(w)[=lx-yl= -P- =p·lz-wl.
Now suppose that z,w E A have the same itinerary {3. Then ¢n(z) and cf;n(w)
are in the same I13n for every n ~ 0, so applying (5.6) to these two points, we find
that
for all n ~ 0.
Hence by induction,
for all n ~ 0.
(5.7)
5.5. The Dynamics of (z2 - z) / p
261
The lefthand side of (5. 7) is bounded as n --t oo, since z and w are in A. Hence we
must have Jz- wl = 0.
(b) Let wE ! 0 U h. We claim that ¢- 1 (w) consists of two points {z 0 , zl}, one of
which is in ! 0 and one of which is in h, and further, if w E Qp, then z0 and z 1 are
in Qp. To see why this is true, we fix w and solve
cp(z) = w,
or equivalently, solve
z 2 - z- pw = 0.
The quadratic formula, the binomial theorem, and some elementary algebra gives the
two solutions explicitly as
1±(1+4pw) 112 =~±~~(1/2)( 4 )k=~±~~(-1)k- 1 (2k)( w)k.
2
2 2 L.....t k
pw
2 2 L.....t 2k - 1 k p
k=O
k=O
s
Note that the series converge for allJwl
1. The plus sign gives a solution satisfying
z 1 = 1 (modp) and the minus sign gives a solution satisfying z2 = 0 (modp), so
one solution is in ! 0 and the other is in h. Further, if w E QP, then z 1 and z2 are
also in Qp.
We are going to apply iliis observation to (nonempty) sets U c ! 0 U h. The
inverse image ¢- 1 ( U) of such a set thus consists of two nonempty disjoint pieces,
one in Io and one in h. And since the inverse images of QP points in ! 0 U h are
again in Qp, we have
In particular, if U contains a point in Qp, then each of the pieces of ¢- 1 (U) contains
a point in Qp.
Let Jo = Io n Qp and J 1 = h n Qp, or equivalently, J0 and J 1 are the
open unit disks in Zp centered at 0 and 1, respectively. For any binary sequence
of aoa1a2 ... an wiili n ~ 1, define a set
Ja 0 a1 ... an = { z E Qp: z E Ja 0 and cp(z) E Ja 1 and ¢ 2 (z) E Ja 2
... and cpn(z) E Jan}
= Qp n Jao n
¢- 1 (JaJ
n
¢- 2 (Ja2)
n ... n ¢-n(JaJ·
Notice that if z E Ja 0 a1 ... an n A, then ilie initial n terms in the itinerary of z
are aoa1 ... an.
The sets Ja 0 .•. an are closed in Qp, since¢ is continuous and Jo and J1 are closed,
and iliey are nested in ilie sense that
We claim iliat they are also nonempty. To prove this by induction, suppose that we
know iliat Jf30 f3 1 ... f3n-l is nonempty for all sequences f3of31 ... f3n-1 oflength n. We
use the equality
262
5. Dynamics over Local Fields: Bad Reduction
and apply the inductive hypothesis to see that Ja 1a 2... an is nonempty. Then from our
earlier remarks we know that ¢- 1 { Ja 1a 2 ... an) consists of two pieces, one in Jo and
one in J1. so its intersection with Ja 0 is nonempty.
Let a= [aoa1a2 ... ] be any binary sequence. Then
Ja ~
n
Jaoal ... an
n?:O
is the intersection of a nested sequence of nonempty closed bounded subsets of Qp.
By compactness, the set Ja is nonempty, and by construction, any point in Ja has
bounded orbit and itinerary a. This proves that the itinerary map
is surjective.
We also note that since every point in Ja has itinerary a and since we know
from (a) that f3 is injective, it follows that Ja consists of a single point.
(c) If z = w, there is nothing to prove. Assume that z "=I wand let k be the first
index at which their itineraries diverge, so by definition, p (/3 (z), f3 (w)) = p- k. This
means that for each 0 ~ n < k, the points <Pn(z) and <Pn(w) are in the same disk
(either J0 or J 1 ), but <Pk(z) and <Pk(w) are in different disks. Repeated application
of(5.6) tells us that
for all 0 ~ n ~ k.
(5.8)
On the other hand, the assumption that <Pk (z) and ¢k (w) are in different disks (one
in J0 and one in J 1 ) implies that
(5.9)
Combining (5.8) and (5.9) yields
Jz- wJ = I<Pk(z) -k ¢k(w)l = ~ = p(f3(z),f3(w)).
p
p
(d) It is clear that the itinerary of <P(z) is the left shift of the itinerary of z.
0
Proposition 5.24 allows us to identify the dynamics of the polynomial map
¢( z) = ( z2 - z) j p with the dynamics of the shift map on the space of binary sequences. It then becomes an easy matter to read off a great deal of information about
the dynamics of <P(z) from elementary properties of the shift map.
Corollary 5.25. Let p ;::: 3 be a prime and let
2
<P(z) = z - z
p
(a) .J(¢) =A c Qp.
and
A= {z E Cp: <Pn(z) is boundedfor all n;::: 0}.
5.6. A Nonarchimedean Montel Theorem
263
(b) # Per n ( ¢) = 2n + 1 for all n 2:: 1, and aside from the fixed point at oo, every
periodic point of¢ is repelling.
(c) The repelling periodic points are dense in .J (¢ ).
(d) There exists a point w E .J(¢) such that the orbit Oq,(w) is dense in .J(¢).
(Thus ¢ is topologically transitive on .J (¢).)
Proof Let f3 : A -t {0, 1}N be the itinerary map. Proposition 5.24 tells us that f3 is
injective, and further that it is surjective even when restricted to An Qp. It follows
that A c Qp, which proves one part of(a).
For the other part, we note that Proposition 5.24 says that f3 is an isomorphism
of metric spaces
f3: A-t {0, 1}N.
(The proposition says that f3 is bijective and respects the metrics, so its inverse also
respects the metrics.) The proposition also tells us that this isomorphism transforms
the map¢ into the left shift map L, i.e., f3o¢ = Lof3. Hence the dynamical properties
of¢ acting on A are identical to the dynamical properties of L acting on the space of
binary sequences.
We proved earlier that .J (¢) c A, so
f3: .J(¢) ~ .J(L).
However, the shift map L is uniformly expanding on each of the disks /3(10 )
and f3(h) (Proposition 5.23(b)). A uniformly expanding map is nowhere equicontinuous, so .J(L) = L, from which we deduce that .J(¢) =A.
(b) The points not in A are attracted to oo, so the only periodic point in .IP'1 ( Cp) " A
is the fixed point at oo. The periodic points in A are determined by the isomorphism
/3: Pern(¢) n A~ Pern(L),
and we proved (Proposition 5.23(c)) that # Pern ( L) = 2n.
Further, we know that the shift map L is uniformly expanding (by a constant
factor p) on each of /3(10 ) and f3(h), so via the metric isomorphism f3 and the identification of¢ with L, we find that l(¢n)'(z)l = pn for every z E Pern(¢) except
z = oo. In particular, every periodic point other than oo is repelling.
(c,d) The density of the (repelling) periodic points in .J(¢) and the topological
transitivity of¢ on .J(¢) = A follow from the corresponding facts for L acting
on {0, 1}N (Proposition 5.23(d,e)).
D
5.6
A Nonarchimedean Montel Theorem
In this section we give an important characterization of the Fatou and Julia sets and
use it to draw a number of conclusions. The analogous results over C are classical and
due mainly to Fatou and Julia. The nonarchimedean results are due to Hsia, whose
paper [208] we follow. We assume throughout that K is complete and algebraically
closed.
264
5. Dynamics over Local Fields: Bad Reduction
Theorem 5.26. (Hsia [208]) Let <I> be a collection of power series that converge
on D( a, r ), and suppose that there is a point a E K such that
U<P(D(a, r)).
a~
</JEif>
Then <I> is uniformly Lipschitz on D( a, r) with respect to the chordal metric p, so in
particular, it is equicontinuous on D( a, r ).
Proof We choose some c E K with lei = r (see Remark 5.7) and replace each
function ¢(z) E <I> by ¢(cz +a). This changes the chordal metric by only a bounded
amount (Theorem 2.14), so we are reduced to the case of power series converging
on the unit disk D(O, 1). By assumption, the function ¢(z) -a does not vanish
on D(O, 1), so Theorem 5.13(b) tells us that
i¢(z)- ai = II¢- ail
is constant for all z E D(O, 1).
(5.10)
Suppose that a =I- 0 (the case a = 0 is similar) and consider the following
decomposition of the set <I>:
<I>1 = {¢ E <I>: II¢- ail~ ial},
<I>2 = {¢ E <I>: II¢- ail> iai}.
If¢ E <I> 1 and z, wE D(O, 1), then
p(</J(z), ¢(w)) ~ J¢(z)- ¢(w)J
= J(¢(z)- a)- (¢(w)- a)J
~ II¢- ail · iz- wl
from Proposition 5.10,
~ ial·lz- wl
since ¢ E <I>1,
= ialp(z,w)
since z, w E D(O, 1).
This shows that the functions in <I> 1 satisfy a uniform Lipschitz inequality.
Next let¢ E <I> 2• Note that the definition of <I> 2 and (5.10) imply that
i¢(z)- ai =II¢- all> ial
for all z E D(O, 1).
(5.11)
Using (5.11) and Lemma 2.3, and then applying (5.1 0) again, we find that
J¢(z)l = l(¢(z)- a)+ al = I<P(z)- al =II¢- ail
for all z E D(O, 1). (5.12)
We use this formula to compute, for z, wE D(O, 1),
I<P(z)- ¢(w)l
· max{1, i¢(w)l}
i¢(z)l}
p(¢(z),¢(w)) = max{1,
i(¢(z)- a)- (¢(w)- a) I
max{1, i¢(z)i} · max{1, i¢(w)l}
5.6. A Nonarchimedean Montel Theorem
265
II¢- ail · iz- wl
2
- max{ 1, II¢- ail}
<
~
iz-wl
from (5.12) and Proposition 5.10,
regardless of the value of II¢- ail,
= p(z, w)
since z, wE D(O, 1).
Thus functions in the set 1> 2 are nonexpanding, so in particular they satisfy a unifonn
Lipschitz condition.
Finally, we note that if a = 0, then the same argument works using the sets
1>1 = {¢ E 1> : 11¢11 ~ 1} and 1>2 = {¢ E 1> : 11¢11 > 1}. We leave the details to the
D
~cr
It is now a simple matter to use Theorem 5.26 to prove a nonarchimedean version
ofMontel's theorem for rational functions.
Theorem 5.27. (Nonarchimedean Monte! Theorem, Hsia [208]) Let 1> be a collection of rational, or more generally meromorphic,functions D(a, r) ----+ JPl 1 (K), and
suppose that the union
U ¢(D(a,r))
(5.13)
c/>Eif!
omits two or more points of JPl 1 ( K). Then 1> satisfies a uniform Lipschitz inequality
on D(a, r), so in particular 1> is an equicontinuous family offunctions on D(a, r).
Proof Let a= [a 1, a 2] and (3 = [(31, (32] be two points ofJPl1(K) that are not in the
union (5.13 ). Consider the family of rational (or merom orphic) functions
\jf
= { a2f(z)- a1g(z) . f(z) E 1>}
(32f(z)
fJ1g(z)
· g(z)
·
By construction, we have
'1/J(z) =1- oo for all '1/J E \jf and all z E D(a,r),
so Proposition 5.8(c) tells us that the functions in \jf are holomorphic on D(a, r)
(i.e., they are given by convergent power series). We also know that '1/J(z) =1- 0 for
all z E D( a, r ), so the functions in \jf omit at least one point in K. It follows from
Theorem 5.26 that there is a constant C 1 such that
p('l/J(z), '1/J(w)) ~ C1 p(z, w)
for all '1/J E \jf and all z, wE D(a, r).
(5.14)
Finally, let A(z) be the linear fractional transfonnation
The assumption that a and (3 are distinct implies that A is invertible. Then a very
special case of Theorem 2.14 (for the rational map A - 1 ) says that there is a constant
C2 = C2(A) = C2(a, (3) > 0 such that
266
5. Dynamics over Local Fields: Bad Reduction
for all z, wE JP1 (K).
(5.15)
Further, by construction we have ¢> E ~ if and only if A o ¢> E w, so for any
z,w E D(a,r),
C1p(z, w) ~ p(A(¢>(z)), A(¢>(w)))
~ C2 1P(¢>(z),¢>(w))
from (5.14),
from (5.15).
This completes the proof of the nonarchimedean version of Montel theorem.
0
Remark 5.28. The proof of Theorem 5.27 is fairly straightforward. Later, in Section 5.10.3.4, we describe a deeper p-adic version ofMontel's theorem on Berkovich
space; see Theorem 5.80.
In the classical setting, there are a number of important properties of the Julia set
that follow more or less formally from Montel's theorem. We conclude this section
with a few instances.
Proposition 5.29. Let¢>: JP 1 (K) ~ JP 1 (K) be a rational map of degree d ~ 2
and let U c JP 1 (K) be an open set such that U n .:1(¢) # 0. In particular, we are
assuming that the Julia set of¢> is nonempty.
(a) The set Un~o ¢>n(U) omits at most one point of JP 1 (K).
(b) Suppose that the set in (a) does omit a point. Then¢> is a polynomial function
and the omitted point is the totally ramifiedfixed point. (In other words, there is
a change of variables f E PGL2(K) such that¢>! (z) E K[z] and the omitted
point has been moved to oo.)
Proof The set U is covered by disks D(a, r), which are both open and closed, so it
suffices to prove the proposition under the assumption that U = D(a, r). If the union
omits two or more points oflP1 (K), then Montel's theorem (Theorem 5.27) implies
that ¢> is equicontinuous on U, contradicting the assumption that U contains a Julia
point. This proves (a).
If the union omits a, then ¢> - l (a) = {a}, so a is a totally ramified fixed point
of ¢. Hence ¢> is a polynomial map by definition (page 17), and after a change of
variables it becomes a polynomial in K[z] (Exercise 1.9(c)).
0
Let ¢> : JP 1 ( K) ~ JP1 ( K) be a rational map of degree d ~ 2. Recall that a
subset E of JP 1 ( K) is said to be completely invariant for ¢> if it is both forward and
backward invariant,
¢- 1 (E) = E =¢>(E).
Proposition 1.24 says that the Fatou set F (¢>) and the Julia set .:J (¢>) are completely
invariant. (The proof in Chapter 1 is over C, but the proof works, mutatis mutandis,
for any complete field.)
In Chapter 1 we used the Riemann-Hurwitz formula to characterize all finite
completely invariant sets (see Theorem 1.6). More precisely, we showed that a finite
completely invariant set E has at most two elements. Further, if #E = 1, then after
a change of variables, ¢(z) E K[z] is a polynomial and E = { oo }, and if#E = 2,
267
5.6. A Nonarchimedean Montel Theorem
then again after a change of variables, ¢(z) = zd or ¢(z) = z-d and E = {0, oo }.
We now show that except for these trivial cases, the Julia set is the smallest closed
completely invariant subset oflP>l (K).
Proposition 5.30. Let ¢ : IP'1 ( K) ----> IP' 1 ( K) be a rational map of degree d ~ 2,
and let E <;::; IP' 1 ( K) be a closed completely invariant subset for ¢ containing at least
three points. Then E is an infinite set and E ;:;> :T( ¢ ).
Proof Theorem 1.6 tells us that a finite completely invariant subset contains at most
two points, so our assumption that #E ~ 3 implies that E is infinite. Notice that the
complete invariance of the closed set E implies the complete invariance of its complement U, which is an open set. It follows that the union Un>o <Pn(U) omits at least
two points, since it in fact omits the infinite set E. Montel's theorem (Theorem 5.27)
tells us that U <;::; F(¢). Hence E ;:;> :T(¢).
D
Remark 5.31. Proposition 5.30 tells us that if the Julia set :T(¢) is nonempty, then it
is the smallest closed completely invariant set containing at least two points. (Notice
that the case of exactly two points is ruled out by the fact that if¢ has a completely
invariant subset containing exactly two points, then¢ is conjugate to either zd or z-d,
in which case its Julia set is empty.)
Corollary 5.32. Let ¢ : IP' 1 ( K) ----> IP' 1 ( K) be a rational map of degree d ~ 2, and
assume that :T(¢) f::. 0.
(a) :T(¢) has empty interior.
(b) Let P E :T(¢) and let
O¢(P) =
U ¢-n(P)
n:::>:O
be the backwardorbitofP. The Julia set :T(¢) is equal to theclosureof0¢(P)
in IP' 1 (K).
(c) :T(¢) is a perfect set, i.e. ,for every point P E :T(¢), the closure of:T(¢) "{P}
contains P.
(d) :T (¢) is an uncountable set.
Proof (a) Let 8:T(¢) denote the boundary of the Julia set :T(¢). Theorem 1.24
tells us that :F(¢) and 8:T(¢) are completely invariant, so the same is true of their
union 8:T(¢) U F(¢). This union is also closed, since its complement is the interior
of :T( ¢).Proposition 5.20 says that the Fatou set F( ¢) is always nonempty, and since
it is open, it must contain infinitely many points. Hence the union 8:T(¢) U F(¢) is
an infinite, closed, completely invariant set, so Proposition 5.30 tells us that
:r(<P) <;::; a:r(¢) u F(¢).
But :T(¢) and F(¢) are disjoint by definition, which proves that :T(¢) = 8:T(¢),
i.e., the Julia set has empty interior.
(b) We know that :T(¢) is completely invariant, so in particular O¢(P) C :T(¢)
for any point P E :T( ¢ ).
5. Dynamics over Local Fields: Bad Reduction
268
Next let U be any open set with U n ..7( ¢) =I= 0. Then Proposition 5.29(a) tells us
that Un>o rj;n(U) omits at most one point, and Proposition 5.29(a) says that if it does
omit a point, that point is a totally ramified fixed point, hence is in the Fatou set. In
particular, the possible omitted point cannot be P, since P E ..7(¢) by assumption.
This proves that P E Un>o rj;n(U), or equivalently, that there is some n 2 0 such
that
This proves that every open set U that intersects ..J (¢) nontrivially also intersects
the backward orbit
P) nontrivially. Hence ..J (¢) is contained in the closure
o; (
of0¢(P).
(c) Let Po E ..J (¢). We claim that the backward orbit
(Po) must contain a nonperiodic point. To see this, suppose instead that 0¢ (P0 ) consists entirely of periodic
points. Then ¢- 1(Po) consists of a single point, so Po is a totally ramified periodic
o;
point and hence in the Fatou set, contrary to assumption. Therefore we can find a
nonperiodic point P1 E 0¢ (Po).
The point P1 is in ..J (¢), since ..J (¢) is completely invariant, so (b) tells us that
Po E closure of 0¢ (PI).
On the other hand, Po is not in
o; (P1), since otherwise P1would be periodic. Hence
P0 E closureof(..J(¢)" {P}).
(d) The Baire category theorem [387, §5.1,5.2] implies that a nonempty perfect
subset of IP'1 ( K) is uncountable.
0
5.7
Periodic Points and the Julia Set
Our goal in this section is to show that the Julia set ..J (¢) of a rational map ¢ is
contained in the closure of the periodic points of ¢. We begin with an elementary
lemma that is obvious in the classical setting by a compactness argument, but which
requires a different proof over a non-locally compact field such as Cp.
Lemma 5.33. Let rPl (z) and ¢2 (z) be power series that converge on D( a, r ), and
suppose that rPl (D( a, r)) n ¢2 ( D( a, r)) = 0. Then
i_nf
zED(a,r)
p(¢1(z),¢2(z)) > 0.
Proof Let
M1 =
syp
zED(a,r)
lr:P1(z)l
and
M2 =
sup
zED(a,r)
lr:P2(z)l.
The maximum modulus principle (Theorem 5.13(a)) says that there are points
z 1 , z2 E D(a, r) such that rPI(zi) = M1 and ¢2(z2) = M2. In particular, M1
and M 2 are finite, since ¢ 1 and ¢ 2 are power series that converge on D(a, r).
269
5. 7. Periodic Points and the Julia Set
Let M = max{M1 , M2 , 1}. Then for any z E D(a, r) we have
On the other hand, the function ¢ 1 - ¢ 2 does not vanish on D( a, r) by assumption,
so Theorem 5.13(b) tells us that
for all z E D(a, r).
Hence
D
The next lemma is used in conjunction with Lemma 5.33 to move a varying set
of pairs of points {a, ,6} to the specific pair {0, 1}.
Lemma 5.34. Let A, B C Cp be bounded sets that are at a positive distance from
one another. In other words, there are constants ~' {J > 0 such that
sup lal :::; ~'
aEA
sup 1,61 :::; ~'
f3EB
and
inf
aEA,{3EB
p(a,,6)=0>0. (5.16)
For each (a, ,6) E A x B, define a linear fractional transformation
La,f3(z) = (,6- a)z +a.
Then there is a constant C > 0, depending only on ~ and o, such that
p(La,{3(z),La',f3'(z')):::; C · max{p(a,a'),p(,6,,6'),p(z,z')}
for all a, a' E A, all ,6, ,6' E B, and all z, z' E !P'1 (Cv).
Remark 5.35. Although Lemma 5.34 appears somewhat technical, it is not saying
anything mysterious. The linear fractional transformation La,/3 is determined by the
three conditions
La,f3(0) = a,
La,f3(1) = ,6,
La,f3(oo) = oo.
The lemma is asserting, roughly, that if we take two nearby (a, ,6) values, then the
associated transformations are close to one another, where we use the chordal sup
norm
p(L,L') =
sup
PEIP'1 (Cp)
{p(L(P),L'(P))}
to measure the closeness of two maps. Thus Lemma 5.34 is equivalent to the assertion
that the map
(a, ,6, z) r----t La,{3(z) = (,6- a)z +a
is Lipschitz.
5. Dynamics over Local Fields: Bad Reduction
270
Proof ofLemma 5.34. To ease notation, for x, y E Cp we write
lx, Yl = max{lxl, IYI}.
We also assume (without loss of generality) that ~ 2:: 1 and o :S: 1. Then for
any a, a' E A and (3, (3' E B we have
Ia- a' I= p(a, a') ·Ia, 1l·la', 11 :S: ~ 2 p(a, a'),
(5.17)
1!3- !3'1 = p((J,(J') ·1(3, 11·1!3', 11::::: ~ 2 p((3,(3').
Leta, a' E A,let(J,(J' E B,andletz,z' E lP' 1 (Cp).Directlyfromthedefinition s
of La,(3 and the chordal metric, we have
1(((3'-a')z'+a')-(((3-a)z +a)l
1
p(La,(3(z), La',(3'(z )) = max{I((J- a)z + al, 1} · max{l(fJ'- a')z' + a'l, 1}.
(5.18)
Assuming for the moment that z -=1- oo and z' -=1- oo, we multiply out the numerator
and estimate it using the triangle inequality:
1((!3' -a')z' +a')- (((3-a)z+a)l
= lf3'z'- (Jz- a'z' + az +a'- al
= lf3'(z'- z) + ((3'- (J)z- a'(z'- z) +(a- a')z +(a'- a)l
:S: max{lfJ'- fJI·Izl, Ia'- al·lz, 11, lz'- zl·la', (3'1}
:S: max{ ~ 2 p((J, (3') · lzl, ~ 2 p(a, a') · lz, 11, lz' - zl · Ia', !3'1}
from(5.17),
:S: max{~ 2 p((J,(J') ·lzl, ~ 2 p(a,a') ·lz, 11, ~ ·lz'- zl}
from (5.16),
:S: max{ ~ 2 p((3, (3') ·lzl, ~ 2 p(a, a') ·lz, 11, ~p(z, z') ·lz, 1l·lz', 11}
definition of p,
:::; ~ 2 ·
max{p((J, (3'), p(a, a'), p(z, z')} ·lz, 11 ·lz', 11.
Substituting this into (5.18) and doing a little bit of algebra yields
max{p((J, (3'), p(a, a'), p(z, z')}
< ~ 2 • ------;-:----'1---'z'----,1"--l-----,-~
max{I((J- a)z + al, 1}
-
lz', 11
max{l(fJ'- a')z' + a'l, 1} ·
(5.19)
We are left to show that the righthand side is bounded in terms of 8 and ~. By
symmetry, it suffices to bound the first fraction.
We consider two cases. First, if Iz I :S: ~ /8, then we have the trivial estimate
< lz 11 < ~.
lz,11
' - 8
max { I((3 - a) z + a I, 1} -
5.7. Periodic Points and the Julia Set
271
Second, suppose that lzl > b./8. Then the fact (5.16) that 1.8- al 2:: 8 implies that
j(,B- a)zj >b. 2:: lal,
Hence
1(,8- a)z + al = j(,B- a)zj2: 8lzl.
so
lz, II
max{l(,8-a)z+al,l}
=
lz, II
< ~ < b..
max{8lzl,l}- 8- 8
Thus b./ 8 serves as an upper bound in both cases, and substituting this bound
into (5.19) yields the estimate
4
p(La,(3(z),La',(3'(z')):::; ~ max{p(,8,,8'),p(a,a'),p(z,z')}.
2
This completes the proof of Lemma 5.34 with explicit dependence on 8 and b. in the
case that z I- oo and z' -:f. oo. The remaining cases are similar and are left to the
0
reader.
We next prove a version ofMontel's theorem in which the two omitted points are
allowed to move. Lemma 5.34 is the key technical tool that allows us to uniformly
replace the two moving points with two particular points, thereby reducing the proof
to our earlier result.
Theorem 5.36. (Montel Theorem with Moving Targets, Hsia [208]) Let c/->1, ¢ 2 be
power series that converge on D( a, r ), and suppose that
¢1(D(a,r)) n¢2(D(a,r)) = 0.
Further let <I> be a collection of rational, or more generally meromorphic, functions
on D( a, r) such that
¢(z) I- ¢1 (z)
and ¢(z) I- ¢2 (z)
for all¢ E <I> and all z E D( a, r ).
Then <I> satisfies a uniform Lipschitz inequality on D( a, r ), so in particular <I> is an
equicontinuousfamily offonctions on D(a, r).
Proof The proof is very similar to the proof of Theorem 5.26, but somewhat more
elaborate.
First we note that since ¢ 1 , ¢ 2 : D( a, r) --t K have disjoint images, there is
at least one point a omitted by both of them. Making a linear change of variables
z f-t z- a, we may assume that ¢ 1 and ¢ 2 omit the value 0. Then Theorem 5.13(b)
tells us that
for all z E D(a, r).
We are going to apply Lemma 5.34 to the disjoint bounded sets ¢ 1 ( D( a, r))
and ¢ 2 ( D( a, r)). Thus for each point w E D( a, r ), if we let
Lw(z) = (¢2(w)- rPl(w))z + ¢1(w) E PGL2(Cp),
272
5. Dynamics over Local Fields: Bad Reduction
then Lemma 5.34 says that there is a constant C such that
p(Lw(z), Lu(z')) :::; C max{p( ¢1 (w), ¢1(u)), p( ¢2(w), ¢2( u)), p(z, z')}
forallw,u E D(a,r) andallz,z' E JP>1(Cp). (5.20)
(In the notation of Lemma 5.34, we have set a= ¢ 1(w), (3 = ¢2(w), a'= ¢ 1(u),
(3' = ¢2 (u).)
We next use the fact that ¢1 and ¢ 2 themselves satisfy a Lipschitz condition.
More precisely, Proposition 5.1 0 says that
for all w, u E D(a, r) and i = 1, 2.
Since w and u are bounded, this implies that there is a constant C' ~ 1 such that
for all w, u E D(a, r) and i = 1, 2.
Substituting this into the inequality (5.20) yields
p(Lw(z), Lu(z')) :::; CC' max{p(w, u), p(z, z')}
for all w, u E D(a, r) and all z, z' E JP>1(Cp)·
(5.21)
We are now ready to prove Theorem 5.36. The idea is that we know that each
¢ E <I> omits at least two values, but the omitted values may vary with ¢, so we use
the linear transformations Lw to move the omitted values to two specific points. Thus
for each ¢ E <P we define a new function
'lj;q,(w) = L;;/(¢(w)),
and we consider the family of functions
Each 'lj;q, can be expressed as a rational function of ¢ 1, ¢ 2, and ¢, so it is a
meromorphic function on D(a, r). Further, since we are given that ¢( z) i- ¢ 1(z)
and ¢(z) i- ¢ 2(z) for all z E D(a, r), it follows that
'lj;q,(w) i- 0
and
'lj;q,(w) i- 1
for all wE D(a, r).
To see why this is true, note that
'lj;q,(w) = 0 <===:::> L;;/(¢(w)) = 0 <===:::> ¢(w) = Lw(O) = ¢1(w),
contradicting the assumption on¢, and similarly for 'lj;q,(w) = 1. Hence the family
of maps W on D( a, r) omits at least the two values 0 and 1, so Montel's theorem
(Theorem 5.27) tells us that wsatisfies a uniform Lipschitz inequality,
p('¢q,(u),'¢q,(w)):::; C"p(u,w)
for all¢ E <I> and all u, wE D(a, r). (5.22)
5.7. Periodic Points and the Julia Set
273
Using this and our earlier estimates, we compute, for¢ E <I> and u, w E D( a, r ),
p(¢(u), ¢(w)) = p(Lu('l/Jq;(u)), Lw('l/Jq;(w)))
::; CC' max{p(u, w), p('l/Jq;(u), 7/Jq;(w))}
::; CC' C" p( u, w)
by definition of 7/Jq;,
from (5.21),
from (5.22).
This completes the proof that the family of maps <I> is uniformly Lipschitz.
D
We now have the tools to prove the main theorem of this section.
Theorem 5.37. (Hsia [208]) Let ¢(z) E K(z) be a rational function of degree d
with d 2: 2. Then
.J(¢) C Per(¢),
i.e., the closure of the periodic points of¢ contains the Julia set of¢.
Proof We may clearly assume that .J (¢) is not empty. Take any open set U having
nontrivial intersection with .J (¢). We must show that U contains a periodic point.
The Julia set is a perfect set (Corollary 5.32), so the open set U actually intersects .J (¢) in infinitely many points. In particular, there is a point P E U n .J (¢) that
is not the image of a ramification point of¢, since ¢ has at most 2d - 2 ramification
points. This implies that there is a neighborhood D(P, r) C U of P such that
consists of d disjoint open sets with the property that the maps
¢: v; ___, D(P,r)
for 1 ::; i ::; d
are bijective. In particular, they have inverses
¢i : D(P, r) ~ v;
given by convergent power series. (This is a p-adic version of the one variable inverse
function theorem. See Exercise 5.5.) We take i = 1 and i = 2 and consider the
maps ¢ 1 and ¢ 2 and the disjoint sets V1 and V2 as illustrated in Figure 5.2.
We now examine the effect of applying the iterates ¢n of¢ to the disk fJ (P, r).
The assumption that D(P, r) contains a point of the Julia set of ¢ means that ¢ is
not equicontinuous on D(P, r), so Theorem 5.36 tells us that the iterates of¢ cannot
avoid both of the moving targets described by the power series ¢ 1 and ¢ 2 . Hence
there exists an iterate ¢n of¢ and a point Q E D(P, r) such that either
or
Applying ¢ to both sides and using the fact that ¢ o ¢ 1 and ¢ o ¢ 2 are both the identity
map on D(P, r) yields
¢n+l(Q) = Q,
so Q E fJ (P, r) c U is a periodic point.
D
274
5. Dynamics over Local Fields: Bad Reduction
Figure 5.2: Inversion of <P over a critical-point-free neighborhood.
Remark 5.38. In the classical setting over C, one can further show that the Julia set is
equal to the closure of the repelling periodic points. This follows from the complex
analogue of Theorem 5.37 combined with the fact that a rational function over C has
only finitely many nonrepelling periodic points. Unfortunately, a rational function
over Cp may well have nonempty Julia set and infinitely many nonrepelling periodic
points. However, one still hopes that the classical result is true in the nonarchimedean
setting.
Conjecture 5.39. (Hsia [208]) Let </>( z) E K (z) be a rational function of degree
d ~ 2. Then the Julia set .:1 (</>) is equal to the closure ofthe repelling periodic points
of¢.
Some evidence for Conjecture 5.39 is provided by the following result ofBezivin.
It says that if the conjecture is false, then there are maps with nonempty Julia set
containing no periodic points.
Theorem 5.40. (Bezivin) If a rational function </>( z) E Cp (z) has at least one repelling periodic point, then J (</>) is the closure of the repelling periodic points of¢.
In particular, one repelling periodic point implies infinitely many repelling periodic
points.
Proof See [71] for the first assertion. The second then follows immediately from
Corollary 5 .32(d), since an uncountable set cannot be the closure of a finite set.
D
However, some evidence against Conjecture 5.39 is provided by Benedetto [58,
Example 9], who shows that it is possible for a rational function to have a sequence
of attracting periodic points whose limit is a repelling periodic point! Further, a
slight variation of [57, Example 3] shows that for every n > 0 there is a polynomial <P(z) E Cp[z] of degree p + 2 that has no repelling periodic points of period
smaller than n, yet ¢( z) does have repelling periodic points of higher periods.
Example 5.41. Consider the polynomial map
zP- z
cp(z) = - - .
p
5.7. Periodic Points and the Julia Set
275
It is clear that the Julia set of¢ is contained in D(O, 1), since if lad > 1, then laP I >
lal, so
I¢( a) I= IaP; a I= p ·laP- al = p ·laPI >pial.
Hence ¢n(a) ---too, so a E F(¢), since a is attracted to the attracting fixed point at
infinity.
We also observe that if a E D(O, 1) n Qp = Zp, then Fermat's little theorem
tells us that aP = a (mod p), so ¢(a) E Zp. Thus Zp is a completely invariant
subset of¢. Hsia [206, Example 4.11] (see also [449]) explains how to identify the
dynamics of¢ on Zp with a shift map on p symbols, similar to the example studied
in Section 5.5, from which one deduces the following facts:
• .7(¢, Qp) = Zp.
• .J (¢, Qp) contains all of the periodic points of ¢ (other than oo), so in particular all of the periodic points of¢ are defined over Qp, and all except oo are
repelling.
• .7(¢, Qp) = .7(¢, Cp), since Theorem 5.37 tells us that .7(¢, Cp) is contained
in the closure of the periodic points of¢. Thus .7(¢, Cp) is compact. (See
also [73].)
Example 5.42. Let p :2: 5 be a prime, and let ¢(z) = pz3 + az 2 + b with a, bE
We first consider the fixed points of¢, which are the roots of the equation
z;.
pz 3 + az 2 - z + b = 0.
z;
The assumption that a, b E
implies that the roots satisfy la1l = p and la2l =
la3l = 1. (Look at the Newton polygon!) We also observe that pa~ and aai have
norm p 2, while a 1 - b has norm p, so a 1 must have the form
a
a1 = - - + c
p
for some c with lei = 1.
This allows us to compute
l¢'(ai)I = l3pai + 2aa1- II
= lal(3pal + 2a) -II= la1( -a+ 3pc)- 11 = la1l = p.
Thus a 1 is a repelling fixed point of¢.
A similar, but more involved, calculation can be used to show that there are repelling periodic points of higher orders. Alternatively, we can invoke Bezivin's Theorem 5.40, which says that the existence of the single repelling fixed point a 1 implies
that¢ has infinitely many repelling points whose closure is .7(¢).
In order to study the periodic points in the Fatou set, we observe that ¢ is nonexpanding on D(O, 1). To see this, note that¢ maps the disk D(O, 1) to itself, so in
particular 11¢11 ::; 1. Applying Proposition 5.10 yields
l¢(z)- ¢(w)l :S lz- wl
for all z, w E D(O, 1).
5. Dynamics over Local Fields: Bad Reduction
276
Hence¢ is uniformly Lipschitz on D(O, 1), so D(O, 1) c F( ¢).Notice that the nth iterate of¢ has the following form (think about the reduction of ¢n modulo p, which
is the same as the nth iterate of the reduction ¢(z) = iiz 2 +b):
¢n(z) = Az 3 n + · · · + Bz 2 n+I + Cz 2 n + Dz 2 n-l + · · · + Ez +F.
coefficients in p'llp
i
c E z;
coefficients in ?lp
Again using the Newton polygon, we see that the polynomial ¢n(z)- z has exactly 2n roots a (counted with multiplicity) satisfying lal :::; 1. These roots are
points of period dividing n for ¢ and they are in the Fatou set, since we showed
that D(O, 1) c F(¢). One can prove that this gives infinitely many distinct periodic
points in D(O, 1). See [206, Example 4.11].
5.8
Nonarchimedean Wandering Domains
We first recall a famous theorem from complex dynamics (see Theorem 1.36).
Theorem 5.43. (Sullivan's No Wandering Domains Theorem [426])
Let ¢(z) E C(z) be a rational function of degree d ;:=: 2 over the complex numbers
and let U C F( ¢) be a connected component of the Fatou set of¢. Then U is preperiodic in the sense that there are integers n > m > 0 such that
In other words, the connected component U does not wander, whence the name of
the theorem.
The first obstacle to translating Sullivan's theorem to the nonarchimedean setting
is the fact that Qp and Cp are totally disconnected. Thus with the classical definition
of connectivity, there is no good way to break up the Fatou set into "connected" components. Various alternatives have been studied, including disk connectivity, rigid
analytic connectivity, and the use of Berkovich spaces.
In this section we consider disk connectivity, which is the simplest of the three to
describe. We state a theorem of Benedetto to the effect that a large class of rational
maps in Qp (z) have no wandering "disk domains" and illustrate the result by proving
it under the somewhat stronger hypothesis that there are no critical points in the Julia
set of¢. We also note that Benedetto has shown that the statement is false over Cp,
that is, there are rational maps over CP that do have wandering disk domains.
Much work has been done on this problem, especially in a series of papers by
Benedetto [54, 56, 57, 58, 59, 63, 62] and Rivera-Letelier [372, 373, 375, 376, 378],
but there are still many open questions. The material that we cover in this section is
due to Benedetto.
5.8. Nonarchimedean Wandering Domains
5.8.1
277
Disk Domains and Disk Components
The ordinary definition of connectivity is not useful for studying the totally disconnected spaces iQp and Cp, nor is the notion of path connectedness helpful. We begin
with an abstract notion of connectivity that uses a chain of "disks" in place of a path.
Definition. Let X be a topological space and let V be a collection of open subsets
of X. (In practice, V will be a base for the top logy of X.) For convenience, we refer
to the sets in Vas disks.
Let U c X be an open subset and let P E U. The disk component of U containing P (relative to V) is the set of Q E U with the property that there is a finite
sequence of disks D1, D2, ... , Dn C U such that
Di n Di+ 1 =1- 0 for alll :::; i < n.
In other words, Q is connected toP by a path of disks, as illustrated in Figure 5.3,
although as we shall see, in the nonarchimedean world, Figure 5.3 is somewhat misleading. Note that we define disk components only for open subsets of X. It is easy
to see that U breaks up into a disjoint union of disk components (Exercise 5.22).
Figure 5.3: A path of disks from P to Q.
Example 5.44. Let X = C and let V be the usual collection of open disks in C.
Then the disk components of an open set U c X are the same as the usual pathconnected components. This is clear, since if r is a path from P to Q, then r can
be covered by open disks contained in U, and the compactness of r shows that it
suffices to take a finite subcover. Thus the definition of disk components and the
related notion of disk connectivity (see Exercise 5.23) are reasonable. For example,
Sullivan's theorem may equally well be stated in terms of the disk components of the
open set :F( ¢ ).
For the purposes of this section, we modify slightly our definition of disks
in lP' 1 ( Cp). The resulting topology is the same, and indeed the disks contained within
the unit disk D(O, 1) are the same, but the alternative definition is more convenient
for working with disks that may contain the point at infinity.
Definition. The standard collection of closed disks in lP' 1 (Cv), denoted by Vclosed, is
given by
278
5. Dynamics over Local Fields: Bad Reduction
~~= {
all closed disks D(a, r) and
.
the complement JPl 1(Cp) "D(a, r) of all open d1sks D(a, r).
Of course, despite the name, all of the disks in Vc1osed are both open and closed sets.
Note that Vc1osed includes all such disks, not only the disks of rational radius (cf.
Remark 5.7). One can show (Exercise 5.24) that the disks of rational radius in Vc1osed
are exactly the images of the unit disk D(O, 1) via elements ofPGL 2(Cp)·
Similarly, the standard collection of open disks in JPl1(Cp), denoted by 'Dopen. is
given by
'Dopen =
{
all open disks D(a, r) and
the complementJPl1(Cp) "D(a, r) of all closed disks D(a, r).
In the nonarchimedean world, every disk component has a very simple form.
More precisely, it is either a disk, the complement of a single point, or all of JPl 1(Cp).
Proposition 5.45. Let 'Dopen and Vc1osed be, respectively, the collections of standard
open and closed disks in JPl 1(Cp) as defined above.
(a) Let D1, D2 E Vclosed· Then one of the following is true:
(i) D1 n D2 = 0.
(ii) D1 U D2 = JPl 1(Cp).
(iii) D1 ~ D2.
(iv) D2 ~ D1.
(b) Let U c JPl1(Cp) be an open set and let V be a disk component of U relative
to 'Dciosed· Then V has one ofthefollowingforms:
(iii) V E Vclosed U 'Dopen·
Proof (a) If D 1 U D2 = JPl 1(Cp), we are done. Otherwise, choose any point in the
complement of D 1 U D 2 and use a linear fractional transformation to move that point
to oo. This reduces us to the case that neither D 1 nor D 2 contains oo, so they have
the form
D1 = D(a1. rl) and D2 = D(a2, r2).
If D 1 and D 2 are disjoint, we are done, so we may assume that there is a point
and switching D1 and D2 if necessary, we may also assume that r1 :::; r 2. Let /3 E D1.
Then
so /3 E D2. Hence D1 ~ D2.
(b) If V = JPl1(Cp), we are done, so we assume that V =f JPl 1(Cp). Using a linear
fractional transformation to move a point not in V to oo, we are reduced to the case
that oo rf: V.
5.8. Nonarchimedean Wandering Domains
279
Let D1, ... , Dn E Vclosed be any path of disks contained in V. Each pair of
adjacent disks (Di, Di+I) has nonempty intersection, so (a) tells us that one of them
is contained within the other. Applying this reasoning to each pair, we see that the
union U~=l Di is itself a closed disk, i.e., the union is in Vc1osect· This shows that
every disk path in V actually consists of a single disk.
Fix some point a E V and let
R = sup{r 2 0: D(a,r) C V}.
Note that R > 0, since a E V and V is open. If R = oo, then V = Cp and we are
done, so we assume that R < oo. We claim that
D(a, R) s;:: V s;:: D(a, R).
(5.23)
The lefthand inclusion is clear from the definition of R. For the righthand inclusion,
suppose that b E V and consider a disk path from a to b lying within V. From our
previous remarks, this disk path consists of a single disk D(a, s). The definition of R
and the fact that D(a, s) s;:: V tell us that s::; R, and then the fact that bE D(a, s)
tells us that bE D(a, R). This gives the other inclusion.
But we get even more. If there is even a single b E V satisfying lb- ai = R,
then s =Rand D(a,R) = D(b,s) s;:: V, so we find that V = D(a,R) E Vclosed·
On the other hand, if lb- ai < R for every bE V, then V C D(a, R), so (5.23) tells
us that V = D( a, R) E Vopen· This completes the proof of Proposition 5.45.
0
5.8.2
Hyperbolic Maps over Nonarchimedean Fields
In this section we prove that the Julia set of a rational map ¢ contains no critical
points if and only if¢ is strictly expanding on its Julia set. This result is used later to
prove that such maps, which we call (p-adically) hyperbolic, satisfy a p-adic version
of Sullivan's no wandering domains theorem. On first reading, the reader may wish
to omit the somewhat technical proof of the main theorem in this section and proceed
directly to the application of the theorem in proving Theorem 5.55 in Section 5.8.3.
Theorem 5.46. (Benedetto [56]) Let Kj!Qp be a.finite extension ofp-adicfields and
let cp(z) E K(z) be a rational function of degree d 2 2. Proposition 5.20(c) tells us
that :F( ¢) f=- 0, so changing variables if necessary, we may assume that oo E F( ¢ ).
Then the following are equivalent:
(a) There are no critical points in .J( ¢ ).
(b) For every finite extension L / K there exists an integer m 2 1 such that
for all a E .J(¢) n L.
In other words, q;m is strictly expanding on .J (¢) n L.
Definition. Let Kj!Qp be a finite extension. A rational function¢ E K(z) will be
called p-adically hyperbolic if it satisfies the equivalent conditions of Theorem 5 .46.
See Exercise 5.25 for the relationship with the classical definition ofhyperbolicity.
5. Dynamics over Local Fields: Bad Reduction
280
Remark 5.47. The classical analogue of Theorem 5.46 over C is much weaker. It
says that some iterate of ¢ is strictly expanding on the Julia set if and only if the
closure of the postcritical set is disjoint from the Julia set. (The postcritical set is the
union of the forward orbits of the critical points.)
Proof of Theorem 5 .46. The implication (b) =? (a) is clear, since if a is a critical
point in :J (¢), we can take L = K (a) and observe that
m-1
(¢m)'(a) =
IJ ¢'(¢i(a)) = 0
(since ¢'(a)= 0).
i=O
Thus the existence of a critical point in :J (¢) implies that (¢m )' has a root in :J (¢)
for every m, so (b) cannot hold.
The other implication is more difficult. Using the assumption that ¢' does not
vanish on.:!(¢), we can apply Proposition 5.16(c) to every point a in.:!(¢) n L to
find a disk D(a, r a) with the property that
for all z, wE D(a, ra).
j¢(z)- ¢(w)j = j¢'(a)j·lz- wl
These disks form an open covering of the compact set :J (¢) n L, so we can find a
finite subcover. Let E be the smallest radius of the disks in this finite subcover. We
then consider a finite covering of :J (¢) n L by disks of radius E, say
The (nonarchimedean) triangle inequality implies that each disk in this covering is
contained in one of the disks of the previous covering, so we conclude that¢ stretches
by a constant factor on each [) (a;, E). In other words, there are constants S; such that
for each 1 :::; i :S n,
for all z, wE D(a;, E).
The next step is to show that for any fixed a E :J (¢) n L, the set of derivatives
n=1,2,3, ... ,
(5.24)
is unbounded. We prove this by contradiction, using the following claim:
Claim 5.48. Let a E .:! (¢) n L and suppose that there is an R > 1 such that
j(¢n)'(a)j :S R
foralln 2': 1.
Then
¢n(D(a,E/R)) C D(¢n(a),E)
foralln 2': 1.
(5.25)
Proof of Claim 5.48. We verify the claim by induction on n. The inclusion (5.25) is
clearly true for n = 0, since R > 1. Suppose the inclusion (5.25) is known for all
0 :::; n < N. Our choice of E ensures that for any (3 E .:!( ¢) n L, the map¢ stretches
by a constant factor on the disk D((J, E), so in particular each of the maps
5.8. Nonarchimedean Wandering Domains
281
O:S n <N,
stretches by a constant factor. It follows that
stretches by a constant factor. In other words, there is a constant S such that
for all z, wE D(o:, E/ R).
Taking w = o: and letting z --+ o:, we see that S =
w = o: and z E D(o:, E/ R) arbitrary, we conclude that
I(q)N )' (o:) I, and then taking
since I ( q)n )' (o:) I :::; R for all n by assumption,
since z E D(o:, E/R).
This shows that the inclusion (5.25) is true for n = N, hence for all n by induction.
0
Recall that we are in the midst of a proof by contradiction that the derivatives (5.24) are unbounded. Under the assumption that the derivatives are bounded,
we have proven the inclusion (5.25), which in turn certainly implies that
u D(/3, E).
(3E.J( ¢ )nL
However, the Julia set is bounded (since we assumed that oo E F( q) )), so an
E-neighborhood of the Julia set is also bounded. In particular, the above union omits
at least two points, so by the nonarchimedean Montel theorem (Theorem 5.27), the
map q) is equicontinuous on D( a, E/ R). This is a contradiction, since o: E .J( q)) by
assumption, which completes the proof that the derivatives (5.24) are unbounded.
We now know that for each point o: E .J (q)) n L there is some integer ma
such that I (¢m"' )' (o:) I 2 2. (There is no significance to the number 2; any number
strictly larger than 1 would suffice.) By continuity, there is a disk D(o:, sa) such that
¢m"'
2 2 at every point in the disk. Taking a finite subcovering, we can cover
.J (q)) n L by disks
I( )'I
with the property that there are integers m 1 , ... , mt 2 1 such that
for alll :::; i :::; t and all z E Di.
For convenience, we may assume that the disks D1 , ... , Dt are pairwise disjoint,
since if two closed disks have a point in common, then one is contained in the other
and may discarded. It remains to show that there is a single iterate q)m that is expanding on all of the disks. In fact, we show that this is true for all sufficiently large m.
5. Dynamics over Local Fields: Bad Reduction
282
To ease notation, we let 7/Ji = ¢m;. Given any point a E :1 (¢) n L, we map
out an orbit and an itinerary for a by applying 7/Ji whenever we land in Di. More
formally, define inductively an orbit a 0 , a 1, ... and a sequence of indices i 0 , i 1, ...
by the following procedure:
a 0 =a
and
io = (index so that ao E Di 0 ),
a1 = 7/Jio (ao)
and
i1 = (index so that a1 E DiJ,
a2 = 7/Jh(al)
and
i 2 = (indexsothata2 E Di 2 ),
and
In other words, if an is in Di, then an+l is obtained by applying 7/Ji = ¢m; to an.
We note that
(5.26)
for all n = 0, 1, 2, ... ,
since by construction, the point ain is in Din and 7/Jin = ¢m;n.
We next observe that the derivative ¢' (z) is continuous and nonvanishing on the
compact set :J( ¢) n L. (Continuity follows from the assumption that oo ~ .J(¢) and
nonvanishing is our assumption (a) that :J( ¢) contains no critical points.) It follows
that there is a constant f-l > 0 such that
for all f3 E .J(¢) n L.
(We may assume that f-l :::; 1.) Using the chain rule and applying this repeatedly
yields
n-1
l(¢n)'(f3)1
=II l¢'(¢if3)12 f-ln
for all f3 E .J(¢) n Land all n 2 0. (5.27)
i=O
Let M = max { m 1, ... , mt} and let N be any integer satisfying
2N > 4MJ-l-M2.
(5.28)
We claim that I(¢N)' I > 1 on :1( ¢) n L, which will complete the proof of the
theorem.
To verify this claim, let a E :1 (¢) n L and let a 0 , a 1, ... and i 0 , i 1, ... be the
orbit and itinerary of a as described earlier. We use the chain rule to compute
We continue this process until the exponent N - mio - mi 1 comes smaller than M. Notice that this implies that
· • • -
M>N-m·to -mt1 -···-m·tk >N-(k+1)M
'
so k + 2 > N / M. We have thus found integers k and r satisfying
mik
first be-
5.8. Nonarchimedean Wandering Domains
283
(qF)'(a) = 'I/J~ 0 (ao)'I/J~ 1 (a1)'1/J~ 2 (a2) · · · '1/J~k (ak) · (q)T)'(ak+I)
with k > N / M - 2 and r < M.
Hence
I(<PN)'(a)l = 1'1/J~o(ao)'I/J~l (a1)'1/J~2(a2) · · · '1/J~k(ak)I·I(<Pr)'(ak+l)l
'---------------------~
~
~ 2k from (5.26)
~ I' r from (5.27)
~ 2N/M- 2 ·p,M
since k > N/M- 2 and r < M,
>1
from the choice (5.28) of N.
This shows that I(¢N)'(a)l > 1 for all N > M 2 log 2 (p,- 1 ) and all a E .J(¢) n L,
which completes the proof of Theorem 5.46.
D
5.8.3
Wandering Disk Domains
If U is a disk component of the Fatou set :F(¢) of a rational map ¢ E Cp(z),
then ¢(U) may not be a full disk component of :F(¢). This situation, which does
not occur in the complex case, prompts the following definition.
Definition. Let ¢(z) E Cv(z) be a rational map defined over a nonarchimedean
field, and let
VC (¢) = {disk components of the Fatou set :F(¢) }
be the collection of disk components of the Fatou set of¢. Then¢ induces a map of
the set VC (¢) to itself according to the rule
VC(¢) ---+ VC(¢),
U ~(disk component of :F(¢) containing ¢(U)).
(5.29)
We say that U E VC(¢) is periodic, preperiodic, or wandering according to the
behavior of U under iteration of the map (5.29).
Example 5.49. Let p be an odd prime and let
2
<P(z)=z -z
p
be the function that we studied in Section 5.5. We proved (Proposition 5.22) that the
Julia set of¢ is contained in the union of two open disks,
.J(¢) c D(O, 1) U D(1, 1),
and that .J (¢) contains the repelling fixed points 0 and 1.
For a E Cv, let B(a) denote the disk component of a in :F(¢). We claim
that B( -1) = D( -1, 1). To see this, we observe that B( -1) cannot contain any
larger disk, since it does not contain 0. On the other hand, D(-1, 1) is in :F(¢),
since it is disjoint from D(O, 1) U D(1, 1). Hence B( -1) = D( -1, 1).
5. Dynamics over Local Fields: Bad Reduction
284
Now consider the image point ¢( -1) = 2p- 1 and the image of the associated
disk component
¢(B(-1)) = ¢(D(-1, 1)) = D(2p- 1 ,p).
The disk D(2p- 1 ,p) is contained in :F(¢), but it is certainly not the largest disk
around 2p -I contained in :F (¢). Indeed,
2p- 1 E JP> 1 (Cp) "D(O, 1) c :F(¢).
It is not hard to check that JP> 1 ( Cp) " D( 0, 1) is the full disk component of :F( ¢)
containing ¢(B( -1)).
Conjecture 5.50. (No Wandering Disk Domains Conjecture) Let K/QP be a finite
extension and let ¢( z) E K ( z) be a rational map of degree d 2: 2. Then the Fatou
set :F( ¢) has no wandering disk components.
Benedetto proves Conjecture 5.50 for a large class of rational maps. In order to
state his result, we give four definitions (some of which we already know):
Definition. Let ¢ E Cp (z) be a rational map of degree d 2: 2 and let P E JP> 1 ( Cp).
We say that Pis:
(i) Julia if P E .J(¢).
(ii) critical if e p (¢) 2: 2.
(iii) wildlycriticalifep(¢)
=0 (modp).
(iv) recurrent if there is a sequence of integers ni -. oo such that c/Jni (P) -. P,
i.e., if Pis in the closure of the set { ¢n(P) : n 2: 1 }.
Theorem 5.51. (Benedetto [54]) Let K /Qp be a finite extension of p-adic fields
and let ¢( z) E K ( z) be a rational map ofdegree d 2: 2. Assume that ¢ has no wildly
critical recurrent Julia points (defined over K). Then the Fatou set :F( ¢) has no
wandering disk components.
We make three short observations concerning Theorem 5.51.
Remark 5.52. Ifp is odd, then Theorem 5.51 is true for "almost all" rational maps
in Cp(z). This is true because if ¢(z) has a wild critical point P, then in particular it
has a point whose ramification index satisfies e p ( ¢) 2: p 2: 3. It is not hard to show
that all of the critical points of a "generic" rational map of degree d have ramification
index equal to 2. (See Exercise 4.23 for a more precise statement.)
Remark 5.53. For a fixed degree d, Theorem 5.51 is true for all primes p > d,
since p > d rules out the existence of wild critical points.
Remark 5.54. If a recurrent critical point P is in the Fatou set :F( ¢ ), then one can
show that P is in fact periodic (Exercise 5.26). On the other hand, if P is critical,
then locally around P the map¢ looks like ¢(z) = ¢(P) + a(z- z(P))k + · · ·
for some a E Cp and some k 2: 2. Thus if Q is sufficiently close to P, then
5.8. Nonarchimedean Wandering Domains
285
p(</J(Q),</J(P)) = p(P,Q)k, so <Pis highly contractive near P. And if Pis also
recurrent, then <Pn(P) gets very close toP infinitely often, so it receives the highly
contractive effect of <P infinitely often. This should cause points near P to remain
near P, and thus force Pinto the Fatou set. The critical recurrent condition and the
Julia condition are thus in opposition to one another, which means that nonperiodic
recurrent critical points should be quite rare. On the other hand, Rivera-Letelier [378]
has shown that there are maps having wildly critical recurrent points (which are then
necessarily in the Julia set) in lP' 1 (1Qv)· It is not known whether Rivera-Letelier's
examples have wandering disk domains.
In the other direction, it is known that there are rational maps defined over Cv
that have wandering disk domains [63, 59, 62, 378]. In these examples the critical
points are in the Fatou set, so Theorem 5.55 implies that the maps cannot be defined
over a finite extension of !Qp.
We now use Theorem 5.46 and a simple compactness argument to prove that
the Fatou sets of p-adically hyperbolic maps over finite extensions of !Qp have no
wandering disk domains. This is not as strong as Theorem 5.51, but still covers an
important class of maps. The proof of Theorem 5.51 uses similar ideas, but is more
complicated; see [54].
Theorem 5.55. (Benedetto [56]) Let K/!Qv be a.finite extension ofp-adicfields, let
¢( z) E K (z) be a rational map ofdegree d 2: 2, and assume that the Julia set .J (¢)
contains no critical points of¢, i.e., assume that¢ is p-adically hyperbolic. Then the
Fatou set F( <P) has no wandering disk components.
Proof Proposition 5.20 assures us that F(¢) is nonempty, and it is open, so it con-
tains an algebraic point. (Note that QP is dense in Cp.) Replacing K by a finite
extension and changing coordinates, we may assume that oo E F (<P), and indeed we
may even assume that .J (<P) c D(0, 1). Equivalently, we may assume that the disk
component of oo contains lP' 1 (<Cv) '- D(O, 1).
We suppose that U c F( <P) is a wandering disk component ofF(¢) and derive a contradiction. Replacing U with the disk component containing <Pn(U) for a
sufficiently large n, we may assume that the orbit of U does not include the disk
component at oo. In particular, </Jn(U) c D(O, 1) for all n 2: 0.
Again taking a finite extension of K if necessary, we can find a point a 0 E K
and a radius ro such that
D(ao, ro) cU.
At this stage we fix the field K and we use Theorem 5 .46 to find an integer m such
that 1(</Jm)'l > 1 on .J(<P) n K. Replacing <P by </Jm, it suffices to consider the case
that WI > 1 on .J(<P) n K.
For each n 2: 0, the image <Pn(.tJ(ao, r 0 )) is a (closed) disk centered at the
point an= </Jn(ao), say
</Jn(D(ao,ro)) = D(an,rn)·
In particular, <P(D(an,rn))
yields
= D(an+l,rn+l), so applying Proposition 5.16(b)
286
5. Dynamics over Local Fields: Bad Reduction
(5.30)
We also know that the disks D(an, rn) are disjoint, since D (an, r n) is contained
in ¢n(U), and further, each disk D(an, rn) contains a point of D(O, 1)nK.It follows
that the radii must satisfy
(5.31)
lim rn=O,
n~oo
since the set D(O, 1) n K has finite volume, so can contain only finitely many
nonempty disjoint disks of any fixed radius.
It follows from (5.30) and (5.31) that there are infinitely many n with the property
that
i.e., since rn - t 0, we must have rn+l < rn infinitely often. Consider the infinite set
of points
{an: I<P'(an)l < 1,
n= 1,2,3, ... }.
It is contained in the compact set D(O, 1) n K, so it contains an accumulation point
(3 E K. The continuity of¢' implies that 1¢'(!3)1 ::; 1, which shows that (3 E :F(¢),
since we used Theorem 5.46 to ensure that I¢'1 > 1 on J (¢).
Let V be the disk component of :F( ¢) containing (3. Then by construction V
contains infinitely many of the iterates an = ¢n(a0 ). Since the radii of ¢n(U) and
¢n(V) shrink to 0 as n - t oo, it follows that some iterate ¢n(U) is contained in V
and that some (nontrivial) iterate of ¢n(V) is contained in V. Therefore U is not
D
wandering, contradicting our original assumption.
5.8.4
Wandering Disk Domains Exist in CP
We have proven that p-adically hyperbolic maps defined over Qp have no wandering
disk domains. More generally, Theorem 5.51 shows that rational maps ¢(z) E Qp(z)
with wandering disk domains are very rare, if they exist at all. And of course, Sullivan's theorem 5.43 says that rational maps ¢(z) E C(z) defined over the complex
numbers never have wandering domains. It is thus somewhat surprising to discover
that there are very simple polynomial maps defined over Cp that have wandering disk
domains.
Theorem 5.56. (Benedetto [59]) ForcE Cp. let <Pc(z) be the polynomial
<Pc(z) = (1- c)zP+l + czP.
Then there exists a value a E Cp such that:
(1) J(</Ja)-/:- 0,
(2) :F( <Pa) has a wandering disk domain,
(3) :F( <Pa) contains every critical point of <Pa·
Proof See [59] for a proof of this specific theorem, and see [63, 62, 373, 378, 380]
for generalizations and related results.
D
5.9. Green Functions and Local Heights
5.9
287
Green Functions and Local Heights
The canonical height h¢ associated to a morphism ¢ : IP' 1 ---+ IP'1 is defined as the
limit of d-nh( cf>n(P)) as n---+ oo. The utility of h</> lies in the two formulas
h¢(P) = h(P) + 0(1)
and
where the first says that h<P contains arithmetic information and the second says
that h</> transforms canonically.
It is tempting to try a similar construction locally and define (say)
(v-adiclocalheightofa)= lim d1 logmax{l¢n(a)l ,1}.
n~oo
n
v
(5.32)
It is clear that if the limit (5.32) exists, then it transforms canonically, and indeed
if ¢(z) is a polynomial, then the limit does exist and everything works quite well
(see Exercise 3.24). Unfortunately, for general rational maps the limit (5.32) may
not exist.
Rather than working directly with a rational map ¢ : IP' 1 ---+ IP' 1 , it turns out to be
easier to develop a theory of local heights by first lifting ¢ to a map <I> : A 2 ---+ A 2 and
then constructing a real-valued function g on A 2 that satisfies the canonical transformation formula g (<I> (x, y)) = dQ (x, y). In this section we construct the Green
function g, prove some of its basic properties, and then use g to construct local
canonical height functions on IP'1 as described in Theorem 3.27.
A point in projective space [x, y] E IP' 1 is given by homogeneous coordinates, so
it is really an equivalence class of pairs (x, y). We make explicit the natural projection map
1r :
(x, y) ~----+ [x, y],
(A 2 '{0, 0}) --tIP'\
that sends a point (x, y) E A 2 to its equivalence class [x, y] E IP' 1 . To ease notation,
we write
A;= A2 , {0,0}
for the affine plane with the origin removed.
Let ¢ : IP'1 ---+ IP' 1 be a rational map of degree d. Then ¢ can be written as usual
in the form ¢ = [F, G] with homogeneous polynomials F and G of degree d having
no common factors. The polynomials F and G then define a map
<I>(x,y) = (F(x,y),G(x,y)),
that fits into the commutative diagram
A2
.ftl*
~
------7
.t-2
FA*
pl ~ pl
We call <I> a lift of¢>. By homogeneity, if <I> = [F, G] is one lift of¢, then every other
lift of¢ has the form c<I> = [cF, cG] for some constant c E K*.
5. Dynamics over Local Fields: Bad Reduction
288
Definition. Let K be a field and let v be an absolute value on K. We denote the
absolute value (or sup norm) ofa point ( x, y) E A2 ( K) by
Similarly, the absolute value (or sup norm) of one or more polynomials is the maximum of the absolute values of their coefficients. (We have already made use of this
convention in the proof of Theorem 3.11.)
We begin by recalling how a map «P : A2 (K) ---t A2 (K) affects the v-adic norm
of a point.
Proposition 5.57. Let K be afield with an absolute value v. Let
«P = (F, G) : A 2 ---t A 2
be given by homogeneous polynomials F, G E K[x, y] ofdegree d 2: 1, and assume
that F and G have no common factors in K[x, y].
(a) There are constants c1, c2 > 0, depending only on «P and v, such that
< II«P(x, y) llv <
Cl -
II (x,y) II vd
- C2
for all (x, y) E A~(K).
(5.33)
(b) If v is nonarchimedean and «P satisfies II«PIIv = 1, then (a) is true with the
explicit constants
IRes(F G)l < II«P(x,y)llv < 1
'
v - ll(x,y)ll~ -
for all (x, y) E A:(K).
(5.34)
Proof (a) We proved inequality (5.33) for general morphisms pN ---t pM during the
course of proving Theorem 3.11. More precisely, see (3.6) on page 92 for the upper
bound with an explicit value for c2 («P, v), and see (3.7) on page 93 for the lower
bound.
(b) By homogeneity, it suffices to prove (5 .34) for points satisfying II (x, y) II v = 1.
Then the upper bound is obvious from the triangle inequality, and the lower bound
was proven during the course of proving Theorem 2.14, see (2.5) on page 57.
D
The next result describes a kind of v-adic canonical height associated to a
map «P : A 2 ---t A 2 • The construction is the same as the one that we used to construct
canonical heights in Section 3.4.
Proposition 5.58. Let K be a field with an absolute value v, let ¢ : IP1 ---t IP1 be a
morphism ofdegree d 2: 2, and let «P = (F, G) : A2 ---t A2 be a lift of¢.
(a) For all (x, y) E A~(K) the following limit exists:
.
1
y;p(x, y) = n~~ dn logii«Pn(x, y)llv·
We call Y<P the Green function of«P.
(5.35)
5.9. Green Functions and Local Heights
289
(b) The Green fonction is the unique fonction A; (K) ----> IR having the following
two properties:
Q.p(<I>(x, y)) = dQ.p(x, y)
Q.p(x, y) = logll(x, y)llv + 0(1)
for all (x,y) E A:(K).
(5.36)
for all (x, y) E A:(K).
(5.37)
(c) lfv is nonarchimedean and <I> satisfies II<I>IIv = 1 and IRes(F, G) lv = 1, i.e., if
the map¢= [F, G] : IP' 1 ----> IP' 1 has good reduction at v, then
Q.p(x,y) = logll(x,y)llv
for all (x, y) E A:(K).
(The converse is also true. See Exercise 5.27.)
(d) For all (x, y) E A; (K) and all c E K*, the Green function Q.p has the following
homogeneity properties:
Q.p(cx, cy) = Q.p(x, y) +log lclv·
1
9c<P(x, y) = Q.p(x, y) + d _ 1 log lclv·
(5.38)
(5.39)
(e) The Green function Q.p : A; (K) ----> IR is continuous. (In fact, Q.p is Holder
continuous, but this is more difficult to prove. See Exercise 5.28.)
Proof We consider the two functions
and
Proposition 5.57(a) tells us that they satisfy
logii<I>(x, y) llv = dlogll (x, y) llv + 0(1)
for all (x, y) E A;(K).
(5.40)
This is exactly the situation needed to apply Theorem 3.20, from which we conclude
that the limit (5.35) exists and satisfies (5.36) and (5.37). Further, Theorem 3.20 says
that Q.p is the unique function satisfying (5.36) and (5.37). This completes the proof
of(a) and (b).
(c) The assumptions II <I> II v = 1 and IRes (F, G) Iv = 1 combine with Proposition 5.57(b) to tell us iliat II<I>(x, y) llv =
Hence by induction we obtain
I (x, y) II~ for all points (x, y) E A; (K).
Then ilie definition of Q.p immediately gives Q.p (x, y) = log II (x, y) II v, which
proves (c).
(d) The map <I>n is homogeneous of degree dn, so <I>n(cx, cy) = cdn <I>n(x, y).
Substituting this into ilie definition of Q.p gives (5.38).
Similarly, if we let <I>c(x, y) = c<I>(x, y), then homogeneity and an easy induction
argument show that
5. Dynamics over Local Fields: Bad Reduction
290
Hence
Ycq,(X, y) = nl~~ d1n logii<I>~(x, y) llv
)
dn-1
1 (
lim -d logii<I>n(x,y)ll v + -d-loglclv
= n---+oo
- 1
n
1
= Qq,(x, y) + d _ 1 log lclv·
(e) Let Yn(x, y) = d-n logii<I>n(x, y) llv· Then for any fixed value of n, the function Yn : A; (K) ----+ IR is continuous, since it is the composition of the continuous
maps <I>n: A;(K)----+ A;(K) and log II · llv: A;(K)----+ R Further, the telescoping
sum argument used in Theorem 3.20 to prove the existence of limn---+oo Yn(x, y)
implies that
c
IQq,(x, Y)- Yn(X, Y)i S (d _ 1)dn
for all n 2 1 and all (x, y) E A;(K),
where C = C(<I>,v) depends only on the 0(1) constant in (5.40). (In the inequality (3.15) on page 98, switch the roles of m and nand then let m ----+ oo.) Hence the
sequence of continuous functions Yn (x, y) converges uniformly to Qq, (x, y), so the
D
limiting function Qq, is also continuous. (See also Exercise 3.14.)
The Green function allows us to decompose the canonical height h¢ into a sum
of local terms.
Theorem 5.59. Let K be a number field, let ¢ : IP' 1 ----+ IP' 1 be a rational function
of degree d 2 2 defined over K, and let <I> be a fixed lift of¢. For each absolute
value v E MK, let Y4>,v be the associated Green function as described in Propo-
sition 5.58, where we now include v in the notation so as to distinguish between
different absolute values. Then the (global) canonical height decomposes as a sum
for all P = [x,y] E IP' 1 (K).
Proof Let
ry(x, y) =
L nvY4>,v(x, y)
for (x, y) E A;(K),
vEMK
so a priori the function "7 is a function on A; ( K). However, applying the Green function homogeneity property (5.38) from Proposition 5.58(d) and the product formula
(Proposition 3.3), we see that
ry(cx,cy)=
L nv(Y4>,v(x,y)+loglclv)= L nvY4>,v(x,y)=ry(x,y),
5.9. Green Functions and Local Heights
291
so "1 gives a well-defined function on JP>1 (K).
Next we use the transformation property (5.36) from Proposition 5.58(b) to compute
Similarly, we use the normalization property (5.37) from Proposition 5.58(b) to estimate
=
L nv(log jjPIIv + Ov(1)) = h(P) + L nv0v(1),
(5.41)
where h(P) is the usual height of the point P E JP> 1 (K) and the Ov(1) are the
bounded functions appearing in (5.37). We further observe that Proposition 5.58(c)
says that we may take Ov (1) = 0 for all but finitely many v E M K. More precisely,
we may take Ov(1) = 0 for all v satisfying
(i) v E Mj},
(ii) II<I>IIv = 1,
and
I
I
(iii) Res( <I>) v = 1.
Hence the final sum in (5.41) is a bounded function.
We have now proven that "1 : JP> 1 ( K) ~ JR. satisfies
ry(¢(P)) = dry(P)
and
ry(P) = h(P) + 0(1).
It follows from the uniqueness ofthe canonical height (Theorem 3.20) that "1 is equal
~~.
D
In Section 3.5 we described a decomposition of the canonical height he/> as a sum
oflocal canonical height functions 5.</>,v• but we deferred the proof. The intuition in
Section 3.5 was that the local canonical height should measure
5.</>,v (P) = -log( v-adic distance from P to oo).
More generally, it is convenient to define a local canonical height that measures the
v-adic distance from P to a collection of points. In the following theorem we describe a set of points, possibly with multiplicities greater than 1, by specifying the
homogeneous polynomial E E K[x, y] at which they vanish. In slightly fancier terminology, we are identifying positive divisors in Div(JP> 1 ) with homogeneous polynomials in K[x, y].
Theorem 5.60. Let K be a field with an absolute value v and let ¢ : JP> 1 ~ JP>1 be
a rational function of degree d ~ 2 defined over K. Fix a lift <I> = ( F, G) of¢ and
let Q.:p be the associated Green function.
292
5. Dynamics over Local Fields: Bad Reduction
(a) For any homogeneous polynomial E (x, y) E K [x, y] of degree e 2: 1 we define
~¢,E([x,yl) = eQ<I>(x,y) -logiE(x,y)lv
for [x, y] E lP' 1 (K) with E(x, y) -=1- 0.
Then ~¢,Eisa well-definedfunction on lP' 1, i.e., the value of~q,,E(P) does not
depend on the choice of homogeneous coordinates [x, y]for P.
The function ~¢,E is called a local canonical height associated to¢ and E.
In the special case that E(x, y) = y, we drop E from the notation and refer
simply to a local canonical height associated to ¢.
(b) For all P E lP' 1 (K) with E(P) -=/- 0 and E( if.>(P)) -=/- 0 we have
'
'
A¢,E(¢(P)) = d>.q,,E(P) -log
I (E 0 if.>) (P) Iv.
E(P)d
(Note that the homoegeneity of E and if.> ensures that the ratio (E o if.>)/ Ed is a
well-defined function on lP' 1.)
(c) The function
[E(P)[v
,
(5.42)
P r---+ >.q,,E(P) +log IIPII~ ,
which a priori is defined only at points P satisfYing E(P) -=1- 0, extends to a
bounded continuous function on all oflP' 1 (K).
(d) Given the particular choice of lift if.>. the function ~¢,E defined in (a) is the
unique real-valued function
satisfYing (b) and (c). If cif.> is a different lift, then with the obvious notation,
In particular, any two local canonical heights differ by a constant.
Proof The Green function satisfies Q<I>(cx, cy) = Q<I>(x, y) +log [c[v, while the
polynomial E satisfies E(cx, cy) = ce E(x, y), so the difference
eQ<I>(x, y) -log [E(x, y)[v
defining ~¢,E does not change if we replace (x, y) by (ex, cy). This proves (a).
(b) We compute
~¢,E(¢(P)) = eQ<I>(if.>(x,y)) -logiE(if.>(x,y))lv
= edQ<I>(x, y) -logiE(if.>(x, y))lv
from Proposition 5.58(b),
= d~q,,E(P) + d log [E(x, y) [v -log IE( if.>(x, y)) lv·
293
5.9. Green Functions and Local Heights
(c) Directly from the definition of ~</>,Ewe see that the righthand side of(5.42) is
equal to
(5.43)
e(Qcp(P) -log IIPIIv)·
The boundedness of(5.43) is exactly (5.37) in Proposition 5.58(b). Further, we know
from Proposition 5.58(e) that Qq,(x,y) is a continuous function on A;(K), and it
is clear that logll(x,y)llv is also a continuous function on A;(K), so the difference (5.43) is a continuous function on A;(K). Further, the difference is invariant
under (x, y) f----+ (ex, cy), so it descends to a continuous function on lP' 1 ( K).
(d) Suppose that ~ and ~~ both satisfy (b) and (c), and let ~~~ = ~ - ~~. Writing
~"(P) = (~(P) +log IE(P)Iv) - (~~(P) +log IE(P)Iv)
IIPII~
IIPII~
'
we see from (c) that ~~~ extends to a continuous bounded function on all of lP' 1 ( K).
Let C be an upper bound for I~"I·
From (b) we find that
~"(¢(P)) = d~"(P)
for all P E lP' 1 (K) with E(P) # 0 and E(¢(P)) # 0,
and iterating this relation yields
~"(¢n(P)) = dn~"(P)
providedE(¢i(P)) # 0 forallO::; i::; n.
Hence
(5.44)
for all points P E lP' 1 ( K) satisfying E (¢i (P)) # 0 for all 0 ::; i ::; n.
But each equation E ( ¢i (P)) = 0 eliminates only finitely many points, so the
inequality (5.44) is true for all but finitely many points oflP' 1 (K). Then the continuity
C d-n is true for all P E lP' 1 ( K). Since n is arbitrary,
of ~ 11 tells us that I~~~ P)
( I ::;
this proves that ~~~ (P) = 0, so ~ = .\ 1 •
Finally, the effect of replacing <I> by c<I> follows immediately from the definition
of ~q,,E in terms of the Green function Qq, and from the corresponding transformation
D
formula for Qq, given in Proposition 5.58(d).
Finally, as promised in Section 3.5, we prove that the global canonical height is
equal to the sum of the local canonical heights.
Theorem 5.61. Let K be a number field, let ¢ : lP' 1 ----+ lP' 1 be a rational function of degree d 2: 2 defined over K, and fix a lift <I> = ( F, G) of¢. Choose a
homogeneous polynomial E(x, y) E K[x, y], and for each absolute value v E MK,
let ~</>,E,v be the associated local canonical height described in Theorem 5.60, where
we now include v in the notation so as to distinguish between different absolute
values. Then the (global) canonical height has a decomposition as a sum of local
canonical heights,
for all P E lP' 1 (K) with E(P) # 0.
5. Dynamics over Local Fields: Bad Reduction
294
Proof We use the definition of ~</>,E,v in terms of the associated Green function Q~,v
from Theorem 5.60(a) to compute
Theorem 5.59 says that the first sum is equal to hq,(P), while the product formula
(Proposition 3.3) tells us that the second sum is 0. (Note that this is where we use the
D
assumption that E(P) -=/= 0.)
Remark 5.62. If v E M~ is nonarchimedean and¢ has good reduction at v, then the
Green function and the local canonical height are given by the simple formulas
(max{ lxlv, IYiv})
)
(
,
.
iE(x, y)iv
Q~,v(x, y) =log max { lxlv, IYiv } and >..q,,E,v [x, y] =log
Thus it is only for maps with bad reduction that the Green and local canonical height
functions are interesting. This should not come as a surprise to the reader, since
bad reduction is the situation in which dynamics itself becomes truly interesting. Of
course, this is said with the understanding that every rational map over C has "bad
reduction," so the dynamics of holomorphic maps on IP' 1 ( q are always interesting
and complicated.
Remark 5.63. For additional material on dynamical Green functions and dynamical
local heights, see [21, 88, 234, 233].
5.10
Dynamics on Berkovich Space
We havte seen that Cp is a natural space over which to study nonarchimedean dynamics, since it is both complete and algebraically closed. However, the field Cp has
various unpleasant properties:
• Cp is totally disconnected.
• Cp is not locally compact, so the unit disk { z E Cp : lzl ~ 1} and projective
line IP' 1 ( Cp) are not compact.
c;
• The value group I I = p!Q consists of the rational powers of p, so it is not
discrete in IR>o, yet neither is it all of!R>o·
This list suggests that it might be better to work in some larger space. There is
a general construction, due to Berkovich [64, 67], that solves these problems for Cp
and other more complicated spaces. The study of dynamics on Berkovich spaces
started during the 1990s and is an area of much current research. In this section we
briefly describe the Berkovich disk and associated affine and projective lines and
5.10. Dynamics on Berkovich Space
295
discuss some very basic dynamical results. In a final subsection we state without
proof some recent results. For further reading, see [26, 51] for an introduction to
dynamics on Berkovich space and see [373, 375, 376,379, 377, 380, 381] for RiveraLetelier's fundamental work in this area.
5.10.1
The Berkovich Disk over CP
The unit Berkovich disk [JB is a compact connected metric space that contains the
totally disconnected non-locally compact unit disk in CP" We describe two constructions of [JB, the first an explicit description as the union of four types of points and
the second as a set ofbounded seminorms on Cp[z].
5.10.1.1
The Four Types ofBerkovich Points
The most concrete description of [JB is as the union of the following four sets of
points:
Type-I Berkovich Points. Each point a in the standard unit disk fJ( 0, 1) of Cp is
associated to a point of the Berkovich disk, which we denote by
Type-11 Berkovich Points. Each closed disk D(a, r) contained in D(O, 1) with radius r E IC; I = pQ is associated to a point of the Berkovich disk, which we
denote by
Type-III Berkovich Points. Similarly, each closed disk D(a, r) C D(O, 1) with
positive radius r ¢. IC;I = pQ is associated to a point of the Berkovich disk,
which we naturally also denote by ~a,r·
Type-IV Berkovich Points. These are the trickiest points in [JB. They are associated to nested sequences of closed disks
with the property that
nD(an, rn) = 0.
n2:1
We denote these points by ~a,r, where, as the notation suggests, the vectors a
and rare a= (a1, a2, .. . ) and r = (r1, r2, ... ).
Remark 5.64. Note that Berkovich points ~a,r of Types II and III are disks D(a, r),
so different values of a may yield the same Berkovich point. Indeed, we have
~a,r = ~b,s
if and only if
r = s and
Ia- bl :::; r,
5. Dynamics over Local Fields: Bad Reduction
296
since these are the conditions for the disks 15( a, r) and D(b, s) to coincide. Similarly,
two Berkovich points of type IV are the same if their sequences of disks can be
suitably intertwined. See Exercise 5.40 for details.
Remark 5.65. A point ~a,r of Type-I, II, or III corresponds to a disk (possibly of
radius 0), so we define the radius of ~a, r to be r. The radii r 0 , r 1 , r 2 , ... of a TypeIV point ~a,r are nonincreasing, so the limit r = limi....,oo ri exists and is called the
radius of~a,r·
We claim that the radius of a Type-IV point is strictly positive. To see this, suppose that ~a,r has radius 0. Then the sequence of points a 1 , a 2 , ... is a Cauchy sequence in Cp, so it converges to a point a E Cp. Let
6i =
_inf
zED(ai,ri)
lz - al
be the distance from a to the i 1h disk. Notice that 0 :::; 61 :::; 62 :::; 63 :::; · · ·, since the
disks form a decreasing sequence. On the other hand, 6i :::; Ia - ai I --+ 0 as i --+ oo.
Hence 6i = 0 for all i, so a E D(ai, ri) for all i and the intersection is nonempty,
contradicting the assumption that ~a,r is a Type-IV point.
5.10.1.2
The Berkovich Disk as a Set of Seminorms
The description of the Berkovich disk 15 6 as a union of points of Types I, II, III,
and IV is very concrete, but it can be awkward to apply, since one must deal with four
different kinds of points. 4 There is an alternative description of 15 6 as a collection of
seminorms on the ring CP [z] that is sometimes easier to use and that also naturally
generalizes to other rings and other spaces.
Definition. Fix R > 0. The Gauss norm I · IIR on Cp[z] is defined by
The maximum modulus principle (Theorem 5.13(a)) tells us that the Gauss norm is
equal to the sup norm,
IIJIIR = sup Jf(z)J.
zED(O,R)
Definition. A (nontrivial) II · IIR-bounded seminorm on Cp[z] is a nonconstant map
with the following properties:
1. Ill 2 0 for all f E Cp[z].
2. lfgl = If I· lgl for all f, g E Cp[z].
4 0ne can unify the four types of points by defining all of them as equivalence classes of nested sequences of closed disks. See Exercise 5.40.
5.10. Dynamics on Berkovich Space
297
3. If+ gl ::; max{lfl, lgl} for all j, g E Cp[z].
4. IJI ::; IIJIIR for all f E Cp[z]. (This is the boundedness condition.)
Thus a seminorm has all of the properties of an absolute value except that there may
be nonzero elements f E Cp[z] with III = 0.
Definition. The (closed unit) Berkovich disk iJB is the set
D 8 = { II · III -bounded seminorms on Cp [zl}.
More generally, the closed Berkovich disk of radius R is the set
D~ = { II · II wbounded seminorms on CP [zl}.
The definition of iJB as a set of bounded seminorms is quite unintuitive, but
its utility becomes clear when one examines Table 5.2 and sees how each of the
four types of Berkovich points naturally defines a seminorm on Cp[z]. A fundamental theorem of Berkovich, whose proof we omit, says that every bounded seminorm on Cp[z], and more generally on certain power series rings containing Cp[z],
comes from one of the four types of Berkovich points. (See [64, page 18] or [26,
Proposition 1.1].) Notice that the seminorm I · lo 1 corresponding to the Berkovich
point ~o,1 is exactly the Gauss norm lflo,I = 11!11{, so following Baker and Rumely,
we call ~o,1 the Gauss point.
I Point I
Seminorm
Type-I
~a,O
lfla,O = IJ(a)l
Types-II & III
~a,r
IJia,r =
Type-IV
~a,r
IJia,r = n-HXJ
lim IJian,Tn
sup
zED(a,r)
lf(z)l
Table 5.2: Seminorms on Cp[z] associated to Berkovich points.
Remark 5.66. It is easy to see that the seminorms associated to points ofType-11, III,
and IV are actually norms. (See Exercise 5.35.) However, the seminorm associated
to a point ~a,o of Type-I is not a norm, since lfla,o = 0 if and only iff vanishes at a.
For example, lz- ala,o = 0. This explains why the Berkovich disk is defined using
seminorms, rather than norms.
Remark 5.67. In order to properly develop function theory on the Berkovich disk
and to glue disks together to create larger spaces, it is advantageous to use sets of
seminorms on more general rings, such as power series rings. For example, let 'f R
be the ring of power series in Cp[z] that converge on the closed disk D(O, R) of
radius R, i.e.,
'fR = {f(z) =
L CnZn E Cp[z]: J~~ lcniRn = 0 }·
n;:.o
298
5. Dynamics over Local Fields: Bad Reduction
The ring 'll' R is a Banach algebra over Cp using the Gauss norm,
which the maximum modulus principle tells us is the same as the sup norm. The
Berkovich disk D~ is often defined to be the set of bounded seminorms on the
ring '][' R· One can show, using the Weierstrass preparation theorem, that this leads
to the same set of points and the same topology as taking the II · llwbounded seminorms on the polynomial ring Cp[z]; see [26). The ring 'll'R is an example of a Tate
algebra. Applying the same construction to more general Tate algebras allows one
to construct Berkovich spaces for the associated rigid analytic spaces.
5.10.1.3
Visualizing the Berkovich Disk
In order to visualize the Berkovich disk, we place the Gauss point ~0 , 1 at the top of
the page and observe that there is a line segment running from any point~=/:. ~0 , 1 up
to the Gauss point. If~= ~a,r is of Type-I, II, or III, then this line segment is the set
of points
La,r = {~a,t: r:::;: t:::;: 1}.
Notice that any two line segments La,r and Lb,s merge with one another at the
point ~a,t = ~b,t determined by
t = max{r, s, lb- al}.
This is the smallest allowable value oft for which the disks D(a, t) and D(b, t)
acquire a common point, hence for which they coincide. Thus one can imagine the
various line segments continually merging as they run upward toward the Gauss point
at the tlop of the tree.
The line segment running up from a Type-IV point ~a,r is slightly more complicated. It is the set of points
U{~a;,t:
00
La,r = {~a,r} U
i=1
U
00
Ti ::S: t ::S: Ti-d= {~a,r} U
La;,r;·
(5.45)
i=1
(Note that r 0 = 1 by definition.) The Type-IV point ~a,r is included in the Berkovich
disk precisely to provide an endpoint for the union of line segments ULa;,r;.
Now imagine starting at the Gauss point and moving downward through the tree.
We claim that at any instant, there are three possible scenarios:
1. If you have reached a point ~a,o of Type I or a point ~a,r of Type IV, then you
have hit the end of a segment and cannot proceed further.
2. If you have reached a point ~a,r of Type II, then r > 0 is in the value group
of Cp and there are countably infinitely many branches along which you can
move down the tree.
3. If you have reached a point ~a,r of Type III, then r > 0 is not in the value
group of Cp and there is only one direction to move down the tree.
5.10. Dynamics on Berkovich Space
299
The picture for points of Type I is clear. In order to understand Types II and III,
suppose that we fix a point ~a,r of Type II or III. Each b E D( a, r) gives a line
segment Lb,o that runs up from the Type-1 point ~b,o and through the point ~r,a·
Two such line segments Lb,o and Lb' ,o merge before reaching ~a,r if and only if
lb- b'l < r, so it is really each open disk D(b, r) inside the closed disk D(a, r) that
gives a line segment running up to ~a,r·
If~a,r is of Type III, then D(b, r) = D(a, r) for any bE D(a, r), so there is only
one segment running downward from ~a,r·
The situation is much more interesting, and complicated, if ~a,r is of Type II. In
this case D(a, r) is covered by a countable union of open disks D(b, r), so there is
a countable set of branches downward from ~a,r· A convenient, although noncanonical, way to describe the branches is as follows. Let s,p = { z E Cp : lzl < 1} denote
the maximal ideal in the ring of integers of Cp and fix some c E Cp with lei = r.
Then the open disks of radius r are in one-to-one correspondence with the residue
field iFP via the map
{disks D(b, r) inside D( a, r)} ---+ iFP•
D(b, r) ~ (bjc) mod s,p.
The surjectivity of this map is clear, and the injectivity follows from the fact
that D(b, r) = D(b', r) if and only if lb- b'l < r = lei.
In order to fit the Type-IV points into the picture, let
be any sequence of nested disks and consider what happens as we move down the
line segments La;,r;. The fact that the disks D(ai, ri) are nested implies that the
line segments La;,r; extend one another downward as i increases. If the intersection nn(ai,ri) is nonempty, then it is equal to D(a,r) for some a E Cp and
some r ?: 0, so the intersection corresponds to a point ~a,r of Type I, II, or III
and the union ULa;,r; together with the endpoint ~a,r forms a closed line segment.
However, if the intersection D(ai, ri) is empty, then there is no actual disk
D( a, r) sitting at the bottom of the union of the line segments ULa; ,r;. Thus as
already noted, the points of Type IV exist precisely to remedy this situation and to
ensure that every downward path has a termination point. Further, this explains why
we defined La,r by (5.45) to be the line segment running from ~a,r to ~0,1· (See
Exercise 5 .41.)
The Berkovich disk fJB is "illustrated" in Figure 5.4. Of course, Figure 5.4 is
at best a pale imitation of the true glory of the Berkovich disk, since in a complete
picture of D8 , every line segment contains a countable number of Type-II points, off
of each of which there is a countable number of downward branches. Unfortunately,
despite advances in modem technology, there are still no computer packages capable
of fully rendering a (countably) infinitely branched broomstick.
n
5.10.1.4 The Gel'fond Topology on the Berkovich Disk
The next step is to put a topology on the Berkovich disk.
5. Dynamics over Local Fields: Bad Reduction
300
~0,1 ~Gauss point I
IType-I point I -
~b,O
~0,0
~a,o
Figure 5.4: The Berkovich disk D8 .
Definition. The Gel'fond topology on IJB is the weakest topology such that for
every f E <Cp[z] and every B > 0, the following sets are open:
U(f,B) = {x E D8 : lflx < B}
and
V(f,B) = {x E D8 : lflx > B}.
Theorem 5.68. (Berkovich) The Berkovich disk D8 with the Gel 'fond topology is
a compact path-connected Hausdorff space.
Proof See [64, Theorem 1.2.1] or [29, Appendix D] for the proof that D8 is compact
0
and Hausdorff and [64, Corollary 3.2.3] for the proof that it is path connected.
Basic open sets for the Gel'fond topology on D 8 , viewed as a tree, can be described by taking (and deleting) branches of the tree as described in the following
definition.
Definition. The closed branch of IJB rooted at (a, r), denoted by Ir-a r• consists of
all points ~b,s such that ~a,r is on the line segment Lb,s running fro~ ~b,s to ~0,1,
together with whatever Type-IV points are needed to finish off the bottom of any
open line segments. Thus
lr-a,r = {~b,s : s :S rand
lb- al :S r} U {appropriate Type-IV points}.
The open branch rooted at (a, r), denoted by ~r,: r is obtained by starting with
the closed branch lr-a,r' removing the point ~a,r. and then taking the connected component that contains ~a,o. If ~a, r is of Type III, this is simply the closed branch at ~a, r
with the single point ~a,r removed; but if ~a,r is of Type II, then there are countably
many branches at ~a,r and we select the one that includes the point ~a,O· It is not hard
to see that
~r-:,r = {~b,s : s < rand lb- al < r} U {appropriate Type-IV points}.
Then a base of open sets on IJB for the Gel'fond topology consists of all sets of
the following three types:
5.10. Dynamics on Berkovich Space
301
• Open branches.
• Open branches with a finite number of closed subbranches removed.
• The entire tree with a finite number of closed branches removed.
Remark 5.69. There is a natural inclusion
-
- B
D(O, 1) '--------+ D ,
a f-----t ~a,O,
that identifies the unit disk D(O, 1) as the set of Type-I points in the Berkovich
disk J)B. It is not hard to check that the Gel 'fond top logy on tJB, restricted
to D(O, 1), is the usual topology induced by the metric on Cp. See Exercise 5.44.
5.10.2
The Berkovich Affine and Projective Lines
It is relatively easy to construct the Berkovich affine line A B and the Berkovich
projective line lP' 6 as topological spaces, which we do in this section. It is more
difficult to construct them as ringed spaces with sheaves of functions appropriate for
doing analysis. See Remark 5.74 for a brief discussion and references.
5.10.2.1
The Berkovich Affine Line A8
Recall that the Berkovich disk J)B consists of four types of points. Each may be
described in terms of disks that are contained in the closed unit disk D(O, 1). Equivalently, J)B is the collection of II · [ 1 -bounded seminorms on Cp[z].
More generally, we define the Berkovich disk D~ of radius R to be the collection
of II · [[wbounded seminorms on Cp[z]. It is given the Gel'fond topology and has
its own Gauss point ~o,R corresponding to the seminorm
It is clear how to define points of Type I, II, III, and IV in D~ using closed disks
contained in D(O, R), just as we did for [JB. Further, there is an inclusion
This is clear from the definition of D~ as a set of seminorms. In terms of the picture
of D~ as a branched tree, we see that D~ 1 is the closed branch of D~ 2 lying below
the Gauss point ~o,R 1 of D~ 1 •
Definition. The Berkovich affine line A B is the union of the increasing collection of
Berkovich disks,
AB =
D~,
u
R>O
with the topology induced by the direct limit topology on the individual Berkovich
disks. It suffices, of course, to take the union over any increasing sequence of radii,
for example, over R = pk with k ---+ oo.
5. Dynamics over Local Fields: Bad Reduction
302
Thus every point a E Cp, every disk D( a, r) c Cp, and every nested sequence of
disks with empty intersection gives a point in the Berkovich affine line A B, and A B is
composed of exactly this collection of points. In particular, there is a natural inclusion
ofCP as the set ofType-I points in A 8 ,
We also note that A B inherits a tree structure from the natural tree structure of the
Berkovich disks D~. However, the tree A 8 extends infinitely far upward; there is no
Gauss point sitting at the top of AB.
5.10.2.2
The Berkovich Projective Line JID 8
The easiest way to construct the Berkovich projective line JID 8 as a topological space
is to glue together two copies of the Berkovich disk jjB along their annuli
- B
: lal = 1 }
Ann B = { ~a,r E D
using the map
We note that the map f is induced from the inversion map f (z) = 1/ z, since it is
easy to check (Exercise 5.37) that ifO rJ_ D(a, r), then
f(D(a,r)) = {z- 1 : lz- ai ~ r} = D(a-l,r/lal 2 ).
In particular, if Ia I = 1, then f (D( a, r)) = D( a- 1 , r ), so f(~a,r) = ~a-l,r·
The full Berkovich disk is the disjoint union of the annulus and the open branch
containing ~o,o,
o
B
-B
D = Ann U A\o,1.
Thus when we glue two copies of jjB along their annuli, the only parts of the two
disks that are not identified are the two open branches ~~ 1 . Hence another way to
construct JID8 is to take one copy of jjB and attach one extra copy of ~~ 0 running
vertically upward from the Gauss point ~0 , 1 . The result is illustrated in Figure 5.5.
It is natural to denote the extra vertical branch by ~"oc, and to label its points
using the reciprocals of the points in D(O, 1),
~"oc, = {~a,r : Ia I > 1 and
r < 1} U {~oo,o}.
The open and closed branches in ~"oc, are defined using the natural identification
For example, a basic open neighborhood of the Gauss point ~0 , 1 is obtained by removing from JID8 a finite number of closed branches, some of which may be in the
vertical branch ~"oc, at infinity.
5.10. Dynamics on Berkovich Space
303
Extra branch
"at infinity"
Figure 5.5: The Berkovich projective line JP>8 .
Remark 5.70. There is a natural embedding of A.8 into JP>8 . However, the inversion
map f (z) = 1/ z used to glue together the two pieces of JP> 8 can cause notational
confusion regarding the "radius" of points of Types II and III. For example, the
point ~p-2 ,p-3 in the branch ~"ex, of JP>8 would be denoted by ~p-1 ,p when viewed
as a point in A 8 . Thus it might be wiser to denote the points in ~"ex, using some
alternative notation, for example ~a,r. but we will not do so.
Remark 5.71. A useful alternative construction oflP8 mimics the construction of the
scheme JP>i as the set of homogeneous prime ideals. It starts with the set of bounded
seminorms on the two-variable polynomial ring Cv[x, y] that extend the usual absolute value on Cp and that do not vanish on the maximal ideal (x, y). Two seminorms I · 11 and I · 12 are considered equivalent if there is a constant c > 0 such
that
for all homogeneous f E Cp[x, y].
Then JP>8 is the set of equivalence classes of such seminorms. For details of this
construction, see [66].
Let P E JP> 1 (1Cp)· To create a seminorm from P, choose homogeneous coordinates P = [a, b] and set Ill = lf(a, b)l. Notice that a different choice of homogeneous coordinates for P gives an equivalent seminorm. This embeds JP> 1 ( Cv) into JP>8 .
5.10.2.3
Properties of A.8 and JP>8
As repayment for the effort required to construct them, Berkovich spaces have many
nice properties that Cv lacks.
304
5. Dynamics over Local Fields: Bad Reduction
Theorem 5.72. (a) The Berkovich disks D~ are compact, Hausdorff, and uniquely
path connected. 5
(b) The Berkovich affine line A 8 is locally compact, Hausdorff, and uniquely path
connected.
(c) The Berkovich projective line IP 8 is compact, Hausdorff, and uniquely path connected.
Proof See [26] and [64].
D
Remark 5.73. As noted earlier, the Berkovich affine line A 8 contains a copy
of A1 (Cp) = Cp, since each a E Cp gives an associated Type-1 point ~a,o in D~
provided R ~ lal. Similarly, the Berkovich projective line JP 8 contains a copy of the
classical projective line JlD 1 ( Cp) via the map
a f----t
{ ~a,o E ~::,
~a,O E D(O, 1)
if lal :::; 1,
~oo,O E ~::,
if a = 00.
if 1 < Ia I < oo,
One can show that the restriction of the Gel'fond topology on A 8 and JP8 to A 1 (Cp)
and JP 1 (Cp), respectively, gives the topology induced by the usual metric on A 1 (Cp)
and the chordal metric on JlD1 (Cp). See Exercise 5.44. This explains why the Gel'fond
topology is the "right" topology to use on Berkovich spaces.
Remark 5.74. We have constructed A 8 and JP8 purely as topological spaces. It is
more difficult, but very important, to refine this construction and make A 8 and JP8
into ringed spaces with structure sheaves built up naturally from rings of functions.
There are two approaches, both due to Berkovich. The first takes unions of open
Berkovich disks, which have a natural structure as analytic spaces, and glues them
along open annuli; see [26, 64]. The second glues affinoids (which are closed) using
nets; see [65]. This second construction is less intuitive, but it allows one to functorially attach a Berkovich analytic space to any reasonable rigid analytic space.
5.10.3
Dynamics on Berkovich Space
Having constructed the Berkovich spaces [JB, A 8 , and JP 8 , we are finally ready to
study iteration of maps on these spaces.
5.10.3.1 Polynomial and Rational Maps on Berkovich Space
Let </J(z) E Cp[z] be a polynomial. There is a natural way to extend the map
</J : A 1 ( Cp) --+ A 1 ( Cp) to a map on Berkovich affine space </J : A8 --+ A 8 . In terms
of seminorms, the action of </J is simply given by composition,
lfl<t>(~l = If o<PI~·
5 Recall that a topological space X is path connected if given any two points xo, XI E X there is a
continuous map f : [0, 1] -->X with /(0) = xo and /(1) = x1. Then X is uniquely path connected if
any two such paths hand hare homotopic to one another, i.e., han be continuously deformed to /2.
5.10. Dynamics on Berkovich Space
305
However, it is perhaps easier to understand the map ¢ : A 6 ---+ A 6 by looking at
the action of¢ on points of Types I-IV. Recall (Proposition 5.16) that¢ maps disks
to disks, say
¢(15(a,r)) = 15(¢(a),R)
for some R = R(¢, a, r).
Then for points of Types I, II, and III we define
</J(~a,r) = ~<jJ(a),R(<jJ,a,r)'
and for points of Type IV we take the usual limit
Remark 5.75. The maximum modulus principle (Theorem 5.13) allows us to explicitly describe the radius R(¢,a,r) ofthe image ¢(15(a,r)). First expand ¢(z) as a
polynomial in powers of z - a, say
d
¢(z) =
2::: ci(¢, a)(z- a)i.
i=O
Then
R(¢,a,r) =max Jci(¢,a)Jri =
1:s;":s;d
sup
zED(a,r)
J¢(z)- ¢(a)J.
A rational function ¢( z) E Cp (z) similarly induces a map on the Berkovich
projective line Jll' 6 extending the usual map on Jll' 1 ( CP). If¢ has no zeros or poles on
the disk 15( a, r ), then it is relatively easy to describe the value of ¢(~a,r ). We know
in this situation that ¢ ( 15 (a, r)) is a disk, say equal to 15 (¢(a), s) . Then
¢(~
a,r
) = {~</J(a),s
~</J(a)-l,sfi</J(a)12
if 1</J(a)l :S 1,
if 1</J(a)l > 1.
Note that the assumption that¢ does not vanish on 15(a, r) is equivalent to the inequality 1¢( a) I > s, so the indicated points are in [JB.
The description of ¢(~a,r) when¢ has zeros and/or poles on 15(a, r) is more
complicated. An explicit description in terms of open annuli is given by RiveraLetelier [373, 375, 376]. (See also [26, Section 2].) A succinct, but less explicit, way
to specify the induced map ¢ : Jll' 6 ---+ Jll' 6 is to use the construction of Jll'6 as a
space of homogeneous seminorms as described in Remark 5.71. Then for a given
seminorm ~ E J11' 6 , the seminorm ¢(~) is determined by writing¢ = [F, G] using
homogeneous polynomials F and G and setting
Jf(x,y)J<P(~) = Jf(F(x,y),G(x,y))J
for all homogeneous f E Cp[x, y].
306
5.10.3.2
5. Dynamics over Local Fields: Bad Reduction
The Julia and Fatou Sets in Berkovich Space
A natural way to put a metric on the Berkovich spaces J)B, A B, and !P'8 is to use
the underlying tree structure and measure distances along line segments. Unfortunately, this path-length metric does not give the Gel'fond topology, and as we
have observed, it is the Gel'fond topology that extends the natural metric topologies on D(O, 1), A 1 (Cp), and IP' 1 (Cp). (See Exercises 5.42 and 5.44.) It is possible
to define a metric that does yield the Gel'fond topology, but the definition of the
"Gel'fond" metric is quite indirect. See [26, Corollary 1.3].
So rather than using a metric, we instead characterize the Fatou and Julia sets in
Berkovich space using an abstract topological version of equicontinuity.
Definition. Let X and Y be topological spaces and let 1> be a collection of continuous maps X -+ Y. The set 1> is (topologically) equicontinuous at x iffor every
point y E Y and every neighborhood V c Y of y there are neighborhoods U c X
of x and W c Y of y such that for every cp E 1>, the following implication is true:
cp(U) n W # 0
==}
cp(U) c V.
Intuition: 1> is equicontinuous at x if for each y E Y, whenever cp E 1> sends some
point close to x to a point that is close to y, then cp sends every point close to x to a
point that is close toy.
One can show that if Y is a compact metric space, then topological equicontinuity agrees with the usual metric definition of equicontinuity. (See [26, Proposition 7 .17].) We say that 1> is (topologically) equicontinuous on X if it is topologically
equicontinuous at every point of X.
Definition. Let cp(z) E Cp(z) be a rational map. The (Berkovich) Fatou set of cp is
the largest open subset of IP' 8 on which ¢ is equicontinuous, or more precisely, on
which the set of iterates {¢n} is equicontinuous. The (Berkovich) Julia set of¢ is the
complement of the Berkovich Fatou set. We denote these sets by :F8 ( ¢) and :J 8 ( ¢ ),
respectively.
Remark 5.76. Recall that the classical points in !P'8 , i.e., the points ofType-1, form
a copy of IP' 1 (Cp) sitting inside !P'8 . As noted earlier in Remark 5.73, the restriction of the Gel'fond topology on IP'8 to the classical points gives the same topology on IP' 1 (Cp) as that induced by the chordal metric. Using this one can show that
equicontinuity at a classical point of!P'8 using the Gel'fond topology is equivalent to
equicontinuity using the chordal metric. Hence the classical Fatou and Julia sets sit
within their Berkovich counterparts:
and
Remark 5.77. Let cp(z) E Cp(z) be a rational map of degree at least 2. Various
authors have shown that there is a unique probability measure p,q, on IP' 8 satisfying
and
5.10. Dynamics on Berkovich Space
307
(Recall that a probability measure is a nonnegative measure of total mass 1.) We
call p,q;, the canonical measure associated to ¢, since the property ¢* p,q;, = d · p,q;,
resembles the analogous property h,p (¢( P)) = d · hq;, (P) of the canonical height.
The reader should be aware that other common names for J.t<t> in the literature include Brolin measure, Lyubich measure, and invariant measure. For the construction
and applications of p,q;,, see Baker and Rumely [26, Theorem 7.14], [29], ChambertLoir [98], Thuillier [434], and Favre and Rivera-Letelier [168], as well as [360].
Theorem 5.78. Let ¢( z) E Cp (z) be a rational map of degree at least 2. The support of the canonical measure p,q;, is equal to the Julia set J B ( ¢). In particular, the
Berkovich Julia set J 8 ( ¢) is not empty.
Proof This theorem is an amalgamation of results due to Baker, Rumely, and RiveraLetelier. We refer the reader to (26, Section 7.5] for the construction of the canonical
measure and to (26, Theorem 7.18], [27, Theorems 8.9 and A.7], and [381] for the
proof that p,q;, is supported exactly on J 8 ( ¢). The last part of the theorem is then
D
clear, since the empty set cannot provide the support for a nontrivial measure.
Example 5.79. Let ¢(z) E Cv(z) be a rational map of degree at least 2 and suppose that ¢ has good reduction. We know (Theorem 2.17) that the classical Julia
set J(¢) c JP> 1 (Cp) is empty. Using the construction of the canonical measure, it is
not hard to show (26, Example 7.2] that for a map of good reduction, the canonical
measure is entirely supported at the Gauss point, i.e.,
p,q;,(U) = 1 if ~o,1 E U
and
p,q;,(U) = 0 if ~o,1 ~ U.
Thus J 8 (¢) = {~o,1}, so the nonempty Julia set guaranteed by Theorem 5.78 is not
very interesting, since it consists of a single point. Hence even in Berkovich space,
the most interesting dynamical behavior occurs for maps of bad reduction. On the
other hand, if the conjugates¢! of¢ have bad reduction for every f E PGL2 (Cp),
then J (¢) is a perfect set, and hence uncountable. (See Theorem 5.82.)
5.10.3.3
The Map ¢(z) = z2 on Berkovich Space
To conclude our brief foray into Berkovich space, we illustrate Berkovich dynamics
by studying the simplest possible map, namely ¢(z) = z 2 • For any a E Cp, we
expand
¢(z)- ¢(a)= z 2 - a2 = 2a(z- a)+ (z- a) 2 •
Assuming henceforth that p ~ 3 and using our convention that all points in JP>8 have
radius satisfying r :::.; 1, we find that
with
s = max{ l2alr, r 2 } = r ·max{ Ia I, r}.
(5.46)
This explicitly gives the action of¢ on points of Types I, II, and III in [JB, and the
action of¢ on Type-IV points is given by the appropriate limit.
5. Dynamics over Local Fields: Bad Reduction
308
The formula (5.46) allows us to compute many orbits O(~a,r ). For example,
suppose that Ia I < 1 and r < 1. Then Icj>n (a) I ----+ 0, so the radius of cj>n (~a, r) also
goes to 0. Thus
Ia I < 1
and
r <1
==?
lim cj>n(~a,r) = ~0,0·
n--->oo
In other words, the open branch li\~, 1 is in the attracting basin of the fixed point ~0 , 0 .
(We leave it to the reader to make this informal argument rigorous using the Gel'fond
topology on D 13 .)
What are the fixed points of¢? For a point ~a,r E D 13 of Type I, II, or III in the
Berkovich disk, we have
cP(~a,r) = ~a,r
r=r·max{lal,r}
{
r= 0
and
and
D(a,r)=D(a2 ,r)
a= a 2 , or
max{ Ia I, r} = 1
and
Ia- a 2
1
::;
r.
There are three cases to consider. First, if r = 0, then a = a 2 , so we see that ~o,o
and 6, 0 are the only fixed points of Type I in D 13 • Second, if r = 1, then ~a,r
is equal to the Gauss point ~0 , 1 , which is clearly fixed by ¢(z). Finally, suppose
that 0 < r < 1. Then ~a,r is fixed if and only if
and
lal = 1
Thus ~a,r is fixed if and only if a E D(1, r ), in which case ~a,r = 6,r· This exhausts
the Type-1,-II, and-III fixed points in D 13 • A similar analysis on the branch leading
up to ~oo,o yields one more fixed point, namely the Type-1 point ~oo,o at the top of
the tree. Hence aside from Type-IV points, the fixed-point set of¢ on lP' 13 consists of
two (attracting) Type- I fixed points ~o,o and ~oo,o and the line segment running from
the Gauss point ~0 , 1 down to the (neutral) Type- I fixed point 6, 0 ,
Fix(¢,1P'13 ) = {~o,o,~oo,o} U {6,r: 0::; r::; 1}.
We leave it as an exercise to show that ¢(z) = z 2 has no fixed points of Type IV
(Exercise 5.36).
We have seen that every point in li\~,o is attracted to ~o,o, and the Gauss point ~0 , 1
is fixed. We next show that every other point in D 13 of Type II or III is preperiodic.
Let ~a,r E D 13 be such a point, which means that
0<r<1
and
lal = 1.
Then the disk D( a, r) has positive radius, so it contains points that are algebraic
over Qp (note that Qp is dense in Cp). Replacing a with such a point, we may assume that a is algebraic over Qp. Next we observe that (5.46) combined with the
assumption that lal = 1 implies
cP(~a,r) = ~c/>(a),ro
so by iteration
5.10. Dynamics on Berkovich Space
309
Hence
Let K = «Jlv (a) and notice that
U D(¢n(a), r) n K c D(O, 1) n K.
n2':0
The disk D(O, 1) n K cannot contain infinitely many disjoint disks of radius r, so
there must exist m > n such that
Then r/Jm(~a,r) = r/Jn(~a,r), so ~a,r is preperiodic.
We conclude this section by using the definition of topological equicontinuity to
directly demonstrate that the Gauss point ~0 , 1 is in the Julia set of¢. The intuition is
that any neighborhood of ~o,1 contains points ~o,r on the line segment connecting ~o,1
to ~0 , 0 . Ifr < 1, then the iterates rj;n(~o,r) approach ~0 , 0 , but the Gauss point ~0 , 1 is
fixed. Hence ¢n(~0 ,1) does not remain close to rj;n(~o,r ). We now make this argument
rigorous.
We suppose that x = ~0 , 1 E :F8 ( ¢) and derive a contradiction. Let y = ~0 , 0 , and
let V = ~~. 1 ; 2 be our chosen neighborhood of ~O,O· The definition of equicontinuity
says that there are neighborhoods ~o. 1 E U and ~o,o E W such that for all n ;::: 0,
(5.47)
The implication (5.47) remains true if we replace U and W by smaller neighborhoods, so we may assume that
with 0 < ri < 1 for all i, and
i=1
with 0 < r < 1.
Choose a value of s satisfying
max ri < s < 1.
1:S:i:S:k
Then ~o,s E U, since ~o,s is not on any of the closed branches ~ai,ri On the other
hand,
rf;n(~o,s) = ~o,sn E W = ~~,r for sufficiently large n,
since we just need to ensure that sn < r. This proves that
so the assumption that¢ is equicontinuous at ~0 , 1 implies that
5. Dynamics over Local Fields: Bad Reduction
310
But this inclusion is clearly not true, since, for example, ~0 , 1 E U is fixed by ¢,
but ~0,1 ~ k-.~, 112 . Therefore¢ is not equicontinuous at ~0,1, so ~0 , 1 E .7 8 (¢).
A similar case-by-case argument using the explicit description (5 .46) of the action of¢ shows that every other point in !P'8 is in the Fatou set. We leave the details
for the reader.
5.10.3.4
Further Results
We briefly describe, without proof, some deeper results on Berkovich dynamics. Our
exposition follows [27], and the author is grateful to Baker and Rumely for making
their preprint available.
Theorem 5.80. (Strong Monte! Theorem on IP' 8 ) Let¢ E Cp(z) be a rational map
ofdegree at least 2, let~ E !P'8 , let U C IP' 8 be an open neighborhood of~, and let V
be the union of cpn (U) for all n ~ 1.
(a) lf!P' 1 (Cp) "V containsatleast3points, then~ E F 8 (¢).
(b) if IP'8 " (V U IP' 1 (Cp)) is nonempty, then~ E F 8 ( ¢ ).
Proof This theorem is due to Baker and Rumely [27, Theorem 7.1] for maps ¢
defined over a finite extension of Qp, and to Rivera-Letelier in the general case;
D
see [27, Theorem A. I] and [381].
The proof is based on Rivera-Letelier's classification of periodic components in
the Fatou set. In order to describe this classification and its applications, we need
to define what it means for a periodic point in Berkovich space to be attracting or
repelling. For Type-I points, i.e., for points in IP' 1 (Cp), we use the usual definition. It
turns out that all attracting periodic points in IP' 8 are Type-I points. (See Exercise 5.45
for an explanation of why this is reasonable.) The definition of repelling periodic
points is more complicated. Repelling periodic points are all of Type I or II [376,
Proposition 5.5], where we use Rivera-Letelier's definition that a Type-11 periodic
point is repelling if its residual degree (see [376, Section 5]) is at least 2.
Definition. Let ¢ E Cp (z) be a rational map of degree at least 2 and let ~ E !P'8 be
an attracting periodic point of period n. The basin of attraction of~ is the set
The connected component of this set is called the immediate basin ofattraction of~.
Definition. Recall that a point P is called recurrent for ¢ if it is in the closure
of { cpn (P) : n ~ 1}. The domain ofquasiperiodicity of¢ is the interior of the set of
points in !P'8 that are recurrent for¢.
We are now ready to state Rivera-Letelier's strong Monte! theorem for rational
maps on the Berkovich projective line.
5.10. Dynamics on Berkovich Space
311
Theorem 5.81. (Rivera-Letelier) Let Uri> be the set ofpoints € E JP>8 with the property
that there is a neighborhood U of € such that
JP>l (Cp) "
U ¢n(U)
n:O::l
contains at least 3 points.
(a) Every periodic connected component ofUrJ> is either an immediate basin of attraction of¢ or a connected component of the domain of quasiperiodicity of¢.
All such components are in :F8 ( ¢).
(b) Every wandering component ofUrJ> is contained in :F8 ( ¢ ).
Proof The proofof(a) is given in [27, Theorem A.2] and the proof of (b) is in [27,
Corollary A.5].
D
As in the classical setting, Montel's theorem has a large number of important
consequences, some of which we state here.
Theorem 5.82. Let ¢ E Cp (z) be a rational map ofdegree at least 2.
(a) If there is some f E PGL2(Cp) such that the conjugate¢! has good reduction,
then the Julia set .J8 ( ¢) consists ofa single point.
(b) If every conjugate ¢! of¢ has bad reduction, then the Julia set .J8 ( ¢) is a
perfect set, and hence in particular it is uncountable.
(c) Let € E .J8 (¢), let U E JP>8 be an open neighborhood of €, and let V be
the union of ¢n(U) for all n 2:: 1. Then (i) V contains JP>8 "JP>1(Cp); (ii) V
contains .J8 (¢); and (iii) JP>1(Cp) "V consists ofat most two points.
(d) The Julia set .J8 ( ¢) is either connected or else it has irifmitely many connected
components. Further, it has empty interior.
(e) Let~ E .J8 (¢). Then the backward orbit of~ is dense in .18 (¢).
(f) Let V c JP>8 be a closed completely ¢-invariant set containing at least 3 points.
Then V 2 .18 (¢).
(g) The Julia set .J8 ( ¢) is exactly equal to the closure of the repelling periodic
points in JP>8 .
Proof (a) See [27, Lemma 8.1].
(b)
(c)
(d)
(e)
See [27, Corollary 8.6].
See [27, Theorem 8.2].
See [27, Corollary 8.3 and Corollary 8.7].
See [27, Corollary 8.5].
(f) See [27, Corollary 8.8].
(g) See [27, Theorem A.7].
D
Exercises
312
Exercises
Section 5.1. Absolute Values and Completions
5.1. Prove that up to equivalence, the only nontrivial absolute values on Q are the usual
archimedean absolute value and the p-adic absolute values. (This result is known as Ostrowski's theorem.)
Section 5.2. A Primer on Nonarchimedean Analysis
5.2. (a) Let cp(z) be a holomorphic function on D(a, r). Prove that the Taylor series coefficients of¢( z) are uniquely determined by ¢.
(b) Let ¢( z) be a function that is represented by a convergent Laurent series on the punctured disk D (a, r) '- {a}. Prove that the coefficients of its Laurent series are uniquely
determined by ¢.
5.3. (a) Let ¢(z) and '1/J(z) be holomorphic functions on D(a,r). Prove that the product cp(z)'lj;(z) is a holomorphic function on D(a, r).
(b) Let ¢( z) and '1/J (z) be meromorphic functions represented by Laurent series on D (a, r).
Prove that the product ¢( z )'1/J( z) is a meromorphic function and is represented by a
Laurent series on D( a, r ).
5.4. Let ¢( z) E Cp [z] be a nonconstant polynomial.
(a) Prove directly (i.e., without using Newton polygons) that¢ sends a closed disk D(a, r)
to a closed disk JJ( ¢(a), s).
(b) Prove that¢ maps D( a, r) bijectively to D (¢(a), s) if and only if
l¢(z)- ¢(a)
al
I= ~lzr
for all z E D(a, r).
5.5. This exercise outlines a prove of a p-adic inverse function theorem. Let ¢ E Cp ( z) be a
rational function of degree d ;::: 1 and let P E lP' 1 ( Cp) be a point that is not a critical value
of¢, i.e., P is not the image of a critical point of¢.
(a) Prove that there is a disk D(P,r) centered at P such that ¢- 1 (D(P,r)) consists
of d disjoint open sets
Show that the Vi are disks.
(b) Let rPi : Vi ---> D (P, r) be the restriction of ¢ to Vi for each 1 :::; i :::; d. Prove that rPi is
bijective.
(c) Possibly after reducing r, prove that the inverse maps ¢i 1 : D(P, r) ---> Vi are given
by convergent power series. (If oo E ¢ - l (P), either change coordinates or instead use
a convergent Laurent series.)
5.6. (a) Let K be an algebraically closed field that is complete with respect to a (nonarchimedean) absolute value and let¢ : lP' 1 (K) ---> lP' 1 (K) be a rational map. Prove that¢
is an open map, i.e., the image of an open set is an open set.
(b) Consider the map¢: lP' 1 (Q 3 )---> lP' 1 (Q 3 ) defined by cp(z) = z2 . Prove that¢ is not an
open map by showing that the image of the open disk D(O, 1) in lP' 1 (Q 3 ) is not open.
Thus rational maps over complete, but not algebraically closed, fields need not be open.
313
Exercises
5.7. Let <f>(z) E Cp[z] be analytic on D(a, r) and assume that <(>(a) f:. 0. Prove that 1/<P(z)
is analytic on some closed disk centered at a. More precisely, prove that 1/<P(z) is analytic
on D(a, t) for any t < rl<f>(a)l/114>11·
Section 5.3. Newton Polygons and the Maximum Modulus Principle
5.8. This exercise generalizes Proposition 5.16(b) by asking you to prove a general Cauchy
estimate for p-adic analytic functions. Let <f>(z) E Cv[z] be a power series that converges
on D(a,r) and let <f>(D(a,r)) = D(<f>(a),s). Prove that
for all n ~ 1.
(Note that as n increases, the estimate becomes worse, since In! I ---+ 0 as n---+ oo.)
Section 5.4. The Nonarchimedean Julia and Fatou Sets
5.9. Prove the implications between uniform Lipschitz, uniform continuity, and equicontinu-
ity stated in Section 5.4. More generally, give a definition of what it means for a family of
functions to be equi-Lipschitz and prove the following implications:
I uniformly Lipschitz I ===* I equi-Lipschitz at every point I
I uniformly continuous I ===* I equicontinuous at every point I
5.10. This exercise develops the abstract theory of residues (on lP' 1) in order to prove Theorem 1.14 in arbitrary characteristic. Let <f>(z) E K(z) and a E K. The function <f>(z) can be
written as a partial Laurent series
a-N
a-N+1
a-1
( )
4> (z ) = (z-a )N + (z-a )N- 1 + ··· + -(z-a
--) + '1/J z
with '!j;(z) E K[z- a],
and we define the residue of¢ at a to be
Res(¢(z) dz) = a-1·
z=et
The residue of¢ at oo is defined by using the substitution z = 1/w; thus
(a) Compute all of the nonzero residues of each of the following functions. In each case,
check that the sum of the residues is 0.
,~. ( )
z2 + 2
(") ,~. ( )
z2 + 2
11
'~' 1 z = z 2 - 3z + 2 ·
'1' 2 z = z 3 - 4z 2 + 5z - 2 ·
(b) If ¢(z) has a simple pole at a, prove that the residue of¢ at a is the value of the
function (z- a)¢(z) evaluated at z =a.
(c) More generally, for any integer n ~ 0, let '1/Jn (z) be the function
(1')
1 dn (
n+1
)
'1/Jn ( z ) = n! dzn (z- a)
¢(z) .
314
Exercises
Here we are taking fonnal derivatives in K(z). Note that in characteristic p, one must
be careful to "cancel" the n! before setting p = 0. (If you have not seen this kind of
computation before, rewrite the expression for 1/Jn (z) to make it clear that it makes sense
in any characteristic.) Suppose that cf>(z) has a pole of exact order n 2: 1 at a, i.e.,
writing cf>(z) = F(z)/G(z) as a ratio of polynomials with no common factors, we have
n = ordz=a (G(z)). Prove that
Res(cf>(z)dz) = 1/Jn-l(a).
z=a
(d) Prove that
{ P E JID 1 (K): Res(cf>(z)
dz) # 0}
p
is a finite set.
(e) Assume that K is algebraically closed and prove the Cauchy residue fonnula
L ~s(cf>(z)dz) = 0.
PEII'1 (K)
(This is a hard exercise. Try it first under the assumption that cf>(z) has simple poles
al!ld ¢>( oo) # oo, which suffices for (f).)
(f) Use (e) to prove that Theorem 1.14 is true in arbitrary characteristic, and use it to deduce
that Corollary 5.19 and Proposition 5.20(c) are true. Hence Fatou sets are nonempty for
all algebraically closed nonarchimedean fields.
5.11. (a) Let p 2: 3 be a prime and let c E Cp satisfY lei > 1. Prove that the function ¢>( z) =
z 2 + c has exactly one nomepelling periodic point, namely the totally ramified fixed
point at infinity.
(b) *Suppose that cf>(z) E Cp(z) has an indifferent periodic point, i.e., a periodic point P
whose multiplier satisfies IAp (¢>)I = 1. Prove that ¢>( z) has infinitely many indifferent
periodic points. (This result is due to Rivera-Letelier [372].)
(c) **Do there exist rational maps cf>(z) E Cp(z) that have at least two, but only finitely
many, nomepelling periodic points? (From (b), the nomepelling periodic points would
have to be attracting.)
Section 5.5. The Dynamics of (z 2 - z)jp
5.12. Let s be a finite set, let sN be the space of sequences on s with the symbolic dynamics
metric, and let L: sN--+ sN be the left shift map. (See Section 5.5.1.)
(a) Let a, ,8 E SN and suppose that limn_,oo p( Ln(a), Ln(,B)) = 0. Prove that there exists
an m such that Lm(a) = Lm(,B).
(b) Prove that the periodic points of L are dense in SN.
(c) Prove that there is an element 'Y E SN with the property that the orbit
Ch('Y) = {C('Y): n EN}
is dense in SN. One says that L is topologically transitive on SN. (Hint. Create 'Y by first
listing all possible blocks oflength I, then all possible blocks oflength 2, and so on.)
(d) Let a E sN. Prove that the backward orbit 0£ (a) = Un;:::o L -n (a) is dense in sN.
Exercises
315
(e) More precisely, prove that the backward orbit of any 0: E sN is equidistributed in the
following sense: For all (3 E SN and all k ::::: 0,
(In fact, prove that the limit stabilizes as soon as n > k.)
5.13. Let ¢( z) = (z2 - z) / p with p ::::: 3. Use Exercise 5.12 and the identification provided
by Proposition 5.24 to prove that the backward orbit of a point a E .:J (¢) is equidistributed in
the following sense: For all b E .:J(¢) and all radii r = p-k,
5.14. Let p::::: 3 and for any c E Cp, let </>c(z) = (z 2 - cz)/p.
(a) If c E
prove that the statement of Corollary 5.25 is true for </>c.
(b) More generally, prove an analogous statement if c is a unit in a finite extension of Zp.
(c) What happens if c E pZp?
z;,
5.15. Let d ::::: 2 be an integer, let p > d be a prime, and consider the dynamics of the map
</>(z) = --'z('--z_-_1-'--')(_z----'2)'--·-··-'-(z_-_d+------'-1)
p
over Cp. Prove that .:J (</>) can be described using symbolic dynamics and use this identification
to prove the following generalization of Corollary 5.25: .:J( </>) C Qp, the set Pern( ¢)consists
of the fixed point at oo and dn repelling points, the Julia set .:J (¢) is the closure of the repelling
periodic points, and ¢ is topologically transitive on .:J (¢).
Section 5.6. A Nonarchimedean Version of Mentel's Theorem
5.16. Complete the proof of Theorem 5.26 (see page 265) by writing down the details in the
case that o: = 0.
5.17. Let ci> be a collection of power series that converge on D(a, r), and suppose that there
is a point o: E K such that o: ~ ¢( D(a, r)) for all¢ E <I>. We proved in Theorem 5.26 that
there is a constant C = C(o:, a, r) such that
p(<f>(z), <f>(w)) ::; Cp(z, w)
for all¢ E ci> and all z, wE D(a, r).
Find an explicit value for the Lipschitz constant C in terms of o:, a, and r.
5.18. Let ci> be a collection of rational, or more generally meromorphic, functions
D(a, r)---> lP' 1 (K),
and suppose that o:, (3 E lP' 1 (K) do not lie in¢( D(a, r)) for all¢ E <I>. Our proof of the nonarchimedean Montel theorem (Theorem 5.27) shows that there is a constant C = C(o:, (3, a, r)
such that
p(¢(z), ¢(w))::; Cp(z, w)
for all¢ E ci> and all z, wE D(a, r).
Find an explicit value of C in terms of o:, (3, a, and r.
316
Exercises
Section 5.7. Periodic Points and the Julia Set
5.19. As in the statement of Lemma 5.34, let A, B C Cp be bounded sets for which there are
constants 0 < 8 -::; 1 -::; ~ such that
sup Ia I -::; ~'
<>EA
sup 1,61 -::; ~'
(3EB
and
inf
<>EA, (3EB
p( a, ,6) = 8 > 0.
For each (a, ,B) E Ax B, let L<>,f3(z) = (,6- a)z +a. DuringtheproofofLemma 5.34we
showed that
4
p(L<>,f3(z),L<>',f3'(z'))-::; ~ ·max{p(a,a'),p(,6,,6'),p(z,z')}
2
for all a, a' E A,a11,6,,6' E B,andallz,z' E IP 1 (Cp)·
Improve this result by reducing the exponent of ~ and/or 8 in the constant. Try to find the
best-possible exponents.
5.20. Let A, B, C c IP 1 ( Cp) be three sets that are at a positive distance from each other, i.e.,
there is a constant 8 > 0 such that
inf
<>EA, (3EB
p( a, ,6) 2: 8,
inf
<>EA,"(EC
p( a, 1) 2: 8,
inf
(3EB,"(EC
p(,6, 1) 2: 8.
For any triple of points (a, ,6, 1) E Ax B x C, define
to be the unique linear fractional transformation satisfying
Prove that the map
is Lipschitz and find an explicit Lipschitz constant depending only on 8. (This exercise generalizes Lemma 5.34, which is the case that C = { oo} is the single point at infinity.)
5.21. This exercise can be used in place of Lemma 5.34 to prove Mantel's theorem with
moving targets (Theorem 5.36).
(a) Let P, Q E IP 1 (K) be distinct points. Prove that there is a linear fractional transformation
A= AP,Q E PGL 2 (K) satisfying
A(O) = P,
A(oo) = Q,
IRes(A) I = p(P, Q).
S)
is normalized to satisfy max{lal, lbl, lei, ldl} = 1, then Exer(Hint. If A = (~
cise 2.8 says that the resultant of A is equal to the determinant ad- be.)
(b) Let A, B C Cp be bounded sets that are at a positive distance from each other. Prove
that the map
(P, Q, R) >---+ Ap,q(R),
is Lipschitz.
Exercises
317
Section 5.8. Nonarchimedean Wandering Domains
5.22. Let X be a topological space and let D be a collection of open subsets of X that form
a base for the toplogy of X. We useD to define disk components as described on page 277.
(Note that the sets in D need not be actual disks, since the space X is merely assumed to be a
topological space.)
(a) Let U c X be an open set, let P1, P2 E U be points, and let V1 and V2 be the
disk components of U containing Pt and P2, respectively. Prove that either V1 = V2
orV1 n V2 = 0.
(b) Prove that U is a disjoint union of disk components.
(c) Prove that the disk components of U are open.
5.23. Let X be a topological space and let D be a collection of open subsets (disks) of X that
form a base for the topology of X. An open set U C X is defined to be disk-connected
(relative to D) if for every pair of points P, Q E U there is a finite sequence of disks
D1, D2, ... , Dn C U such that
D; n Di+l =J 0 for all1 :::; i < n.
(a) Let U1 and U2 be disk-connected subsets of X. Prove that either
(b) Let U C X be an open set and let P E U. Prove that there is a maximal disk-connected
open subset of U containing P.
(c) Prove that the maximal disk-connected open subset of U containing P described in (b)
is in fact the disk component of U containing P. This gives an alternative definition of
disk component.
5.24. Prove that the standard closed disks with rational radius (cf. Remark 5. 7) in lP' 1(Cv) are
exactly the images via elements of PG L2 (Cp) of the unit disk D(0, 1). Similarly, the standard
open disks of rational radius are the images of D(O, 1).
5.25. Let K/Qv be a finite extension and let cf>(z) E K(z) be a rational function of degree
d ~ 2 with oo E F(¢>). Prove that¢> is hyperbolic, i.e.,¢> satisfies the conditions of Theorem 5.46, if and only if¢> has the following property:
For every finite extension L / K there are a set U c L containing .J (¢>) n L, positive
constants b > a > 0, and a continuous function a : U----> [a, b] such that
for all a E U.
This shows that p-adic hyperbolicity and classical (complex) hyperbolicity can both be characterized as saying that a map is everywhere expanding on the Julia set with respect to some
reasonable metric. For further information about hyperbolic maps in the classical setting; see,
for example [95] or [302].
5.26. * Let ¢> : lP' 1 ----> lP' 1 be a rational map of degree at least 2 and let P E lP' 1(Cp) be a
recurrent critical point that is in the Fatou set F( ¢>). Prove that P is a periodic point.
318
Exercises
Section 5.9. Green Functions and Local Heights
5.27. Let K be a field with a nonarchimedean absolute value v, let ¢ : lP' 1 --+ lP' 1 be a
morphism of degree d ;::: 2, let <I> = (F, G) : A? --+ A? be a lift of ¢ satisfying II <I> II v = 1,
and let Y<I> be the associated Green function. If !Res(F, G)lv =1- 1, or equivalently if¢ has
bad reduction at¢, prove that there exists a point (x, y) E A; (K) such that
This is the converse to Proposition 5.58(c).
5.28. Let K be a field with an absolute value v, let ¢ : lP' 1 --+ lP' 1 be a morphism of degree d ;::: 2, and let <I> : A 2 --+ A 2 be a lift of¢.
(a) Prove that the map
g(x, y) = logii<I>(x, y) llv - dlogil (x, y) llv
induces a well-defined function g : lP' 1 ( K) --+ JR.
(b) Prove that g is Lipschitz, i.e., prove that there is a constant C = C( ¢) such that
lg(P)-g(Q)I :S:Cpv(P,Q) forallP,QElP' 1 (K).
(c) If v is nonarchimedean, prove fuat g is locally constant. More precisely, if the lift <I> is
chosen to satisfy II<I>IIv = 1, prove that
g(P) = g(Q)
for all P, Q E 1P' 1 (K) with Pv(P, Q) < iRes( <I>) lv·
(d) Define a modified Green function O<t> by
Prove that O<t> is a Holder continuous function on lP' 1 ( K). In other words, prove that O<t>
is well-defined on lP' 1 ( K) and that there are positive constants C and 8 such that
for all P, Q E lP' 1 (K).
(Hint. Show that O<t> (P) is given by the telescoping series 'L':=o d-n g( ¢n(P)). For
small n, say n :::; N, estimate the difference lg( ¢n(P)) - g( ¢n( Q)) I using (b) and
the fact that ¢ is Lipschitz, and for large n use an elementary bound. Then make an
appropriate choice for N.)
5.29. The definition of the canonical height hq, (P) as the limit of d-n h ( cpn (P)) is not practical for numerical calculations, even for P E lP' 1 (Q), because one would need to compute
the exact value of points cpn(P) whose coordinates have O(dn) digits. A better method to
compute hq,(P) is as the sum ofthe Green functions 9<t>,v(P) as described in Theorem 5.59.
Let ¢ : lP' 1 --+ lP' 1 be a morphism of degree d ;::: 2 and fix a lift <I> of¢. If v E MK
is a nonarchimedean absolute value such that II<I>IIv = 1 and I Res(<I>)Iv = 1, then Proposition 5.58(c) says that
Y<t>,v(x,y) = logil(x,y)llv·
This covers all but finitely many absolute values, so it remains to devise an efficient method
to compute Y<I>,v (x, y) in the remaining cases.
Exercises
319
Let P = [x,y] E IP' 1 (K) and use the algorithm described in Figure 5.6 to define a
sequence of triples
This exercise asks you to prove that N iterations of the algorithm gives the value of gil>, v ( x, y)
to within O(d-N).
(a) Prove that luilv ~ 1 for all i 2: 0. Hence the computation of quantities such as F( Ui, 1)
in the algorithm do not involve excessively large numbers.
(b) Prove that for all i 2: 1 we have
(c) As in Exercise 5.28, we let
g(X, Y) = logii<P(X, Y)llv- dlogiiCX, Y)llvThe homogeneity of<P shows that g is a well-defined function on IP'1 (K), and Proposition 5.57 says that g is bounded. Prove that
Q<~>(x,y) = logiiCx,y)llv + L ~ig(<Pi-l(x,y)).
00
i=l
(d) Prove that
N
9<~>(x, y) = logiiCx, y)llv + L ~i logmax{lxilv, IYilv} + O(d-N),
(5.48)
i=l
where the big-0 constant depends only on <P. Deduce that the algorithm described in
Figure 5.6 computes Q<~>(x, y) to within O(d-N).
(e) Find an explicit value for the big-0 constant in (5.48) in terms ofthe quantities d, II <P II v,
and !Res(<P)Iv·
5.30. Implement the algorithm described in Figure 5.6 to compute the Green function Q.p,oo
for the archimedean absolute value on IP'1 (IR).
(a) Let c/J(z) = z + 1/z and compute Q.p,oo(x, y) to (say) 8 decimal places for each of the
points (1, 1), (2, 1), and (5, 2). Check your program by verifying that your values satisfy
Q.p,oo(5, 2) ~ 2Q.p, 00 (2, 1) ~ 4Q.p, 00 (1, 1).
Compute the canonical height h.p(1) and compare it with the value that you obtained in
Exercise 3.20. (Hint. The map c/J has good reduction at all primes.)
(b) Let c/J(z) = (3z 2 - 1)/(z 2 - 1) and note that c/J2 (1) = 3 and c/J3 (1) = lf. Compute Q.p,oo(x, y) for each of the points (1, 1), (3, 1), and (13, 4). Why does Q.p,oo(3, 1)
not equal4Q.p,oo(1, 1)? What is the difference between these two values?
5.31. Let c/J E Q(z) be a rational map of degree at least 2 and let P E IP' 1 (Q). Write an efficient computer program to compute h.p (P) as a sum of Green functions. (For primes p not dividing Res(<P ), use Proposition 5.58 to compute Q.p,p(x, y). For primes dividing Res( <P) and
for the archimedean place, use the Green function algorithm in Figure 5.6.) Compute h.p(a)
to 8 decimal places for each of the following maps and points.
(a) c/J(z) = z 2 - 1, a= ~-
320
Exercises
INITIALIZATION
Write !I> as !I>= (F, G) with F, G E K[X, Y]
Set N = Desired number of iterations
Set x 0 = x and y0 = y and Green = 0
MAIN LOOP: i = 0, 1, 2, ... , N
Increment Green By d-ilogmax{lxilv, IYilv}
I f lxilv ~ IYilv
Set ui = xi/yi
Compute Xi+l = F(ui, 1) and Yi+l = G(ui, 1)
Else IYilv < lxilv
Set Ui = Yi/ Xi
Compute xi+l = F(1, ui) and Yi+l = G(1, ui)
END MAIN LOOP
Return the Value Green
Figure 5.6: An algorithm to compute the Green function 9<t>(x, y).
(b) cf>(z) = z 2 + 1,
a= ~·
(c) cf>(z) = 3z 2 - 4, a= 1.
1
(d) cf>(z) = z + -, a= 1.
z
3z 2 -1
(e) cf>(z) = z 2 _ 1 , a= 1 and a= 3. Check that h¢(3) ~ 4h¢(1).
5.32. Let K be an algebraically closed field with an absolute value v, let ¢> : IP' 1 ---+ IP' 1
be a m~rphism of degree d ?: 2, fix a lift <P of ¢>, and for each homogeneous polynomial E E K[x, y], let 5..</>,E be the associated local canonical height function. (See Theorem 5.60.)
(a) If E1, E 2 E K [x, y] are homogeneous polynomials, prove that
A¢,E1E2 (P)
= A¢,El (P) + A¢,E2 (P)
at all points such that E1 (P) "/= 0 and E 2 (P) "/= 0.
(b) Let D = n1(Ql) + n 2(Q 2) + .. · + nr(Qr) be a divisor on IP' 1, i.e., Dis a formal sum
of points with n1, ... , nr E Z. Use (a) to associate to D a local height function
(c) Prove that there is a rational function f v E K ( z) with the property that
at all points where 5..</>,D ( cf>(P)) and A¢,v(P) are defined.
(d) Let g(z) E K(z) be a rational function and let D 9 be the divisor ofg, which by definition
is the formal sum
Exercises
321
D9 =
L
ordQ(g)(Q).
QEI'l(K)
(Here ordQ (g) is the order of vanishing of g at the point Q; see Example 2.2.) Prove that
the function
P f------> 5-.</>,Dg (P) + loglg(P) lv
extends to a bounded continuous function on all of 1P' 1 ( K).
Section 5.1 0. Berkovich Space and Dynamics
5.33. Let c/J(z) E Cp[z] be a polynomial and let a E D(O, 1). Write c/J(z) as a polynomial
in z- a, say
Prove that
c/J(D(a,r)) = D(c/J(a),s),
where the radius s of the image disk is given by
(Hint. We already proved that c/J( D(a, r)) is a disk; see Proposition 5.16. Now use the maximum modulus principle to find the radius.)
5.34. This exercise develops some elementary properties of bounded seminorms as defined
on page 296.
(a) Let I · I be a nonconstant seminorm. Prove that I0 I = 0 and Ill = 1.
(b) Suppose that I · I is an II · lin-bounded seminorm. Let f(z) = c be a constant polynomial. Prove that If I = lei is the usual absolute value on Cp.
(c) Suppose that we replace property (3) of bounded seminorm with the usual triangle inequality If+ gl ::::: If I+ IYI· Prove that I · I satisfies (3).
(d) Suppose that we replace property (4) of bounded seminorm with the weaker statement
that there is a constant"" such that If I :::; ""II!IIR for all f E Cp[z]. Prove that (4) is true,
i.e., we can take "" = 1.
5.35. Prove that the seminorms associated to points of Types II, III, and IV are actually norms.
(Hint. For Type IV, use the fact that the limiting radius is positive. See Remark 5.65.)
5.36. Let d 2:: 2. Prove that the map c/J(z) = zd has no Type- IV fixed points in A 8 . Does c/J(z)
have any Type-IV periodic or preperiodic points?
5.37. Let D( a, r) be a disk with 0 f/_ D( a, r) and let f(z) = 1/ z. Prove that
f(D(a,r)) = {z- 1 : lz-al :S::r} =D(a-I,rflal 2 ).
5.38. Let
L
d
c/J(z) =
c;zi E Cp[z]
be a polynomial with lc;l :::; 1 and led I = 1.
i=O
The polynomial c/J induces a map c/J : D8 ........, D8 on the Berkovich disk.
(a) Prove that the Gauss point ~o,1 is fixed by c/J.
(5.49)
322
Exercises
(b) Let a E Cp be a fixed point of¢ : Cp --+ Cp. Prove that ~a,o is fixed by¢.
(c) Let a E Cp be a neutral fixed point, i.e., l¢'(a)l = 1, and let 0 ~ T ~ 1. Prove
that ~a,r is fixed by ¢. (In other words, if a is a neutral fixed point, then the entire line
segment La,o C jjB is fixed by¢.)
(d) Prove that every Berkovich fixed point of ¢ of Type I, II, or III is one of the three types
listed in (a), (b), (c). (Hint. Use the explicit description of ¢(~a,r) given in Exercise 5.33.)
(e) Can ¢ have fixed points of Type IV?
I:
5.39. Let ¢(z) =
CiZi E Cp[z] be a polynomial as in (5.49). Prove directly that the Julia
set of¢ is .:JB(q;) = {6, 0 }. (This is a special case of the result described in Example 5.79,
since the conditions on¢ imply that it has good reduction.)
5.40. For any nested sequence of closed disks
regardless of whether the intersection is empty, we define a seminorm in the usual way,
n
(a) Let T = lim Ti. If the intersection
D(ai, Ti) is nonempty and a is a point in the
intersection, prove that lfla,r = lfla,r· Thus every point in jjB, regardless of whether it
is of Type I, II, III, or IV, can be represented by a nested sequence of closed disks.
(b) Two nested sequences of closed disks
D(O, 1) :J D(a1, T!) :J D(a 2 , T 2 ) :J
D(o, 1) :J .D(b1, sl) :J D(b2, s2) :J
00
00
•
,
•
,
are defined to be equivalent if the following two conditions are true:
• For every i :2: 1 there exists a j :2: 1 such that D(bi, Sj) C D(ai, Ti).
• Foreveryi :2:1 thereexistsaj :2:1 suchthatD(aj,Tj) C D(bi,si)·
Prove that two sequences of disks are equivalent if and only if their seminorms I · la,r
and I · lb,s are equal.
5.41. Let ~a,r E jjB be a Type-IV point. We defined the "line segment" running from ~a,r to
the Gauss point ~o,1 to be the set
u
00
La,r = {~a,r} u
{~ai,t: Ti ~ t ~Ti-l}
0
i=l
(Note that To = 1 by definition.) LetT = limi__,oo Ti. Prove that the map
h: [T, 1] ---> La,r,
h(t) = {~a,r
~ai,t
~ft = T,
IfTi ~ t ~Ti-l,
is a homeomorphism, where [T, 1] C lR has the usual topology and [JB has the Gel'fond
topology. Thus La,r is indeed a line segment.
Exercises
323
5.42. There is a natural metric on the Berkovich disk D5 coming from the tree structure. We
first identify the interval [0, 1] with each of the line segments
-s
La,o = {~a,r : 0 .,:; r .,:; 1} E D ,
so each La,o has length 1. We then define the distance between two points ~a,r and ~b,s
to be the length of the shortest path in the tree connecting them. Denote this distance
by ~~:(~a,r, ~b,s). (For Type-IV points, take the limit.)
(a) If ~a,r and ~b,s both lie on some line segment Lc,o, prove that ~~:(~a,r, ~b,s) = Jr- sJ is
simply the distance between them on that line segment.
(b) In general, prove that
~~:(~a,r, ~b,s) =max{ Jr- sJ, 2Jb- aJ - r- s }.
(Hint. How far above ~a,r and ~b,s do the line segments La,r and Lb,s merge? Go up one
line segment and then down the other.)
(c) Prove that the set
{~ E D5 : ~~:(~,~0,1) < ~}
contains no Type-I points. Prove that every neighborhood of ~o,1 in the Gel'fond topology contains infinitely many Type-I points. Deduce that the path metric ~~:does not define
the Gel'fond topology, and indeed that~~: is not even continuous in the Gel'fond topology!
In Baker and Rumely's terminology, the set D5 with the metric ~~: is called the small model.
There is also a big model, in which the edges are reparameterized so that if D(a, r) C D(b, s ),
then the distance from ~a,r to ~b,s is log( r / s) In particular, points of Type I are at infinite
distance from each other and from all points of Types II, III, and IV. It is the big model that is
best adapted for doing potential theory on Berkovich space; see [26] for details.
I
I·
5.43. For points of Type I, II, and III in D5 , the Hsia kernel is defined to be
o(ea,r, 6 .• ) = max(r, s, Ia- bl).
(For points of Type IV, one takes the appropriate limit.) The Hsia kernel is used in studying
potential theory on [JB; see [26, 209].
(a) Prove that 8(~a,r, ~a,r) = r.
(b) Prove that 8(~a,o,6,o) = Ja- bJ. Thus 8 extends the usual norm on ICp, where we
identify the unit disk in ICp with the Type-I points in [JB.
(c) Let ~c,t be the point where the line segments La,r and Lb,s first meet. Prove that
8(~a,r, ~b,s) =f.
(d) Prove that
8(~,(').,:; max{8(~,(),8((,(')}
5.44. There are natural inclusions
In each case, prove that the restriction of the Gel'fond topology on the Berkovich space
yields the usual metric topology on the smaller space, where we use the usual p-adic metric on D(0, 1) and A 1 ( ICp) and the p-adic chordal metric on lP' 1 ( ICp).
324
Exercises
5.45. Let ¢ be a rational map of degree at least 2, let P E lP' 8 , and let U C lP' 8 be a neighborhood of P with the property that
¢(U) ~ U
and
n
¢n(U) = {P}.
n;::o
Prove that Pis a Type-! point and that, considered as a point in lP' 1 (Cp), the point Pis an
attracting fixed point for ¢.
Chapter 6
Dynamics Associated to
Algebraic Groups
In the forest of untamed rational maps live a select few whose additional structure
allows them to be more easily domesticated. They are the power maps, Chebyshev
polynomials, and Lattes maps, whose complex dynamics were briefly discussed in
Section 1.6. The underlying structure that they possess comes from an algebraic
group, namely the multiplicative group for the power maps and Chebyshev polynomials and elliptic curves for the Lattes maps. Although such maps are special in
many ways, they yet provide important examples, testing grounds, and boundary
conditions for general results in dynamics. In this chapter we investigate some of
the algebraic and arithmetic properties of the rational maps associated to algebraic
groups.
6.1
Power Maps and the Multiplicative Group
The simplest rational maps are the power maps given by monic monomials,
where the integer d may be positive or negative. These maps obviously commute
with one another under composition,
A more intrinsic description of the maps associated to the monic monomials Md is
that they are endomorphisms of the multiplicative group,
More precisely, there is an isomorphism of rings,
325
326
6. Dynamics Associated to Algebraic Groups
The fact that the Md are endomorphisms of the multiplicative group Gm makes it
quite easy to describe their preperiodic points.
Proposition 6.1. Let d E Z with ldl 2 2 and let Md : IP' 1 -+ IP' 1 be the power
map Md(z) = zd. Then
PrePer(Md) = (Gm)tors = {( E Gm: C = lforsomen 21} =
UJLn,
where recall that JLn denotes the group of n 1h roots of unity.
Proof We proved this long ago for any abelian group G and homomorphism z ~----> zd
with d 2 2; see Proposition 0.3. The prooffor d :S -2 is similar and left to the
reader.
D
The iterates of Md are given by
so the periodic points of Md are also easy to characterize,
Proposition 6.2. Let ldl 2 2 and let ( E Per~*(Md) be a point of exact period n ~ 2. Then the multiplier of Md at ( is given by
Proof Using M:J:(z) = zdn, we can directly compute
In particular, if we are working over C, then every periodic point of Md inC* is
repelling. On the other hand, over a p-adic field with p f d, the multiplier dn is a unit,
so the periodic points are indifferent. And if p I d, then all of the periodic points are
attracting. Of course, there are also the two superattracting fixed points 0 and oo.
It is not hard to find all rational maps that commute with the power maps. In
particular, we can compute the automorphism group Aut(Md).
Proposition 6.3. Let K be afield and let Md(z) = zd be the d1h-power map for
some ldl ~ 2. Further, if K has finite characteristic p, assume that p f d.
(a) The set ofrational maps that commute with Md(z) is given by
{<P(z) E K(z): ¢oMd = Mdo¢} = {cze: c E JLd-l ande E Z}.
327
6.1. Power Maps and the Multiplicative Group
(b) The automorphism group of Md is
Aut(Md) = {az: a E 1-Ld-d U {bz- 1 : bE 1-Ld- 1},
where we recall that the automorphism group Aut(¢) of any rational map ¢ is
the setoff E PGL2(k) satisfying¢! =¢.In particular, Aut(Md) is a dihedral group oforder 2n, where n is the number of (d - 1yr roots of unity in K*.
Proof (a) It is clear by a direct computation that the indicated maps cze commute
with Md(z), so it suffices to prove that they are the only commuting maps. Suppose
that ¢(z) E K(z) commutes with Md(z), so
¢(zd) = ¢(z)d.
Let ( E k be a primitive rfh root of unity. (This is where we use the assumption
that p f d if K has positive characteristic p.) Then ¢((z)d = ¢(z)d, so
¢( ( z) = (k ¢( z)
for some cf' root of unity (k.
Consider the function 'ljJ(z) = z-k¢(z). It satisfies
'ljJ((z) = ((z)-k¢((z) = ((z)-k(k¢(z) = z-k¢(z) = 'ljJ(z).
Hence 'ljJ is a function of zd. In other words, there is a rational function 'lj; 1 ( z) E k (z)
such that 'ljJ(z) = 'lj; 1 (zd), and thus ¢(z) = zk'ljJ 1 (zd).
More generally, for any n 2': 1 we have ¢ o M:J = M:J o ¢, so ¢( zdn) =
¢(z)dn. The above argument then yields an integer 0 ::::; kn < dn and a rational
function 'l/Jn (z) E k (z) such that
¢(z) = zkn'l/Jn(zdn).
We write '1/Jn(z) = zj An(z) for some integer j and some An(z) E K(z) satisfying An(O) -::/:. 0 and An(O)-::/:. oo. Then
deg( ¢(z)) = deg( zkn'l/Jn(Zdn))
= deg(zkn+)dn An(Zdn)) 2': deg(.An(Zdn)) = dndeg(.An(z)).
Hence deg(.An) = 0 for sufficiently large n, which proves that ¢(z) has the
form ¢( z) = cze for some c E K* and some e E Z. With this information in hand,
it remains only to observe that ¢(zd) = ¢(z)d if and only if c = cd, i.e., if and only
if either c = 0 or c E 1-Ld- 1 ·
(b) By definition, Aut(¢) is the set of rational maps of degree 1 that commute
with ¢. It follows from (a) that Aut(¢) is the set of all cz± 1 with c E 1-Ld- 1 . In
particular, if we let fa(z) = az and g(z) = z- 1 , then
Aut(Md) ={fa: a E 1-Ld-d U Ua o g: a E 1-Ld-1},
and the dihedral nature of the group law is evident from the identities
fa o g =go fa-1.
D
6. Dynamics Associated to Algebraic Groups
328
Example 6.4. We can use the map Md(z) = zd to illustrate the construction of
dynamical units in Section 3.11. First we use Theorem 3.66, which says that if o: has
exact order nand gcd(i- j, n) = 1, then
0:
di
0:
d
-o:
dj
0:
-
.
.
1s a un1t.
Taking j = 0 and gcd(i, n) = 1, this implies that
di-1
o: d- 1 - 1 is a unit for all primitive (dn 0:
-1
1) 81 roots of unity o:.
These are examples of classical cyclotomic units.
Similarly, let m and n be positive integers with m f n and n f m. Then Theorem 3.68 says that if o: is a primitive (dm - 1) 81 root of unity and if (3 is a primitive (dn - 1)81 root of unity, then o: - (3 is a unit. These are again classical examples
of cyclotomic units.
Example 6.5. Recall that a nontrivial twist of a rational map rjy( z) E K (z) is a
rational map 'lj;(z) E K(z) such that 'lj;(z) is PGL 2 (K)-conjugate to rjJ(z), but 'lj;(z)
is not PGL 2 (K)-conjugate to rjJ(z). See Section 4.9 for the general theory. Since
the power maps Md(z) = zd have a large automorphism group, they tend to have
many twists. For example, the twists associated to the subgroup { cz : c E JLd-d
of Aut(rjy) are given by (cf. Example 4.81)
a
K* /(K*)d- 1 ____, Twist(Md),
1-----)
azd.
There are also some rather complicated-looking twists associated to the subgroup
{z, z- 1 } C Aut(Md)·
Each bE K* /(K*) 2 leads to a twist
O<O~d/2 (2:) b' z"
I
o,,J;:_,)/2
ck ~
!)b'z2k+k .
(6.1)
See Example 4.82 for the derivation of this formula.
6.2
Chebyshev Polynomials
The multiplicative group Gm has a nontrivial automorphism given by inversion
z ~---+ z- 1 , and the quotient of Gm by this automorphism is isomorphic to the affine
line A 1 via the map
The inversion automorphism commutes with the cfh-power map Md(z) = zd, so
when we take the quotient ofGm, we find that Md(z) descends to give a map on
the quotient space A 1 . This leads to the following definition (cf. Section 1.6.2).
6.2. Chebyshev Polynomials
329
Definition. The cfh Chebyshev polynomial is the polynomial Td( w) E Z[w] satisfying the identity
in the field Q(z).
(6.2)
Of course, we need to show that that Td exists. In the next proposition we prove
the existence of the Chebyshev polynomials and describe some of their algebraic
properties.
Proposition 6.6. For each integer d ?: 0 there exists a unique polynomial Td (w) E
Q[w] satisfYing
in thefieldQ(z).
(6.3)
We call Td the d1h Chebyshev polynomial.
(a) Td( w) is a monic polynomial of degree din Z[w].
(b) Td(Te( w)) = Tde( w )for all d, e ?: 0.
(c) Td( -w) = ( -l)dTd(w). Thus Td is an oddfunction ifd is odd and it is an even
function if d is even.
(d) The Chebyshev polynomials satisfY the recurrence relation
(e) For all d ?: 1, the d 1h Chebyshev polynomial is given by the explicit formula
Td (w) =
k
"\;"'
d (d-k k) w
( -1) d - k
L
d-2k
.
(6.5)
O~k~d/2
Proof Suppose first that there do exist polynomials Td( w) satisfying (6.3). Then
To(z + z- 1 ) = z 0 + z- 0 = 2,
+ z - 1) = z + z - 1,
T2(z + z- 1) = z 2 + z- 2 = (z + z- 1) 2 - 2,
T1 ( z
so we see that T 0 ( w) = 2, T1 ( w) = w, and T2 ( w) = w2 - 2 are uniquely determined
monic polynomials with integer coefficients. Still assuming that the set of Chebyshev
polynomials exists, we next observe that they satisfy
(z + z- 1)Td+1 (z + z- 1) - Td(z + z- 1)
= (z + 2 -1 )(zd+1 + 2 -d-1) _ (zd + 2 -d)
= 2 d+1 + 2 -d-2
= Td+2(z + z- 1 ).
Putting z + z- 1 = w, this means that Td+ 2(w) = wTdH(w)- Td(w). Hence ifthe
Chebyshev polynomials exist, they are unique, because they are completely determined by the recurrence
330
6. Dynamics Associated to Algebraic Groups
To(w) = 2,
T1(w) = w,
and
Td+2(w) = wTd+l(w)- Td(w)
ford~ 0.
(6.6)
And from this recurrence we see immediately by induction that Td (w) is a monic
polynomial of degree din Z[w].
We now turn this argument around and use the recurrence (6.6) to define a sequence of polynomials Td( w). We claim that Td( w) then satisfies (6.3). This is clear
for T0 and T1 , so we assume that (6.3) is true for T0, T1 , ... , Td+ 1 and use (6.6)
with w = z + z- 1 to compute
Td+2(z + z- 1) = (z + z- 1)Td+1 (z + z- 1) - Td(z + z- 1)
= (z + z-1 )(zd+1 + z-d-1) - (zd + z-d) = zd+2 + z-d-2.
Hence (6.3) is true for Td+ 2, so by induction we have
for all d 2: 0.
This proves that the recurrence defines polynomials satisfying (6.3), so Chebyshev
polynomials of every degree exist. We have now shown that Chebyshev polynomials
exist, are unique, and have the properties in (a) and (d).
Next we make repeated use of (6.3) to compute
Td (Te(z + z- 1)) = Td(ze + z-e) = (ze)d + (ze)-d = zde + z-de = Tde(z + z- 1).
Hence Td(Te(w)) = Tde(w), which proves (b).
To prove (c), we replace z by -z in (6.3) to obtain
Td (- (z + z - 1 )) = Td ( - z + (- z) - 1 )
= (-z)d + (-z)-d = (-1)d(zd + z-d) = (-1)dTd(z + z- 1 ).
Therefore Td( -w) = ( -1)dTd(w), which is (c).
Next we prove the explicit summation formula (6.5) given in (e). Substituting d = 1 and d = 2 into the formula yields the correct values T1 (w) = w and
T2(w) = w 2 - 2. We now assume that the formula is correct up to Td+ 1 and use the
recurrence (6.4) to check it ford+ 2. Thus
Td+2(w) = wTd+1(w)- Td(w)
=w
(-1)k
"
(-1)k_d_ (d- k)wd-2k
~
o::;k::;(d+l)/2
~
o::;k::;d/2
"
~
o::;k::;(d+l)/2
_
(d+1-k)wd+1-2k
k
"
( -1)k
d+1
d+1-k
d-k
k
d + 1 (d + 1- k)wd+2-2k
d+1-k
k
"
(-l)k-1
d
(d-k+1)wd-2k+2
~
d-k+1
k-1
1::;k::;d/2+1
6.2. Chebyshev Polynomials
331
T2 = w 2 - 2
T3 = w 3 -
3w
T4 = w 4 -
4w 2 + 2
Ts = w5 - 5w 3 + 5w
T6 =
w 6 - 6w 4 + 9w 2 - 2
7w 5 + 14w3 - 7w
T7 = w 7 -
Ts = w 8 - 8w 6 + 20w4 - 16w2 + 2
T9 = w9 - 9w 7 + 27w 5 - 30w 3 + 9w
T10 = w10 - 10w8 + 35w 6 - 50w 4 + 25w 2 - 2
Tn = w11 - llw 9 + 44w 7 - 77w 5 + 55w 3 - llw
T12 = w12 -
12w 10 + 54w 8 - 112w6 + 105w4 - 36w 2 + 2
Table 6.1: The first few Chebyshev polynomials.
(d + 1) (d + 1 - k) + d (d + 1 - k)
"
LJ
k
(-l)k
d+l-k
O:::;k::Sd/2+1
k-1
wd+2-2k
'
where we use the standard convention that (~) = 0 if n < m or if m < 0. A simple
algebraic calculation that we leave for the reader (Exercise 6.4) shows that
(d + 1)
(d+ k1 - k) + d (d +k -1 -1 k) = d + 2 (d + 2 - k) .
d+l-k
d+2-k
k
Hence
T.
( )=
d+ 2 w
"
LJ
O::;k::S(d+2)/2
which completes the proof of (e).
(-l)k
d
d+2 -2 k (d + k2 - k) w
+
d+2-2k
'
D
Remark 6. 7. As mentioned in Section 1.6.2, the classical normalization for the
Chebyshev polynomials is
or equivalently,
332
6. Dynamics Associated to Algebraic Groups
Td(cosO) = cos(dO)
for all (J E ~-
The two nonnalizations are related by the simple fonnula Td(w) = ~Td(2w). We
have chosen the alternative nonnalization because it has better arithmetic properties.
In particular, the map Td : IP' 1 ~ IP'1 has good reduction at all primes. The classical
nonnalization Td has bad reduction at 2.
As with the power maps, it is not difficult to describe the periodic points of the
Chebyshev polynomials and to compute their multipliers. We state the result and
leave the computation as an exercise.
Proposition 6.8. Let Td (w) be the d1h Chebyshev polynomial for some d 2 2.
(a) The fixed points ofTd in A 1 (C) are
27rj ) :O~J~.
d +-1} U { 2cos ( d27rj ) :O<j<d- 1}
{ 2cos ( d+
1
2
1
2 .
(b) The multipliers ofTd at its fixed points are given by
:j
d+1
Ard (2cos (d2 1 )) = -d
forO< j < - 2 -,
Ard (2cos (d2 1 )) = d
forO <j < - 2-,
=j
d-1
Ard(±2) = d2 .
(Note that -2 E Fix(Td) if and only ifd is odd.)
In general, the periodic points and multipliers ofTd can be derived from the above
formulasusingT:J = Tdn andPern(Td) = Fix(T:J).
Proof See Exercise 6.5.
D
We now prove an analogue of Proposition 6.3 for Chebyshev polynomials.
Theorem 6.9. Let K be afield and let Td(w) be the d1h Chebyshev polynomial for
some d 2 2. Further, if K has finite characteristic p, assume that p f d.
(a) The automorphism group ofTd is given by
Aut(Td) = { 1
J.tz
if d is even,
if d is odd.
(b) Assume that K does not have characteristic 2. Let ¢( w) E K (w) be a rational
map that commutes with Td(w), i.e., ¢(Td(w)) = Td(¢(w)). Then
¢(w) = ±Te(w)
for some e 2 1.
The minus sign is allowed if and only if d is odd. (See Theorem 6. 79 for a
stronger result.)
333
6.2. Chebyshev Polynomials
Proof (a) The assertion that Aut(Td) C JL 2 is an immediate consequence of (b),
since (b) implies that any f E Aut(Td) satisfies f( w) = ±T1 ( w) = ±w. However,
since the proof of (b) is somewhat intricate, we give a direct and elementary proof
of(a).
Suppose that f E PGL2(K) satisfies TJ = Td. The polynomial Td has a unique
totally ramified fixed point at oo (cf. Exercise 6.8), and TJ similarly has a unique
totally ramified fixed point at f- 1 ( oo), so the equality rJ = Td tells us that
f- 1 (oo) = oo. Hence f(w) = aw + b is an affine transformation. (The same argument applies to any polynomial not of the form awd.)
Proposition 6.6(c) says that Td(w) satisfies Td( -w) = ( -1)dTd(w), so in particular,
Td(w) = wd +(terms of degree at most d- 2).
(6.7)
The identity rJ (w) = Td (w) with f (w) = aw + b can be written as
We evaluate both sides using (6.7) and look at the top degree terms. This gives
adwd + dad- 1 bwd- 1 +(terms of degree at most d- 2)
= awd + (terms of degree at most d -
2).
Hence
and
By assumption, d i= 0 in the field K, so we conclude that ad- 1 = 1 and b = 0.
In order to pin down the value of a, we use the explicit formula for Td (w) given
in Proposition 6.6(e). In fact, we need only the top two terms,
Td(w) = wd- dwd- 2 +(terms of degree at most d- 4).
By assumption we have Td(aw) = aTd(w), so
Hence ad = a and ad- 2 = a, where we again use the assumption that d i= 0 in the
field K. It follows that a 2 = 1. Further, a = -1 is possible only if ( -1 )d = -1,
so when dis odd. This completes the proof that Aut(Td) is trivial if dis even and is
equal to JL 2 if d is odd.
(b) It is easy to verify that the Chebyshev polynomial Td (w) cannot be conjugated
to a polynomial of the form cwd (Exercise 6.8). It follows that any rational map
commuting with Td( w) is necessarily a polynomial, a fact whose proof we defer until
later in this chapter; see Theorem 6.80. We now describe a proof due to Bertram [69]
that the only polynomials commuting with Td are ±Te. We begin with two lemmas.
The first characterizes the Chebyshev polynomials as the solutions of a nonlinear
differential equation, and the second explains how to exploit such equations.
6. Dynamics Associated to Algebraic Groups
334
LemJDa 6.10. Assume that K does not have characteristic 2. Let d ~ 1 and let F( w)
be a polynomial solution to the differential equation
(6.8)
Then F(w) = ±Td(w).
Proof We first check that ±Td (w) are solutions. We differentiate the functional
equation (6.2) defining the Chebyshev polynomials to obtain the identity
T~(z + z- 1 )(1- z- 2 ) = dzd- 1 -
dz-d- 1 ,
and then solve for T~,
Putting w = z + z- 1 as usual and noting that w 2 - 4 = ( z - z- 1 ) 2 , we compute
This proves that ±Td(w) are solutions to (6.8).
Next suppose that F(w) is any polynomial solution to (6.8). If F'(w) is identically 0, then (6.8) implies that F(w) = ±2 = ±T0 (w), so we are done. We may thus
assume that F' (w) =/=- 0. We differentiate both sides of (6.8) and divide by 2F' (w) to
obtain
(6.9)
(4- w 2 )F"(w)- wF'(w) + d2 F(w) = 0.
In particular, Td(w) is a solution to (6.9). Suppose now that F is any polynomial of
degree k that is a solution to (6.9). We write F( w) = awk + bwk- 1 + · · · with a =/=- 0
and substitute into (6.9). The leading term is
so we must have k = d. In other words, we have shown that every nonzero polynomial solution of (6.9) has degree d. But F(w) - aTd(w) is a polynomial of
degree strictly less than d that satisfies (6.9); hence F(w) = aTd(w). Finally,
substituting w = 2 into (6.8) yields F(2) = ±2, while we know that Td(2) =
Td(1 + 1- 1 ) = 1d + 1d = 2. Hence F(w) = ±Td(w), which completes the proof
D
of Lemma 6.10.
Lemma 6.11. Let A( w) be a polynomial of degree r ~ 1 and suppose that F( w) is
a polynomial of degree d ~ 2 satisfying
A(w)F'(wr = dr A(F(w)).
(6.10)
335
6.2. Chebyshev Polynomials
Suppose further that G (w) is a polynomial of degree e 2: 0 that commutes with F,
i.e., F(G(w)) = G(F(w)). Then
A(w)G'(wt = er A(G(w)).
Proof Consider the polynomial
B(w) = A(w)G'(wt- er A(G(w)).
We assume that B( w) -1- 0 and derive a contradiction, which will prove the desired
result. First we observe that the leading coefficients of A( w )G' (wt and erA (G( w))
cancel, so
(strict inequality).
degB < re
Next we use the various definitions and given relations to compute
dr B(F(w))
= dr A(F(w))G'(F(w)r- drer A(G(F(w)))
definition of B,
= dr A(F(w) )G' (F(w) r - dr er A(F( G(w)))
using FoG= Go F,
= A(w)F'(wtG'(F(w)r- er A(G(w))F'(G(w)r
using (6.10) twice,
= A(w)(G o F)'(wt- er A(G(w))F'(G(w)r
= A(w)(F o G)'(wr- er A(G(w))F'(G(w)r
chain rule,
using FoG= Go F,
= A(w)F'(G(w)rG'(wt- er A(G(w))F'(G(w)r
chain rule,
= F'(G(w)r [A(w)G'(wt- er A(G(w))]
= F' (G(w)r B(w)
definition of B.
Taking degrees of both sides gives
(degB)(degF) = r(degF -l)(degG) + (degB),
and the assumption that deg F 2: 2 means that we can solve for
degB = r(degG) = re.
This contradicts the earlier strict inequality deg B < re. Hence B must be the zero
D
polynomial, which completes the proof of Lemma 6.11.
We now resume the proof of Theorem 6.9(b). Let¢( w) be as in the statement (c)
with ¢( w) a polynomial and let e = deg( ¢). Lemma 6.10 tells us that
(4- w 2 )T~(w) 2 = d2 (4- Td(w) 2 ).
Hence we can apply Lemma 6.11 with A( w) = 4 - w2 and the commuting polynomials ¢ and Td to deduce that
Then another application of Lemma 6.10 implies that ¢( w) = ±Te (w).
D
336
6. Dynamics Associated to Algebraic Groups
Using Theorem 6.9, it is easy to describe all of the twists of the Chebyshev polynomials.
Corollary 6.12. Continuing with the notation and assumptions from Theorem 6.9,
if d is even, then Td has no nontrivial K / K -twists, and if d is odd, then each a E K*
yields a twist
1
Td,a(w) = foTd( y'aw).
Two such twists Td,a and Td,b are K -conjugate if and only if ajb is a square inK*.
Proof We use the description of Aut(¢) from (a). If dis even, then the automorphism group Aut(¢) is trivial, so Proposition 4.73 says that¢ has no nontrivial
twists. For odd d we have Aut(¢) = {±z }, so the desired result follows from ExD
ample 4.81 (see also Example 4.75).
Remark 6.13. Over C, there is a short proof that Aut(Td) c J.L 2 using the fact
(Exercise 1.31) that the Julia set ofTd is :J(Td) = [-2,2]. Then the assumption
that Tj = Td implies that f maps the interval [- 2, 2] to itself. Since f is bijective on IP' 1 (C), it follows in particular that f permutes the endpoints of the interval [-2, 2]. Hence f(2) = ±2 and f( -2) = =f2. Writing f as f(w) = aw + b, this
gives two equations to solve for a and b, yielding b = 0 and a = ±1. Note that this
proof does not carry over to characteristic p, since, for example, working over lFP we
have Aut(Tp) = PGL 2 (1Fp); see Exercise 6.10.
6.3
A Primer on Elliptic Curves
The remainder of this chapter is devoted to rational maps associated to elliptic curves.
In this section we give some basic definitions and review, without proof, some of the
properties of elliptic curves that will be needed later. The reader should also review
the summary of elliptic curves over C given in Section 1.6.3. For further reading on
elliptic curves and for the proofs omitted in this section, see for example [96, 198,
248,250,254,257,410,412,420].
6.3.1
Elliptic Curves and Weierstrass Equations
Definition. An elliptic curve E over a field K (of characteristic different from 2
and 3) is described by a Weierstrass equation, which is an equation of the form
(6.11)
with a, b E K and 4a3 + 27b 2 =/= 0. Of course, we really mean that E is the projective curve obtained by homogenizing equation (6.11), so E has one extra point "at
infinity," which we denote by 0. If K has characteristic 2 or 3, then equations of the
form (6.11) are insufficient, and indeed they are always singular in characteristic 2,
so one uses the generalized Weierstrass equation
(6.12)
337
6.3. A Primer on Elliptic Curves
Remark 6.14. Let E I K be an elliptic curve defined over a field K. When we write E,
we mean the geometric points of E, i.e., the points in E(K) for some chosen algebraic closure of E. If we want to refer to points defined over K, we always explicitly
write E (K), and similarly we write E (L) for the points defined over some extension
field L of K.
More intrinsically, an elliptic curve is a pair (E, 0) consisting of a smooth algebraic curve E of genus 1 and a point 0 E E. For convenience we often call E an
elliptic curve, with the understanding that there is a specified point 0. We say that E
is defined over a field K if the curve E is given by equations with K -coefficients and
the point 0 is in E(K).
Using the Riemann-Roch theorem, one can prove that every elliptic curve E I K
can be embedded in lP'2 by a cubic equation of the form (6.12) with 0 mapping to the
point at infinity. (See [410, III §3].) Then, if the characteristic of K is neither 2 nor 3,
we can complete the square on the left and the cube on the right to obtain the simpler
Weierstrass equation (6.11). In order to simplify our discussion, we will generally
make this assumption.
The discriminant Ll(E) and j-invariant j(E) of the elliptic curve E given
by (6.11) are defined by the formulas
4a 3
j(E) = 1728 4a3 + 27b2.
Proposition 6.15. (a) Let a, bE K and let E be the curve given by the Weierstrass
equation (6.11 ). Then E is nonsingular, and thus is an elliptic curve, if and only
if Ll(E) =/= 0.
(b) Two elliptic curves E and E' are isomorphic over k ifand only if j (E) = j (E').
More precisely, E and E' are isomorphic if and only if there is a u E K* such
that a' = u 4 a and b' = u 6 b
Proof See [410, III §1].
6.3.2
D
Geometry and the Group Law
There is a natural group structure on the points of E that may be described as follows.
Let L be any line in lP'2 . Then counted with appropriate multiplicities, the cubic
curve E and the line L intersect at three points, say
EnL={P,Q,R},
where P, Q, and R need not be distinct. The group law onE is determined by the
requirement that the sum of the points P, Q, R be equal to 0,
P+Q+R=O.
The point 0 serves as the identity element of the group. The inverse of a point P,
which we denote by -P, is the third point on the intersection of E with the line
through P and 0.
338
6. Dynamics Associated to Algebraic Groups
Theorem 6.16. Let Ej K be an elliptic curve defined over afield K.
(a) The addition law described above gives E = E(K) the structure ofan abelian
group.
(b) The group law is algebraic, in the sense that the addition and inversion maps,
E XE
(P,Q)>--+P+Q
E,
E
P>--+-P
E,
are morphisms, i.e., are given by everywhere defined rational JUnctions.
(c) The subset E(K) consistingofpointsofE thataredefinedover K is a subgroup
ofE(K).
Proof See [410, III §§2,3].
D
It is not hard to give explicit formulas for the group law on an elliptic curve, as
in the following algorithm.
Proposition 6.17. (Elliptic Curve Group Law Algorithm) Let E be an elliptic curve
given by a Weierstrass equation
E : y 2 = x 3 + ax + b,
and let P1 = (x1, yl) and P2 = (x2, Y2) be points on E.
If X1 = x2 and Yl = -y2, then P1 + P2 = 0. Otherwise, define quantities
A= Y2- Yl ,
X2- Xl
A_ 3x~ +a
-
2yl
X2- X1
-x~ + ax1 + 2b
v - --=-----2yl
,
,
Then y = AX + v is the line through P 1 and P2, or tangent to E if P 1 = P 2, and the
sum of P1 and P2 is given by
As a special case, the duplication formula for P = (x, y) is
x([ 2]P) = x 4 - 2ax 2 - 8bx + a 2
4x 3 + 4ax + 4b
Proof See [410, III.2.3]
6.3.3
D
Divisors and Divisor Classes
Definition. A divisor on E is a formal sum of points
D=
L np(P),
PEE
339
6.3. A Primer on Elliptic Curves
with np E Z and all but finitely many np = 0. The set of divisors under addition
forms the divisor group Div( E). The degree of a divisor d is
deg(D) =
L np.
PEE
The degree map deg: Div(E)-+ Z is a group homomorphism.
There is a natural summation map from Div(E) toE defined by
sum: Div(E) ---> E,
L np(P) ~ L [np](P).
PEE
PEE
(N.B. The two summation signs mean very different things. The first is a formal sum
of points in Div(E). The second is a sum using the complicated addition law on E.)
The zeros and poles of a rational function f on E define a divisor
div(f) =
L ordp(f)(P),
PEE
where ordp(f) is the order of zero off at P if f(P) = 0, and ordp(f) is negative
the order of the pole off at P if f(P) = oo. A divisor of the form div(f) is called a
principal divisor. The principal divisors form a subgroup ofDiv( E), and the quotient
group is the Picard group Pic( E). Within Pic(E) is the important subgroup Pic0 (E)
generated by divisors of degree 0.
The next proposition describes the basic properties of divisors on E.
Proposition 6.18. Let E be an elliptic curve.
(a) Every principal divisor onE has degree 0.
(b) A divisor D E Div(E) is principal if and only if both deg(D)
sum(D) = 0.
0 and
(c) The summation map induces a group isomorphism
sum: Pic0 (E) --->E.
Proof See [410, 111.3.4 and 111.3.5].
D
6.3.4 lsogenies, Endomorphisms, and Automorphisms
Definition. An isogeny between two elliptic curves E 1 and E 2 is a surjective
morphism 'ljJ: E 1 -+ E 2 satisfying 'lj;(O) = 0. (Note that any nonconstant morphism E 1 -+ E 2 is automatically finite and surjective.) The curves E 1 and E 2 are
said to be isogenous if there is an isogeny between them.
Remark 6.19. Every nonconstant morphism 'ljJ : E 1 -+ E 2 is the composition of an
isogeny and a translation (cf. [410, III.4.7]). To see this, let cp(P) = 'lj;(P)- 'lj;(O).
Then the map¢: E-+ Eisa morphism, and ¢(0) = 0, so¢ is an isogeny. Hence
'lj;(P) = cp(P) + 'lj;(O)
is the composition of an isogeny and a translation.
6. Dynamics Associated to Algebraic Groups
340
Remark 6.20. We observe that an isogeny is unramified at all points. This follows from the general Riemann-Hurwitz formula (Theorem 1.5) applied to the map
7/J : El ----+ E2,
2g(El)- 2 = (deg7/J)(2g(E2 ) - 2) +
2::: (ep('ljJ) -1).
PEEt
The elliptic curves E 1 and E 2 both have genus 1, and the ramification indices satisfy ep('ljJ) 2 1, so it follows that every ep('ljJ) is equal to 1, so 7/J is unramified.
Theorem 6.21. An isogeny 7/J : E 1 ----+ E2 is a homomorphism ofgroups, i.e.,
7/J(P + Q) = 7/J(P) + 7/J( Q)
for all P, Q E E1 (K).
0
Proof See [410, 111.4.8]
The degree of an isogeny 7/J : E 1 ----+ E 2 is the number of points in the inverse
image 7/J- 1 (Q) for any point Q E E 2 . This number is independent of the point Q,
since, as noted earlier, 7/J is an unramified map. It is clear that if deg( 7/J) > 1, then 7/J
is not invertible, since it is not one-to-one. However, there does exist a dual isogeny
that provides a kind of "inverse" for 7/J.
Theorem 6.22. Let 7/J : E1 ----+ E2 be an isogeny of degree d. Then there is a unique
isogeny {;; : E 2 ----+ E 1, called the dual isogeny of 7/J, with the property that
{/;(7/J(P)) = [d]P
and
7/J( {;;( Q)) = [d]Q
for all P E E1 andQ E E2.
0
Proof See [410, III §6].
Definition. Let E be an elliptic curve. The endomorphism ring of E, which is de-
noted by End( E), is the set ofisogenies from E to itself with addition and multiplication given by the rules
(In order to make End( E) into a ring, we also include the constant map that sends
every point to 0.) The automorphism group of E, denoted by Aut( E), is the set of
endomorphisms that have inverses. Equivalently, Aut( E) = End( E)* is the group
of units in the ring End( E).
Every integer m gives a multiplication-by-m morphism in End( E). Form> 0
this is defined in the natural way as
m terms
[m]:E~E,
[m](P)=P+P+···+P.
Form< 0 we set [m](P) = -[-m](P), and of course [O](P) = 0. This gives an
embedding ofZ into End( E), and for most elliptic curves (in characteristic 0), there
are no other endomorphisms.
341
6.3. A Primer on Elliptic Curves
Definition. An elliptic curve E is said to have complex multiplication if End(E)
is strictly larger than Z. The phrase "complex multiplication" is often abbreviated
byCM.
Example 6.23. The elliptic curve E : y2 = x 3 + x has CM, since the endomorphism
'1/J(x,y) = (-x,iy),
'ljJ : E -----+ E,
is not in Z. An easy way to verify this assertion is to note that
'lj; 2 (x, y) = (x, -y) = -(x, y),
so 'lj; 2 = [-1]. This gives an embedding ofthe Gaussian integers Z[i] into End(E)
via the association m + ni f---t [m] + [n] o '1/J, and in fact it is not hard to show
that End( E) is isomorphic to Z[i].
Example 6.24. More generally, there are two special families of elliptic curves that
have CM, namely those with a = 0 and those with b = 0. These are the curves
E~ : y 2 = x 3 +ax,
j(E~) = 1728,
End(E~) = Z[i],
Aut(E~) = 11 4 ,
E~ : y 2 = x 3 + b,
j(E~) = 0,
End(E~) = Z[p],
Aut(E~) = 11 6 .
Here p = ( -1 +H) 12 denotes a cube root of unity and 11n is the group of n 1h roots
of unity.
Of course, all of the E~ are isomorphic over an algebraically closed field, since
However, the curves
they have the same j-invariant, and similarly for all of the
in each family may not be isomorphic over a field K that is not algebraically closed.
This is an example of the phenomenon of twisting as described in Section 4.8 (see
also [410, X §5]).
Er
Proposition 6.25. Let E I K be an elliptic curve. Then the endomorphism ring of E
is one of the following three kinds of rings:
(a) End(E) = Z.
(b) End( E) is an order in a quadratic imaginary field F. This means that End( E)
is a subring ofF and satisfies End(E) ® Q = F. In particular, End( E) is a
sub ring offinite index in the ring of integers of F.
(c) End(E) is a maximal order in a quaternion algebra. (This case can occur only
if E is defined over a finite field.)
0
Proof See [410, III §9].
The automorphisms of an elliptic curve are very easy to describe.
Proposition 6.26. Let K be a field whose characteristic is not equal to 2 or 3 and
let E I K be an elliptic curve. Then
11 2
{
Aut(E) = 114
~-t 6
if j(E) :/= 0 and j(E) :f 1728,
ifj(E) = 1728,
if j(E) = 0.
6. Dynamics Associated to Algebraic Groups
342
Proof See [410, III §10].
D
Remark 6.27. It is easy to make the description of Aut(E) in Proposition 6.26 completely explicit. Assuming that E is given by a Weierstrass equation (6.11) as usual,
for an appropriate choice of n there is an isomorphism
[ ·] : JLn ____.Aut( E),
(6.13)
Here we taken = 4 if j(E) = 1728, we taken = 6 if j(E) = 0, and we take
n = 2 otherwise. Of course, for n = 2 and n = 4 the formula simplifies somewhat
to (x, ~y) and (x, ~- 1 y), respectively.
6.3.5
Minimal Equations and Reduction Modulo p
Let K be a local field with ring of integers R, maximal ideal p, and residue field k =
Rip. As in Section 2.3, we write x for the reduction of x modulo p.
Definition. Let E I K be an elliptic curve defined over a local field K. A minimal
Weierstrass equation forE is a Weierstrass equation whose discriminant ~(E) has
minimal valuation subject to the condition that the coefficients of the Weierstrass
equation are all in R.
Example 6.28. If k does not have characteristic 2 or 3, then a Weierstrass equation
(6.14)
for E is minimal if and only if
a, bE R
and
min{3ordp(a), 2ordp(b)} < 12.
In general, if the residue field k does not have characteristic 2 or 3, then any Weierstrass equation (6.14) can be transformed into a minimal equation by a substitution
of the form (x, y) ~---> (u 2 x, u3 y) for an appropriate u E K*.
If k has characteristic 2 or 3, then a minimal Weierstrass equation may require
the general form (6.12). There is an algorithm of Tate [412, IV §9] that transforms a
given Weierstrass equation into a minimal one.
Definition. Fix a minimal Weierstrass equation forE I K. Then we can reduce the
coefficients of E to obtain a (possibly singular) curve EIk. We say that E has good
reduction if E is nonsingular, which is equivalent to the condition that ~ (E) E R*.
In any case, we obtain a reduction modulo p map on points,
E(K) ____. E(k),
p ~-----+ P.
Proposition 6.29. If E has good reduction, then the reduction modulo p map
E(K)-+ E(k) is a homomorphism.
Proof See [410, VII.2.1].
D
6.3. A Primer on Elliptic Curves
343
Remark 6.30. For elliptic curves defined over a number field K, we say that E has
good reduction at a prime p of K if it has a Weierstrass equation whose coefficients
are p-adic integers and whose discriminant is a p-adic unit. Note that one is allowed
to use different Weierstrass equations for different primes. If there is a single Weierstrass equation that is simultaneously minimal for all primes, then we say that E I K
has a global minimal Weierstrass equation. Global minimal equations exist for elliptic curves over Q, and more generally for elliptic curves over any number field of
class number 1, but in general the existence of global minimal equations is somewhat
subtle; see [410, VIII §8] and [48]. We discussed a related notion of global minimal
models of rational maps in Section 4.11.
6.3.6
Torsion Points and Reduction Modulo p
The kernels of endomorphisms help to determine the arithmetic properties of elliptic
curves.
Definition. Let E be an elliptic curve. For any endomorphism 'ljJ E End(E) we
write
E['t/J] = Ker('t/J) ={PEE: '1/J(P) = 0}.
Of particular importance is the kernel of the multiplication-by-m map,
E[m] ={PEE: [m]P = 0}.
The group E[m] is called them-torsion subgroup of E. The union of all E[m] is the
torsion subgroup of E,
Etors =
U E[m].
m?:l
Theorem 6.31. Let E / K be an elliptic curve and assume that either K has characteristic 0 or else that K has characteristic p > 0 and p f m. Then as an abstract
group,
E[m] = ZlmZ x ZlmZ.
In other words, E[m] is the product of two cyclic groups of order m.
Proof See [410, 111.6.4].
0
The next result gives conditions that ensure that the reduction modulo p map
respects the m-torsion points. It may be compared with Theorem 2.21, which tells us
what reduction modulo p does to periodic points of a good-reduction rational map.
Theorem 6.32. Let K be a local field whose residue field has characteristic p,
let E I K be an elliptic curve with good reduction, and let m ~ 1 be an integer
with p f m. Let E(K)[m] denote the subgroup of E[m] consisting ofpoints defined
over K, i.e., E(K)[m] = E[m] n E(K). Then the reduction map
E(K)[m] ~ E(k)
is injective. In other words, distinct m-torsion points have distinct reductions modulop.
6. Dynamics Associated to Algebraic Groups
344
D
Proof See [410, VII.3.1].
Let E I K be an elliptic curve defined over the field K. Then the points in E[m]
are algebraic over K, so their coordinates generate algebraic extensions of K. An immediate corollary of the preceding theorem limits the possible ramification of these
extensions.
Corollary 6.33. Let K be a local field whose residue field has characteristic p,
let E I K be an elliptic curve with good reduction, and let m 2: 1 be an integer
with p f m. Then the field K ( E [m l) obtained by acijoining to K the coordinates
of them-torsion points of E is unramified over K.
Proof Sketch. Let K' = K( E[m]), let p' be the maximal ideal of the ring of integers
of K', and let k' be the residue field. Suppose that CY E Gal( K' I K) is in the inertia
group. Then CY fixes everything modulo p', so in particular,
CY(P) = P
(mod p')
for all P E E[m].
(6.15)
But from Theorem 6.32, the reduction map E[m] ---+ E(k') is injective, so (6.15)
implies that CY(P) = P for all P E E[m]. The points in E[m] generate K' I K, so CY
fixes K'. Hence Gal(K' I K) has trivial inertia group, so K' I K is unramified. (For
D
further details, see [410, VII.4.1].)
Remark 6.34. The coordinates of the points in E[m] are algebraic over K, so the
absolute Galois group G K = Gal(R I K) acts on E[m] compatibly with the group
structure. In this way we obtain a representation
p: GK------> Aut(E[m]) ~ GL 2 (ZimZ).
In order to create a characteristic-0 representation, we fix a prime £ and combine all
of the £-power torsion to form the Tate module
Tg(E) = limE[£n] ~ Zg X Zg.
+--
(Here Z£ denotes the ring of £-adic integers.) Then the £-adic representation of E is
the homomorphism
PE,£: GK------> Aut(Tg(E)) ~ GLz(Zg).
These representations are of fundamental importance in the study of the arithmetic
properties of elliptic curves.
6.3. 7
The Invariant Differential
Definition. Let E : y 2 = x 3 + ax + b be an elliptic curve given by a Weierstrass
equation. The invariant differential onE (associated to the given Weierstrass equation) is the differentiall-form
dx
dy
2y
3x 2
WE=-=
+a
.
6.3. A Primer on Elliptic Curves
345
The next result explains why the invariant differential is so named and shows that
it linearizes the group law in a useful way.
Theorem 6.35. Let E be an elliptic curve given by a Weierstrass equation and let wE
be the associated invariant differential on E.
(a) For any given point Q E E, let TQ : E --t E be the translation-by-Q map
defined by TQ(P) = P + Q. The differentia/form WE is translation-invariant in
the sense that
Tq(WE) =WE
for every Q E E.
(b) The differentia/form WE is holomorphic at every point of E.
(c) Up to multiplication by a nonzero constant, WE is the only holomorphic translation-invariant 1-form on E.
(d) For every m E .Z the differential form wE satisfies
[m]*wE = mwE.
D
Proof See [410, III §5].
The invariant differential can also be used to fix an embedding of the endomorphism ring of E into C. Of course, this is of interest only when E has CM, since
there is only one way to embed .Z into C.
Proposition 6.36. Let E /C be an elliptic curve, write E(C) ~ C/ L for some lattice L as described in Section 1.6.3, and let
R = {a E C : aL <;;;; L}.
Then for each a E R there is a unique endomorphism [a] E End( E) satisfying
[a]*(w) = aw,
(6.16)
and this association defines a unique ring isomorphism
[·]: R ~ End(E).
(Without the normalization (6.16), the isomorphism R ~ End(E) is unique only up
to complex conjugation of R.)
Proof We fix the isomorphism E(C) ~ C/ L so that the invariant differential w
onE corresponds to dz on Cj L. More formally, we use the Weierstrass g;:> function
to define an isomorphism
F : C/ L ---t E(C),
F(z) = (g;:>(z),
~g::>1 (z));
see Section 1.6.3 and [410, VI §3]. Then
F*(w) = F* (dx) = dg::>(z) = dz.
2y
g::>'(z)
346
6. Dynamics Associated to Algebraic Groups
Let 'ljJ E End( E), so using the identification E(C) ~ Cj L, the endomorphism 'ljJ
defines a holomorphic map 'ljJ : C/ L ---+ Cj L satisfying 'lj;(O) = 0. We claim that
the analyticity implies that 'ljJ lifts to a map 1f; : C ---+ C of the form 'ljJ ( z) = az for
a unique a E C, and hence in particular that 'ljJ is a homomorphism. (Compare with
the algebraic statement of this fact given in Theorem 6.21.) To prove this, we first
observe that the covering map C ---+ C/ L is the universal cover of Cj L, so we can
lift 'ljJ to some holomorphic map 1f; : C ---+ C. Further, the fact that 1f; lifts 'ljJ means
that 1f; satisfies
1f;(z + w) -1f;(z) E L
for all z E C and all w E L.
Fixing w E L, we find that the map z f--+ 1f;(z + w) -1f;(z) is a holomorphic map
from C to the discrete set L, so it must be constant. Thus for each w E L there is a
number c( w) E C such that
1f;(z + w) = 1f;(z) + c(w)
for all z E C.
(6.17)
Writing 1{;( z)
E aizi as a convergent power series, one easily checks that the
relation (6.17) forces 1f; to be linear, say 1f;(z) = az+(3. Then the assumption 'lj;(O) =
0 tells us that (3 E L, so az and az + (3 descend to the same map on C/ L. Hence 'ljJ
lifts to a map of the form 1f; (z) = az. Further, a is unique, since if 'ljJ ( z) also lifts
to a' z, then the map z f--+ (a -a') z sends C to L, hence must be constant, so a = a'.
Finally, we observe that in order for az to descend to Cj L, the complex number a
must satisfy aL ~ L. Thus the association 'ljJ f--+ a gives a map End(E) ---+ R, and it
is clear that the map 1/J(z) = az satisfies 1/J* (dz) = adz, so with our identifications,
we have 'lj;*(w) = aw.
Next we check that the resulting map End( E) ---+ R is a ring homomorphism.
Let 'lj; 1 , 'I/J2 E End( E). Then on C/ L we have 'lj;I(z) = a1z and 'I/J2(z) = a2z, so
It remains to check that every a E R comes from some 1/J E End(E). By definition any a E R induces a map z f--+ az on C that descends to a holomorphic
homomorphism C/ L---+ Cj L. Using the theory of elliptic functions (see, e.g., [410,
Theorem VI.4.1]), one can show that every such holomorphic map E(C) ---+ E(C)
is given by rational functions, which shows that End(R) ---+ R is surjective.
D
6.3.8
Maps from E to 1P'1
The quotient of E by a finite group of automorphisms gives a map from E to JPl 1 .
These quotient maps play an important role in dynamics.
Proposition 6.37. Let r be a nontrivial subgroup of Aut(E). Then the quotient
curve E /f is isomorphic to JPl 1 and the projection map 1r : E ---+ E jf ~ JPl 1 is given
explicitly by
347
6.3. A Primer on Elliptic Curves
(j(E) arbitrary),
(j(E) = 1728 only),
(j(E) = 0 only),
(j(E) = 0 only).
Proof. By definition, the quotient curve E ;r is the curve whose function field is the
subfield of K(E) = K(x, y) fixed by r. Using the explicit description (6.13) of the
action of Aut(E) on the coordinates of E, it is easy to find this subfield. For example,
ifr = f..£ 2 , it consists of those functions that are invariant under (x, y) f-+ (x, -y), so
the fixed field K(E)r is K(x, y 2 ) = K(x). As a second example, ifr = f..£ 6 , then
we need functions invariant under (x, y) f-+ (px, -y), where pis a primitive cube
root of 1. This fixed field is K(E)r = K(x 3 , y 2 ) = K(x 3 ), since in this case the
elliptic curve is given by an equation of the form y 2 = x 3 + b. The other cases are
similar.
D
For later use, we prove that the isomorphism class of an elliptic curve E is determined by the critical values of any double cover E ---t lP' 1 .
Lemma 6.38. Let E be an elliptic curve defined over a field of characteristic not
equal to 2 and let 71' : E ---t lP' 1 be a rational map of degree 2. Then 71' has exactly
four critical values and they determine the isomorphism class of E.
Proof. The Riemann-Hurwitz formula (Theorem 1.5) for the map 71' : E ---t lP' 1 says
that
2g(E)- 2 = (deg11')(2g(lP' 1 ) - 2) +
L (ep(¢) -1).
PEE
The map 71' has degree 2, the elliptic curve E has genus 1, and lP' 1 has genus 0, so we
find that
(ep(¢)- 1) = 4.
L
PEE
The ramification indices satisfy 1 :=:; e p ( ¢) :=:; deg 71' = 2, so we conclude that there
are exactly four critical points, i.e., four points with ep(¢) = 2 and all other points
satisfy e p ( ¢) = 1. Further, these four critical points must have distinct images in lP' 1 ,
since for any point t E lP' 1 we have
L
ep(¢) = deg(11').
PE11'- 1 (t)
Let t 1 , t 2 , t 3 , t 4 E lP' 1 be the four critical values of 71', i.e., the images of the critical
points, and let f E PGL 2 be the unique linear fractional transformation satisfying
Explicitly,
348
6. Dynamics Associated to Algebraic Groups
(If any ofh, t 2, t 3 equals oo, take the appropriate limit.) The quantity
is called the cross-ratio oft 1 , t 2 , t3, t4; cf. Section 2.7, page 71.
We let x = f o 1r, so x is a rational function of degree 2 on E with critical
values 0, 1, oo, and "'· To ease notation, we let
Taking 0 to be the identity element for the group law onE, we see that
div(x) = 2(T0 )-2(0),
div(x-1) = 2(Tl)-2(0),
div(x-"') = 2(T,.)-2(0),
so T0 , T 1 , T,. are in E[2], i.e., they are points of order 2. The sum of the three nontrivial 2-torsion points on any elliptic curve is equal to 0, so Proposition 6.18 tells
us that there is a rational function y on E with divisor
div(y) =(To)+ (Tl) + (T,.)- 3(0).
After multiplying y by an appropriate constant, it follows that x and y are Weierstrass
coordinates forE (cf. [198, IV.4.6] or [410, 111.3.1]). More precisely, the rational
functions x and y map E isomorphically to the curve with Weierstrass equation
E: y 2 = x(x- 1)(x- "')·
It is easy to compute the j-invariant of E in terms of"' (cf. [410, 111.1.7(b)]). We find
that
"(E) = 2s ("'2- "'+ 1)3
J
/'\,2(/'\,-1)2'
so in particular j(E) is uniquely determined by h, t 2 , t 3, t 4. Then we apply Proposition 6.15 (or [410, 111.1.4(b)]), which says that the isomorphism class of E is determined by its j-invariant. This completes the proof of Lemma 6.38.
D
6.3.9
Complex Multiplication
Let E j K be an elliptic curve with complex multiplication defined over a number
field. As described in Proposition 6.25, the endomorphism ring of E is isomorphic
to a subring of the ring of integers of a quadratic imaginary field. We briefly recall an
analytic proof of this important fact and then discuss the relationship between complex multiplication and the ideal class group of the associated quadratic imaginary
field. This material is used only in Section 6.6, so may be omitted at first reading.
Proposition 6.39. Let E jC be an elliptic curve with complex multiplication, i.e., the
endomorphism ring End( E) is strictly larger than Z. Choose a lattice L C C such
that E(C) 9:' C/ L, let
349
6.3. A Primer on Elliptic Curves
R = {a E C: aL <;;; L },
and let
[·]: R ~ End(E)
be the isomorphism described in Proposition 6.36. Then R is a sub ring of the ring
of integers of a quadratic imaginary field F, and for any a E R, the degree of [a] is
given by
deg[a] = aa = NF;IQI(a),
where 0: denotes the complex conjugate of a.
Proof To describe the ring R, we choose a basis for L, say L = Zw 1 + Zw 2 . We
have R of. Z by assumption, so there exists an a E R with a ~ Z. Write
aw1 = aw1 + bw2
The numbers
and
aw2 = cw1 + dw2
with
a, b, e, d E Z.
(6.18)
w1 and w2 are JR-linearly independent, so the relation
( a- a -b )
-e a-d
(w1) =
w2
(0)
0
implies that the matrix has determinant 0,
a 2 - (a + d) a + (ad - be) = 0.
(6.19)
Hence a is an algebraic integer in a quadratic field. Further, we must have a ~ JR,
since if a were real, then the relation (a -a )w 1 = bw 2 (and a ~ Z) would contradict
the JR-linear independence of w1 and w2 . This proves that every element of R is an
algebraic integer in a quadratic imaginary field, and hence R is a subring of the ring
of integers of such a field.
Finally, in order to compute the degree of the endomorphism [a] : E ----+ E
corresponding to a E R, we observe that
deg(a) = #Ker ( C/L ~ C/L) = (L: aL).
If a E Z, then it is clear that ( L : aL) = a 2 , since L = Z 2 as an abstract group.
Suppose now that a ~ Z. Then continuing with the earlier notation, the transformation formulas (6.18) imply that the index of aL in L is ( L : aL) = ad - be. On the
other hand, the product aa is the constant term in the minimal equation (6.19) for a
0
over Q, hence also equal to ad- be.
The theory of complex multiplication uses elliptic curves to describe the abelian
extensions of a quadratic imaginary field F in a manner analogous to the description of abelian extensions of Q using torsion points in Gm, i.e., using roots of unity.
For a complete introduction to the theory of complex multiplication, see, for example, [257, Part II], [399, Chapter 5], or [412, Chapter II]. We now describe the tiny
piece of the theory that will be needed in Section 6.6 in order to prove Theorem 6.62.
6. Dynamics Associated to Algebraic Groups
350
Let F be a quadratic imaginary field, let Rp be the ring of integers ofF, and
let IF be the group of fractional ideals ofF. If we fix an embedding F c C, then
each fractional ideal a E IF is a lattice a c C; hence it determines an elliptic
curve Ea whose complex points are
(6.20)
We observe that Ea has complex multiplication by Rp, since any a E Rp has the
property that aa c a, hence it induces a holomorphic map
[a] : Cl a ~----> Cl a,
z ~----> az,
which in tum yields an isogeny [a] : Ea ---> Ea. In fact, since Rp is the maximal
order in F, we have End(Ea) = Rp. We denote by E.eC(RF) the set
E.eC(RF) = {isomorphism classes of elliptic curves E with End(E) ~ Rp}.
We thus have a natural map
a~----> (isomorphism class of Ea)·
(6.21)
We also observe that if we multiply a by a principal ideal, then the isomorphism
class of Ea does not change, since for any c E K* there is an obvious isomorphism
CIa ___:::__. C Ica,
z 1----> cz.
Hence the map (6.21) induces a natural map from the ideal class group CF = IF IF*
to elliptic curves with complex multiplication by Rp,
Proposition 6.40. Let F be a quadratic imaginary field with ring of integers Rp
and ideal class group Cp, and let hp = #CF be the class number of F. Then with
notation as above, the natural map
is a bijection. In particular, there are exactly hp isomorphism classes of elliptic
curves whose endomorphism ring is Rp.
Proof See [412, 11.1.2].
6.4
0
General Properties of Latt(~s Maps
Chebyshev polynomials arise by restricting the power map zn to the quotient of JID 1
by the finite group of automorphisms { z, z- 1 }. As already briefly described in Section 1.6.3, quotients of elliptic curves lead similarly to rational maps called Lattes
maps. In this section we define and discuss general properties of these Lattes maps.
For an excellent introduction to Lattes maps over C, including historical remarks and
proofs oftheir basic geometric and analytic properties, see [300].
351
6.4. General Properties of Lattes Maps
Definition. A rational map ¢ : JPl 1 ---t JPl 1 of degree d 2: 2 is called a Lattes map
if there are an elliptic curve E, a morphism 7/J : E ---t E, and a finite separable'
covering 7r : E ---t JPl 1 such that the following diagram is commutative:
(6.22)
p1 ~ JPl1.
Example 6.41. Let E : y2 = x 3 + ax + b be an elliptic curve. Then the classical
formula for x(2P) (Proposition 6.17) and the isomorphism x : E / { ±1} ---t JPl 1 yield
the Lattes map
cp(x) = x( 2P) = x 4 - 2ax 2 - 8bx + a2
4x3 + 4ax + 4b
Here 7/J is the duplication map 7/J(P) = [2]P, and the projection 7r is given by
7r(P) = 7r(x, y) = x.
Example 6.42. Let E be the elliptic curve E: y 2 = x 3 +ax with j(E) = 1728 and
again let 7/J(P) = [2]P be the doubling map. If we take 7r(x, y) = x, then we are in
the b = 0 case of Example 6.41, and we obtain the Lattes map
(x2 - a)2
cp(x) = x(2P) = 4 ( 2
)"
X X +a
However, for this curve we may instead take 7r(x, y) = x 2 . This gives a new Lattes
map ¢ 1 . We find a formula for ¢ 1 using the relation
¢ ( ) _ ¢(v'x)2 _ ( (x- a) 2 ) 2 _
1 x-
x
-
4ylx(x+a)
(x- a) 4
-16x(x+a) 2 .
Note that the map 7r(x, y) = x2 corresponds to taking the quotient of E by its
automorphism group Aut( E) ~ J..t 4 via the association described in Remark 6.27.
Example 6.43. In a similar manner, the doubling map on the elliptic curve
E: y 2 = x 3 + 1
with j(E) = 0 and Aut(E) = J..t 6 gives various Lattes maps corresponding to
taking the quotient of E by the different subgroups of J..t 6 . Explicitly, the Lattes maps
corresponding, respectively, to J..t2 , J..t3 , and J..t 6 are
z(z 3 - 8b)
¢1(z) = 4(z3 +b) '
¢ 3 (z) =
z(z- 8b) 3
64(z + b) 3
We leave the verification of these formulas to the reader; see Exercise 6.12.
1The assumption that 1r is separable is relevant only when one is working over a field of characteristic p, in which case it is equivalent to the assumption that 1r does not factor through the p-power Frobenius
map.
352
6. Dynamics Associated to Algebraic Groups
We begin with an elementary, but useful, characterization of the preperiodic
points of a Lattes map (cf. Proposition 1.42).
Proposition 6.44. Let ¢ be a Lattes map associated to an elliptic curve E. Then
PrePer(¢) = 7r(Etors)·
Proof Let ( E 1P'1 and let P E E be any point satisfying 1r(P) =(.We consider the
orbits of ( and P. Thus
1r(O.p(P)) = 1r ({¢n(P): n 2 0}) = {1r¢n(P): n 2 0}
= {¢n1r(P): n 2 0} = {c/Jn((): n 2 0} = Oq,(().
The map 1r is finite, so this shows that O.p(P) is finite if and only ifOq,(() is finite.
Hence
PrePer(¢) = 1r(PrePer(¢)),
and it is left to prove that PrePer(¢) = Etors·
We observe that the map ¢ : E ___. E has the form ¢(P) = ¢o(P) + T for
some ¢ 0 E End(E) and some point T E E. (See [410, III.4.7].) We are going to
proveProposition6.44inthecasethat¢ = ¢ 0 E End(E),i.e.,assumingthatT = 0.
For the general case, which requires knowing that the point Tis a point of finite order, see Exercise 6.14.
Suppose first that P E Etors. say [n]P = 0 for some n 2 1. Consider the images
of the iterates¢, ¢ 2 ,¢3 , ... in the quotient ring End(E)/n End(E). It follows from
the description ofEnd(E) in Proposition 6.25 that this quotient ring is finite, so we
can find iterates i > j 2 1 such that
¢i = ¢i
(mod nEnd(E)).
In other words, there is an endomorphism {3 E End(E) such that
¢i = ¢i + {Jn.
Evaluating both sides at P and using the fact that [n]P = 0 allows us to conclude
that¢i(P) = ¢i(P). Hence P E PrePer(¢), which proves that Etors C PrePer(¢).
Next suppose that P E PrePer(¢), say ¢i(P) = ¢i(P) for some i > j. We
--
rewrite this as (¢i - ¢i )( P) = 0 and apply the dual isogeny ¢i - ¢i described in
Theoretn 6.22 to obtain
[deg(¢i- ¢i)](P) = 0.
We know that ¢i =f. ¢i, since deg(¢) = deg(¢) 2 2 and i > j, so ¢i- ¢i
has positive degree. This proves that P E E 10rs, which gives the other inclusion PrePer(¢) C Etors·
0
Many dynamical properties of a rational map can be analyzed by studying the
behavior of the critical points under iteration of the map. This is certainly true for
Lattes maps, whose postcritical orbits have a simple characterization, which we give
after setting some notation.
353
6.4. General Properties of Lattes Maps
Definition. Let ¢ : C 1 ----+ C2 be a nonconstant rational map between smooth
projective curves. The set of critical points (also called ramification points) of¢ is
denoted by
CritPt¢ = {P E C 1 :¢is ramified at P} = {P E C1 : ep(¢) 2': 2}.
The set of critical values of¢ is the image of the set of critical points and is denoted
by
CritVal¢ = ¢(CritPt¢)·
If¢ : C ----+ C is a map from a curve to itself, the postcritical set is the full forward
orbit of the critical values and is denoted by
00
00
n=O
n=l
(See Exercise 6.15.)
Proposition 6.45. Let ¢ : IP'1 ----+ IP' 1 be a Lattes map that fits into a commutative
diagram (6.22). Then
CritVal71' = PostCrit<t> .
In particular, a Lattes map is postcritically finite.
Proof The key to the proof of this proposition is the fact that the map 1/J : E ----+ E
is unramified, i.e., it has no critical points, see Remark 6.20. (In the language of
modem algebraic geometry, the map 1/J is etale.) More precisely, the map 1/J is the
composition of an endomorphism of E and a translation (Remark 6.19), both of
which are unramified.
For any n 2': 1 we compute
CritVal71' = CritVa171',p"
= CritVal¢"71'
= CritVal¢" U ¢n (CritVal71')
because 1/J is unramified,
from the commutativity of (6.22),
from the definition of critical value,
~ CritVal¢" .
This holds for all n 2': 1, which gives the inclusion
U¢n(CritVa1¢) = PostCrit<l>.
00
CritVal71' ~
n=O
In order to prove the opposite inclusion, suppose that there exists a point Po E E
satisfying
(6.23)
and
1r(P0 ) ~ PostCrit<t>.
PoE CritPt71'
Consider any point Q E 'lj;- 1 (P0 ). Then Q is a critical point of Jr'lj;, since 1/J is
unramified and 1r is ramified at 7/J (Q) by assumption. But 1r'lj; = ¢1r, so we see
that Q is a critical point for ¢1r.
354
6. Dynamics Associated to Algebraic Groups
On the other hand,
<f>(n(Q)) = n(Po) tf: CritVal<l> = ¢(CritPt¢),
son( Q) is not a critical point for¢. It follows that Q is a critical point of 1r. Further,
we claim that no iterate of 4> is ramified at n(Q). To see this, we use the given fact
that n(Po) is not in the postcritical set of 4> to compute
n(Po) tf: PostCrit<l> ===} n(P0 ) tf: ¢n(CritPt¢)
===} n( 'lj;( Q)) tj: ¢n(CritPt¢)
===} ¢(n(Q)) tj: ¢n(CritPt¢)
===} n(Q) tj: ¢n(CritPt¢)
===} n( Q) tf: PostCrit<l>.
for all n 2: 1,
for all n 2: 1,
for all n 2: 1,
for all n 2: 0,
To recapitulate, we have now proven that every Q E 1j; - l (Po) satisfies
Q E CritPt1r
n( Q) tf: PostCrit,p .
and
In other words, every point Q E 'lj;- 1 (P0 ) satisfies the same two conditions (6.23)
that are satisfied by P0 . Hence by induction we find that if there is any point Po
satisfying (6.23), then the full backward orbit of 1j; is contained in the set of critical
points of n, i.e.,
00
CritPt1r :::>
U 'lj;-n(P
0 ).
n=l
But 1j; is unramified and has degree at least 2 (note that deg 1j; = deg 4>), so
This is a contradiction, since 1r has only finitely many critical points, so we conclude
that there are no points Po satisfying (6.23). Hence
P0 E CritPt1r ===} n(Po) E PostCrit<l>,
which gives the other inclusion CritVal1r ~ PostCrit<l>.
0
As an application of Proposition 6.45, we show that Lattf$ maps associated to
distinct elliptic curves are not conjugate to one another.
Theorem 6.46. Let K be an algebraically closed field of characteristic not equal
to 2 and let 4> and 4>' be Lattes maps defined over K that are associated, respectively, to elliptic curves E and E'. Assume further that the projection maps 1r and 1r1
associated to 4> and 4>' both have degree 2. If 4> and 4>' are PG L2 (K)-conjugate to
one another, then E and E' are isomorphic.
355
6.5. Flexible Lattes Maps
Proof Let f E PGL 2 (K) be a linear fractional transformation conjugating¢' to¢.
Then we have a commutative diagram
7r'
+-----
E'
11/J'
E ~ pl ~ pl ~ E'
We let 1r 11 = f o 1r1 • Note that since f is an isomorphism, the map 1r 11 still has
degree 2. This yields the simplified commutative diagram
E~IP' 1 ~E'
showing that ¢ is a Lattes map associated to both elliptic curves E and E'.
Applying Proposition 6.45, first to E and then to E', we find that
CritVal1r = PostCrit.p = CritVal1r" .
(6.24)
In other words, the degree-2 maps 1r : E --+ IP' 1 and 7r 11 : E' --+ IP' 1 have the exact
same set of critical values. Then Lemma 6.38 tells us that E and E' are isomorphic.
D
6.5
Flexible Latti.~s Maps
A Lattes map is a rational map that is obtained by projecting an elliptic curve
endomorphism down to IP' 1 . For any integer m ~ 2, every elliptic curve has a
multiplication-by-m map and a projection E --+ E j { ±1} ~ IP' 1 , so every elliptic
curve has a corresponding Lattes map. As E varies, this collection of Lattes maps
varies continuously, which prompts the following definition.
Definition. A .flexible Lattes map is a Lattes map¢: IP' 1 --+ IP' 1 that fits into a Lattes
commutative diagram (6.22) in which the map 'ljJ: E--+ E has the form
'lj;(P) = [m](P) + T
for some m E Z and some T E E
and such that the projection map 1r : E --+ IP' 1 satisfies
deg(1r) = 2
and
1r(P) = 1r(-P) for all PEE.
Remark 6.47. The condition that 1r be even, i.e., that it satisfy 1r(- P) = 1r(P), is
included for convenience. In general, if 1r : E --+ IP' 1 is any map of degree 2, then
there exists a point Po E E such that 1r( -(P +Po)) = 1r(P +Po) for all PEE.
Thus 1r becomes an even function if we use Po as the identity element for the group
law on E. See Exercise 6.16.
356
6. Dynamics Associated to Algebraic Groups
Remark 6.48. We show in this section that the Lattes maps of a given degree have
identical multiplier spectra. This is one reason that these Lattes maps are called "flexible," since they vary in continuous families whose periodic points have identical sets
of multipliers. We saw in Section 4.5 that symmetric polynomials in the multipliers
give rational functions on the moduli space Md of rational maps modulo PGL 2conjugation. Flexible families of rational maps thus cannot be distinguished from
one another in Md solely through the values of their multipliers.
Example 6.49. We saw in Example 6.41 that the Lattes function associated to the
duplication map 'lj;(P) = [2](P) on the elliptic curve E: y 2 = x 3 +ax+ b is given
by the formula
c/Ja b(x) = x(2P) = x4- 2ax2- 8bx + a2
'
4x3 + 4ax + 4b
It is clear that if a and b vary continuously, subject to 4a 3 + 27b2 -=1- 0, then the Lattes
maps ifla,b vary continuously in the space of rational maps of degree 4.
More precisely, the set of maps c/Ja,b is a two-dimensional algebraic family of
points in the space Rat 4 , given explicitly by
A2 ---t Rat 4 c JID9 '
(a, b) f----+ [1, 0, -2a, -8b, a 2, 0, 4, 0, 4a, 4b].
Ifwe conjugate by fu(x) = ux, the Lattes map c/Ja,b transforms into
c/J~~b(x) = u- 1 c/Ja,b(ux) = c/Ju-2a,u-3b(x).
Thus assuming (say) that ab -=1- 0, we can take u = bja to transform c/Ja,b into
,~,fb/a = 'l'c,c
,1,
'+' a,b
"th C = a3/b2 .
Wl
In other words, the two-dimensional family ofLattes maps { c/Ja,b} in Rat 4 becomes
the one~dimensional family of dynamical systems
(6.25)
Of course, it is not clear a priori that the map (6.25) is nonconstant. But if the Lattes
maps c/Jc,c and c/Jc' ,c' are PGL2-conjugate, then Theorem 6.46 tells us that their
associated elliptic curves E and E' are isomorphic. The j-invariant of the elliptic
curve Ec : y2 = x 3 + ex + c is
so we see that j(Ec) = j(Ec') if and only if c = c'. This proves that the map (6.25)
is injective, so these flexible Lattes maps do indeed form a one-parameter family of
nonconjugate rational maps with identical multiplier spectra, i.e., they are a nontrivial
isospectral family.
357
6.5. Flexible Lattes Maps
Example 6.50. The elliptic curve E : y2 = x 3 + ax 2 + bx has the 2-torsion
point T = (0, 0). To compute the Lattes function ¢ : lP' 1 -----* lP' 1 associated to the
translated duplication map 7./J(P) = [2] (P) + T, we first use the classical duplication
formula to compute
_ (x 4 - 2bx 2 + b2 x 6 + 2ax 5 + 5bx 4 - 5b2 x 2 - 2ab 2x- b3 )
.
8y3
'
4y2
2P -
Then the addition formula and some algebra yield
¢(x) = x(2P + T) = 4b(x3 + ax2 + bx).
x 4 - 2bx 2 + b2
As in the previous example, these Lattes maps form a one-dimensional family in M 4 .
We begin with a few elementary, but useful, properties of flexible Lattes maps.
Proposition 6.51. Let ¢ : lP' 1 -----* lP' 1 be a flexible Lattes map whose associated map
7./J: E-----* E has the form 7./J(P) = [m]P + T.
(a) The map¢ has degree m 2 .
(b) The point T satisfies [2]T = 0.
(c) Fix a Weierstrass equation (6.11) for E. Then there is a linear fractional transformation f E PGL2 such that 1r = fox. Hence¢! fits into a commutative
diagram
(6.26)
lP'l~ lP'l.
Proof (a) The commutativity of the diagram (6.22) tells us that
deg(¢) deg(n) = deg(n) deg(?jJ).
The map 7./J has degree m 2 , since multiplication-by-m has degree m 2 and translationby-T has degree 1. Therefore deg(¢) = m 2 .
(b) We are given that the map 1r : E -----* lP' 1 has degree 2 and satisfies n(P) =
n(- P). It follows that n(P) = n( Q) if and only if P = Q or P = -Q. We use the
commutativity of (6.22) to compute
n( -[m]P- T) = n([m]P + T) = n(?jJ(P))
= ¢(n(P)) = ¢(n( -P)) = n(?jJ( -P)) = n( -[m]P + T).
Hence for every P E E we have either
-[m]P- T = -[m]P + T
or
- [m]P- T = -( -[m]P + T) = [m]P- T.
Simplifying these expressions, we find that every point P E E satisfies either
[2]T=0
or
[2m]P=0.
6. Dynamics Associated to Algebraic Groups
358
But there are only finitely many points P E E satisfying [2m]P = 0; hence we
must ~ve [2]T = 0.
(c) The map 1r is a rational function onE satisfying rr( -P) = rr(P). It follows
that 1r is in the subfield K(x) of the function field K(E); see [410, 111.2.3.1]. In other
words, there is a rational function f (z) E K (z) such that 1r = f (x). Equivalently,
the map 1r : E -+ JP>1 factors as
In particular,
2 = deg( 1r) = deg(f o x) = deg(f) deg( x) = 2 deg(f),
so we see that deg(f) = 1. Hence f is a linear fractional transformation, which
proves the first part of (c). Finally, we compute
q/ oX = j- 1 o c/J of oX = j- 1 o c/J 0 7r = j- 1 o 7r o '1/J = X o '1/J,
which proves the commutativity of (6.26).
D
Our next task is to compute the periodic points and multipliers of flexible Lattes
maps. For ease of exposition, we do the pure multiplication case, i.e., for maps of the
form 'lj;(P) = [m](P), and leave the general case for the reader.
Proposition 6.52. Let¢ : JP>1 -+ JP>1 be a flexible Lattes map and assume that T = 0,
so 'lj;(P) = [m](P). (See Exercise 6.18for the case T =I= 0.)
(a) The set ofn-periodic points of¢ is
Pern(¢) = rr(E[mn- ll) U rr(E[mn + ll).
(b) Let ( be a periodic point of¢ of exact period n. Then
if( E rr(E[mn -ll) and(~ rr(E[2l),
if( E rr(E[mn + ll) and(~ rr(E[2l),
if( E rr(E[mn + ll) n rr(E[2l).
(Notice that 1r ( E[2l) is the set ofcritical values of1r.)
Proof. (a) Let( E JP> 1 beafixedpointof¢andchooseapointP E Ewithrr(P) = (.
Note that there are generally two choices for P, so we simply choose either one of
them. Then
rr(P) = ( = ¢(() = ¢{rr{P)) = rr('lj;(P)) = rr{[m]P).
As noted during the proof of Proposition 6.5l(b), we have rr{P) = rr(Q) if and only
if P = ±Q, so we conclude that either
[m]P=P
or
[m]P=-P.
6.5. Flexible Lattes Maps
359
Conversely, if [m]P = ±P, then
¢(1r(P)) = 1r('l/;(P)) = 1r([m]P) = 1r(±P) = 1r(P),
so 1r(P) is fixed by¢. This proves that ( = 1r(P) E Fix(¢) if and only if
[m-l]P=O
or
[m+l]P=O,
and hence
Fix(¢)= 1r(E[m- ll) U 1r(E[m + 1]).
(6.27)
In order to find the points of period n, we observe that
so cpn is also a flexible Lattes map. It is associated to the map [mn] : E ----t E. (For a
generalization ofthis observation, see Exercise 6.17.) Applying (6.27) to the Lattes
map cpn yields the desired result,
(b) The multipliers of ¢ are invariant under PGL 2 -conjugation, so we can use
Proposition 6.51 (c) to replace ¢ by a conjugate satisfying
¢ox= xo'lj;,
where x is the x-coordinate on a Weierstrass equation
E : y 2 = x 3 + ax + b.
In order to compute the multipliers of¢, we are going to use the translation-invariant
differential form
dx
dy
w - - - -,----;::-=--(6.28)
- 2y- 3x 2 +a
onE described in Theorem 6.35. The invariant differential satisfies the formula
'lj;*(w) = [m]*(w) = mw.
(6.29)
Substituting w = dxj2y and doing some algebra yields
'lj;*(dx)
yo 'ljJ
--=m--.
dx
y
(6.30)
Let¢ be the Lattes map associated to 'lj;, let ( E Fix(¢) with ( =1- oo, and lett
be a coordinate function on IP1 . Then the multiplier >..q:, ( () of¢ at ( can be computed
using the differential form dt via the equation
¢*(dt) I
= dcp(t) I
= ¢'(() = >..q:,(().
dt t=<
dt t=<
(6.31)
360
6. Dynamics Associated to Algebraic Groups
Using the relation¢ ox= x o 'lj; and formula (6.30), we compute
x* (<P*(dt)) = x*¢*(dt) = (¢ o x)*(dt)
x*(dt)
x*(dt)
dt
= (x o 'lj;)*(dt) = 'lj;*x*(dt) = 'lj;*(dx) =my o 'lj;
dx
x*(dt)
x*(dt)
y ·
(6 .32 )
We lift (to a point P E E satisfying x(P) = ( and evaluate both sides of (6.32)
at P to obtain
¢*(dt) I = m (yo 'lj;) (P).
y
dt t=(
(6.33)
Equating (6.31) and (6.33) gives the useful formula
yo'lj;) (P)
.Aq,(() = m ( -y-
for ( = x(P) E Fix(¢) with ( # oo.
(6.34)
Assume first that [2]P # 0, which ensures that y(P) # 0 and y(P) # oo. Then
we can directly evaluate the fraction in (6.34) and conclude that
.A (() =
¢
m
(yo 'ij;)(P)
y(P) .
We are assuming that ( = x(P) is a fixed point of¢, so
x(P) = ¢(x(P)) = x('lj;(P)).
Thus 'lj;(P) = ±P, and hence y('lj;(P)) = ±y(P), which proves that .Aq,(() = ±m.
More precisely, .Aq,(() = m if mP = P and .Aq,(() = -m if mP = -P. To
summarize, we have proven that if ( ~ x (E[2l), then
( E Fix(¢)
.Aq,(() = { m if( E x(E[m -1]),
-m if( E x(E[m + 1]).
(6.35)
Next suppose that y(P) = 0, so [2]P = 0, but P # 0. We also have [m]P =
±P from the assumption that ( = x(P) E Fix(¢), somis odd and 'lj; fixes P. The
functions y andy o 'lj; both vanish at P, so we can use l'Hopital's rule to compute
( y ~ 'lj;) (P) = ( d(yd~
'lj;)) (P),
assuming that the righthand side has a finite value. (Note that this formula is valid
algebraically, since what we are really doing is looking at the linear terms in the local
expansions of y andy o 'lj; at P.) We now use the chain rule to compute
361
6.5. Flexible Lattes Maps
d('lj;*y)
dy
d('lj;*y)
'lj;*(3x 2 +a) 3x 2 +a
2
'lj;*(3x +a)
3x 2 +a
dy
='lj;* (
dy
) . 'lj;*(3x 2 +a). 3x 2 +a
3x 2 + a
3x 2 + a
dy
'lj;*(w) 'lj;*(3x 2 +a)
from (6.28),
--·
w
3x 2 +a
'lj;*(3x 2 +a)
= m -'---=-'---::-------'-from (6.29),
3x2 +a
3(x o 'lj;) 2 +a
- m ___:.._-::--;o-:--3x 2 +a ·
-
We evaluate both sides at P = ((, 0). Note that the quantity 3( 2 +a is nonzero, since
otherwise P would be a singular point of E. Also, x('lj;(P)) = x(P) =(.Hence
+a =m3( +a =m.
( yo'lj;) (P) = d('lj;*y)(P) =m3(xo'lj;(P))
y
dy
3x(P) 2 +a
3( 2 +a
2
2
Substituting this value into (6.34) yields the desired result A¢(()= m 2 .
It remains to deal with the case P = 0. There are several ways to do this case.
First, we could perform an explicit calculation using local coordinates around 0.
Second, since we are missing only one multiplier, we could use Theorem 1.14, although this would require knowing a priori that A =1- 1. Third, at least for odd m, we
could observe that 'lj;(P) = [m]P looks the same locally around each of the points
in E[2], and we already computed A= m 2 for the nonzero points in E[2]. We leave
as an exercise for the reader (Exercise 6.20) to complete the proof using whichever
argument he or she prefers.
Finally, to compute the multiplier of a periodic point ( E Pern(¢), we apply the
results that we have just derived to the fixed points of the Lattes map <j>n satisfying <j>n ox= x o [mn].
0
Remark 6.53. Let¢ : IP' 1 --t IP' 1 be a flexible Lattes map associated to 'lj;(P) = [m]P
as in Proposition 6.52. If we work over C, then the multiplier of every periodic
point ( E Per(¢) satisfies
for some e = e( () 2: 1.
Hence ( is repelling, so Per(¢) is contained in the Julia set (cf. Exercise 1.27).
Choosing a lattice L and a complex uniformization C/ L --t E(C) as described
in Section 1.6.3, it is clear that E(C)tors is dense in E(C). Therefore Per(¢) =
x(E(C) 10rs) is dense in IP'1 (C) and is contained in J(¢). Further, the Julia set is
closed. This proves that J(¢) = IP' 1 (C) and F(¢) = 0, which is Lattes's Theorem 1.43 discussed in Section 1.6.3.
On the other hand, if we work over a p-adic field such as Qp or Cp, then every
periodic point ( E Per(¢) satisfies
for some e = e(() 2: 1.
362
6. Dynamics Associated to Algebraic Groups
Thus every periodic point is nonrepelling, so Per(¢) c F( ¢) from Proposition 5.20.
Further, if vim, then [A¢(()[ = lmle < 1, so in this case every point in Per(¢) is
attracting.
Remark 6.54. Recall that the multiplier spectrum of a rational map ¢ : JP> 1 -+ JP> 1 of
degree d is the map that associates to each integer n 2:: 1 the set
where we treat Pern(¢) as the set of dn + 1 (not necessarily distinct) fixed points
of ¢n. Two maps with the same multiplier spectrum are called isospectral. (See Section 4.5, page 187.) Proposition 6.52(b) shows that flexible Lattes maps of degree m 2
are isospectral, since their multiplier spectrum depends only on m. A deep theorem
of McMullen [294, §2] (Theorem 4.53) says that these are the only isospectral rational maps that vary in a continuous family.
Not surprisingly, good reduction ofLattes maps is closely related to good reduction of the associated elliptic curve.
Proposition 6.55. Let K be a local field, let R be the ring of integers of K, and
let ¢ : JP>k -+ JP>k be a flexible Lattes map of degree m 2 associated to an elliptic
curve E / K. Suppose that E has good reduction and that m E R*. Then there exists
an f E PG L2 ( K) such that ¢! has good reduction.
Proof Since E has good reduction, we can find a Weierstrass equation for E with
coefficients in R and discriminant in R*. We then use Proposition 6.51 to replace ¢
by¢! so that it fits into a Lattes diagram (6.26). In other words, the Lattes projection
map 1r is the x-coordinate function on a minimal Weierstrass equation for E.
For any n 2:: 1 we can find polynomials Fn(X), Gn(X) E R[X] such that
([ l ) = Fn(x(P))
x nP
Gn(x(P)) ·
This is easily proven by writing out the first few polynomials explicitly and then
computing the subsequent ones by a recurrence formula. The recurrence also shows
that the leading terms of Fn and Gn are
and
(See [96, page 133], [410, Exercise 111.3.7], or Exercise 6.23.)
We note that the roots of Gn (X) are the x-coordinates of the n-torsion points,
and one can check that Gn(X) factors as
II
(X- x(P)).
PEE[n], P-j.O
We are assuming that 'ljJ(P) = [m](P) + T for some fixed integer m and
some T E E[2]. For simplicity we prove here the case T = 0 and leave the general
case as an exercise. Then
6.5. Flexible Lattes Maps
363
cf>(x(P)) = x(1/J(P)) = x([m](P)),
so
Fm(X)
cf>(X) = Gm(X).
We fix an auxiliary prime .e satisfying .e f m and .e E R* and consider the polynomial
II
H(X) =
(Fm(X)- x(Q)Gm(X)).
QEE[£], Qf-CJ
Notice that H(X) is a monic polynomial, since Fm(X) is monic and deg(Fm) >
deg(Gm). We also note that all of the x(Q) are integral over R, since they are roots
ofGt(X) and .e E R*. It follows that H(X) E R[X]. Further, Proposition 2.13(b)
(see also Exercise 2.6) tells us that the resultant of H(X) and Gm(X) is
Res(H(X), Gm(X)) = ±m2 degH
IT H(()
Gm(()=O
= ±m2degH
II Fm((t-1
Gm(()=O
= Res(Fm(X), Gm(X))
£2-1
.
Hence in order to show that cf> has good reduction, it suffices to prove that
Res(H(X), Gm(X)) E R*.
Let K' = K(E[mil) be the field extension obtained by adjoining the coordinates of the points of order mi to K, let R' be the ring of integers of K', let p'
be the maximal ideal in R', and let k' = R' jp' be the residue field of R'. The extension K' j K is unramified, because we have assumed that E has good reduction
and m£ is a unit in R; see Corollary 6.33.
Note that H(X) and Gm(X) factor completely in K', and in fact their roots
are in R'. This is clear for Gm(X), since its roots are the x-coordinates of the points
in E[m] and its leading coefficient is m 2, which is a unit in R. We now analyze H(X)
more closely.
Claim 6.56. The roots of H (X) are given by
{roots of H(X)} = x(E[mi] "E[ml) c R'.
Proof of Claim. The roots of H(X) are the solutions to
Fm(X) = (Q)
Gm(X)
X
for some Q E E[£].
Writing a root of H(X) as x(P) for some PEE, this means that
Fm(x(P))
x([m]P) = Gm(x(P)) = x(Q),
6. Dynamics Associated to Algebraic Groups
364
and hence [m]P = ±Q. But Q E E[£], soP E E[m£]. This shows that the roots
of H(X) are contained in x(E[mil). Further, if P E E[m], then Gm(x(P)) = 0
andFm(x(P)) :f:O,so
t2
1
H(x(P)) = Fm(x(P)) - :f: 0.
This gives the inclusion
{roots of H(X)} c x(E[m£] "E[ml).
The other inclusion is clear from the definition of H(X), since
P E E[m£] "- E[m]
=}
[m]P E E[£] "{0}.
Thus x(P) is a root of Fm(X)- x([m]P)Gm(X), which is one ofthe factors in the
product defining H (X).
Finally, we note that K' contains the x-coordinates of the points in E[m£] by
construction. Further, these x-coordinates are the roots of the polynomial Fmt(X) E
R[X] whose leading coefficient is m 2£2 E R*, so the roots are integral over R, hence
are in R'.
D
We now resume the proof of Proposition 6.55. We assume that the resultant
Res(H(X),Gm(X)) is not a unit in Rand derive a contradiction. This assumption means that H(X) and Gm(X) have a common root modulo p', so we can
find x1, x2 E R' such that
H(xl) = 0,
and
x1 = x2 (mod p').
From our description of the roots of H(X) and Gm(X), this means that we can find
points
P 1 E E[m£] "E[m]
and P2 E E[m] "{0}
=-
satisfying P1
=P2 (modp').
(In principle, we might get P 1
P 2, but if that happens, then just replace P2
by-P2.) Since clearly P1 :f: P2, this proves that the reduction modulo p' map
E(K') ~ E(k')
is not injective on E[m£]. This is a contradiction, since Theorem 6.32 tells us that
the prime-to-p torsion injects on elliptic curves having good reduction.
D
6.6 Rigid Lattes Maps
In general, a Lattes map ¢ : JP>1 ---+ JP>1 is defined via the commutativity of a diagram
E~E
(6.36)
6.6. Rigid Lattes Maps
365
where 'ljJ is a morphism of degree d > 2 and 1r is a finite separable map. Every morphism of an elliptic curve to itself is the composition of an endomorphism and a translation (Remark 6.19), so 'ljJ has the form 'ljJ(P) = a(P) + T for
some a E End(E) and some TEE. However, it turns out that the commutativity
of(6.36) puts additional constraints on¢, '1/J, and 1r. More precisely, it forces the existence of a similar diagram in which 1r has a special form. We state this important
result and refer the reader to [300] for the analytic proof.
Theorem 6.57. Let K be afield ofcharacteristic 0 and let¢ be a Lattes map defined
over K. Then there exists a commutative diagram of the form (6.36) such that the
map 7r has the form
1r: E---+ Ejr ~ IP' 1
for some nontrivial finite subgroup r c Aut(E).
Proof For a proof over C, see [300, Theorem 3.1]. The general case for character0
istic-0 fields follows by the Lefschetz principle, cf. [410, VI §6].
Definition. Let ¢ be a Lattes map. A reduced Lattes diagram for ¢ is a commutative
diagram of the form
E
E
(6.37)
E/r ~ JP>1 ~ JP> 1 ~ E/r
Theorem 6.57 says that every Lattes map fits into a reduced Lattes diagram.
Corollary 6.58. Let¢ be a Lattes map given by a reduced diagram (6.37). Then the
point '¢(0) is .fixed by every element ojr, so in particular, '¢(0) E Etors·
Iffurther j (E) -/:- 0 and j (E) -/:- 1728, then
r = JL 2 ,
deg1r = 2,
and
'ljJ(O) E E[2].
Proof We defer the proof that '1/J( 0) is fixed by every e
E
r until Proposi-
tion 6.77(b), where we prove it in a much more general setting. (Cf. the proof for
flexible Lattes maps in Proposition 6.51(b).) To see that '¢(0) is a torsion point,
let e E r be a nontrivial element of r. Then e ('1/J( 0)) = '1/J( 0), so applying Theorem 6.22 to the isogeny e - 1, we find that
[deg(e- 1)] ('1/J(O)) = ((-:1) o (e- 1)('1/J(O)) =
o.
For the final statement of the corollary, we note that if j(E) is not equal
to 0 or 1728, then Proposition 6.26 tells us that Aut(E) = JL 2 . Hence r =
JL 2 and deg1r = 2. Further, since '¢(0) is fixed by every element of r, we
0
have [-1]'¢(0) = '¢(0), so [2]'¢(0) = 0.
366
6. Dynamics Associated to Algebraic Groups
Remark 6.59. The proof of Theorem 6.57 in [300] actually shows something a bit
stronger. Suppose that¢ is a Lattes map fitting into the commutative diagram (6.36).
It need not be true that the map 1r : E ----> IP' 1 is of the form E ----> E /f, i.e., the given
diagram need not be reduced, and indeed the map 1r may have arbitrarily large degree.
However, what is true is that there are an elliptic curve E', an isogeny E ----. E', and
a finite subgroup r' c Aut(E') such that 1r factors as
E ------r E' ------r E' /f ~ IP' 1 .
Further, this factorization is essentially unique. See [300, Remark 3.3].
Remark 6.60. The proof of Theorem 6.57 is analytic and does not readily generalize
to characteristic p. A full description ofLattes maps in characteristic pis still lacking.
Aside from the curves having j-invariant 0 or 1728, every Lattes map has
r = J.t 2 = Aut(E) and deg 1r = 2, so after a change of coordinates, the projec-
tion 1r : E ----. IP' 1 is 1r(x, y) = x. For simplicity, we will concentrate on this situation,
although we note that the two special cases with Aut( E) = J.t 4 and Aut( E) = JL 6
have attracted much attention over the years for their interesting geometric, dynamical, and arithmetic properties.
Our next task is to describe the periodic points of (rigid) Lattes maps and to
compute their multipliers.
Proposition 6.61. Let ¢ : IP'1 ----> IP'1 be a Lattes map and fix a reduced Lattes
diagram (6.3 7) for ¢. We assume that j (E) =J 0 and j (E) =J 1728. We further
assume that 7/J is an isogeny, i.e., with our usual notation 7/J(P) = [a](P) + T, we
are assuming that T = 0. (See Exercise 6.24 for the other cases.)
(a) The set offixed points of¢ is given by
Fix(¢)= 1r(E[n: + 1] U E[n:- 1]).
(6.38)
(b) The intersection satisfies
E[a + 1] n E[n:- 1] c E[2].
If deg( a - 1) is odd, then the intersection is 0.
(c) Let 1r(P) E Fix(¢). The multiplier of¢ at 1r(P) is
ifP E E[a -1] and P ~ E[a + 1],
if P E E[n: + 1] and P ~ E[a- 1],
(6.39)
if P E E[n: + 1] n E[n:- 1].
Proof (a) We have 1r(P) E Fix(¢) if and only if
1r(P) = ¢(1r(P)) = 1r(¢(P)).
Our assumption on j(E) means that r = Aut( E) = JL 2 , so 1r(P) is fixed by¢ if
and only if¢(P) = ±P. Since we are also assuming that 7/J(P) = [a](P), this is the
desired result.
6.6. Rigid Lattes Maps
367
(b) Let P E E[a -1] n E[a + 1]. Adding [a- 1](P) = 0 to [a+ 1](P) = 0
yields [2]P = 0, soP E E[2].
To ease notation, let m = deg(a- 1). Then using Theorem 6.22, we find that
[m](P) = [a-=1:] o [a- 1](P) = [a-=1:](0) = 0,
soP E E[m]. Hence P E E[2] n E[m], so ifm is odd, then P = 0.
(c) The proof is identical to the proof of Proposition 6.52. The only difference is
that 'ljJ = [a] may no longer be multiplication by an integer, but we still have the key
formula
'ljJ*(w) = [a]*(w) = aw
giving the effect of 'ljJ on the invariant differential w of E. Using this relation in place
offormula (6.29) used in proving Proposition 6.52 and tracing through the argument
yields the desired result.
0
We recall from Section 4.5 that a~n)(cf>) denotes the ith symmetric polynomial
of the multipliers of the points in Pern(¢), taken with appropriate multiplicities.
Ford 2: 2 and each N 2: 1, we write
t:Td,N: Md---+ Ak
(6.40)
for the map defined using all of the functions a}n) with 1 :::; n:::; N. McMullen's
Theorem 4.53 says that for sufficiently large N, the map u d,N is finite-to-one away
from the locus of the flexible Lattes maps. As noted by McMullen in his paper and
stated in Theorem 4.54, rigid Lattes maps can be used to prove that u d,N has large
degree. For the convenience of the reader, we restate the theorem before giving the
proof.
Theorem 6.62. Define the degree ofu d,N to be the number ofpoints in u ;t,k, (P) for
a generic point Pin the image u d,N(Md)· One can show that the degree of u d,N
stabilizes as N --+ oo. We write deg( u d) for this value. Then for every E > 0 there is
a constant Ce such that
for all d.
In particular, the multiplier spectrum ofa rational function cf> E Ratd determines the
conjugacy class of cf> only up to Oe(d~-<) possibilities.
Proof We prove the theorem in the case that d is squarefree and leave the general
case for the reader.
Let F = Q( H), let Rp be the ring of integers ofF, and let at, ... , ah be
fractional ideals ofF representing the distinct ideal classes of Rp. Consider the
elliptic curves E 1 , ... , Eh whose complex points are given by
1 :::; i :::; h.
Each Ei has End(Ri) ~ Rp (Proposition 6.40), and we normalize an isomorphism
6. Dynamics Associated to Algebraic Groups
368
as described in Proposition 6.36. We fix a Weierstrass equation for each Ei and we
define a Lattes map ¢i by
c/Ji 0 X = X o [ J=d].
Then deg(¢i) = d from Proposition 6.39, and Proposition 6.61 tells us that the
multipliers of ¢i are given by (6.39). In particular, they are the same for every ¢i,
i.e., the set of maps {¢1, ... , ¢h} is isospectral, so we see that
Next we observe that ¢ 1 , ... , ¢h give distinct points in Md, because Proposition 6.40 says that E 1 , ... , Eh are pairwise nonisomorphic, and then Theorem 6.46
tells us that ¢ 1 , ... , ¢hare pairwise nonconjugate. This proves that ad,N is generically at least h-to-1, where his the class number of the ring of integers ofQl( R).
(Note that the ¢i are not flexible Lattes maps, and McMullen's Theorem 4.53 tells
us that u d,N is finite-to-one away from the flexible Lattes locus.)
To complete the proof we need an estimate for this class number. Such an estimate is given by the Brauer-Siegel theorem [258, Chapter XVI], which for quadratic
imaginary fields says that
lim
d---too
d squarefree
log (class number of Ql (R))
logd
1
2
(Note that this is where we use the assumption that d is squarefree, since it implies
that the discriminant ofQl( R) is equal to either d or 4d.) In particular, the class
number is larger than d112 - ' for all sufficiently large squarefree d, which completes
the proof of Theorem 6.62 for squarefree d.
In the general case, there are two ways to proceed. The first, which is sketched in
Exercise 6.25, is to find a quadratic imaginary field F whose discriminant is 0( d 1 -')
and whose ring of integers contains an element of norm d. The second is to write d =
ab 2 with a squarefree and use elliptic curves whose endomorphism rings are isomorphic to the order Rb = Z + bRp in the field F = Ql (Fa). The class number of Rb
is equal to hpb times a small correction factor; see [399, Exercise 4.12].
D
6. 7
Uniform Bounds for Lattes Maps
A fundamental conjecture in arithmetic dynamics asserts that there is a constant
C = C(d, D) such that for all number fields K/Ql of degree D and all rational
maps <jJ(z) E K(z) of degree d;::: 2, the number of K-rational preperiodic points
of ¢ satisfies
# PrePer(¢, IP 1 (K)) S:: C(d, D).
(See Conjecture 3.15 on page 96.) Aside from monomials and Chebyshev polynomials, the only nontrivial family of rational maps for which Conjecture 3.15 is known
6.7. Uniform Bounds for Lattes Maps
369
is the collection of Lattes maps. The proof uses the following deep theorem, whose
demonstration is unfortunately far beyond the scope of this book.
Theorem 6.63. (Mazur-Kamienny-Merel) For all integers D ~ 1 there is a constant B(D) such that for all number fields KIQ ofdegree at most D and all elliptic
curves E I K we have
#E(K)tors :S B(D).
Discussion. This deep result was first proven by Mazur [292] for K = Q, then
by Kamienny [225] for [K : Q] = 2, and then was extended to various specific
larger degrees before the proof was completed for all degrees by Merel [297]. The
proof uses the theory of modular curves and Jacobians, which do have counterparts
in arithmetic dynamics (cf. Sections 4.2-4.6). However, the proof also relies in a
fundamental way on the fact that E is a group, and hence that there exist a large
number of commuting maps E ---> E. This is in marked contrast to the situation
for a general rational map ¢ : IP' 1 ---> IP' 1 , for which only the iterates of ¢ commute
with¢. The inclusion Z c End( E) leads to the existence ofHecke correspondences
on elliptic modular curves, and these correspondences provide an essential tool in
the proof of Theorem 6.63. Unfortunately, there does not appear to be an analogous
theory of correspondences for the dynamical modular curves and varieties attached
to non-Lattes maps on IP' 1 .
0
Corollary 6.64. For all integers n > 1, all number fields K IQ, and all elliptic
curves E I K we have
#( U E(L)tors) :S B(n[K: Ql)
3
,
(6.41)
(L:K]~n
where B(D) is the constant appearing in Theorem 6.63.
Proof To ease notation, we let D = [K : Q]. Every field L appearing in the union
in (6.41) satisfies
[L : Q] = [L : K][K : Q] :::; nD,
so Theorem 6.63 tells us that #E(L)tors :::; B(nD). In particular, E(L) contains
no points of order strictly larger than B(nD). This is true for every such L, so we
conclude that
u
E[b].
#E[b] =
L
U E(L)tors C
[L:K]~n
19~B(nD)
Then using #E[b] = b2 yields
#(
U E(L)tors) :S L
[L:K]~n
lSbSB(nD)
lSbSB(nD)
b2 :'S B(nD) 3 .
0
6. Dynamics Associated to Algebraic Groups
370
We now use Theorem 6.63 to prove uniform boundedness of preperiodic points
for Lattes maps. This bound is in fact independent of the degree of the Lattes map ¢,
which may be surprising at first glance. However, it is easily explained by the fact
that Lattes maps associated to the same elliptic curve all commute with one another,
so they have identical sets of preperiodic points.
Theorem 6.65. Let D 2: 1 be an integer. There is a constant C (D) such that for all
number fields K /Q ofdegree D and all Lattes maps ¢ : JP>1 ---> JP> 1 defined over K we
have
#PrePer(¢,JP> 1 (K))::; C(D).
Proof Without loss of generality we fix a reduced Lattes diagram (6.37) for ¢.
Then Proposition 6.26 says that the projection map 1r : E ---> JP> 1 has degree at
most 6, and indeed if j (E) -I- 0 and j (E) -:f. 1728, then deg( 1r) = 2.
Proposition 6.44 tells us that
PrePer(¢,JP> 1 ) = 7r(Etors),
so the fact that deg (1r) ::; 6 yields
PrePer(¢,JP> 1 (K)) C
U 7r(E(L)tors)·
(6.42)
[L:K]::S6
Corollary 6.64 says that the set on the righthand side of (6.42) has size bounded
solely in terms of D, hence the same is true of# PrePer( ¢, JP>1 ( K)).
D
Example 6.66. The rational map
x 4 - 2ax2 - 8bx + a2
c/Ja,b(x) =
4x3 + 4ax + 4b
is the Lattes map associated to multiplication-by-2 on the elliptic curve
Ea,b : y 2 = x 3 + ax + b.
The j-invariant and discriminant of Ea,b are given by the usual formulas
4a 3
j(Ea) = 1728 4a3 + 27b2
and
Theorem 6.65 tells us that# PrePer( c/Ja,b, JP> 1 (K)) is bounded solely in terms of the
degree d = [K : Q]. In general, the best known bounds are exponential in d, but
if j ( Ea,b) is an algebraic integer, then much stronger bounds can be proven as in the
following result.
Theorem 6.67. Let K be a number field of degree D 2: 2, let E / K be an elliptic
curve whose j-invariant is an algebraic integer, and let¢ be a Lattes map associated
to E. Then there is an absolute constant c such that
# PrePer( ¢, JP> 1 (K)) ::; c(D log D) 3 .
371
6.7. Uniform Bounds for Lattes Maps
Proof The assumption that the elliptic curve E has integral j-invariant means that it
has everywhere potential good reduction. Replacing K by an extension of bounded
degree, we may assume that E has everywhere good reduction. (In fact, it suffices
to go to the field K(E[3l), a field of degree at most 48 over K.) Then a result of
Hindry-Silverman [204] implies a bound slightly stronger than
#E(K)tors :=;: 221 D log D.
Finally, we note that as in the proof of Corollary 6.64, a bound for E(K)tors of the
form #E(K)tors::; B([K: Ql) for all number fields K implies a bound of the form
#( U E(L)tors) B(n[K: QJ(
:=;:
[L:K]:::;n
Hence as in the proof of Theorem 6.65 we have
PrePer(¢,lP' 1 (K)) C
U 7r(E(L)tors) :=;: c(DlogD)
3
[L:K]:=;6
0
for an absolute constant c.
Theorem 6.65 proves uniformity for rational preperiodic points of Latti~s maps.
In the other direction, recall that we proved (Theorem 3.43) that the orbits of rational
wandering points contain only finitely many integers except in a few precisely specified situations. In particular, Lattes orbits contain only finitely many integer points,
since Lattes maps are not polynomial maps. Using deep results from the theory of
elliptic curves, it is possible to obtain strong uniformity estimates for the number of
integer points lying in Lattes orbits.
For simplicity we state results over Q and Z, but we note that an appropriately
formulated version is true for rings of S-integers in number fields.
Definition. Let E /Q be an elliptic curve. Recall that a global minimal Weierstrass
equation for E is a Weierstrass equation that is simultaneously minimal at all primes
(Remark 6.30). We define a quasiminimal Weierstrass equation for E jQ to be an
equation of the form
E : y 2 = x 3 + ax + b,
a, bE Z,
(6.43)
such that l4a3 + 27b 2 l is as small as possible. Equivalently, the equation (6.43) is
quasiminimal if there are no primes p such that p 4 1a and p 6 l b.
Remark 6.68. Given an arbitrary Weierstrass equation
E : y2 = x 3 + ax + b,
a,b E Z,
it is easy to create a quasiminimal equation. Simply let u be the largest integer such
that u 12 divides gcd(a 3 , b2 ), and then
6. Dynamics Associated to Algebraic Groups
372
is a quasiminimal equation for E jQ. Elliptic curves over Q have global minimal
Weier:strass equations, see Remark 6.30, and it is not hard to show that a quasiminimal equation is minimal at every prime p ~ 5, and that it is almost minimal at 2
and 3; see Exercise 6.26.
The following theorem is a conditional resolution of a conjecture of Lang [254,
page 140].
Theorem 6.69. (Hindry-Silverman) Let E jQ be an elliptic curve given by a quasiminimal Weierstrass equation and let E(Z) be the set of points in E(Q) having
integer coordinates. Also let v( E) be the number ofprimes dividing the denominator
of the ]-invariant of E.
(a) There is an absolute constant C such that for any subgroup r C E(Q),
# (r n E(Z)) ::; cv(E)+rankr.
(b) If the "ABC conjecture" is true, 2 then there is an absolute constant C such that
for any subgroup r c E(Q),
#(f n E(Z)) ::; crankr.
Proof The proof of (a) is given in [407] and the proof of (b) is in [202].
D
We can use Theorem 6.69 to prove a uniform bound for integer points in orbits
of flexible Lattes maps (cf. Conjecture 3.47).
Theorem 6. 70. Let E j Q be an elliptic curve given by a Weierstrass equation with
integer coefficients, let m ~ 2 be an integer, and let ¢(z) E Q(z) be the Lattes map
satisfYing
¢(x(P)) = x([m]P)
for all PEE.
Assume further that ¢ is affine minimal in the sense that
Res(¢)=
min
JEPGL2(<Q>)
Res(¢!).
(6.44)
f(z)=az+b
(See page 112 for the definition of the resultant Res(¢) of a rational map.) Thus the
assumption (6.44) says that we cannot reduce the resultant of¢ by conjugation by an
affine linear transformation f(z) = az +b. Let ( E Q and consider the orbit Oq,(()
of( by¢.
(a) There is an absolute constant C such that
#(Oq,(() nz)::; cv(E),
where v( E) is the number ofprimes dividing the denominator of the ]-invariant
of E.
2 The ABC conjecture of Masser and Oesterle says that if A, B, C > 0 are pairwise relatively prime
integers satisfying A+ B = C, then C «< f]piABC pl+<.
373
6.7. Uniform Bounds for Latti~s Maps
(b) If the ABC conjecture is true, then the number of integer points in 0 q, (() is
bounded by an absolute constant independent of E and(.
Proof Write the given Weierstrass equation for E as
with a, bE Z.
We begin by showing that the minimality assumption (6.44) implies that there are no
primes p with p 2 la and p3 lb. The rational function ¢(x) = F(x )/G(x) is associated
to the multiplication-by-m map, so it is given by polynomials
F(a,b;x),G(a,b;x) E Z[a,b,x]
that are weighted homogeneous in the sense that
For example, if m = 2, then
¢(x) =
x 4 - 2ax 2 - 8bx + a 2
4x3 + 4ax + 4b
Hence if p 2 la and p3lb, then conjugating ¢(x) by f(x) = px yields
¢f (x) = p_ 1 F(a, b;px) = F(p- 2 a,p- 3b; x).
G(a, b;px)
G(p- 2 a,p- 3b; x)
The assumption that p 2 la and p3 lb implies that these polynomials have integer coefficients, and then homogeneity yields
Res(¢f) = Res(F(p- 2 a,p- 3b; x), G(p- 2 a, p- 3b; x))
= p-m 2 (rn 2 -l) Res(F(a, b; x), G(a, b; x))
= p_m2(m2-1) Res(¢).
This contradicts (6.44), so we have proven that there are no primes p satisfying p 2 la
andp3 lb.
We would like to apply Theorem 6.69 to the rank-1 subgroup generated by a
point P = ((, ry) of E lying above(. Unfortunately, although ( E Ql, there is no
reason that ry need be rational. So it is necessary to move to a twist of E.
We are given a point ( E Ql and we choose a point P = ( (, ry) E E lying above (.
3 + a( + b is rational. We write
We do not assume that ry =
J(
then we factor
with u 1 squarefree,
and consider the elliptic curve
6. Dynamics Associated to Algebraic Groups
374
(6.45)
(In the terminology of Section 4.7, E' is a twist of E; cf. Example 4.71.) Notice that
the point
P{ = (u1x1, uiv1) E E'(Q)
is a point of E' having integer coordinates.
W'e claim that the Weierstrass equation (6.45) forE' is quasiminimal. To prove
this claim, let p be any prime. We showed earlier that either
or
ordp(b) < 3.
Since u1 is squarefree by construction, it follows that either
or
which shows that the Weierstrass equation (6.45) is quasiminimal.
This means that we can apply Theorem 6.69 toE' and the rank-1 subgroup generated by P' to conclude that
#{n 2: 1: [n]P' E E'(Z)} :S: Cv(E'l+l.
(6.46)
Further, if the ABC conjecture is true, then the upper bound may be replaced by C.
The two elliptic curves E and E' are isomorphic, although the isomorphism is
defined only over Q( y'u1). This isomorphism, which we denote by F, is given
explicitly by
E : y 2 = x 3 + ax + b
(x,y)
In particular, j(E) = j(E'), so v(E) = v(E').
In order to relate integers in 0¢(() to integer points in E'(Q), we write
and
P~ = [n]P{ = (x~, y~).
Since the isomorphism F respects multiplication by n, we have
F(Pn) = F([n]P1) = [n](F(PI)) = [n]P{ = P~.
In particular, since P{ E E' (Q), it follows that [n]P{ = P~ E E'(Q), so F maps the
multiples of P 1 , which, note, are not in E(Q), to points in E'(Q). Further, if Xn E Z,
then it is clear from the definition ofF that x~ = u 1 xn E Z, and hence y~ is also
in Z, since we just showed that y~ is in Q and the equation of E' shows that y~ is
the square root of an integer.
To summarize, we have proven that
Xn E Z
==}
P~ = F(Pn) E E'(Z).
(6.47)
375
6.8. Affine Morphisms and Commuting Families
By construction we have rjJk(() = x([mk]P) = Xmk, and hence
#(Oq,(P) nz) = #{k 2:0: Xmk E Z}
::; #{ k 2: 0 : P:nk E E' (Z)}
:S: #{n 2:0: P~ E E'(Z)}
from (6.47),
:S: Cv(E')+l
from (6.46),
= cv(E)+l
since E and E' are isomorphic.
Further, if the ABC conjecture is true, then we may replace the upper bound by C.
D
Remark 6.71. We note that something like the affine minimality of rjJ is necessary in
the statement of Theorem 6.70. Indeed, without some kind ofminimality condition,
we saw in Proposition 3.46 that we can make #(Oq,(() nz) arbitrarily large by
replacing rjJ(z) with BrjJ(B- 1 z). This conjugation has the effect of multiplying every
point in the orbit by B, hence allows us to clear an arbitrary number of denominators.
Remark 6.72. Continuing with notation from the statement of Theorem 6.70, we
note that there is a cutoff value k0 such that
rjJk(() E Z
for
0::; k::; ko
and
rjJk(() ~ Z
for
k > k0 .
This reflects the more general fact that if x([n]P) E Z and if rln, then x([r]P) E Z
([410, Exercise 9.12]). Note that no such cutoff statement holds for general rational
maps that are not non-Lattes maps.
6.8
Affine Morphisms, Algebraic Groups, and
Commuting Families of Rational Maps
Power maps and Chebyshev maps are attached to endomorphisms of the multiplicative group Gm and its quotient Gm/{z = z- 1 }, and similarly Lattes maps are
attached to maps of quotients of elliptic curves. In this section we put these constructions into a general context and state a classical theorem on commutativity of
one-variable rational maps.
Definition. Let G be a commutative algebraic group. An affine morphism of G is
the composition of a finite endomorphism of degree at least 2 and a translation.
Remark 6.73. The reason for this terminology is as follows. Let G jC be a connected
commutative algebraic group of dimension g. Then its universal cover is C9 and
every affine morphism 'ljJ : G ---+ G lifts to an affine map C9 ---+ Cf_9, i.e., there are a
matrix A and vector a such that the following diagram commutes:
z >-+ Az+a
1
1
G
G
376
6. Dynamics Associated to Algebraic Groups
Example 6.74. Every affine morphism of the multiplicative group Gm has the form
'lj;(z) := azd for some nonzero a and some dE Z. More generally, for any commutative group G, any a E G, and any dE Z there is an affine morphism 'lj;(z) = azd.
Notice that it is easy to compute the iterates of this map,
'lj;n(z) = al+d+·+dn-1 zdn
0
Proposition 6.75. Let 'lj; : G ----t G be an affine morphism of an algebraic group G,
so 'lj; has the form 'lj;(z) =a· a(z)for some a E End(G) and some a E G.
(a) The endomorphism a and translation a are uniquely determined by 'lj;.
(b) Let a and a be as in (a). Then the iterates of 'lj; have the form
Proof The definition of affine morphism tells us that there are an element a E G
and an endomorphism a of G such that the map 'lj; has the form 'lj;(z) = aa(z).
Evaluating at the identity element e E G yields 'lj;(e) = aa(e) =a, so a is uniquely
determined by 'lj;. Then a(z) = a- 1'lj;(z) is also uniquely determined by 'lj;. This
proves (a). The proof of (b) is an easy induction, using the commutativity of G and
the fact that a is a homomorphism.
D
Definition. A self-morphism of an algebraic variety ¢ : V ----t V is dynamically
affine if it is a finite quotient of an affine morphism. What we mean by this is that
there are a connected commutative algebraic group G, an affine morphism 'lj; : G ----t
G, a finite subgroup c Aut( G), and a morphism G
V that identifies G
with a Zariski dense open subset of V (possibly all of V) such that the following
diagram is commutative:
r
;r ___,
G
'lj;
-------)
G
G
1
1
;r -------) G ;r
11
v
q,
-------)
;r
(6.48)
11
v
Example 6.76. Examples of dynamically affine rational maps¢ : lP' 1 ___, lP' 1 include
the power maps ¢( z) = zn with G = Gm and r = {1}, the Chebyshev polynomials Tn(z) with G = Gm and r = {z, z- 1 }, and Lattes maps with G an elliptic
curve E and r a nontrivial subgroup of Aut(E).
Proposition 6.77. Let¢: V ___, V be a dynamically affine map and let 'lj;: G ___, G
and r c Aut( G) be the associated quantities fitting into the commutative diagram (6.48).
(a) For every~ E f there exists a unique ( E f with the property that 'lj;o~ = ( o'lj;.
(b) Write 'lj;(z) = a· a(z) with a E G and a E End( G) as in Proposition 6.75.
Then ~(a)= a for every~ E r.
6.8. Affine Morphisms and Commuting Families
377
(c) Assume that #f 2:: 2 and that G is simple. (An algebraic group is simple if its
only connected algebraic subgroups are {1} and G.) Then a E G10rs, i.e., the
translation used to define 'lj; is translation by a point offinite order.
Proof (a) The uniqueness is clear, since if 'lj; o ~ = 6 o 'lj; = 6 o '¢,then 6 = 6
because the finite map 'lj; : G--+ G is sutjective.
We now prove the existence. The commutativity of (6.48) tells us that for all z E
G and all~ E f,
(11' o 'lj; o ~)(z) = (</> o 7l' o ~)(z) = (</> o 11')(z) = (11' o 'lj;)(z).
Thus ('lj;o~)(z) and 'lj;(z) have the same image for the projection map 7l' : G--+ G /f,
so there is an automorphism E f satisfying
e
w(~(z)) = ~'(¢(z)).
e.
We claim that the automorphism
which a priori might depend on both~ and z,
is in fact independent of z. To see this we fix~ and write ¢(~(z)) = ~~('¢(z)) to
indicate the possible dependence of eon z. In this way we obtain a map (of sets)
G ______, f,
Since f is finite, there exists some (' E f such that ~~ = (' for a Zariski dense
subset of z E G. (Note that a variety cannot be a finite union of Zariski closed proper
subsets.) It follows that 'lj; o ~ is equal to (' o 'lj; on a Zariski dense subset of G, and
hence they are equal on all of G.
(b) From (a) we see that there is a permutation off defined by the rule
T:f---->f,
Evaluating both sides of 'lj; o ~ = T( ~) o 'lj; at the identity element 1 E G and using
the fact that ~(1) = 1 and '¢(1) =a· a(1) =a, we find that
a= '¢(~(1)) = T(~)('¢(1)) = T(~)(a).
ButT is a permutation off, so as~ runs over f, so does T(~). Hence a is fixed by
every element off.
(c) From (b) and the assumption that #f 2:: 2, there exists a nontrivial ~ E f
with ~(a) =a. It follows that a is in the kernel of the endomorphism
G ______, G,
The kernel is not all of G, since ~ is not the identity map, so the simplicity of G tells
us that the kernel is a finite subgroup of G. Hence a has finite order.
D
Remark 6. 78. In this book we are primarily interested in dynamically affine maps
of IP' 1 , but higher-dimensional analogues, especially of Lattes maps, have also been
studied. See for example [68, 134, 145, 439].
378
6. Dynamics Associated to Algebraic Groups
The commutativity of (6.48) implies that deg( </>) = deg( 1jJ). It follows that all
dynamically affine maps for the additive group Ga have degree 1, since every affine
morphism ofGa has the form ¢(z) = az +b. Hence nonlinear dynamically affine
maps on 1P' 1 are attached to either the multiplicative group Gm or to an elliptic curve,
since these are the only other algebraic groups of dimension 1.
We note that over a field of characteristic 0, the endomorphism ring End( G)
of a one-dimensional algebraic group G is commutative. 3 More precisely, the multiplicative group has endomorphism ring End(Gm) = Z, and the endomorphism
ring End( E) of an elliptic curve E is either Z or an order in a quadratic imaginary
field. The commutativity of End( G) means that dynamically affine maps commute
with many other maps. An appropriately formulated converse of this statement is a
classical theorem of Ritt.
Theorem 6.79. (Ritt and Eremenko) Let¢>, 1jJ E C(z) be rational maps ofdegree at
least 2 with the property that </> o 1jJ = 1jJ o ¢. Then one ofthe following two conditions
is true:
(a) There are integers m, n 2:: 1 such that <f>n = 1/Jm.
(b) Both </> and 1jJ are dynamically affine maps, hence they are either power maps,
Chebyshev polynomials, or Lattes maps.
In all cases, the commuting maps </> and 1jJ satisfy
F(</>) = F(¢),
.J(<I>) = .J(¢),
and
PrePer(¢) = PrePer(¢).
Proof The first part of the theorem, in somewhat different language, is due to
Ritt [371]. See Eremenko's paper [152] for a proof of both parts of the theorem
and some additional geometric dynamical properties shared by commuting</> and¢.
A higher-dimensional analogue is discussed in [135]. We remark that the equality
PrePer( </>) = PrePer( 1jJ) is a formal consequence of the commutativity of</> and 1jJ
and the fact that the preperiodic points of a nonlinear rational map are isolated; see
Exercise 1.15.
D
Although we do not give a proof of Ritt's theorem, we conclude this section by
proving the easier statement that only polynomial maps can commute with polynomial maps. This result was used in our description of the rational maps commuting
with the Chebyshev polynomials (Theorem 6.9).
Theorem 6.80. Let K be a field, let </>( z) E K [z] be a polynomial of degree d 2:: 2,
and let ¢(z) E K(z) be a nonconstant rational map. We assume that both</> and 1jJ
are separable, i.e., neither ofthe derivatives </>' (z) and¢' (z) is identically 0. Suppose
further that </> and 1jJ commute under composition, </> o 1jJ = 1jJ o ¢. Then one of the
following is true:
(a) 7/J(z) E K[z], i.e., 1jJ is also a polynomial.
3 Even in characteristic p, most elliptic curves have commutative endomorphism ring. However, there
are a finite number of elliptic curves whose endomorphism ring is a maximal order in a quatemion algebra.
These supers ingular curves are all defined over IFp2 . See [410, V §3].
379
6.8. Affine Morphisms and Commuting Families
(b) After simultaneous conjugation by an affine map f (z)
z + {3, the polynomial ¢(z) has the form ¢(z) = azd and the rational map 'lj;(z) has the
form 'lj;(z) = bzr for some r < 0.
Proof The proof is an application of ramification theory and the Riemann-Hurwitz
formula (Theorem 1.1). By assumption, the map¢ is a polynomial, so oo is a totally
ramified fixed point of¢. Suppose that 'lj;( z) is not a polynomial. This means that we
can find a point a: E 'lj;- 1 ( oo) with a: =/:- oo. We use the commutativity of¢ and 'ljJ to
compute
n-1
e,:.(¢n o 'lj;) = ea('l/J)
IT erf>i,P(a)(¢) ea('l/J)eoo(¢t ea('l/J)dn
=
=
i=O
(6.49)
Hence
IT
i=O
erf>i(a)(¢) =
d
e 0 ('1j;) > _1_
e</>n(a)(W) - deg'lj;
for all n ~ 1.
(6.50)
Every ramification index e<f>i (a) (¢) is an integer between 1 and d, so letting n ->
oo, we see that erf>i(a)(¢) = d for all sufficiently large i. On the other hand, ¢is
a polynomial and a: =1- oo, so ¢i (a:) =1- oo for all i. Hence there is at least one
point {3 =1- oo with e13 (¢) = d. The Riemann-Hurwitz formula then implies that {3
and oo are the only two points at which ¢ is ramified. It follows that ¢i (a:) = {3 for
all sufficiently large i, which implies that ¢({3) = {3. In other words, {3 and oo are
both totally ramified fixed points of¢, i.e.,
¢- 1 ((3) = {{3}
and
¢- 1 (oo) = {oo},
(6.51)
and ¢ has no other ramification points. In particular, since by construction we
have ¢i(o:) = {3 for some i, it follows that a: = {3. But a: =1- oo was an arbitrary
point in 'lj;- 1 ( oo ), so we have also proven that 'lj;- 1 ( oo) c {{3, oo }.
Next let 'Y E 'lj;- 1 ({3). We use the fact that ¢i'lj;('y) = {3 and e13(¢) = d to repeat
the calculation (6.49) with a: replaced by 'Y· This again leads to the inequality (6.50),
but with 'Y in place of a:, and hence to the conclusion that ¢ is totally ramified at
some iterate ¢i('y). It follows that ¢i('y) E {{3, oo} for some i, and hence from (6.51)
that 'Y E {{3, oo }.
We have now proven that
'lj;- 1 ({{3,oo}) C {{3,oo}.
Thus 'ljJ is totally ramified at {3 and oo, and since 'lj;(f3) =1- oo by assumption, the
map 'ljJ must switch {3 and oo. Since we also know that {3 and oo are totally ramified
fixed points of¢, it follows that ¢ and 'ljJ have the form
¢(z) = {3 + a(z- {3)d
and
'lj;(z) = {3 + b(z-
f3r for some r < 0.
Then conjugation by f(z) = z + {3 puts them into the desired form.
D
380
Exercises
Exercises
Section 6.1. Power Maps and the Multiplicative Group
6.1. Let K be a field of positive characteristic p.
(a) Let Mp(z) = zP. Prove that the automorphism group of Mp over K equals PGL 2(1Fp).
(b) More generally, if q is a power ofp, prove that Aut(Mq) = PGL2(lFq).
(c) Again let q be a power of p, and let d be an integer with p f d. Describe Aut(Mqd).
6.2. Let K be an algebraically closed field, let d E Z, and let a E K*. Further, if K has
positive characteristic p, assume that p f d. Describe all rational functions ¢( z) E K (z) that
commute with azd under composition.
6.3. **Let Md(z) = zd be a power map for some ldl ~ 2, and if K has positive characteristic p, assume that p f d. Example 6.5 describes two types of twists of Md (z). The first type has
the form <Pa(z) = azd and the second type 'l/Jb(z) is given by the complicated formula (6.1).
Does Md(z) have any other twists? If so, describe all of the twists of Md(z).
Section 6.2. Chebyshev Polynomials
6.4. Complete the proof of Proposition 6.6(e) by verifying the identity
1 [(d + 1) (d + 1k - k) + d(d + 1- k)] = d+2-k
d + 2 (d + 2- k).
k
d+1-k
k-1
6.5. Let Td (w) be the rf' Chebyshev polynomial for some d ~ 2.
(a) Prci>Ve that the fixed points ofTd(w) are as described in Proposition 6.8(a).
(b) Prove that the multipliers of Td(w) at its fixed points are as described in Proposition 6.8(b).
(Hint. For (a) use the trigonometric identity
cos(
A) -cos( B) =sin ( B; A) sin ( B; A) ,
and for Cb) differentiate Td(z + z- 1 ) = zd + z-d to obtain the identity
,
Td(z + z
-1
zd- z-d
) = d z _ z- 1 . )
6.6. Proposition 6.8 describes the multipliers of the fixed points of the Chebyshev polynomial Td(w). Prove directly that the multipliers satisfy the summation formula described in
Theorem 1.14,
I:
1
(EFix(Td) 1 - ATd ( ()
=1.
6.7. Let K be a field of characteristic p ~ 3, let n ~ 1 be an integer with p f n, and
let 1-'n C K* be the nth roots of unity. There is no "cosine function" for the field K, but we
can define a set of cosine values by
Cosn = {
a+a- 1
}
: a E 1-'n .
2
We also let 2 Cosn = {2(: ( E Cosn}·
Let d ~ 2 be an integer with p f d( d 2 - 1) and let Td (w) be the rf' Chebyshev polynomial.
381
Exercises
(a) Prove that
Fix(Td) = 2 Cosd+l U2 Cosd-1.
Also compute the intersection 2 Cosd+ 1 n2 Cosd- 1.
(b) Prove that the multipliers ofTd at its fixed points are given by
if ( E 2 Cosd+1 and ( of ±2,
if ( E 2 Cosd-1 and ( of ±2,
if(= ±2.
(c) Give a similar description of the fixed points and their multipliers in the case that
d = ±1 (modp).
6.8. We stated during the proof of Theorem 6.9 that for d ~ 2, the Chebyshev polynomial Td(w) is not equivalent to a monomial, i.e., no conjugate (f- 1 o Td o f)(w) has the
form cwd. Prove this assertion.
6.9. Prove that the (formal) derivatives of the Chebyshev polynomials satisfy the following
identities:
(a) (4- w 2 )T~(w) + dwTd(w) = 2dTd-1(w).
(b)
(4- w 2 )T~'(w)- wT~(w) + d 2 Td(w) = 0.
6.10. Let K be a field of positive characteristic p.
(a) Prove that the pth Chebyshev polynomial Tp( w) is equal to wP in K[w].
(b) In general, if q is a power ofp, prove that Tqd(w) = Td(wF = Td(wq) for all d ~ 1.
(c) Again letting q be a power of p, deduce that Aut(Tq) = PGL2 (IF q ). (Cf. Exercise 6.1.)
6.11. Let K be a field of characteristic 2 and let d ~ 1 be an odd integer. Prove that
wT~ (w) = Td (w). What is the derivative T~ (w) if d is an even integer?
Section 6.4. LattE~s Maps- General Properties
6.12. Let E be the elliptic curve E: y 2 = x 3 + 1 with j(E) = 0, so Aut( E) = p, 6 is cyclic
of order 6. Let 'lj;(P) = [2]P be the doubling map.
(a) Let 1r : E --> E I p, 2 ~ lP' 1. Prove that we can take 1r( x, y) = x and that the Lattes map
corresponding to 'lj; is
z(z 3 - 8b)
¢1(z)= 4(z3+b)'
(b) Let1r: E--> Elp, 3 ~ lP' 1. Provethatwecantake1r(x,y) = yandthattheLattesmap
corresponding to 'lj; is
z 4 + 18bz 2 - 27b 2
8z3
¢2(z) =
(c) Let 1r : E --> E I p, 6 ~ 1P' 1 • Prove that we can take 1r(x, y) = x 3 and that the Lattes map
corresponding to 'lj; is
z(z- 8b) 3
¢ 3 (z) = 64(z + b) 3
Compute the conjugate ¢3 ( z - b) + b of ¢3 ( z ), compare it to ¢2 ( z), and explain.
6.13. Let ¢be a Lattes map. Prove that there does not exist a linear fractional transformation f E PGL2 such that the conjugate¢! is a polynomial. (Cf. Exercise 6.8.)
382
Exercises
6.14. Complete the proof of Proposition 6.44 in the general case that 'lj;(P) = o:(P) + T
with 01 E End(E) and T E E not necessarily equal to 0. However, you may assume that
T E Etors, i.e., Tis a point of finite order.
6.15. Let ¢ : C ---+ C be a nonconstant rational map from a smooth curve to itself. Recall
that CritVal.p denotes the set of critical values of¢. Prove that
U¢n(CritVal.p) = UCritVal.pn .
00
00
n=O
n=l
Section 6.5. Elliptic Curves and Flexible Lattes Maps
6.16. Let E be an elliptic curve and let 1r : E ----> lP' 1 be a map of degree 2.
(a) Let R be any point on E. Show that we can define a new group law (call it*) onE by
the rule
P*Q = P+Q-R.
Show that R is the identity element for the group (E, *).
(b) Prove that there exists a point Po E E such that 1r( -(P +Po)) = 1r(P +Po) for
all PEE.
(c) Conclude that after choosing a new identity element forE, the map 1r is even, i.e., satisfies 1r(P) = 1r(-P) for all PEE.
6.17. FixanellipticcurveEandandadegree-2map1r: E---+ lP' 1 satisfying1r(P) = 1r(-P).
For any integer m and any point T E E[2], let ¢m,T : lP' 1 ----> lP' 1 be the flexible Lattes map
associated to the map 'lj;(P) = [m](P) + T as in the commutative diagram (6.22).
(a) Prove that ¢m,T o¢m',T' = ¢mm',mT'+T· In particular, the maps ¢m,o commute under
composition.
(b) Prove that ¢':n,r is either ¢mn,r or ¢mn,o. More precisely, ifm is odd and n is even,
prove that ¢':n,r = ¢mn,o, and prove in all other cases that ¢':n,r = ¢mn,r. (Of course,
ifT = 0, the cases are all the same.)
(c) It follows from (a) that the collection of maps { ¢m,T : m ~ 1, T E E[2]} is closed
under composition. Prove that ¢ 1 ,o is the identity element and that the associative law
holds. Thus this set of flexible Lattes maps forE is a noncommutative monoid.
6.18. Let¢ : lP' 1 ----> lP' 1 be a flexible Lattes map associated to 'lj;(P) = [m]P + T, where the
point T E E[2] is not necessarily equal to 0.
(a) Prove that the set of fixed points of¢ is given by
Fix(¢)= x([m-
1r (T)) Ux([m + 1r (T)).
1
1
(b) Compute the multiplier of¢ at each point in Fix(¢).
(c) Use the results from (a) and (b) and the formula for the composition ofLattes maps in
Exercise 6.17 to describe the periodic points of¢ and to compute their multipliers.
(Hint. Mimic the proof of Proposition 6.52, which dealt with the case T = 0.)
6.19. Proposition 6.52 describes the multipliers of a flexible Lattes map. Using these values,
verify directly that the formula
""""
~
(EPern(</>)
1...--,.((-:-) = 1
-1-_-,-->.
'+'
from Theorem 1.14 is true for flexible Lattes maps.
Exercises
383
6.20. Complete the proof of Proposition 6.52(b) by computing the multiplier >.q, (oo) at the
fixed point oo = x(O). (Hint. Move 0 to (0, 0) using the change of variables z = xjy
and w = 1/y. Then write the invariant differential in terms of z and w and mimic the proof in
the text.)
6.21. Let K be an algebraically closed field and let¢ and¢' be flexible Lattes maps defined
over K that are associated, respectively, to elliptic curves E and E'. Suppose that ¢ and ¢'
are PGL 2(K)-conjugate to one another. We proved (Theorem 6.46) that if the characteristic
of K is not equal to 2, then E and E' are isomorphic. What can be said in the case that K has
characteristic 2? (Note that in characteristic 2 it is necessary to use a generalized Weierstrass
equation (6.12) to define E.)
6.22. We proved Proposition 6.55 in the case that 'lj;(P) = [m](P).
(a) Prove Proposition 6.55 for general flexible Lattes maps, i.e., Lattes maps associated to
maps ofthe form 'lj;(P) = [m](P) + T with T E E[2].
(b) Formulate and prove a version of Proposition 6.55 for rigid Lattes maps.
(c) ** To what extent is the converse of Proposition 6.55 true? More precisely, if¢ is a
Lattes map fitting into a reduced Lattes diagram (6.37) and if ¢1 has bad reduction for
every f E PGL2(K), does the elliptic curve E necessarily also have bad reduction?
6.23. Let E be an elliptic curve given by a Weierstrass equation
E :l
= x 3 +ax + b.
Let m 2: 1 be an integer and write x([mJP) as a quotient of polynomials
(
) _ Fm(x(P))
(
).
Gm x(P)
(6.52)
x [m]P -
(a) Prove that Fm and Gm can be taken to be polynomials in x, a, and b. More precisely,
prove that there are polynomials Fm, Gm E Z[a, b, x] satisfying (6.52) and that they are
uniquely determined by the requirement that Fm be monic in the variable x.
(b) Prove that deg(Fm) = m 2 and deg(Gm) = m 2 - 1 and that their leading terms
are Fm(x) = xm 2 + · · · and Gm(x) = m 2xm 2 - 1 + · · ·.
(c) If m is odd, prove that there is a polynomial 'lj;m(x) E Z[a, b, x] such that Gm(x) =
'lj;m(x) 2. Similarly, ifm is even, provethatthere is a polynomial 'lj;m(x, y) E Z[a, b, x, y]
such that Gm (x) = 'lj;m (x, y) 2, where in the computation we replace y 2 by x 3 +ax+ b.
The polynomial 'lj;m is called the m 1h division polynomial for E, since its roots are the
nontrivial points of order m.
(d) Prove that Fm and Gm satisfy
2
Fm ( t 2 a, t 3 b; tx ) = t m Fm(a, b; x)
and
2
Gm(t 2 a, t 3 b; tx) = tm - 1 Gm(a, b; x).
Thus Fm and Gm are homogeneous if x, a, and bare respectively assigned weights 2, 4,
and6.
(e) Let b.(E) = -16( 4a 3 + 27b2). Prove that the resultant of Fm and Gm with respect to
the variable x is given by
Res(Fm, Gm) = ±b.(E)m2 (m 2 -l)/G.
384
Exercises
Section 6.6. Elliptic Curves and Rigid LattE~s Maps
6.24. This exercise extends Proposition 6.61. Let ¢> : IP'1 -+ IP' 1 be a Lattes map and fix a
reduced Lattes diagram (6.37) for¢>. Write 1/;(P) = [a](P) + T as usual, where we use
the standard normalization described in Proposition 6.36 to identify End( E) with a subring
of C.
(a) Prove that the fixed points of¢> are given by
Fix(¢>)=
U{1r(P) : [a- ~](P) = -T}.
(6.53)
~Er
(b) Let 1r(P) E Fix(¢>). Prove that Pis a critical point for 1r if and only if Pis fixed by a
nontrivial element of~- More generally, prove that the ramification index is given by
ep(1r) = {~ E f: [~]P = P}.
Itt particular, if Pis not a critical point, then there is a unique~ E f that fixes P.
(c) Assume that T = 0. Let 1r(P) E Fix(¢>) and choose some automorphism~ E f such
that 1/;(P) = [~](P). Compute the multiplier of¢> at 1r(P) as in the following table (we
have given you the first four values):
C 1 a ifep(1r) = 1,
Atr(P)(c/>) =
a2
a3
iff= J-t 2 and ep(7r) = 2,
iff= ~-t 3 and ep(7r) = 3,
ea 2
iff= /-t4 andep(7r) = 2,
iff= J-t 4 and ep(7r) = 4,
iff= ~-t 6 and ep(1r) = 2,
iff= J-t 6 and ep(1r) = 3.
6.25. In the text we proved Theorem 6.62 under the assumption that d is squarefree. This
exercise sketches an argument to eliminate the squarefree hypothesis. We set the notation S(b)
for the squarefree part of the integer b ;:::: 1.
(a) For each integer d ;:::: 2, let Dd ;:::: 1 be an integer with the property that dis a norm from
the ring Z [v'- Dd] down to Z. In other words, there are integers u and v such that
u 2 +Ddv 2 =d.
T4en with notation as in the statement of Theorem 6.62, prove that for every f > 0 there
is a constant C. such that
for all d.
(Hint. Use elliptic curves with CM by the ring Z [v'- D d ] and Lattes maps associated
to the endomorphism [u + vv'-Dd] and follow the proof of Theorem 6.62.)
(b) Prove that for every f > 0 there is a constant C~ > 0 such that
max S(d- u 2 ) ;:::: C~dl-<
O~u<Vd
for all d.
(Hint. It suffices to prove that for sufficiently large d the average satisfies
Exercises
385
~
L logS(d-u2)2:(1-E)logd.
O:C::u<Vd
Write this as two sums using ordp (S(b)) = ordp(b)- 2l! ordp(b) J and show that the
first sum is asymptotic to log( d) and the second is bounded as d-+ oo.)
(c) Combine (a) and (b) to complete the proof of Theorem 6.62.
Section 6.7. Uniform Boundedness for Lattes Maps
6.26. Let E /Q be given by a quasiminimal Weierstrass equation
E : y 2 = x 3 + ax + b,
i.e., the discriminant 116( 4a 3 + 27b 2) I is minimized subject to the condition that a and b are
integers.
(a) Show that the equation forE is minimal at every prime p 2: 5.
(b) Let l13(E) be the discriminant of a general Weierstrass equation (6.12) forE that is
minimal at 3. Prove that
(c) Let l12(E) be the discriminant of a general Weierstrass equation (6.12) for E that is
minimal at 2. Prove that
6.27. Theorem 6.70 suggests that there should be an absolute upper bound for the number of
integer points in the orbits of affine minimal Lattes maps defined over Q.
(a) Let E be the elliptic curve
E : y 2 = x 3 - 48907 + 8481094
and let rjJ(x) be the Lattes map associated to multiplication-by-2, i.e., fox = x o [2].
Verity that the orbit V.p(2363) contains five integer points, but that rjJ 5 (2363) tJ. Z. Also
verity that there is a point in E(Q) with x-coordinate 2363.
(b) Let E be the elliptic curve
E: y 2 = x 3 - 40467 + 4120274
and again let rjJ(x) be the Lattes map associated to multiplication-by-2. Verity that the
orbit V.p(193) contains five integer points, but that rjJ 5 (193) tJ. Z. In this case E(Q) does
not contain a point with x-coordinate equal to 193.
(c) **Find an affine minimal Lattes map rjJ and an initial point ( E Z such that rjJ 5 (() E Z,
or prove that none exist. (This will force rpk (() E Z for all 0 ~ k ~ 5; see Remark 6. 72.)
6.28. Prove a version of Theorem 6.70 for a Lattes map satisfYing
r/J(x(P)) = x([m]P+T)
for all PEE,
where T is a fixed 2-torsion point of E. (You may need a more general version of Theorem 6.69; see [202, 407].)
Chapter 7
Dynamics in Dimension
Greater Than One
Up to this point our primary focus has been on arithmetic dynamics of rational maps
on JID 1 . In this chapter we take a look at dynamics in higher dimensions. Even over <C,
although there is now a significant body of knowledge, it seems fair to say that complex dynamics on JIDN is still in its infancy. And arithmetic dynamics in higher dimensions is at present a patchwork of results from which a general theory is yet
to emerge. Our goal in this chapter is to provide a glimpse into two aspects of this
developing theory by highlighting two ways in which higher-dimensional dynamics
differs significantly from the one-dimensional case.
The first difference arises from the fact that rational maps JIDN --> JIDN for N ~ 2
need not be everywhere defined, i.e., they need not be morphisms. In Section 7.1 we
study the dynamics of rational maps ¢ : JIDN --> JIDN having the property that they
restrict to automorphisms on AN. The geometry and arithmetic of such maps can be
quite complicated, despite the fact that they are bijective on AN.
The second difference arises due to the far greater variety of varieties in higher
dimensions. Thus in dimension 1, only self-maps of JID 1 and of elliptic curves are
dynamically interesting, since the only self-maps of a curve of genus greater than 1
are automorphisms of finite order. But even in dimension 2 there is an abundance
of varieties that admit self-maps of infinite order, and the dynamical properties of
these maps are extremely interesting, although as yet imperfectly understood. In
Section 7.4 we study the arithmetic dynamics of certain surfaces that admit a pair
ofnoncommuting involutions L1 and L2 . The composition¢= L1 o L2 is an automorphism of infinite order.
The theory of height functions that we developed in Sections 3.1-3.5 provides a
powerful tool for studying arithmetic dynamics on JID 1 . One of the recurring themes
of this chapter is the use of height functions in a higher-dimensional setting and on
varieties other than projective spaces.
387
388
7.1
7. Dynamics in Dimension Greater Than One
Dynamics of Rational Maps on Projective Space
Recall that a rational map ¢ : IP'N -> IP'N is described by homogeneous polynomials
with no common factor, and that ¢ is a morphism if the polynomials have no common root in IP'N (K). (See page 89 in Chapter 3 for the precise definition.) As noted
in the introduction to this chapter, height functions are a powerful tool for studying
the arithmetic of morphisms ¢ : IP'N -> IP'N. The situation is considerably more complicated if the map¢: IP'N -> IP'N is required to be only a rational map. Notice that
we did not run into this situation when studying rational functions ¢(z) E K(z) of
one variable, since every rational map ¢ : IP'1 -> IP'1 is automatically a morphism. But
in dimensions 2 and higher, there are many rational maps that are not morphisms.
Example 7.1. The rational map
(7.1)
is not a morphism, since it is not defined at the point [0, 1, OJ. Notice that if we
discard [0, 1, OJ, then ¢ fixes every point on the line X 0 = X2 , and ¢ sends every
point on the line X 0 = 0 to the single point [0, 0, 1]. This kind of behavior is not
possible for morphisms IP'2 -> IP'2 .
Continuing with this example, recall that if ¢ were a morphism, then Theorem 3.11 would tell us that h(¢(P)) = 2h(P) + 0(1) for all P E IP'2 (Q). But this
is clearly false for the map (7 .1 ), since for all a, b E Q* we have
¢([a, b, a]) = [a 2 , ab, a2 ] = [a, b, a].
Thus
h( ¢([a, b, al)) = h([a, b, al),
so we cannot use Theorem 3.7 to conclude that¢ has only finitely many Q-rational
periodic points. Of course, that's good, since in fact ¢ has infinitely many Q-rational
fixed points!
An initial difficulty in studying the dynamics of a rational map ¢ : IP'N -> IP'N
arises from the fact that the orbit O¢(P) of a point may "terminate" if some iterate ¢n(P) arrives at a point where¢ is not defined. This suggests looking first at
maps¢ for which there is a large uncomplicated (e.g., quasiprojective, or even affine)
subset U c IP'N with the property that ¢(U) c U and studying the dynamics of¢
on U. As a further simplification, we might require that ¢ be an automorphism of U,
since quasiprojective varieties often allow interesting automorphisms.
7.1.1
Affine Morphisms and the Locus of Indeterminacy
In this section we study rational maps IP'N -> IP'N with the property that they induce
morphisms of affine space AN -> AN. Concretely, an affine morphism
¢:AN-tAN
is a map of the form
389
7.1. Dynamics of Rational Maps on Projective Space
To avoid trivial cases, we generally assume that at least one of the Fi is not the zero
polynomial.
Definition. The degree of a polynomial
it, ... ,iN
is defined to be
In other words, the degree of F is the largest total degree of the monomials that appear in F. (By convention the zero polynomial is assigned degree -oo.) The degree
ofa morphism <P = (F1, ... , FN) :AN-+ AN is defined to be
deg <P = max{deg F 1, ... , deg FN}.
Homogenization ofthe coordinates of an affine morphism <P: AN-+ AN of degreed yields a rational map ([>: pN -+ IP'N of degree d. For each coordinate function Fi of¢, we let
Notice that each Pi is a homogeneous polynomial of degree d (or the zero polynomial), so the map
¢ = [X0 , F1, F2, ... , FN] : IP'
-
d--
-
N
-+ IP'
N
is a rational map of degree d. We call ¢ the rational map induced by ¢. A rational
map need not be everywhere defined.
Definition. Let <P : AN -+ AN be an affine morphism of degree d and let
be the rational map that it induces. The locus of indeterminacy of <P is the set
Z(¢) = {P = [O,x1, ... ,xN] E IP'N: F1(P) = · · · = FN(P) = 0}.
(To ease notation, we write <P and Z(¢) instead of([> and Z(([>).) This is the set of
points at which ([> is not defined. Notice that Z(¢) lies in the hyperplane H 0
{Xo = 0} at infinity, since <Pis well-defined on AN.
The polynomials F1 , ... , PN can be used to define a morphism
The map <I> is called a lift of([>. If we let 1r be the natural projection map,
7. Dynamics in Dimension Greater Than One
390
then 1r, <I>, and ¢ fit together into the commutative diagram
Example 7.2. The map
induces the rational map
and has indeterminacy locus Z(¢) = {[0, 0, ll} consisting of a single point.
7.1.2
Affine Automorphisms
Of particular interest are affine morphisms that admit an inverse.
Definition. An affine morphism ¢ : AN ----+ AN is an automorphism if it has an
inverse morphism. In other words, ¢ is an affine automorphism if there is an affine
morphism ¢- 1 : AN ----+ AN such that
Somewhat surprisingly,¢ and ¢- 1 need not have the same degree, nor does deg( <Pn)
have to equal (deg ¢) n
0
Example 7.3. Consider the map ¢(x, y) = (x, y + x 2 ). It has degree 2 and is an
automorphism, since it has the inverse ¢- 1 ( x, y) = ( x, y - x 2 ). The composition ¢ 2
is
¢ 2 (x, y) = ¢(x, y + x 2 ) = (x, y + 2x 2 ),
so deg(¢ 2 ) = 2 = deg(¢). More generally, <Pn(x,y) = (x,y + nx 2 ) has degree 2,
so the degree of <Pn does not grow. This contrasts sharply with what happens for
morphisms of lP'N.
Example 7.4. Let a E K* and let f(y) E K[y] be a polynomial of degree d ~ 2.
The map
¢(x,y) = (y,ax+f(y)),
is called a Henan map. It is an automorphism of A 2 , since one easily checks that it
has an inverse ¢- 1 given by
7.1. Dynamics of Rational Maps on Projective Space
391
Henon maps, especially those with deg(f) = 2, have been extensively studied since
Henon [200] introduced them as examples of maps JR 2 ---. JR2 having strange attractors. There are many open questions regarding the real and complex dynamics
ofHenon maps; see, for example, [132, §2.9] or [211], as well as [212, 413] for a
compactification of the Henon map.
The rational maps 1?2 ---+ 1?2 induced by¢ and ¢- 1 are
where we write f (u, v) = ud f (v / u) for the homogenization of f. It is easy to see
that the loci of indeterminacy of¢ and ¢- 1 are
Z(¢) = {[0, 1,0]}
and
Z(¢- 1 ) = {[0,0, 1]}.
In particular, the locus of indeterminacy of¢ is disjoint from the locus of indeterminacy of¢ - 1 . Maps with this property are called regular; see Section 7.1.3.
Example 7.5. Consider the very simple Henon map
¢(x, y) = (y, -x + y 2 ).
The extension¢= [XJ, X0 X 2 , - X 0 X 1 +Xi] of¢ to 1?2 has degree 2, but it is not
a morphism, since it is not defined at the point [0, 1, 0]. And just as in Example 7.2,
there is no height estimate of the form h(¢(P)) = 2h(P) + 0(1) for¢. We can see
this by noting that
¢-( [b, a, b] ) = [b 2 , b2 , -ab + b2 ] = [b, b, -a + b],
so if a, b, E Z with gcd(a, b) = 1 and b > a > 0, then [b, a, b] and ¢([b, a, bl) have
the same height. Hence for every E > 0 even the weaker statement
h((/J(P)) ~ (1 + E)h(P) + 0(1)
for all P = (x, y) E A 2 (Q)
is false. It turns out that ¢ has only finitely many Q-rational periodic points (Theorem 7 .19), but the proof does not follow directly from a simple height argument.
Example 7.6. More generally, if¢: AN ---+AN is an affine automorphism, then it is
not possible to have simultaneous estimates of the form
h(¢(P)) ~ (1 + E)h(P) + 0(1),
h(¢- 1 (P)) ~ (1 + E)h(P) + 0(1),
(7.2)
for some E > 0 and all P E AN (K). To see this, suppose that (7.2) were true. Then
we would have for all P E AN (K),
7. Dynamics in Dimension Greater Than One
392
Thus h(P) would be bounded, leading to the untenable conclusion that _AN (K) is
finite, So it is too much to require that both cp(P) and ¢- 1 (P) have heights larger
than the height of P. However, as we shall see, it is often possible to show that some
combination of h( cp(P)) and h( ¢- 1 (P)) is large, which is then sufficient to prove
that Per(¢) is a set of bounded height.
We conclude this section with two useful geometric lemmas. The first relates the
locus of indeterminacy of an affine automorphism and its inverse, and the second
characterizes when the degree of a composition is smaller than the product of the
degrees.
Lemma 7. 7. Let ¢ : AN ----+ AN be an affine automorphism of degree at least 2 and
denote the hyperplane at infinity by H0 = {Xo = 0} = lP'N "-AN. Then
Proof Let
d
-
-
-
<P = (X0 ,Fl,F2, ... ,FN) and
<!>
-1
=
e
-
-
-
(X0 ,Gl,G2, ... ,GN)
be the lifts of¢ and {fy- 1 , respectively. The fact that¢ and ¢- 1 are inverses of one
another implies that there is a homogeneous polynomial f of degree de - 1 with the
property that
But the first coordinate of the composition is xge, so we see that f = xge-l. Thus
(<P- 1 o<P)(Xo, ... ,XN) = (Xge,xge-lXl,Xge-lXl,···,Xge-lXN),
or equivalently,
-
d
-
-
de-l
Gj(X0 ,F1, ... ,FN) = X 0
Xj
foralll::::; j::::; N.
(7.3)
Now let P = [0, x 1 , ... , xN] E H 0 " Z(¢), so (/J(P) = [0, P\ (P), ... , PN(P)]
with at least one Pi(P) =1- 0. From (7.3) we see that
Hence
so {fy- 1 is not defined at (/J(P). Therefore (/J(P) E Z( ¢- 1 ).
0
Lemma 7.8. Let¢ : _AN ----+ AN and 'ljJ : _AN ----+ AN be affine morphisms, and
let Ho = {Xo = 0} = lP'N "-_AN be the usual hyperplane at infinity. Then
deg('I/J o ¢) < deg('I/J) deg(¢)
if and only if {/Y(Ho "- Z(¢)) c Z('ljJ).
7.1. Dynamics of Rational Maps on Projective Space
393
Proof Let d = deg( ¢ ), let e = deg( '1/J ), and let <I> and \II be lifts of (/> and 1{;,
respectively. We write <I> explicitly as
The composition \II o <I> has the form
where E 1 , ... , EN are homogeneous polynomials of degree de. The degree of'¢ o ¢
will be strictly less than de if and only ifthere is some cancellation in the coordinate
this shows that
polynomials of \II o <I>. Since the first coordinate is
xge,
deg( '1/J o ¢) < deg( '1/J) deg( ¢)
{::::::::}
X 0 divides Ej for every 1 5: j 5: N.
Suppose now that X 0 1Ei for every j and let P = [0, x1, ... , XN] E H0 "Z(¢).
Since (/> is defined at P, some coordinate of
<I>(P) = (o, FHP), ... , PN(P))
is nonzero. On the other hand, the assumption that X 0 lEi implies that
(\llo <I>)(P) = (O,E1(<I>(P)),E2(<I>(P)), ... ,EN(<I>(P))) = (0,0,0, ... ,0).
Hence -If; is not defined at (/>(P), so (/>(P) E Z('¢). This completes the proof that
if deg ('1/J o ¢) < de, then (/> (H 0 " Z (¢)) c Z ('1/J).
For the other direction, suppose that (/>(Ho "Z(¢)) c Z('¢). This implies that
for (almost all) points of the form (0, x 1, ... , XN ), the map -If; is not defined at the
point ¢([0, x1, ... , xNl). Hence
so Ej(O, XI, x2, ... 'XN) = 0 for all j. Therefore XoiEj for all j.
D
Example 7.9. Let¢ be the map ¢(x, y) = (x, y+x 2) that we studied in Example 7.3.
Dehomogenizing ¢ yields
so the locus of indeterminacy for ¢ is Z (¢) = { [0, 0, 1] } . Notice that
(/>([O,X1,X2l) = [O,O,XrJ = [0,0, 1] E Z(¢).
Hence (fi(Ho "Z( ¢)) = Z( ¢),so Lemma 7.8 tells us that deg( ¢ 2 ) < deg( ¢ ) 2 . This
is in agreement with Example 7.3, where we computed that deg(¢ 2 ) = 2.
7. Dynamics in Dimension Greater Than One
394
7.1.3
The Geometry of Regular Automorphisms of A_N
In this section we briefly discuss the geometric properties of an important class of
affine automorphisms.
Definition. An affine automorphism ¢ : AN ----+ AN is said to be regular if the
indeterminacy loci of¢ and ¢- 1 have no points in common,
The following theorem summarizes some of the geometric properties enjoyed
by regular automorphisms of AN. We sketch the proof of (a) and refer the reader
to [401] for (b) and (c).
Theorem 7.10. Let¢: AN ----+AN be a regular affine automorphism.
(a) For all n ;::: 1,
cpn is regular,
and
(b) Let
d1 = deg¢,
d2 = deg¢-1,
£1 = dimZ(¢)+1,
£2 = dimZ(¢- 1)+1.
Then
£1+£2=N
and
d~' =d12 •
(c) For all n 2: 1 the set ofn-periodic points Pern( ¢)is a discrete subset ofAN (C).
Counted with appropriate multiplicities,
#PerNn(¢) = d~'Nn = df2Nn.
Proof (a) We first prove by induction on n that
This is trivally true for n = 1, so assume now that it is true for n- 1. Let P E Z (cpn),
so in particular P E H 0 . Suppose that P ~ Z(¢). The induction hypothesis tells us
that P ~ Z( cpn- 1 ), so applying Lemma 7.7 to the map ¢n- 1, we deduce that
(For the last equality we have again used the induction hypothesis.) On the other
hand, we have that ¢n- 1 is defined at P and ¢n is not defined at P, which implies
that ¢n- 1(P) E Z(¢). This proves that ¢n- 1(P) is in both Z(¢- 1 ) and Z(¢),
contradicting the assumption that ¢ is regular. Hence P E Z (¢), which completes
the proof that Z(¢n) c Z(¢). Similarly, we find that Z(¢-n) c Z(¢- 1).
Having shown that Z(¢n) c Z(¢) and Z(¢-n) C Z(¢- 1), we see that the
regularity of¢ implies that
7.1. Dynamics of Rational Maps on Projective Space
395
so qP is also regular.
Next suppose that deg(¢n) < deg(¢)n for some n ~ 2. We taken to be the
smallest value for which this is true, so in particular deg( ¢n- 1 ) = deg( ¢) n- 1 , and
hence
deg(¢n) < deg(¢n- 1 )deg(¢).
We apply Lemma 7.8 with '1/J = ¢n- 1 to conclude that
¢(Ho "Z(¢)) c Z(¢n- 1 ) c Z(¢),
where the last inclusion was proven earlier. On the other hand, Lemma 7. 7 says that
¢(H0 "Z(¢)) c Z(¢- 1 ). Hence
¢(Ho "Z(¢)) c Z(¢) n Z(¢- 1 ) = 0.
This is a contradiction, which completes the proof that deg(¢n) = deg(¢)n.
It remains to show that Z (¢) C Z (<Pn). Let
be a lift of¢, so
Z(¢) = {P E Ho: F1(P) = · · · = FN(P) = 0}.
By a slight abuse of notation, we say that P E Z(¢) if and only if<I>(P) = 0. (To be
precise, we should lift P to AN+ 1 .)
We proved that deg(¢n) = deg(¢), which implies that the coordinate functions
of<J>n have no common factor. Thus <Pn can be computed by evaluating <I>n and mapping down to JP>N. Hence just as above we have P E Z (<Pn) if and only if <J>n (P) = 0.
Therefore
This proves that Z(¢) c Z(¢n) and completes the proofof(a).
(b) See [401, Proposition 2.3.2].
(c) See [401, Theorem 2.3.4].
0
Remark 7 .11. If ¢ : A 2 --. A 2 is a regular automorphism of the affine plane, then
Theorem 7.10(b) tells us that £1 = £2 = 1 (which is clear anyway since the indeterminacy locus of a rational map has codimension at least 2) and that d 1 = d2 • Thus
planar regular automorphisms satisfy deg( ¢) = deg( ¢- 1 ). In the opposite direction,
if d 1 = d2 , then Theorem 7.10(b) says that £1 = £2 , and hence that N = £1 + £2
is even. In other words, a regular automorphism ¢ : AN --. AN with N odd always
satisfies deg( ¢) ¥: deg( ¢- 1 ).
7. Dynamics in Dimension Greater Than One
396
Example 7.12. Let¢: A3 ~ A3 be given by
One can check that the inverse of ¢ is
Homogenizing x = Xd X 0 , y = X 2 / X 0 , z = X 3 / Xo, we have the formulas
from which it is easy to check that
Z(¢) = {Xo = X2 = X3 = O} = {[o, 1, o, o]},
Z(¢- 1) = {Xo = X1 = 0} = {[O,O,u,v]}.
Thus Z (¢) consists of a single point, while Z (¢- 1 ) is a line. In the notation of
Theorem 7.10, we haveN= 3 and
d1 = deg¢ = 2,
d2 = deg¢- 1 = 4,
£1 = dimZ(¢) + 1 = 1,
£2 = dimZ(¢- 1 ) = 1 = 2.
The map¢ is regular, since Z(¢) n Z(¢- 1 ) = 0.
Remark 7 .13. Let ¢ : AN ~ AN be an affine morphism and let <P : AN +1 ~ AN +1
be a lift of¢. The map ¢ is called algebraically stable if
<Pn({Xo = 0}) of. {0}
foralln;::: 1.
In other words, ¢ is algebraically stable if for every n ;::: 1, some coordinate
of <Pn(Xo, ... , XN) is not divisible by X 0 . Since the first coordinate of q,n is a
power of X 0 , this implies that there can be no cancellation among the coordinates,
so an algebraically stable map ¢ satisfies
Further, an adaptation of the proof of Theorem 7.1 0(a) shows that
Regular automorphisms are algebraically stable, but there are algebraically stable
automorphisms that are not regular. For a discussion of the complex dynamics of
algebraically stable maps, see [174, 187, 401].
7.1. Dynamics of Rational Maps on Projective Space
397
Remark 7 .14. For arbitrary rational maps rjJ : lP'N --> lP'N, the dynamical degree of rjJ
is defined to be the quantity
and its logarithm log dyndeg( rjJ) is called the algebraic entropy of rp. (One can show
that the dynamical degree is in fact the infimum of deg(cpn)l/n.) The dynamical
degree provides a coarse measure of the stable complexity of the map cp, and presumably it has a major impact on the arithmetic properties of rp. See [10, 199, 290]
for an indication of this effect in certain cases.
The dynamical degree need not be an integer, or even a rational number; see
Exercise 7.4 for an example. However, Bellon and Viallet [49] have conjectured that
it is always an algebraic integer.
The dynamical degree, and more generally the sequence of integers
dn = deg(cpn),
n = 0, 1, 2, ... ,
can be quite difficult to describe. See [10, 45, 46, 49, 77, 133, 199] for work on
this problem. In many cases the sequence (dn )n20 satisfies a linear recurrence with
rational coefficients, or equivalently, the generating function Ln>o dnTn is in Q(T).
However, see [46] for an example of a birational map rjJ : lP'N __:: lP'N whose degree
generating function is not in Q(T).
7.1.4
A Height Bound for Jointly Regular Affine Morphisms
In this section we prove a nontrivial lower bound for the height of points under
regular affine automorphisms. The theorem is an amalgamation of results due to
Denis [131], Kawaguchi [230, 231], Marcello [287, 288, 289, 290], and Silverman [413, 418]. Before stating the theorem, we need to define what is meant by
the height of a point in affine space.
Definition. The height h( P) of a point P = ( x 1 , ... , x N) E AN (Q) in affine space
is defined to be the height of the associated point in projective space using the natural
embedding AN --> lP'N,
Eventually we will apply the following height estimate to a regular affine automorphism rjJ and its inverse rjJ -l, but it is no harder to prove the result for any pair
of jointly regular maps, and working in a general setting helps clarify the underlying
structure of the proof.
Theorem 7.15. Let rp 1 :AN --> A_N and r/J2 : A_N -->AN be affine morphisms with
the property that
(We say that rp 1 and rp 2 are jointly regular.) Let
398
7. Dynamics in Dimension Greater Than One
There is a constant C = C(¢1, <h) such that for all P E AN (Q),
1
1
d 1 h(¢1(P)) + d 2 h(¢2(P)) ?: h(P)- C.
(7.4)
Remark 7.16. We recall that the upper bound
h('¢(P)) ::; (deg'ljJ)h(P) + 0(1)
(7.5)
is valid even for rational maps '¢ : J!DN ---t J!DN (see Theorem 3.11 ), since the proof
of (7.5) uses only the triangle inequality. Thus Theorem 7.15 may be viewed as
providing a nontrivial lower bound complementary to the elementary upper bound
ProojQjTheorem 7.15. Write the rational functions J!DN
and ¢2 as
---t
J!DN induced by ¢ 1
and
where the Pi are homogeneous polynomials of degree d1 and the Gi are homogeneous polynomials of degree d 2 . The loci of indeterminacy of ¢ 1 and ¢ 2 are given
by
Z(¢1) = {Xo = F1 = .. · = PN = 0},
Z(¢2) = {Xo = G1 = .. · = GN = 0}.
We define a rational map '¢ : J!D2N ---t J!D2N of degree d1d2 by
The locus of indeterminacy of'¢ is the set
since by assumption Z(¢1) and Z(¢2) are disjoint. Hence'¢ is a morphism, so we
can apply the fundamental height estimate for morphisms (Theorem 3.11) to deduce
that
for all P E JID 2 N (Q).
(7.6)
The following lemma will give us an upper bound for the height of 1/J(P).
Lemma 7.17. Let u, a1, ... , aN, b1, ... , bN E Qwith u i= 0. Then
7.1. Dynamics of Rational Maps on Projective Space
399
Proof Let ai = ad u and f3i = bd u for 1 :::; i :::; N. Then for any absolute value v
we have the trivial estimate
max{ 1, la1lv, ···,iaN lv, lfJ1Iv, · · ·, lf3N lv}
:S max{ 1, la1lv, · · ·, laNiv} ·max{ 1, lfJ1Iv, · · ·, lf3Niv }.
Raising to an appropriate power, multiplying over all absolute values, and taking
logarithms yields
This is the desired result, since the height does not depend on the choice of homogeD
neous coordinates of a point.
We apply Lemma 7.17 to the point
'1/J(P) = [Xo(P)dld2' F\ (P)d2' ... 'PN(P)d2' Gl (P)dl' ... 'G N(P)dl]
with P E AN (Ql), which ensures that X 0 (P) i- 0. The lemma tells us that
h('1/J(P)) :::; h ( [X0(P)d 1 d2, P1(P)d 2, ... , PN(P)d 2])
+ h ([Xo(P)dld2' Gl (P)dl' ... 'G N(P)dl])
= d2h ( [Xo(P)d 1 , F1(P), ... , PN(P)])
+ d1h ([Xo(P)d2, G1(P), ... , GN(P)])
= d2h( ¢I(P)) + d1h( ¢2(P)).
We combine this with (7.6) to obtain
Dividing both sides by d 1 d2 completes the proof of Theorem 7.15.
D
For regular affine automorphisms, it is conjectured that the height inequality (7 .4)
in Theorem 7.15 may be replaced by a stronger estimate.
Conjecture 7.18. Let¢: AN ----+AN be a regular affine automorphism. Then there
is a constant C = C( ¢) such that for all P E AN (QJ),
(7.7)
Kawaguchi [230] proves Conjecture 7.18 in dimension 2, i.e., for regular affine
automorphisms ¢ : A 2 ----+ A 2 ; see also [413]. However, for general jointly regular
affine morphisms, it is easy to see that (7.4) cannot be improved; see Exercise 7.8.
Kawaguchi also constructs canonical heights for maps that satisfy (7.7); see [230]
and Exercises 7.17-7.22.
400
7.1.5
7. Dynamics in Dimension Greater Than One
Roundedness of Periodic Points for Regular
Automorphisms of AN
Theorem 7.15 applied to a regular affine automorphism ¢ and its inverse implies that
at least one of ¢(P) and ¢- 1 (P) has reasonably large height. This suffices to prove
that the periodic points of¢ form a set of bounded height, a result first demonstrated
by Marcello [287, 288] (see also [ 131, 418]) using a height bound slightly weaker
than the one in Theorem 7.15.
Theorem 7.19. (Marcello) Let¢: AN ---*AN be a regular affine automorphism of
degree at least 2 defined over Q. Then Per(¢) is a set of bounded height in AN (Q).
In particular,
Per(¢) n AN (K) is finite for all number fields K.
Proof Let
and
Applying Theorem 7.15 with ¢ 1 = ¢ and ¢ 2 = ¢- 1 yields the basic inequality
(7.8)
where C is a constant depending on ¢, but not on P E AN (Q).
We prove the theorem initially under the assumption that d 1 d 2 > 4. Define a
function
)
1 ( -1
1
(7.9)
f(P) = d1 h(P)- ad2 h ¢ (P) - a- 1'
c
where the real number a > 1 will be specified later. Then f satisfies
c)
1
1
h(P)-h(¢(P))- -d
f(¢(P))- af(P) = ( -d
a-1
a2
1
-a (.!_h(P)- - 1 h(¢- 1 (P))- ____2___)
a- 1
ad2
d1
= (:1 h(¢(P)) + : 2 h(cp- 1
2':
(1- ~- -1-) h(P)
d1
(P)))- (~ + a~J h(P) + C
ad2
from (7.8).
Hence if we take
then
1
a
1----=0
d1
ad2
'
and our assumption that d1 d2 > 4 ensures that a > 1, so for this choice of a we
conclude that
401
7.1. Dynamics of Rational Maps on Projective Space
f(¢(P)) 2: o:f(P)
Applying this estimate to the points P, ¢( P), ¢ 2 ( P), 000, ¢n-l (P), we obtain the
fundamental inequality
for all P E AN (Q) and all n 2: 00
(7.1 0)
Similarly, we define
c
1
1
g(P) = -h(P)- -h(¢(P))-(3- 1
(Jdl
d2
and take
(3 = dld2 + J(dld2) 2 - 4dld2
2dl
(7011)
0
Then an analogous calculation, which we leave to the reader, shows that g satisfies
for all P E AN (Q),
from which we deduce that
for all P E AN (Q) and all n 2: 00
(7012)
We compute
0:-n J(cpn+l(P)) + (3-ng(¢-n-l(P))
2: f(¢(P)) + g(¢- 1 (P))
=
from (7.10) and (7012),
(11 h(¢(P))- o:~2 h(P)- o: ~ 1)
+ (2_h(¢- 1 (P))- - 1 h(P)- ____!!___)
>
-
(1- -1-- -1-)
o:d2
(3- 1
from the definition (7 09) and (7 oll) off and g,
(Jdl
d2
(Jdl
h(P)-
(1 + - 1- + - 1-)
(}; - 1
(3 - 1
C
from(7°8)o
Using the definition of f and g and rearranging the terms, we have proven the inequality
(7013)
Now suppose that P E AN(Q) is a periodic point for ¢0 Then h(¢k(P)) is
bounded independently of k, so letting n----+ oo in (7013) yields
7. Dynamics in Dimension Greater Than One
402
(a(3 -1)C
> ( 1 - _1 _ _
1 ) h(P)
(a- 1)((3- 1) -
ad2
(3d1
'
where we are using the fact that a > 1 and (3 > 1. Our assumption that d 1 d2 > 4
also ensures that
1- _1_- _1_- J1- 4 > 0
ad2 (3d1 d1 d2
'
so the height of P is bounded by a constant depending only on ¢. This completes
the pr<i>of of the first assertion of Theorem 7.19 under the assumption that d1 d2 > 4,
and the second is immediate from Theorem 7.29(±), which says that for any given
number field, pN (K) contains only finitely many points of bounded height.
In order to deal with the case d1d2 ::; 4, i.e., d1 = d2 = 2, we use Theorem 7.1 0,
which tells us that 1JJ2 is regular and has degree di. Similarly, deg( ¢- 2) = d~. Hence
from what we have already proven, the periodic points of ¢ 2 form a set of bounded
height, and since it is easy to see that Per(¢) = Per( ¢ 2 ), this completes the proof in
all cases.
D
Remark 7.20. We observe that Theorem 7.19 applies only to regular maps. It cannot
be true for all affine automorphisms, since there are affine automorphisms whose
fixed (or periodic) points include components of positive dimension. For example,
the affine automorphism ¢(x, y) = (x, y + f(x)) fixes all points of the form (a, b)
satisfying f(a) = 0. Of course, this map¢ is not regular, since one easily checks
that
Z(¢) = Z(¢- 1 ) = {[0, 1, OJ}.
Definition. Let¢ : V ---t V be a morphism of a (not necessarily projective) variety V. A point P E Per (¢) is isolated if P is not in the closure of Per n ( ¢) " {P}
for all n ~ 0. In particular, if Per n ( ¢) is finite for all n, then every periodic point is
isolated.
Conjecture 7.21. Let¢: A.N ---t A_N be an affine automorphism ofdegree at least 2
defined over Q. Then the set ofisolated periodic points of¢ is a set ofbounded height
in A.N (Q).
A classification theorem of Friedland and Milnor [176] says that every automorphism ¢ : A. 2 ---t A.2 of the affine plane is conjugate to a composition of elementary
maps and Henon maps. Using this classification, Denis [131] proved Conjecture 7.21
in dimension 2. (See also [287, 288].)
7.2
Primer on Algebraic Geometry
In this section we summarize basic material from algebraic geometry, primarily having to do with the theory of divisors, linear equivalence, and the divisor class group
(Picard group). This theory is used to describe the geometry of algebraic varieties
and the geometry of the maps between them. We assume that the reader is familiar
403
7.2. Primer on Algebraic Geometry
with basic material on algebraic varieties as may be found in any standard textbook,
such as [186, 197, 198, 205].
This section deals with geometry, so we work over an algebraically closed field.
Let
K = an algebraically closed field,
V = a nonsingular irreducible projective variety defined over K,
K(V) =the field of rational functions on V.
7.2.1
Divisors, Linear Equivalence, and the Picard Group
In this section we recall the theory of divisors, linear equivalence, and the divisor
class group (Picard group).
Definition. A prime divisor on V is an irreducible subvariety W C V of codimension 1. The divisor group of V, denoted by Div(V), is the free abelian group
generated by the prime divisors on V. Thus Div(V) consists of all formal sums
l.::nwW,
w
where the sum is over prime divisors W c V, the coefficients n w are integers, and
only finitely many nw are nonzero. The support of a divisor D = 2.::: nw W is
IDI =
u w.
Wwith
nw#O
If W is a prime divisor of V, then the local ring at W is the ring
Ov,w = {f E K(V) : f is defined at some point ofW}.
It is a discrete valuation ring whose fraction field is K (V). Normalizing the valuation
so that ord w ( K (V) *) = Z, we say that
ordw(f) =order of vanishing off along W.
Then f vanishes on W if ordw (f) :::: 1, and f has a pole on W if ordw (f) :::; -1.
Definition. Let f E K(V)* be a nonzero rational function on V. The divisor off is
the divisor
(f)=
2.::: ordw(f)W E Div(V).
w
A principal divisor is a divisor of the form (f) for some f E K (V). The principal
divisors form a subgroup ofDiv(V). The divisor class group (or Picard group) of V
is the quotient group
Div(V)
p· (V) _
- (principal divisors)
lC
404
7. Dynamics in Dimension Greater Than One
Two divisors D 1 , D2 E Div(V) are linearly equivalent if they differ by a principal
divisor, D 1 = D 2 +(f), i.e., if their difference is in the kernel of the natural map
Div(V) -----+ Pic(V).
We write D 1 "' D2 to denote linear equivalence.
The next proposition follows directly from the definitions and the fact that every
nonconstant function on a projective variety V has nontrivial zeros and poles.
Proposition 7.22. There is an exact sequence
1 -----+ K* -----+ K(V)* ~ Div(V) -----+ Pic(V) -----+ 0.
Remark 7.23. The exact sequence in Proposition 7.22 is analogous to the fundamental exact sequence in algebraic number theory,
. ) -----+ (multiplicative) -----+ (fractional)
(ideal class) -----+ 1.
1 -----+ (umts
.
-----+
group
Ideals
group
Definition. Let ¢ : V ---+ V' be a morphism of nonsingular projective varieties
and let W' c V' be a prime divisor such that ¢(V) is not contained in W'.
Then ¢- 1 (W') breaks up into a disjoint union of prime divisors, say
Letf E K(V')beauniformizeratW',i.e.,ordw'(f) = 1. ThenthepullbackofW'
by ¢ is defined to be the divisor
T
¢*W' = L:ordw,(fo¢)Wi E Div(V).
i=l
More generally, if D' = L nw' W' E Div(V'), the pullback of D' is the divisor
¢* D' =
L nw,¢*(W'),
W'
provided that all of the terms with nw' =1- 0 are well-defined. Thus¢* D' is defined
ifandonlyif¢(V) r.f_ ID'I·
There is also a way to push divisors forward.
Definition. Let ¢ : V ---+ V' be a morphism of nonsingular projective varieties,
let W c V be a prime divisor, and let W' = ¢(W). If dim W' =dim W, then
the function field K(W) is a finite extension of the function field K(W') via the
inclusion
¢* : K(W') '--------> K(W),
¢*(f)=fo¢,
and we define the pushforward of W by ¢ to be the divisor
405
7.2. Primer on Algebraic Geometry
¢*W = [K(W): K(W')]W' E Div(V').
If dim W' < dim W, we define¢* W = 0. And in general, for an arbitrary divisor
D = I: nw W E Div(V), the pushforward of D is
¢*D = 2:nw¢*(W).
w
Example 7.24. If¢ : V ---+ V' is a finite map, then
¢*¢* D' = deg( ¢ )D'
for all D' E Div(V').
Proposition 7.25. Let¢ : V ---+ V' be a morphism ofnonsingular projective vari-
eties.
(a) Every D' E Div(V') is linearly equivalent to a divisor D" E Div(V') satisfYing ¢(V) C/. ID" 1(b) If D' and D" are linearly equivalent divisors on V' such that ¢* D' and¢* D"
are both defined, then ¢* D' and¢* D" are linearly equivalent.
(c) Using (a) and (b), the map
¢*: {D' E Div(V'): ¢(V) C/.ID'I} ---7 Div(V)
extends uniquely to a homomorphism
¢* : Pic(V') ---7 Pic(V).
Example 7 .26. A prime divisor W of IP'N is the zero set of an irreducible homogeneous polynomial FE K[X 0 , ... , XN]. We define the degree ofW to be the degree
of the polynomial F and extend this to obtain a homomorphism
deg(2:nww) = 2:nwdeg(W).
w
w
It is not hard to see that a divisor on IP'N is principal if and only if it has degree 0, so
the degree map gives an isomorphism
deg : Pic(IP'N)
Z.
Any hyperplane H c IP'N is a generator ofPic(IP'N).
Example 7.27. A prime divisor of IP'N x IP'M is the zero set of an irreducible bihomogeneous polynomial FE K[Xo, ... , XN, Yo, ... , YM]· We say that F and W have
bidegree (d, e) ifF satisfies
F(o:Xo, ... , o:XN, ,8Yo, ... , ,8YM) =ad ,Be F(Xo, ... , XN, Yo, ... , YM ).
The bidegree map can be extended linearly to give an isomorphism
bideg : Pic(IP'N x IP'M)
Z x Z.
Let p 1 : IP'N x IP'M ---+ IP'N and p 2 : IP'N x IP'M ---+ IP'M be the two projections and
let H 1 be a hyperplane in IP'N and H 2 a hyperplane in IP'M. Then Pic(IP'N x IP'M) is
generated by the divisors
and
406
7. Dynamics in Dimension Greater Than One
7.2.2
Ample Divisors and Effective Divisors
L::
Definition. A divisor D =
nw W is said to be effective (or positive) if nw 2:: 0
for all W. We write D 2:: 0 to indicate that Dis effective.
The base locus of a divisor D, denoted by Base( D), is the intersection of the
support of all of the effective divisors in the divisor class of D,
Base(D) =
n lEI.
E~D
E?_O
Notice that any divisor is a difference of effective divisors,
D=
2:: nwW- 2:: (-nw)W.
W with
nw>O
W with
nw<O
Definition. Let D E Div(V). Associated to Dis the finite-dimensional K-vector
space
L(D) = {f E K(V): (f)+ D 2:: 0} U {0}.
We write C(D) =dim L(D) for the dimension of L(D).
Let D E Div(V) be a divisor with C(D) 2:: 1. We choose a basis h, ... , fc(D)
for L(D) and use it to define a rational map
cPD = [fi, ·. ·, fc(D)]: V--+ pl(D)- 1 .
The map ¢D is well-defined up to a linear change of coordinates on JP'e(D)- 1 , i.e.,
up to composition by an element ofPGLt(D)(K). Further, if D and D' are linearly
equivalent, then ¢ D and ¢ D' differ by a change of coordinates.
Conversely, let i : V '--+ lP'N be a morphism (or even a rational map) and let
H c lP'N be a hyperplane with i(V) rf_ H. Then i is equal to the composition of ¢i• H
with a change of coordinates and a projection.
Definition. A divisor D E Div(V) is very ample if the map ¢D : V--> JP'e(D)- 1 is an
embedding, i.e., an isomorphism onto its image. A divisor D is ample if some multiple nD with n :?: 1 is very ample. Ampleness and very ampleness are properties of
the divisor class of D. Notice that if i : V <--+ lP'N is an embedding and H E Div(lP'N)
is a hyperplane, then¢* His a very ample divisor on V.
Example 7.28. Let pr H 1 and pzH2 be the generators of Pic(lP'N x lP'M) described
in Example 7.27. Then prH1 + pzH2 is a very ample divisor on lP'N x lP'M. The
associated embedding is called the Segre embedding. It is given explicitly by the
formula
JP'N X JP'M
--+
JP'NM+N+M
([Xo, ... , XN], [Yo, ... , YMl) f--...* [XoYo, XoY1, ... , Xi}j, ... , XN YM]·
Now let V be a subvariety of lP'N x JIDM, say ¢ : V '--+ lP'N x lP'M. Then
¢*(pi H1 + p;H2) = (P1 o ¢)* H1 + (P2 o ¢)* H2
is a very ample divisor on V.
7.3. The Weil Height Machine
7.3
407
The Weil Height Machine
The theory of height functions that we developed in Sections 3.1-3.5 provides a
powerful tool for studying the arithmetic of morphisms ¢ : JP>N ___, JP>N of projective
space. For example, if¢ has degree d ;::: 2 and is defined over a number field K, then
the fundamental estimate (Theorem 3 .11)
h(¢(P)) = d · h(P) + 0(1)
for all P E JP>N (Q)
(7.14)
and the fact (Theorem 3.7) that there are only finitely many points in JP>N (K) of
bounded height lead immediately to a proof of Northcott's theorem (Theorem 3.12)
stating that ¢ has only finitely many K -rational preperiodic points.
Recall that the height h( P) of a point P E JP>N (Q) is a measure of the arithmetic
complexity of P. Similarly, the degree of a finite morphism¢ measures the geometric
complexity of¢. Thus an enlightening interpretation of(7.14) is that it translates the
geometric statement"¢ has degree d" into the arithmetic statement "h(¢(P)) is
approximately equal to dh(P)."
A natural way to define a height function on an arbitrary projective variety is to
fix an embedding¢: V '---' JP>N and define hv(P) to equal h(¢(P)). Unfortunately,
different projective embeddings yield different height functions. But letting H denote
a hyperplane in JP>N, one can show that if the divisors ¢* H and 'lj;* H are linearly
equivalent, then the height functions attached to ¢ : V ,___. ]p>N and 1j; : V '---' JP>M
differ by a bounded amount.
More intrinsically, the projective embedding¢ determines the divisor class of the
very ample divisor¢* H. This suggests assigning a height function to every divisor
on V. The Weil height machine provides such a construction. It is a powerful tool that
translates geometric facts described by divisor class relations into arithmetic facts
described by height relations. As such, the Weil height machine is of fundamental
importance in the study of arithmetic geometry and arithmetic dynamics on algebraic
varieties of dimension greater than 1.
Theorem 7.29. (Weil Height Machine) For every nonsingular variety VjQ there
exists a map
hv: Div(V)-+ {functions V(Q) ___, JR},
D f--.> hv,v,
with the following properties:
(a) (Normalization) Let H c JP>N be a hyperplane and let h : JP>N (Q) ___, lR be the
absolute logarithmic height jUnction on projective space defined in Section 3.1.
Then
hrN,H(P) = h(P) + 0(1)
for all P E JP>N (Q).
(b) (Functoriality) Let¢ : V ___, V' be a morphism of nonsingular varieties defined
over Q and let D E Div(V'). Then
hv,q,•v(P) = hv,,v(¢(P)) + 0(1)
for all P E V(Q).
7. Dynamics in Dimension Greater Than One
408
(c) (Additivity) Let D, E E Div(V). Then
hv,D+E(P) = hv,D(P) + hv,E(P) + 0(1)
for all P E V(Q).
(d) (Linear Equivalence) Let D, E E Div(V) with D linearly equivalent to E. Then
hv,D(P) = hv,E(P) + 0(1)
for all P E V(Q).
(e) (Positivity) Let D E Div(V) be an effective divisor. Then
hv,D(P) 2: 0(1)
for all P E V(Q) "Base( D).
That is, hv,D is bounded below for all points not in the base locus of D.
(f) (Finiteness) Let D E Div(V) be ample. Then for all constants A and B, the set
{ P E V(Q): [Q(P): Q]::::; A and hv,D(P)::::; B}
is finite. In particular, if V is defined over a number field K and if L / K is a
finite extension, then
{ P E V(L): hv,D(P)::::; B}
is a finite set.
(g) (Uniqueness) The height functions hv,D are determined, up to 0(1), by the
properties of (a) normalization, (b) functoriality, and (c) additivity. (It suffices
to assume functoriality for projective embeddings V '-+ lP'N.)
Proof See [76, Chapter 2], [205, Theorem B.3.2], or [256, Chapter 4].
D
Remark 7.30. All of the 0(1) constants appearing in the Weil height machine (Theorem 7.29) depend on the various varieties, divisors, and morphisms. The key fact
is that the 0 (1) constants are independent of the points on the varieties. More precisely, Theorem 7.29 says that it is possible to choose functions hv,D, one for each
smooth projective variety V and each divisor D E Div(V), such that certain properties hold, where those properties involve constants that depend on the particular
choice of functions hv,D· In principle, one can write down particular functions hv,D
and determine specific values for the associated 0 (1) constants, so the Weil height
machine is effective. In practice, the constants often depend on making the Nullstellensatz effective, so they tend to be rather large.
Remark 7 .31. Many of the properties of the Weil height machine may be succinctly
summarized by the statement that there is a unique homomorphism
{functions V(K) --+ JR}
h v : P 1.c (V) ----) -:------"----,---.:........,'-c-:-=-:-_:__-=-:
:{bounded functions V (K) --+ lR}
such that if¢ : V '-+ lP'N is a projective embedding, then hv,¢• H = h + 0(1 ).
7.3. The Weil Height Machine
409
Example 7.32. Let¢: IP'N ___., IP'N be a morphism of degree d and let H E Div(IP'N)
be a hyperplane. Then¢* H"' dH, so Theorem 7.29 allows us to compute
hwN,H(¢(P)) = hwN,q,•H(P)+0(1) = hwN,dH(P)+0(1) = dhwN,H(P)+0(1).
This formula is Theorem 3.11.
Example 7.33. Let V be a subvariety of IP'N x IP'M, say ¢ : V '---' IP'N x IP'M.
Continuing with the notation from Examples 7.27 and 7.28, the height of a point
P = [x, y] E V with respect to the divisors ¢*pr H 1 and ¢*p2H2 is given by
hv,q,·p~ H 1 (P) = hwN ,H1 (Pl c/J(P)) = h(x),
hv,q,•p~H2 (P) = hwM,H2(P2¢(P)) = h(y).
Example 7.34. This example uses properties of elliptic curves; see Sections 1.6.3
and 6.3. Let E be an elliptic curve given by a Weierstrass equation. Then the x-
coordinate onE, considered as a map x : E ___., IP' 1 , satisfies
x*(oo) = 2(0),
so we have
1
he,(o)(P) = 2hwl,(oo)(x(P)) + 0(1).
Note that the height hw1 ,( 00 ) is just the usual height on IP' 1 from Theorem 7.29(a).
Now let d ~ 2, let [d] : E ___., E denote the multiplication-by-d map, and let
E[d] ={PEE: [d]P = 0}.
The map [d] is unramified and
¢*((0)) =
2:= (T) E Div(E).
TEE[d]
The group E [d] is isomorphic as an abstract group to Z I dZ x ZI dZ, so the sum of the
points in E[d] is O.It follows from Proposition 6.18 that there is a linear equivalence
of divisors
¢*((0)) "'d 2 (0).
Hence we can apply Theorem 7.29 to compute
he,(o) ([d]P) = he,[d]*(O) (P) + 0(1)
from functoriality (b),
= he,d2(0)(P) + 0(1)
linear equivalence property (d),
d2he,(o)(P) + 0(1)
from additivity (c).
=
Theorem 3.20 then tells us that there exists a function he,( a) onE satisfying
and
he,(o)(P) = he,(o)(P) + 0(1).
The function he,( a) is called the canonical height on E. It has many applications,
ranging from counting rational points to evaluating L-series. For further information about canonical heights on elliptic curves and abelian varieties, see, for example [205, B.5], [256, Chapter 5], or [410, VIII §9].
410
7. Dynamics in Dimension Greater Than One
Remark 7.35. The reader should be aware that the theory of heights is often
rephrased in the language of metrized line bundles, which offers greater flexibility,
albeit at the cost of additional work to set up the general theory.
7.4
Dynamics on Surfaces with Noncommuting
Involutions
An involution ~ of a variety V is a rational map ~ : V - t V with the property that ~ 2
is the identity map on V. If we look at the quotient variety
W = Vf{~(P) = P},
then the natural projection p : V - t W is a double cover , and the effect of ~ on V
is to switch the two sheets of the cover. Conversely, any double cover p : V - t W
induces an involution ~ : V - t V.
The dynamics of a single involution is not very exciting, but some varieties have
two (or more) noncommuting involutions ~ 1 and ~ 2 whose composition ¢ = ~ 1 o ~ 2 is
an automorphism of V of infinite order. The dynamics of such maps ¢ can be quite
interesting. In this section we study in detail an example of this type. The material in
this section is taken from [409].
7.4.1
K3 Surfaces in IP2 x IP2
We consider a surface S contained in JP>2 x JP>2 defined by two bihomogeneous equations, one ofbidegree (1, 1) and the other ofbidegree (2, 2). Thus
s = {(x,y) E JP>2 X JP>2 : L(x,y) = Q(x,y) = o}
for bihomogeneous polynomials
2
2
L(x,y) = LLAijXiYj,
i=O j=O
Q(x,y) =
(7.15)
L
The surface S is determined by the coefficients
A= [Aoo, A01, ... , A22] E JP>8
and B = [Boooo, Booob ... , B2222] E IP35
of the polynomials Land Q. To indicate this dependence, we write SA,B·
There are two natural projections from S to JP>2, which we denote by
P1(x,y) = x,
P2(x,y) = y.
These projections are maps of degree 2. To see this, choose a generic point a E JP>2.
Then
7.4. Dynamics on Surfaces with Noncommuting Involutions
411
P1 1 (a) = {(a,y) E IP' 2 X IP' 2 : L(a,y) = Q(a,y) = 0}
consists of two points (counted with multiplicity), since it is the intersection of the
line L( a, y) = 0 and the conic Q( a, y) = 0 in lP' 2 . And similarly, p 2 is a map of
degree 2.
In general, a degree-2 map between varieties induces an involution on the domain given by switching the two sheets of the cover. In our situation the maps p 1
and p 2 induce involutions t1 and t2 on SA,B· Explicitly, if P = [a, b] E SA,B,
then t 1 (P) = [a, b'] is the point satisfying
and similarly, t 2 (P) = [a', b] is the point satisfying
These involutions are uniquely determined as nonidentity maps SA,B ---+ SA,B satisfying
and
P2 o t2 = P2·
We note that t 1 and t 2 are rational maps on S, i.e., they are given by rational
functions. To see why this is true, observe that b and b' are the intersection points
in JP'2 of the line and the conic
L(a,y)=O
and
Q(a,y) = 0.
Thus each of b and b' can be expressed as a rational function in the coordinates of
the other. The following example will help make this clear, or see Exercise 7.25 for
explicit formulas to compute t 1 and t 2 .
Example 7.36. We illustrate the involutions on SA,B using the example
L(x,y) = XoYo + X1Y1 + X2Y2,
Q(x, y) = x6y6 + 4x6YoYl - x6yi + 7x6YlY2 + 3xoXlY6 + 3xoXlYOYl
+ XoXlY~ + xiy6 + 2xiyi + 4xiYlY2 - XoX2Yi
+ 5xoX2YOY2 - 4X1X2YI - 4X1X2YoY2 - 2x~YoYl + 3x~y~.
The point P = ([1, 0, 0], [0, 7, 1]) is in S(Q). In order to compute t 1 (P), we substitute the value x = [1, 0, 0] into Land Q and solve for y. Thus
£([1, 0, 0], y) =Yo= 0 and Q([1, 0, 0], y) = Y6 + 4YoYl - Yi + 7YlY2 = 0,
so the solutions are y = [0, y1 , Y2], where y 1 and y 2 are the roots of the polynomial -yr + 7y 1y2 = 0. One solution is y1 = 7, which gives the original point P,
and the other solution is y 1 = 0, which gives t 1 (P) = ([1, 0, 0], [0, 0, 1]).
Next we compute t 2 (P). To do this, we substitute y = [0, 7, 1] into Land Q to
obtain
7. Dynamics in Dimension Greater Than One
412
L(x, [0, 7, 1]) = 7x1 + x2 = 0,
Q(x, [0, 7, 1]) = xox1- 49xax2 + 126xi- 196xlx2 + 3x§ = 0.
Substituting x 2 = -7 x 1 into the second equation gives
Q([xo,xl, -7xl], [0, 7, 1]) = 344xox1 + 1645xi = 0.
The solution x 1 = 0 gives back the original point P. The other solution is [x 0 , x 1 ] =
[1645, -344], and then setting x 2 = -7x 1 = 2408 gives
L2(P) = L2 ([1, 0, 0], [0, 7, 1]) = ([1645, -344, 2408], [0, 7, 1]).
We could continue this process, but the size of the coordinates grows very rapidly.
Indeed, they-coordinates of L1 (L 2 (P)) are already integers with 12 to 13 digits.
Remark 7.37. The surfaceS described by (7.15) is an example of a K3 surface.
Formally, a K3 surface is a surface S of Kodaira dimension 0 with the property
that H 1 (S, Os) = 0. However, all of the information that we will need is contained
in the explicit equations (7.15) defining S. The reader desiring more information
about the geometric properties of K3 surfaces might consult [40, 44, 178, 298). The
dynamics of K3 surfaces with nontrivial automorphisms are studied by Cantat [93]
and McMullen [296).
Remark 7.38. The collection ofK3 surfaces S A,B is a 43-parameter family, since the
coefficients (A, B) vary over JID8 x JID 35 . However, many of the surfaces are isomorphic. For example, we can use elements ofPGL3 to change variables in each of two
factors ofJID2. This reduces the dimension of the parameter space by 16, since PGL 3
has dimension 8. Further, the surface SA,B really depends only on the ideal generated by the bilinear form L(x, y) and the biquadratic form Q(x, y ), so the surface
does not change if we replace Q(x, y) by
Q(x, y) + L(x, y) · M(x, y)
for an arbitrary bilinear form M(x, y). The space of such M is 9-dimensional, so
we see that the isomorphism classes of K3 surfaces SA,B constitute a family of
dimension at most
8
+ 35 -
AEIP' 8
BEJP' 35
"'-v-"
"'-v-"
8
"'-v-"
PGL3
-
8
"'-v-"
PGL3
-
9
"'-v-"
= 18.
M
One can prove that these are the only isomorphisms between the various SA,B,
so there is an 18-parameter family of isomorphism classes of nonsingular surfaces SA,B·
Definition. There are several linear, quadratic, and quartic forms that come up naturally when one is working with the surface SA,B· We define linear and quadratic
forms by setting
7.4. Dynamics on Surfaces with Noncommuting Involutions
413
Lj(x) =the coefficient ofyj in L(x, y),
Lf (y) = the coefficient of Xi in L(x, y),
Q%t(x) =the coefficient of YkY£ in Q(x, y ),
Qfj (y) = the coefficient of Xi Xj in Q(x, y).
(7.16)
This notation allows us to write the bilinear form L and the biquadratic form Q as
2
2
L(x,y) = LLj(x)yj = LLf(y)xi,
j=O
i=O
Then for each triple of distinct indices i, j, k E {0, 1, 2} we define quartic forms
G% = (Lj) 2 Qfi- Lf LjQfj + (Lf) 2 Qjj,
G% = (L~) 2 Qfi- Lf L~Qfj + (Lf) 2 Q~J'
Hij = 2Lf LjQ%k- Lf L%Qjk- LjL%Qfk + (L%) 2 Qfj,
(7.17)
Ht = 2Lf L~Q%k- Lf L%Q~k- L~L%Qfk + (L%) 2 Qfj·
For some choices of A and B, there may be points on the surface SA,B at
which ~ 1 or ~ 2 is not well-defined. The next proposition, which provides a criterion for checking whether ~ 1 and ~ 2 are defined at a point, shows how the quartic
forms (7 .17) naturally appear.
Proposition 7.39. Let P =[a, b] E SA,B·
(a) The involution ~1 is defined at P unless
G~(a) = Gf(a) = G2(a) = H 01 (a) = H 02 (a) =
Hf2(a) = o.
(b) The involution ~2 is defined at P unless
Gg(b) = Gf(b) = G~(b) = H81 (b) = H82 (b) = Hf2 (b) = 0.
Proof By symmetry, it is enough to prove (a). The map ~ 1 is defined at P = [a, b]
if and only if the fiber p1 1 (a) consists of exactly two points. That fiber is the set of
points [a, y] satisfying
L(a,y) = Q(a,y) = 0,
so as long as these two polynomials are not zero, the y values are given by the
intersection of a line and a conic in 1P'2 . If a line and a conic intersect properly, then
they intersect in exactly two points, counted with multiplicity. Further, given one
solution y = b, the coordinates of the second solution b' are rational functions of b
and the coefficients of L (a, y) and Q (a, y). Hence ~ 1 is a morphism 1 except in the
following two situations:
1We leave for the reader to check that everything works in a neighborhood of points where the
line L(a, y) = 0 is tangent to the conic Q(a, y) = 0.
7. Dynamics in Dimension Greater Than One
414
• L(a, y) is identically 0.
• L(a, y) = 0 is a line that is contained in the set where Q(a, y) = 0.
With the notation defined by (7 .16) and (7 .1 7), we use the bihomogeneity of Q to
write
and then we eliminate the variable y0 by substituting L5Yo = L - Lfyl - L2Y2·
After some algebra, we obtain an identity of the form
(Lo) 2Q = G2y~ + Hf2Y1Y2 + Gfy~
+ L{ QooL + (LoQol- 2LfQoo)Yl + (LoQo2- 2L2Qoo)Y2}·
Since we will be interested in studying points [x, y] satisfying L(x, y) = 0, we write
this identity, and the analogous ones obtained by eliminating y 1 or y2 , as congruences
in the polynomial ring Z[Aij, Bijk£, xi, Y1l· Thus
L 0(x) 2Q(x,y) = G2(x)y~ + Hf2(x)Y1Y2 + Gf(x)y~
Lf(x) 2Q(x,y) = G2(x)y5 + Ht2(x)YoY2 + G0 (x)y~
(mod L(x,y)), (7.18)
L2(x) 2Q(x, y) = Gf(x)y5 + Ht1(x)YoYl + G0 (x)y~
(mod L(x, y)). (7.20)
(mod L(x, y)), (7.19)
Suppose first that L(a, y) is identically 0. Substituting x =a into (7.18), we find
that the quadratic form
is identically 0. Hence G2(a) = Hf2(a) = Gl(a) = 0. Similarly, substituting x =a into (7.19) and (7.20) shows that all of the other values G%(a) and Hij(a)
are equal to 0, which completes the proof in this case.
We may now suppose that L(a, y) is not identically 0. Then the assumption
that ~ 1 is not defined at [a, b] implies that the line L(a, y) = 0 is contained in
the zero set ofQ(a, y).
If Lf(a) = L2(a) = 0, then the definition (7.17) ofG0 shows that G 0(a) = 0.
And if Ll(a) and L2(a) are both nonzero, then we let
b' = [o, L2(a), -Lf(a)]
and note that [a, b'] E Sa,b· Hence
A similar argument shows that also Gl(a) = G2(a) = 0.
Next we evaluate (7 .18), (7 .19), and (7 .20) at x = a and use the fact that we now
know that G5(a) = Gl(a) = G2(a) = 0. This yields
7.4. Dynamics on Surfaces with Noncommuting Involutions
415
Hf2(a)YlY2 = H~2(a)YoY2 = H~l(a)YoYl = 0
for ally= [yo, Yl, Y2J satisfying L(a, y) = 0. (7.21)
We will prove that Hf2 (a) = 0; the others are done similarly. If there is a point on the
line L(a,y) = 0 with Y1Y2 -1 0, then (7.21) immediately implies that Hf2(a) = 0.
So we are reduced to the cases in which the line L (a, y) = 0 is either y1 = 0
or Y2 = 0. If it is the line Y1 = 0, then L( a, y) = cy1 for some constant c -1 0,
so L0(a) = Uj(a) = 0, and similarly if it is the line y2 = 0, then L0(a) = L'f(a) =
0. In either case, the definition (7 .17) of Hf2 yields
Hf2 (a) = 2L~(a)L~(a)Q~0 (a)
- Lf(a)L~(a)Q~0 (a)- L~(a)L~(a)Q~0 (a) + L~(a) 2 Q~2 (a) = 0. D
Example 7.40. We illustrate Proposition 7.39 using the surface described in Example 7.36. The polynomials G'k and HiJ for this example are given in Table 7.1. Proposition 7.39 says that ~ 1 is defined at P = [a, b] provided that at least one of the six
polynomials G0, Gf, G~, Hg1 , Hg2 , Hf2 does not vanish at a. For convenience we
say that a point a E JID 2 is degenerate if
Our first observation is that
and
Hence there are no degenerate points with a0 = 0. (We assume that K does not have
characteristic 2.) Thus if there exists a degenerate point a, we can dehomogenize a
and write it as a = [1, a 1 , a 2 ]. We use a tilde to indicate the dehomogenization x 0 =
1 of the G'k and Ht1 polynomials. So for example,
Now suppose that ii = (a 1,a 2) is a degenerate point. Then x 1 = a 1 is a common
root of the polynomials
Hence if we take resultants with respect to the x 2 variable, then x 1 = a 1 is a root of
both of the polynomials
R1 (x2) = Resx 1 ( G~(x1, x2), H~1 (xl, x2)),
R2(x2) = Resx 1 (G~(x1,x2),H~1 (xl,x2)).
Explicitly, these polynomials are
416
7. Dynamics in Dimension Greater Than One
2
G Ox = XoX 31 - 7XoX1X2
- 4X31X2 - x 02x 22 + 5X12X22 - XoX 32 - 4 X1X 32
Gf = x~x1- x6x~ + 7xox1x~ + xix~
G~ = -x6- 4x~x1 + 3xoxi + xf- x~x2- 4x6x1X2 + 2xox1x~
Hg1 = 2x6xi- 7x~x2- 4xoxix2 + 4x6x~ + 4xox1x~ + 4xix~- 2x~
H 02 = -7x~x1- 4xoxi- 2x~x2- 4x6x1x2 + 6xoxix2- 4xix2
- 2x6x~- 8xox1x~ + 2x1x~
Hf2 = 7x6 + 4x6xi- 4x~x2- 6x6x1x2 + 10xoxix2 + 2xix2 + 2xox~
G5 = -2YoYi + 4YiY2 + Y5Y~ + 4YoY1Y~ + 5yry~ + 4Y1Y~
Gi = -2Y5Y1 + YoYiY2 - Y5Y~ + 4YoY1Y~ - YiY~ + 7Y1Y~
G~ = Y6 - 3ygy1 + 4yoyi - yf + 4Y5Y1Y2 + 7YiY2 - YoY1Y~
H(n = -4y5yr + 4YoYiY2 + YiY2 + 7y5y~ + 4YoY1Y~ + Yi
H1{2 = 4yoyi- yf + 2Y5Y2 + Y6Y1Y2 + 6YoYiY2 + 8yoY1Y~- Y1Y~
H"/2 = -4y5yr + YoYi- 7y~y2 - 6y6Y1Y2 + 8yoYrY2- 2yry2 + 14yry~- YoY~
Table 7.1: The polynomials G'k and H;j for the surface in Example 7.36.
R1 (xi) = 4xF + 112xP + 1160x~ 0 + 5112xi + 7052x~ - 2271xi + 18573x~
+ 216oxr + 16053xf- 7304xi + 1045xi- 49x1,
R 2( xl) = 4x~ 6 + 80x~ 5 + 600x~ 4 + 2064xP + 2548x~ 2 - 3616x~ 1 - 14216x~ 0
- 10892x{ + 9856x~ + 21708xi + 15648x~ + 1000xf- 13986xf
- 10462xi - 3124xi - 412x 1 - 18,
and the assumption that R 1 and R 2 have a common root implies that their resultant
must vanish. However, when we compute it, we find that
Res(R1,R2) = 198929 ... 3830147072 ~ 1.99 ·10 87 .
Hence ~1 is defined at every point of SA,B unless the characteristic of K divides this
(large) nonzero integer Res(R1, R2).
We can use other resultants to reduce the list of possible bad characteristics.
For example, let Ro(x2) = Resx 1 (G 0, H01 ). Then ~1 is everywhere defined unless
both Res(Ro, R2) and Res(R1, R2) vanish. We compute
gcd(Res(Ro, R2), Res(R1, R2)) = 439853213743020234882809856
= 2 17 . 3 6 . 317. 14521485737273461,
(7.22)
7.4. Dynamics on Surfaces with Noncommuting Involutions
417
which proves that 1, 1 is everywhere defined unless pis one of the four primes appearing in (7 .22).
We sketch a similar calculation for 1,2 • Let
To(Y2) = Resy 1 ( G6 (Yl, Y2), H61 (Yl, Y2)),
T1(y2) = Resy 1 (Gf(y1, Y2), H61(y1, Y2)),
T2(y2) = Resy 1 ( G~(y1, Y2), H61(y1, Y2)).
Then Res(T0 , T2) ~ 2.57 · 1097 and Res(T1 , T2) ~ 2. 75 · 10114 , and
gcd(Res(To, T2), Res(T1 , T2 )) = 2 16 · 507593 · 2895545793631.
Hence ~, 2 is everywhere defined unless the characteristic p of K is one of the three
primes appearing in this factorization.
A map p : V ---> W of degree 2 between varieties always induces an involution 1, : V ---> V, but in general 1, is only a rational map, it need not be a morphism.
This distinction is quite important. For example, height functions transform well for
morphisms, but not for rational maps. We now show that for most choices of (A, B),
the involutions on S A,B are morphisms.
Proposition 7.41. There is a proper Zariski closed set Z C lP'8 x lP'35 such that
if(A, B) ~ Z, then the involutions
are morphisms.
Proof According to Proposition 7.39, the involution 1, 1 is defined on all of SA,B
provided that the system of equations
G()(x) = Gf(x) = G~(x) = H~1 (x) = H 02 (x) = Hf2(x) = 0
(7.23)
has no solutions in lP' 2 • A general result from elimination theory (see [198, 1.5.7A])
says that there are polynomials h, ... , fr in the coefficients of
such that the equations (7.23) have a solution if and only if h = · · · = fr = 0.
The coefficients of G0, ... , Hf2 are themselves polynomials in the coefficients
of L(x, y) and Q(x, y), so we can write each fi as a polynomial in the variables A
and B. There are then two possibilities:
• The set of fi consists only of the zero polynomial, and hence every surface S A,B has some point at which 1, 1 is not defined.
• The set of fi contains at least one nonzero polynomial f1, and then 1, 1 is welldefined on S A,B provided that h (A, B) =/= 0.
418
7. Dynamics in Dimension Greater Than One
In order to eliminate the first case, it suffices to write down a single surface S A,B
for which the system of equations has no solutions in IP'2 . We gave such a surface in
Example 7.40, at least provided that the characteristic of K is not equal to 2, 3, 317,
or 14521485737273461. Similarly, the above argument and Example 7.40 show
that i 2 is defined on a Zariski open set of (A, B) as long as the characteristic of K
is not equal to 2, 507593, or 2895545793631. This completes the proof of Proposition 7.41 except for fields having one of these six characteristics. We leave as an
exercise for the reader to find other examples to cover the remaining cases.
D
Remark 7.42. See Exercise 7.28 for a less computational proof of Proposition 7.41.
7.4.2
Divisors and Involutions on SA B
'
In this section we study how the involutions i 1 and i2 act on divisors on SA,B· Later
we use this information to study how iterates of the involutions act on points. This
prompts the following definitions.
Definition. Let i 1 and i2 be the involutions of the surface SA,B defined by (7.15).
These involutions generate a subgroup (possibly all) of Aut(SA,a). We denote this
subgroup by A. Then for any point P E SA,B, the A-orbit of Pis the set
A(P) = {1/l(P): ¢ E A}.
Let H E Div(IP'2 ) be a line. As described in Example 7.28, pulling back using
the two projections gives divisors on IP'2 x IP'2 ,
and
and the Picard group of IP'2 x IP'2 is isomorphic to '!} via
(By abuse of notation, we write n 1 H 1 + n 2 H 2 for its divisor class.) We note
that n 1 H 1 + n 2 H 2 is a very ample divisor on IP'2 x IP'2 if and only ifboth n 1 and n 2
are positive.
Next we define two divisors D1, D 2 E Div(SA,B) using the two projections
of S A,B to IP' 2 ,
and
Wehler [448] has proven that the Picard group of a general surface SA,B satisfies
Pic(SA,B) ~ Z2 and that D 1 and D 2 are generators, 2 but for our purposes it will
not matter if Pic( S A,B) is larger than Z 2 , we will simply use the part of Pic( S A,B)
generated by D 1 and D2.
We now compute the action of i 1 and i 2 on D 1 and D 2 .
2What this means is that the set of coefficients (A, B) E JI>8 x JI>35 for which Pic(SA,B) is strictly
larger than 7L. 2 forms a countable union of proper Zariski closed subsets of JI>8 x JI> 35 . This implies that
most (A, B) in JI>B(IC) x JI>35 (1C) have Picard group 7L. 2 , but it does not directly imply that there are any
such values in JI>8 ( Ql) x JI>35 ( Ql), since IQl is countable.
7.4. Dynamics on Surfaces with Noncommuting Involutions
419
Proposition 7.43. Let D1 = Pi H and D2 = p2H. The involutions i1 and i2 act
on the subspace ofPic(SA,B) generated by D1 and D2 according to the following
rules:
i'2D1 = -D1 +4D2,
i'2D2 = D2.
i;'D1 = D1,
iiD2 = 4Dl- D2,
(7.24)
(7.25)
Proof The involution il switches the sheets of the projection p 1, so it is clear
that P1 o i1 = Pl· This allows us to compute
This proves the first formula in (7.24).
Next we observe that for any P E SA,B, the two points in the set p1 1 (p1 ( P))
are P and i 1 (P). Thus if we start with a divisor on SA,B, use p 1 to push it down
to JID 2 , and then use p 1 to pull it back to SA,B, we get back the original divisor plus
its translation by i 1 . In other words,
PiPhD = D + iiD for all DE Pic(SA,a).
Using this formula with D = D 2 allows us to compute
(7.26)
The divisor p1 *p2H on JID2 is linearly equivalent to some multiple of H. For simplicity, let H be the line y 2 = 0. Then p2H is the curve in JID2 x JID2 (lying on SA,B)
given by the equations
Y2 = 0.
We solve the linear equation L = 0 to express y0 and y 1 as linear functions
of [x0, x 1, x 2], and then substituting into the quadratic equation Q = 0 yields a
homogeneous equation of degree 4 in xo, x1, x2. So when we use Ph to push p2H
down to JID2 , we get a curve of degree 4 in JID2 . Hence
PhP2H =4H,
where this is an equality in Pic(JID2 ). Substituting into (7.26) yields the second formula in (7.24).
By symmetry, or by repeating the above argument, the two formulas in (7 .25) are
also true.
D
Remark 7.44. It is not hard to prove that the only relations satisfied by compositions
of il and i2 are i~ = 1 and i~ = 1. In other words, A is isomorphic to the free
product of the groups of order 2 generated by i 1 and i 2 . An alternative description is
that A is isomorphic to an infinite dihedral group. See Exercise 7 .31.
420
7. Dynamics in Dimension Greater Than One
7.4.3
Height Functions on S A,B
In this section we use the Weil height machine to translate the divisor relations and
transformation formulas from Proposition 7.43 into relations among height functions. We recall from Example 7.33 that the height functions associated to the divisors D 1 and D 2 are given at a point P = [x, y] E SA,B by
hn 1 (P) = hp~ H(P) = hH(PlP) = h(x),
hn 2 (P) = hp2H(P) = hH(P2P) = h(y).
(7.27)
Proposition 7.45. Assume that SA,B is defined over a number field K. Let
a=2+v'3
and define functions h+, h- : SA,a(K) - t IR by the formulas
h+ ([x, yl) = -h(x) + ah(y)
and
h-([x,yl) = ah(x)- h(y).
Then h+ and h- transform according to the following rules:
h+ o t 1 = ah- + 0(1),
h- o t 1 = a- 1 h+ + 0(1),
h+ o t2 = a- 1 h- + 0(1),
h- o t2 = ah+ + 0(1).
Remark 7.46. Before starting the proof of Proposition 7.45, we pause to explain
why the number a and the functions h+ and h- arise naturally. Consider a twodimensional real vector space V with basis elements D 1 and D 2 , where we view V
as a subspace of Pic(SA,B) Q9 JR. Then the formulas (7.24) and (7.25) in Proposition 7.43 tell us how ti and t2 act on V. In terms of the given basis, they are linear
transformations that act via the matrices
*=
Ll
(1 4)
0 -1
and
*
t2=
(-1 0)
41.
We now look for a new basis {E 1 , E 2 } for V with the property that ti and t2 interchange the basis. More precisely, we ask that
for some constants a, b, c, d. This problem can be solved directly, but it is easier
to observe that t2ti E1 = adE1 and t2ti E2 = bcE2. Thus E1 and E 2 must be
eigenvectors for the linear transformation t2ti, whose matrix is
* *
L2 L1
4 ) (-1
15 -1
4 ) .
= (10 -1
4 0)
1 = ( -4
It is easy to check that
and
7.4. Dynamics on Surfaces with Noncommuting Involutions
421
are a pair of independent eigenvectors with eigenvalues a 2 and a- 2 respectively.
This explains the appearance of a, and then one checks that these eigenvectors satisfy
t *E
1 2 =a -
1E
t *E
2 1 =a -
1,
1E
2,
(7.28)
It is then natural to define height functions h+ and h- corresponding to the divisors E 1 = -D 1 + aD 2 and E 2 = aD 1 - D 2 , since the divisor relations (7.28)
and the Weil height machine should then yield corresponding relations for the height
functions.
Proof ofProposition 7 .45. Having given the motivation, we commence the proof of
Proposition 7.45, which is a formal calculation using the additivity and functoriality of height functions (Theorem 7.29(b,c)) and the transformation formulas (7.24)
and (7.25) in Proposition 7.43. Note that
from (7.27). We compute
h+ o t 1 = -hD 1 o t1 + ahD 2 o t1
by definition of h +,
+ ah,~D 2 + 0(1)
= -hD + ah4D -D + 0(1)
= (-1 + 4a)hD ahD + 0(1)
= a 2 hD 1 - ahD + 0(1)
= ah- + 0(1).
= -h,~D 1
1
2
1
1 -
2
2
from Theorem 7.29(b),
from Proposition 7.43,
from Theorem 7.29(c),
since a 2 = 4a- 1,
Similarly
h+ o t 2 = -hD 1 o t 2 + ahD 2 o t 2
by definition of h-,
= -h,2D 1 + ah,2D 2
from Theorem 7 .29(b),
+ ahD
= hD + (-4 + a)hD
= hD
a- 1 hD + 0(1)
= a- 1 h- + 0(1).
= -h-D 1 +4D 2
1
1 -
2
2
from Proposition 7.43,
2
from Theorem 7.29(c),
since a 2 = 4a- 1,
This proves the transformation formulas for h+. The proof for h- is similar and is
left for the reader.
0
We can use Proposition 7.45 and the general theory of canonical heights (Theorem 3.20) to construct two heights on SA,B that are canonical with respect to both t 1
and t2.
Proposition 7.47. Let SA,B be defined over a number field K. There exist unique
JUnctions
422
7. Dynamics in Dimension Greater Than One
and
satisfying both the normalization conditions
and
h,- = ahD 1 - hD 2 + 0(1)
(7.29)
and the canonical transformation formulas
ft+o~ 1 =ah-,
h,- o ~1 = a-1h,+,
h+ o ~2 = a - 1 h-,
h,-o~2=ah+.
(7.30)
Proof Let¢>= ~1 o ~2 be the composition of the two involutions on SA,B and let
and
be the functions defined in Proposition 7.45. Then the transformation formulas in
Proposition 7.45 allow us to compute
The constant a 2 satisfies a 2 ;::::j 13.93 > 1, so we may apply Theorem 3.20 to the
functions ¢> and h + to deduce the existence of a unique function h+ satisfying
and
h,+ = h+ + 0(1).
Repeating this construction with ¢>- 1 = ~ 2 o ~ 1 , we find that
h- o ¢>- 1 = h- o ~ 2 o ~ 1 + 0(1) = ah+ o ~ 1 + 0(1) = a 2 h- + 0(1).
Applying Theorem 3.20 to the functions ¢>- 1 and h-, we find that there is a function h- satisfying
and
h,- = h- + 0(1).
The functions h,+ and h,- that we have just constructed satisfy (7.29). In order to
check the transformation formulas (7.30), we first note that
In order to get rid of the 0(1 ), we compose both sides with ¢>-n and use the formula
to compute
anft+o~1 = h,+o¢>no~1 = h,+o~1 o¢>-n = ah- o¢>-n+0(1) = an+lh,- +0(1).
Divide both sides by an and let n ____, oo to obtain the desired result h + o ~ 1 = ah-.
This proves the first of the transformation formulas (7.30). The others are proven
similarly.
7.4. Dynamics on Surfaces with Noncommuting Involutions
423
Finally, in order to prove uniqueness, suppose that g+ and g- are functions satisfying (7.29) and (7.30). Then
and similarly g- o cjJ- 1 = a 2 g-. Hence g+ and g- have the same canonical properties as 'h+ and 'h-, so the uniqueness assertion in Theorem 3.20 tells us that g+ = 'h+
0
and§-= 'h-.
Remark 7.48. In practice, it is infeasible to compute the canonical heights h+ and h-
to more than a few decimal places using their definition as a limit. As with the other
canonical heights studied in Sections 3.4, 3.5, and 5.9, it is possible to decompose 'h+
and h- as sums oflocal heights that may then be computed using rapidly convergent
series (cf. Exercise 5.29). See [89] for details.
7.4.4
Properties and Applications of Canonical Heights
The next proposition describes various useful properties of the canonical height
functions h+ and h- and their sum. As an application, we prove that there are
only finitely many K -rational points with finite A-orbit. This is the analogue for
the K3 surfaces SA,B of Northcott's Theorem 3.12 on preperiodic points ofmorphisms on lP'N and of Theorem 7.19 on periodic points of regular affine automorphisms.
Proposition 7.49. Let SA,B be defined over a number field K, let 'h+ and 'h- be the
canonical height functions constructed in Proposition 7.47, and let
(a) The set
{ P E SA,B(K): h(P) S C}
is finite. (N.B. This is not true if we replace h by either ofthe heights h+ and h-,
see Exercise 7.35.)
(b) Let P E SA,B(K). Then
'h+(P) = 0 -<====> 'h-(P) = 0 -<====> h(P) = 0 -<====> P has finite A-orbit.
(c) There are only finitely many points P E S A,B ( K) with finite A-orbit.
Proof (a) Using the properties of 'h+ and 'h-, we find that
h = h+ + h= ( -hD 1 + ahD 2 ) + (ahD 1 - hD 2 ) + 0(1)
=(a- 1)(hD1 + hD 2 ) + 0(1).
As noted earlier, the heights hD 1 and hD 2 are given by
by definition of h,
from Proposition 7.47,
424
7. Dynamics in Dimension Greater Than One
hv, ([x, yl) = h(x)
and
hv 2 ([x,yl) = h(y),
where h( x) and h(y) are the standard heights of x and y in lP'2 • Hence
h([x,yl) =(a -1)(h(x) + h(y)) + 0(1),
so if h([x, yl) is bounded, then both h(x) and h(y) are bounded. (Note that a satisfies a > 1, which is crucial for the argument to work.) This completes the proof
of (a), since Theorem 3.7 tells us that lP'2 (K) contains only finitely many points of
bounded height.
(b) Since h = h+ + h- and both h+ and h- are nonnegative, it is clear that
Suppose next that h+(P) = 0. Let¢= ~ 1 o ~ 2 as usual. Then
h(¢n(P)) = h+(¢n(P)) + h-(¢n(P))
= a 2nh+(P) + a- 2nh-(P)
= a- 2 nh-(P).
The righthand side is bounded (indeed, it goes to 0) as n ----+ oo, so we see
that { ¢n(P) : n 2 0} is a set of bounded h-height. It follows from (a) that it is
a finite set. Since ¢ is an automorphism, we deduce that P is periodic for¢. We now
perform a similar calculation using ¢ -n,
h(rp-n(P)) = h+(r_p-n(P)) + h- (rp-n(P))
= a- 2 nh+(P) + a 2 nh-(P)
=
a 2 nh-(P).
The lefthand side is bounded, since P is periodic for ¢, so letting n ----+ oo implies
that h-(P) = 0.
This proves that h+ (P) = 0 implies h- (P) = 0, and a similar argument gives
the reverse implication, which completes the proof that
In order to study A-orbits of points, we make further use of the formula
Suppose first that P has finite A-orbit. Then h(¢n (P)) is bounded, since it takes on
only finitely many values. Letting n ----+ oo in (7 .31) and using the fact that a > 1,
we deduce that h+(P) = 0.
Finally, suppose that h(P) = 0. Then h+(P) = h-(P) = 0, so (7.31) tells
us that h( ¢n(P)) = 0 for all n E Z. In particular, { ¢n(P) : n E Z} is a set of
bounded h-height, so (a) tells us that it is a finite set. But the A-orbit of Pis equal to
7.4. Dynamics on Surfaces with Noncommuting Involutions
425
so A(P) is also finite.
D
The canonical height functions on S A,B can also be used to count the number of
points of bounded height in an A-orbit, as in our next result. See Exercise 7.21 for
an analogous (conditional) estimate for regular affine automorphisms of pN.
Proposition 7.50. Let SA,B be defined over a numberfield K, andfor any point P =
[x,y] E SA,s(K), leth(P) betheheightfunction
h(P) = hv,+D 2 (P) = h(x) + h(y).
Also let a = 2 + V3 as usual. Fix a point Q E S A,B ( K) with infinite A-orbit and
let
J-L(Q) = #{7/J E A: 7/J(Q) = Q}
be the order of the stabilizer ofQ. Then
#{ P E A(Q): h(P)::; B} =
J-L
1 loga ( ,
B2,
) +0(1)
(Q)
h+(Q)h-(Q)
as B ____, oo,
where the 0 ( 1) constant is independent of both B and Q.
The key to proving Proposition 7.50 is the following elementary counting lemma.
Lemma 7.51. Let a, b > 0 and u > 1 be real numbers. Then
as t ____, oo,
where the 0(1) constant depends only on u.
Proof We start by writing the real number logu ( y1JfQ,) as the sum of an integer
and a fractional part,
logu
If= +
m
r
with mE Z and lrl ::;
~·
(The reason that we do this is because the function aux + bu-x has a minimum
at x = logu ( y1JfQ,) .) Then replacing n by n + m in the expression aun +bu-n
yields
Hence
It thus suffices to prove that if c, dE lR are both between u- 1 12 and u 1 12 , then
426
7. Dynamics in Dimension Greater Than One
#{ n E Z: cun +du-n::; t} = log(t 2 ) + 0(1)
as t--> oo.
(7.32)
We note that if n ~ 0, then
and similarly if n ::; 0, then
Here the 0(1) bounds depend only on u, since by assumption c and dare bounded
in terms of u. Therefore
#{n E Z: cun +du-n::; t} = #{n E Z: logu(cun +du-n)::; logu(t)}
= #{n E Z: lnl + 0(1)::; logu(t)}
= 2logu(t) + 0(1).
This is the desired inequality (7.32), which completes the proof of Lemma 7.51.
0
Proof ofProposition 7.50. We do the case that f.l(Q) = 1 and leave the similar
case 11( Q) = 2 to the reader. (It is easy to check that 11( Q) ::; 2; see Exercise 7.29.)
Let¢ = t 1 o t 2 . Every element of A is given uniquely as an alternating composition of t 1 's and t 2 's, so A splits up as a disjoint union
Our assumption that 11( Q) = 1 then implies that the A-orbit of Q is a disjoint union
(7.33)
Let h,+ and h,- be the canonical height functions constructed in Proposition 7.47
and let h = h,+ + h,-. We note that Proposition 7.47 tells us that h,+ o ¢ = ah+
and h- o ¢ = a - l h-. This allows us to compute
#{P E Oq,(Q): h(P)::; B}
= #{ n E Z: h(¢nQ)::; B}
= #{ n E Z: h,+(¢nQ) + h,-(¢nQ)::; B}
= #{ n E Z: a 2 nh+(Q) + a- 2 nh-(Q)::; B}
from Proposition 7.47,
~ ~log" ( h+ (Q~~~ (Q)) + 0(1)
from Lemma 7.51.
since f.l(Q) = 1,
definition of h,
Further, if we replace Q with t 1 ( Q), then we get exactly the same estimate, since
Hence using the decomposition (7.33), we find that
427
Exercises
#{P E A(Q): h(P):::; B}
= #{ P E 0¢(Q): h(P):::; B} + #{ P E 0¢(ilQ): h(P):::; B}
~ log h+(Q~~-(Q)) + 0(1).
0
(
Finally, in order to replace the canonical height h with the naive height h, we note
that
h = h,+ + h,- = (ahn 1 - hn 2 ) + (-hn 1 + ahn 2 ) + 0(1)
=(a- 1)(hn1 + hn 2 ) + 0(1) = (a- 1)h + 0(1).
Thus
#{P E A(Q): h(P):::; B} = #{P E A(Q): h(P):::; (a -1)B + 0(1)},
and replacing B with (a- 1)B affects only the 0(1), since a> 1.
D
Exercises
Section 7.1. Dynamics of Rational Maps on Projective Space
7.1. Let a, b, c, d, e E C and let</>: A 3 --> A 3 be given by
<f>(x, y, z) = (ax+ by 2 + (cx 2 + dz) 2 , ey +(ax+ by 2 ) 2 , dz + cx 2 ).
(7.34)
(a) Prove that</> is invertible if and only if ade i- 0.
(b) Prove that </> is a regular automorphism if and only if abcde i- 0.
(c) Clearly (0, 0, 0) is a fixed point of</>. Let b = -1, d = 1, and e = 1 - t 3 . Prove
that (0, t, t) is also a fixed point of¢. Hence there are infinitely many maps</> E IQl[x, y, z]
of the form (7.34) such that Fix(</>) n A 3 (1Ql) contains at least two points.
(d) Let b = -1 and d = 1. Find all of the (complex) fixed points of¢. If a, b, c, d, e E K,
describe the field K (Fix(¢)). What are its possible Galois groups over K? (Hint. It is
easier to do the computations if you set e = 1 - t 3 .)
(e) Suppose that</> E JR(x, y, z] and that d = 1 and b > 0. Prove that</> has only one real
fixed point, i.e., show that Fix(</>) n A3 (JR) = {(0, 0, 0)}. In particular,</> has only one
rational fixed point.
7.2. Let¢: A3 --> A3 be the map
<f>(x,y,z) = (x 2 z,xy,yz).
(a) Calculate the indeterminacy locus of¢.
(b) What are the values of
.. f
I1mm
(x,y,z)EZ3
h(x,y,z)-oo
h(<f>(x,y,z))
h(x, y, z)
and
.
hmsup
(x,y,z)EZ3
h(x,y,z)-oo
h(</>(x,y,z))
h(x, y, z)
?
Exercises
428
(c) Same question as (b), but with the points (x, y, z) E Z3 restricted to satisfy xyz #- 0.
7.3. Let¢: AN--+ AN be a regular affine automorphism and let n 2: 1. Prove that Pern(¢)
is a discrete subset of AN (C), and that counted with appropriate multiplicities,
(This is Theorem 7.10(c). Hint. Rewrite rj>Nn(P) =Pas ¢£ 2 n(P) = ¢-(N-£ 2 )n(P), show
that the homogenizations of ¢£ 2 n and ¢CN -£ 2 )n have the same degree, and use Bezout's theorem to count the number of solutions.)
7.4. Let¢: A 3 --+ A 3 be the map ¢(x, y, z) = (y, z, x 2 ).
(a) Find an explicit expression for rj>n(x, y, z). (There may be more than one case.)
(b) Calculate the dynamical degree of¢,
dyndeg(¢) = lim deg(¢n)l/n.
n~oo
(See Remark 7.14 for a discussion of the dynamical degree.)
(c) Let dn = deg(¢n). Compute the generating function I:n>o dnTn and prove that it is
in Q(T).
(d) Prove that PrePer(¢) c {P E A 3 (Q): h(P) = 0}.
(e) Let 8 = dyndeg(¢) and P E A3 (Q). Find real numbers b >a> 0 such that
for all (sufficiently large) integers n.
(f) With notation as in (d), if P ~PreFer(¢), prove that
1
#{n2:0:h(¢n(P))~T}
.
= --.
hm
log 8
log T
T~oo
7.5. Let¢ : lP'3 --+ lP'3 be the rational map
(a) Prove that¢ is a birational map, i.e., find a rational map 'ljJ so that¢ o 'ljJ and 'ljJ o ¢are
the identity map at all points where they are defined.
(b) Compute Z (¢) and Z (¢ -l). Where do they intersect?
(c) *Let dn = deg(¢n). Prove that the generating function I:n>o dnTn is not in Q(T).
The map in this exercise and the map in the previous exercise are examples of monomial maps,
see [199].
7.6. Let u, a 1 , ... , aN, b1 , ... , bN E Q with u #- 0. We proved in Lemma 7.17 that
Prove that this inequality need not be true if u = 0.
Exercises
429
7.7. This exercise generalizes Theorem 7.15. Let(/;!, ... , rPt: AN--> AN be affine automorphisms with the property that
Let di = deg( rPi) for 1 ~ i ~ t. Prove that there is a constant C = C (¢1, ... , rPt) so that for
all P E AN (Q),
1
1
1
d 1 h(¢1(P)) + d 2 h(¢2(P)) + · · · + dt h(r/Yt(P)) ~ h(P)- C.
7.8. Let ¢1, ¢2 : A 2 --> A 2 be the maps
and
(a) Prove that ¢1 and ¢2 are jointly regular.
(b) Let a a positive integer and P = (0, a) E A 2(Q). Prove that
This proves that the lower bound in Theorem 7.15 cannot be improved in general for
jointly regular affine morphisms.
(c) ** Can the lower bound in Theorem 7.15 be improved for jointly regular affine automorphisms, i.e., if we add the requirement that ¢1 and ¢2 be invertible, although not
necessarily inverses of one another?
7.9. Let ¢ : AN --> AN be an affine automorphism (not necessarily regular) and let d 1
deg ¢ and d2 = deg ¢- 1. Prove that
Exercises on Integrability and Reversibility
The notions of integrability and reversibility play an important role in classical real and complex dynamics. Their algebraic analogues lead to dynamical systems with interesting arithmetic properties, which we explore in Exercises 7.10-7.14.
Definition. An affine automorphism ¢ : AN --> AN is said to be algebraically reversible if
there is linear transformation g E GLN satisfying
det(g) = -1,
and
The terminology is meant to reflect the idea that conjugation by the involution g has the effect
of reversing the flow of the map¢.
7.10. Assume that¢ is reversible, say ¢ 9 = ¢- 1. Let 1 = ¢ o g. Prove that 1 2 is the
identity map. Thus g and 1 are both involutions, so a reversible map can always be written as
a composition ¢ = 1 o g of two, generally noncommuting, involutions.
Exercises
430
7.11. Let a =1- 0, let f(y) be a polynomial of degree d ~ 2, and let
¢(x, y) = (y, ax+ f(y))
be the associated Henon map. Suppose that ¢ is reversible. Prove that ¢ and its reversing
involution g E GL 2 have one of the following forms:
g(x,y) = (y,x).
(a) a= 1,
f satisfies f( -y) = f(y).
g(x, y) = (-y, -x),
(b) a= 1,
f satisfies f( -y) = - f(y).
g(x,y)=(-y,-x),
(c) a=-1,
7.12. The real and complex dynamics of reversible maps are in some ways less chaotic than
nonreversible maps. Similarly, reversibility (and integrability) appear to have a significant
effect on arithmetic dynamics. For an affine automorphism¢ : AN -> AN, we let
Cp ( ¢) = number of distinct orbits of¢ in A2 (JFP).
For each of the following Henon maps, compute Cp( ¢) for all primes 2 < p < 100 (or
further) and make a graph of p versus Cp( ¢ ):
(a) ¢(x, y) = (y, x + y 2 ).
(b) ¢(x, y) = (y, 2x + y 2 ).
(c) ¢(x, y) = (y, -x + y 3 ).
Do you see a difference in behavior? Try plotting the ratio Cp (¢) / p. (Notice that Exercise 7.11
says that the maps in (a) and (c) are reversible, while the map in (b) is not reversible.)
Definition. Let ¢ : AN -> AN be a rational automorphism, by which we mean that ¢
is a rational map (but not necessarily a morphism) and that there is an inverse rational map
¢- 1 :AN-> AN such that¢ o ¢- 1 is the identity map wherever it is defined. The map¢
is said to be algebraically integrable if there is a nonconstant rational function I : AN -> A1
satisfying I o ¢=I.
7.13. Let¢: A2 -> A2 be the rational map
¢(x,y) = (y,-x- y::11).
(a) Prove that¢ is a rational automorphism.
(b) Let ¢ and ¢- 1 be the extensions of ¢ and ¢- 1 to maps IF2 -> IF2 • Compute Z (¢)
am\ Z (¢- 1 ), the sets of point( s) where ¢ and ¢- 1 are not defined, and verify that
Z(¢) n Z(¢- 1 ) = 0.
(c) Prove that¢ is integrable by the function
I(x, y) = x 2 y + xy 2 + x 2 + y 2 + x + y.
In other words, verify that I o ¢(x, y) = I(x, y).
(d) Prove that for all but finitely many values of c E C, the level curve I(x, y) = cis an
elliptic curve. Find the exceptional values of c for which the level curve is singular.
7.14. This exercise generalizes Exercise 7.13. Let a, b, c, d, e E K and define a rational
map ¢ : P2 -> IF2 (using dehomogenized coordinates on A 2 ) by
¢(x, y) = ( y, -x-
e)
by 2 + dy+
b
2
ay + y + c
.
Prove that ¢ is integrable by the function
I(x, y) = ax 2 y 2 + b(x 2 y + xy 2 ) + c(x 2 + y 2 ) + dxy + e(x + y).
431
Exercises
7.15. Assume that¢ is integrable by the function I. For each c E K, the set J(x) = cis
called a level set of¢.
(a) Prove that¢ maps each level set to itself. Thus the dynamics of¢ may be studied by
investigating the behavior of the iterates of ¢ on the lower-dimensional invariant level
sets that give a foliation of IP'N.
(b) * Let N = 2 and assume that ¢ has infinite order, i.e., no iterate of¢ is the identity map.
Prove that the level sets of¢ are curves of genus 0 or 1.
The following result will be helpful in doing Exercise 7.16.
Theorem 7.52. ([172, Proposition 4.2]) Let¢ : IP'N -> lP'N be a morphism of degree d ~ 2
and let V C lP'N be a completely invariant hypersurface, i.e., ¢- 1 (V) = V = ¢(V). Then V
has at most N + 1 irreducible components.
7.16. Let¢= [¢0 , ... , ¢N] be amorphism¢: IP'N-> IP'N ofdegreedgiven by homogeneous
polynomials ¢i E IC[Xo, ... , XN]. We say that such a map is a polynomial map if its last
coordinate function is equal to X'f,. Equivalently,¢ is a polynomial map if the inverse image
of the hyperplane H = {XN = 0} is simply the hyperplane H with multiplicity d.
(a) Assume that¢ is a morphism and suppose that there is ann ~ 1 such that the iterate <Pn
is a polynomial map. Prove that <Pn is already a polynomial map for some n :::; N + 1.
This generalizes Theorem 1.7. (Hint. Use Theorem 7.52.)
(b) Show that (a) need not be true if we assume only that the map¢: IP'N -> IP'N is a rational
map of degree d. More precisely, prove that for all d ~ 2 and all n ~ 2, there exists a
finite rational map ¢ : IP'N -> IP'N of degree d such that <Pn is a polynomial map, but ¢i
is not a polynomial map for all 1 :::; i < n.
Exercises on Canonical Heights for Regular Affine Automorphisms
Exercises 7.17-7.22 describe Kawaguchi's construction [230] of canonical heights for regular
affine automorphisms assuming the validity of Conjecture 7.18, which is presently known only
in dimension 2 [230, 413]. Let ¢ : AN -> AN be a regular affine automorphism of degree at
least 2 defined over tQ and let
dt = deg(¢)
and
We assume that Conjecture 7.18 is true, i.e., we assume that there is a constant C = C(¢) ~ 0
such that for all P E AN (Q),
Assumption:
: 1 h{¢(P))+ : 2 h{¢- 1 (P)) ~
(1+ dt1dJh(P)-C.
(7.35)
For any point P E AN (Q), Kawaguchi defines canonical heightjimctions by the formulas
it+(P) = limsup :nh(¢n(P)),
n--+oo
1
it-(P) = limsup d~h(rn(P)),
n--+oo
it(P) = it+(P) + it-(P).
2
(7.36)
(7.37)
7.17. Assuming that (7.35) is true, prove that the canonical height functions it+, it-, and it
defined by (7.36) and (7.37) have the following properties:
(a) it+(P):::; h(P) + 0(1)
and it-(P):::; h(P) + 0(1).
(b) h(P) + 0(1) :::; it(P) :::; 2h(P) + 0(1).
432
Exercises
(c) h,+(P)?: 0
and h,-(P)?: 0
(d) h,+(P) = 0
{:=:::}
and h(P)?: 0.
h,-(P) = 0
{:=:::}
h(P) = 0
{:=:::}
P E Per(cf>).
(Hint. Before proving (d), you may find it advantageous to do the next exercise.)
7.18. Assuming that (7.35) is true, prove that the canonical height functions satisfy the following transformation formulas:
(7.38)
: 1 h( cj>(P))
+ : 2 h( 4> - 1 (P)) = ( 1 + d 1~2 ) h(P).
(7.39)
7.19. Suppose that
and
are two functions satisfying (7.39), and suppose further that
h' = h" + 0(1).
Prove that h' = h".
7.20. Let 4> : AN --+ AN be a regular affine automorphism satisfying (7.35) and let
P E AN (Q) be a wandering point for cj>, i.e., P is not a periodic point. Prove that
Hence the constant 1 + d 11d 2 appearing in the inequality (7.35) cannot be replaced by any
larger co111stant.
7.21. Let 4> : AN --+ AN be a regular affine automorphism satisfying (7.35) and let
P E AN (Q) be a wandering point for cf>. We define the (two-sided) orbit-counting jUnction
of Pto be
Prove that
N p(T) =
</>,
(-1- + _1_)
log d1
log d2
logT- (logh+(P) + logh-(P)) +0(
log d1
log d2
1),
where the 0(1) constant depends only on the map 4> and is independent of both the point P
and the number T.
7.22. Let 4>: AN --+AN be a regular affine automorphism satisfying (7.35). Define sequences
(An) and (En) by the formulas
and
Prove that
Exercises
433
(Hint. Verify that An and Bn satisfy the linear recurrences
Ao = 0,
Bo = 0,
d1Ai - (1 + d1d2)Ai-1 + d2Ai-2 = 0,
d2Bi - (1 + d1d2)Bi-1 + dlBi-2 = 0,
and use a telescoping sum argument.)
7.23. **Let ¢J: AN --+AN be an automorphism defined over Q and denote the dynamical
degree of ¢J by
8(¢J) = lim deg(¢Jn)l/n.
n->oo
We associate to ¢J the number
Remark 7.16 tells us that S (¢J) satisfies
S(¢J):::; 2.
If ¢J is regular, then 8(¢J) = deg(¢J) and 8(¢J- 1) = deg(ifJ- 1). If in addition ¢J satisfies
assumption (7.35), then Exercises 7.18 and 7.20 imply that
1
(a) Do there exist automorphisms ¢J : AN --+ AN of degree at least 2 satisfying S (¢J) = 1?
What ifwe require that ¢J be algebraically stable? (See Remark 7.13.)
(b) Do there exist automorphisms ¢J : AN --+ AN of degree at least 2 satisfying S (¢J) = 2?
(c) What are the possible values of S(¢J) for automorphisms of AN?
(d) What are the possible values of S( ¢J) for algebraically stable automorphisms of AN?
7.24. Let K be a field that is complete with respect to a nonarchimedean absolute value and
let ¢J : IP'N ( K) --+ IP'N ( K) be a morphism. Prove that ¢J is an open map, i.e., the image of an
open set is an open set.
Section 7.4. Dynamics on Surfaces with Involutions
7.25. Let G~ and Hti be the quartic forms defined by (7.17). Prove that the following algorithm computes t 1 and L2.
(a) Let [x, y] E SA,B and write LI ([x, yl) = [x, y']. Then
y' =
{
[yoG()(x), -yoH6I(x)- y1Go(x), -yoHo2(x)- y2G()(x)]
if yo =I= 0,
[-y1H61 (x) - yoGf (x), Y1 Gf (x), -y1H12(x) - y2Gf (x)]
if Y1 =/= 0,
[-y2H62(x)- yoG2(x), -y2Hf2(x)- y1G2(x), y2G2(x)]
ify2 =/= 0.
(b) Let[x,y] E SA,B andwritet1([x,yl) = [x',y]. Then
x' =
{
[xaGg(y), -xoH81(y)- x1Gg(y), -xoHo2(y)- x2Gg(y)]
ifxo =/= 0,
[-x1H81(y)- xaGt(y),xiGt(y), -x1H12(y)- x2Gt(y)]
ifx1 =/= 0,
[ -x2H82(y)- xaGHy), -x2Hf2 (y)- x1GHy), x2GHy)]
if x2 =/= 0.
434
Exercises
7.26. The K3 surface given in Example 7.36 contains the following 12 points of small height:
Pt = ([0,1,1],[1,1,-1J), P5 = ([0,0,1],[0,1,0J),
Pg = ([8,6,9],[-6,5,2]),
= ([1,0,0],[0,0,1J), P6 = ([o,0,1],[1,0,0J), Pw = ([1,0,-1],[9,1,9J),
Pa = ([o, 1, 0], [0, 0, 1J), P1 = ([3, 1, 3], [-3, 3, 2J), Pn = ([3, 8, 11], [1, 1, -1J),
P4 = ([1,0,-1], [0, 1,0J), Ps = ([1,0,0], [0, 7, 1J), P12 = ([12, 1,-20], [2, -4, 1J).
P2
(a) Which of these 12 points lie in the same A orbit? How many distinct A orbits do they
generate?
(b) Which of the points in the list are fixed by a nontrivial element of A?
(c) The list includes all points in S(Q) having integer coordinates at most 40. Extend the
computation to find all points in S(Q) having integer coordinates at most 100. (Hint.
Loop over x with lx;l :::; 100, substitute into Land Q, eliminate a variable, and check
whether the resulting quadratic equation has a rational solution.)
7.27. For each of the primes in the set
{2,3,317,507593,2895545793631,14521485737273461}
find an example of a surface S A,B defined over lFP such that t 1 and t2 are defined at every
point of SA,B(JFp). (These examples can be used to complete the proof of Proposition 7.41.)
7.28. This exercise sketches a noncomputational proof of Proposition 7.41 using more advanced methods from algebraic geometry.
(a) LetS and S' be nonsingular projective K3 surfaces and let¢ : S -+ S' be a birational
map, i.e., a rational map with a rational inverse. Prove that ¢ is a morphism. (Hint. Find
a surface T and birational morphisms 'ljJ : T -+ S and '1/J' : T -+ S' so that ¢ o 'ljJ = '1/J'
[198, V.5.5]. Do this so that 'ljJ is a minimal number of blowups and let E be an exceptional curve of the last blowup. Deduce that q(E) is a curve ConS'. Then show that the
intersection of C with the canonical divisor on S' satisfies C · K S' :::; E · K s = -1,
which contradicts the fact that Ks' = 0, since S' is a K3 surface.)
(b) Prove that there is a proper Zariski closed set Z c lP'8 x lP'35 such that for all (A, bf B) tJ.
Z, the surface BA,B is nonsingular. (Hint. Elimination theory says that the set of
(A, bf B) E lP'8 x lP'35 such that BA,B is singular is a Zariski closed set. Thus it suffices
to find a single (A, bf B) for which SA,bf a is nonsingular.)
(c) Combine (a) and (b) to prove Proposition 7.41.
7.29. Let P E SA,B with infinite A-orbit. Prove that the A-stabilizer of P,
{'ljJ E A: '1/J(P) = P},
has order either 1 or 2.
7.30. This exercise describes intersections on the surface SA,B· For the basics of intersection
theory on surfaces, see, for example, [198, V §1].
(a) Let D 1 = piH and D 2 = p2H be the usual divisors in Pic(SA,a). Prove that
and
435
Exercises
(b) Let a= 2 + v'3 and define divisors E+ and E- in Pic(SA,a) l8l R by the formulas
E+ = -D1 + aD2
and
E- = aD1- D2.
Prove that tiE± = a±l E'f and t2E± = a'f 1 E'f.
(c) Prove that E+ · E+ = E- · E- = 0 and E+ · E- = 12a.
7.31. (a) Prove that under composition, the involutions t1, t2 E Aut(SA,B) satisfy no relations other than ti = t~ = 1. Thus A is the free product of the subgroups generated
by t1 and t2. (Hint. Use Exercise 7.30. Apply a composition of t1 'sand t2 's toE++ Eand intersect with E+.)
(b) Show that A is isomorphic to the infinite (discrete) dihedral group
D
_
00
-
{
{ siti : i, j E Z}
t2 = 1 and ts = s- 1t}
via the map
7.32. Let P E SA,B be a point whose A-orbit A(P) is an infinite set. Prove that A(P) is
Zariski dense in SA,B· (Hint. If A(P) is not dense, find a curve C C SA,B fixed by some
nontrivial element 'lj; E A and consider the intersection of C with the divisors E+ and Edefined in Exercise 7.30.)
7.33. Let¢>= t 1 o t 2 and fix a nonzero integer n. Prove that the set
is a finite set. (Hint. If the set is infinite, find a curve C C S A,B fixed by rf>n and consider the
intersection of C with the divisors E+ and E- defined in Exercise 7.30.)
7.34. Let D E Pic(SA,a) be the divisor D = D1 + D2 = piH + p2H. Let¢ = t1 o t2
and a = 2 + v'3 as usual. Prove that
hv(r/>n P) + hv(r/>-n P) = hv(P) + 0(1)
a2n + a-2n
for all P E SA,a(K) and all n ~ 0.
(The 0(1) constant depends on the surface SA,B, but is independent ofboth P and n.)
7.35. Let SA,B be defined over a number field K and let h,+ and h,- be the canonical height
functions constructed in Proposition 7.47. Assuming that SA,a(K) is an infinite set, prove
that there is a constant C such that both of the sets
{ P E SA,a(K): h+(P) ~ C}
and
are infinite. This shows that Proposition 7.49 is not true if his replaced by either h,+ or h,-.
7.36. Let SA,B be definedoveranumberfield K, let h,+ and h,- be the canonical height functions constructed in Proposition 7.47, and let h = h,+ + h,-. Fix a point Q E SA,a(K).
(a) Prove that the product h,+(P)h- (P) is the same for every point P E A(Q). This product measures, in a certain sense, the arithmetic complexity of the A-orbit of Q. Notice
how h+ (Q) h- (Q) naturally appears in Proposition 7.50 counting points of bounded
height in the A-orbit of Q.
436
Exercises
(b) Prove that
2Vh+(Q)h-(Q)::;
min h(P)::; 2aJh+(Q)h-(Q).
PEA(Q)
(Here a = 2 + v'3 as usual.)
7.37. Let SA,B be defined over a number field K and let Q E SA,a(K) be a point whose Aorbit A( Q) is infinite. Further, let h = h+ + h-, and define a height zeta function for the Aorbit of Q by the series
Z(A(Q), s) =
L -,1- .
PEA(Q) h(Q)•
(a) Prove that the series defining Z(A(Q), s) converges on the half-plane Real(s) > 0.
(b) Protve that Z(A(Q), s) has a meromorphic continuation to the entire complex plane.
(c) FindthepolesofZ(A(Q),s).
(d) Find the residues of Z(A(Q), s) at its poles.
7.38. Let SA,B be defined over a number field K and let P E SA,a(K) be a point whose Aorbit A(P) is Galois-invariant, i.e., ifQ E A(P) and a E Gal(K/ K), then a(Q) E A(P).
Prove that P satisfies one of the following conditions:
(a) P E SA,a(K).
(b) A(P) is finite.
(c) [K(P): K] = 2.
If P satisfies condition (c), prove that there exist a 'ljJ E A and an index j E { 1, 2} such that
Pi('t/J(P)) ElP'2 (K).
7.39. Let V c IP'N x IP'N be a variety given by the vanishing of N- 1 bilinear forms and one
biquadratic form,
LI(x,y) = · · · = LN-I(x,y) = Q(x,y) = 0,
and let PI : V ---+ IP'N and P2 : V ---+ JP>N be the usual projection maps PI (x, y) = x and
P2(x,y) = y.
(a) Prove that PI and P2 are generically 2-to-1, so they induce involutions LI : V ---+ V
and t2 : V ---+ V. In other words, there are rational maps LI and t 2 such that ti and t~
are the identity map wherever they are defined.
(b) If N ~ 3, prove that LI and t 2 are not morphisms.
7.40. Let a E K*. The Markoff equation
Ma : x 2 + y 2 + z 2 = axyz
defines an affine surface in A 3.
(a) Prove that there are involutions LI2, ti 3, and t 23 of Ma defined by the formulas
t12(x, y, z) = (x, y, axy- z),
LI3(x, y, z) = (x, axz- y, z),
t23(x, y, z) = (ayz- x, y, z).
Explain how these involutions correspond to natural double covers Ma ---+ A 2 induced
by projection maps A 3 ---+ A 2.
Exercises
437
(b) Prove that the involutions ~ij do not extend to morphisms on the projective variety
Ma = {x 2 w + y2 w + z 2 w = axyz} C JP>3 ,
and determine the points at which they fail to be defined.
(c) Find a birational map JP>2 ----> Ma defined over IQ. (A birational map between projective
varieties V and W is a rational map from V to W that is an isomorphism from a Zariski
open subset of V to a Zariski open subset of W.) In particular, this implies that M a ( IQ)
contains many points.
(d) Prove that every point in M 3 (Z) with positive coordinates can be obtained by starting with the point ( 1, 1, 1) and applying the involutions ~ij. (Hint. Define the size of a
positive integral point P = (x, y, z) to be the largest of its coordinates and prove that
if P =f. (1, 1, 1), then at least one of ~ij (P) has size strictly smaller than the size of P.)
(e) Let a be a positive integer. Prove that if a =f. 1 and a =f. 3, then Ma(Z) = 0. (Hint. Use
the same type of descent argument as suggested in (d).)
(f) A normalized Markoff triple is a point (x, y, z) E M 3 (Z) with x:::; y:::; z. Let
N(T) =#{normalized Markoff triples (x, y, z) with z :::; T}.
Prove that there are positive constants c1 and c2 such that
(7.41)
More precisely, prove that there is a constant c such that
N(T) = c(logT) 2 + O((logT)(loglogT) 2 ).
(7.42)
(g) **Let (x1, y1, zl) and (x2, y2, z2) be normalized Markoff triples. Prove that if z 1 = z 2 ,
then also x1 = x2 and y1 = Y2· {This is known as the unicity conjecture for Markoff
numbers.)
Exercises on K3 Surfaces with Three Involutions
Exercises 7.41-7.44 ask you to explore a family ofK3 surfaces that admit three noncommuting
involutions. These hypersurfaces Sc c JP> 1 x JP> 1 x JP>1 are described by the vanishing of a
trihomogeneous polynomial of degree 2,
Q(x, y, z) =
L
CijklmnXiXjYkYRZmZn
= 0.
O$i$j$1
O<k<£<1
O~m;;n~1
The surface Sc admits three maps of degree 2 to JP> 1 x JP> 1 ,
P12(x,y,z) = (x,y),
P13(x,y,z) = (x,z),
P23(x,y,z) = (y,z),
and these maps induce corresponding involutions
~12: Sc----> Sc,
~13: Sc----> Sc,
~23: Sc----> Sc.
We also fix a point to E JP>l, let 1r1, 1r2, 1r3 : Sc ----> JP> 1 be the maps induced by the three
projections from JP> 1 x 1P' 1 x JP> 1 to JP> 1 , and define divisors on Sc by setting
438
Exercises
7.41. The surface Sc is specified by the 27-tuple
C = [coooo, coo01, ... , cnn] E lP'26
of coefficients of the trihomogeneous polynomial defining Sc.
(a) What is the dimension of the family of surfaces Sc after we identify surfaces that are
isomorphic via the action of PGL2 on each of the three copies of lP'1 in lP'1 x lP'1 x lP'1.
(See Remark 7.38 for a similar calculation for the family SA,B·)
(b) Prove that there is a Zariski closed subset Z C lP'26 such that if C fl. Z, then the
involutions t 12 , L1 3, and t 23 are defined at every point of Sc.
7.42. Let V be the subspace ofPic(Sc) generated by D1, D 2 , and Da.
(a) Prove that the action of ti 2 on Vis given by
Devise analogous formulas for the action of tia and t; 3 on D.
(b) Let Mii be the 3 x 3 matrix of tii acting on V and let J be the matrix ( g~ ~ ) . Prove
2
t -1
-1
220
that Mii = 1 and that Mii J Mii = J .
(c) Prove that double products such at M12M1 3 have all of their eigenvalues equal to 1.
(d) Let¢ be the map¢= L12ot1aot23 and let !3 = H3+v'5). Prove that /3 3 is an eigenvalue
of¢* acting on V and that a corresponding eigenvector is !32D1 + j3D 2 + D 3.
7.43. Assume that Sc is defined over a number field K. Let¢ = L12 o L13 o L23 : Sc ---> Sc
and let !3 = H3 + v'5).
(a) Define a real-valued function f : SeCK) ---> lR by
j(P) = /3 2 hv 1 (P) + /3hv 2 (P) + hv3 (P).
Prove that
f(¢(P)) =/33 f(P)+0(1)
for all P E Sc(K).
(b) Prove that there exists a unique function j : Sc (K) ---> R satisfying
j(P)=f(P)+0(1)
and f(¢(P))=J3 3 j(P)
forallPESc(K).
7.44. Let Sc be the surface given by the equation
x~y~z~ + X~YOY1Z~ + 4x~y~z~ + xox1y~z~ + XoX1Y~Z~
+ X~YOY1Z~ + x~y~zoz1 + 2x~y~z~ = 0.
(a) Prove that the surface Sc is nonsingular. (Hint. Check that it is nonsingular over lF2.)
(b) Prove that the involutions L12, L1a, and L2a are defined at every point of Sc(C).
(c) Verifiy that the point Po = ([0, 1], [-1, 1], [-1, 1l) is in Sc(Q). Then compute the
''tree'' of points starting from Po and generated by applying the involutions in various
orden;:
Exercises
439
Po
In particular, find two branches of the tree that loop around and reconnect with the top.
Notes on Exercises
Many of the exercises in this book are standard, or in some cases not so standard,
results. These notes thus have a dual purpose: to give credit where due, and to point
the reader toward the relevant literature. However, since any attempt to assign credit
is bound to be incomplete, the author tenders his apologies to anyone who feels that
he or she has been slighted.
Chapter 1. An Introduction to Classical Dynamics
1.11. See [415].
1.12. See [415].
1.18. (c) is proven in Corollary 4.7.
(d) This is due to LN. Baker [19], or see [43, §6.8].
1.24. See [43, Theorem 3.2.5].
1.30. Most of this exercise is proven in Proposition 6.6.
1.31. See [43, Section 1.4].
Chapter 2. Dynamics over Local Fields: Good Reduction
2.6. See [436, Section 5.9].
2.17. This special case of a theorem of Rivera-Letelier was suggested to the author
by Rafe Jones.
2.19. See [312, Proposition 3.1].
2.22. The first example of this phenomenon is due to Poonen (unpublished). It appears in Zieve's thesis [454, Lemma 6].
2.24. This is in Jones's thesis [220].
Chapter 3. Dynamics over Global Fields
3.2. Schanual [391] (or see [256, Theorem 5.3]) proves a general formula for anumber field K:
lim #{P EIP'N(K): HK(P) :S B} = hKRK/WK (2r 1 (2'nY2 )N+(N+lr 1 +r2 _ 1
B->oo
BN+l
(K(N + 1)
Dl/2
'
K
where hK, RK, WK, (K,DK, r1. and r 2 are, respectively, the class number, regulator, number of roots of unity, zeta function, absolute discriminant, number of real
embeddings, and number of complex embeddings of K.
441
442
Notes on Exercises
3.4. See [410, Theorem III.5.9] or [256, Lemma 2.2 in §3.2). For better estimates,
see [256, §3.2] and the references cited there.
3.9. See [309).
3.14. This exercise was suggested to the author by Rob Benedetto.
3.38. See [411).
3.40. (b) See [411, Proposition 1.2).
(c) See [411, Theorem B).
3.46. See [411].
3.21. Sylvester's original article is [428]. See [2, 215) for additional material on
Sylvester's and other related sequences.
3.22. This exercise was inspired by [2] and [184, Exercise 4.37).
3.49. (a) See [306).
(b) ge(z) = z3 - (e- l)z 2/2- (e 2 + 2e + 9)z/4 + (e 3 + e2 + 7e- 1)/8 and
Disc(ge) = (e 2 + e + 7) 2 .
Chapter 4. Families of Dynamical Systems
4.4. See [313, Proposition 3.2 and Lemma 3.4).
4.6. This exercise was suggested to the author by Michelle Manes.
4.7. (b,c) These formulas are due to Morton and Vivaldi [314].
4.8. See [313, Theorem 2.1]. For a generalization to morphisms of higher-dimensional varieties, see [214).
4.12. (e) See [314).
4.13. See [307].
4.15. See [132, Proposition 8.6].
4.20. (a,b) See [305, 309).
(c,d) See [171).
4.30. This exercise was inspired by Milnor's paper [303], which studies the geometry and topology of the spaces that we have denoted by BiCritd and M~iCrit
4.36. (a) See [414, example in Section 7].
(b) (a1(¢),a2(¢)) = (-6, 12).
4.45. See [414, Section 6).
4.49. This result is due to Szpiro and Tucker [431). It is a dynamical version of
Faltings' theorem (Shafarevich conjecture) [165, 164] that there are only finitely
many principally polarized abelian varieties of given dimension with good reduction
outside of a finite set of primes.
Chapter 5. Dynamics over Local Fields: Bad Reduction
5.1. See, for example, [324, §10.1] or [78].
5.6. (a) This is wellknown. See [233] for a more general result.
5.10. See [436, §19.9] for a proof of the residue theorem over algebraically closed
base fields due to Roquette.
5.11. (b) Rivera-Letelier shows in his thesis [372] that every indifferent periodic
point is contained in a "domain of quasiperiodicity" that contains infinitely many
(indifferent) periodic points.
Notes on Exercises
443
5.21. Hsia [208] uses a version of this result in his proof ofMontel's theorem with
moving targets.
5.25. See [56].
5.26. This exercise was suggested to the author by Rob Benedetto.
5.27. See [234] for the analogous result on lP'N.
5.28. See [233] for the analogous result on lP'N, including explicit values for the
Holder constants.
5.29. For elliptic curves, this is due to Tate (unpublished letter to Serre) if Kv is not
algebraically closed and to the author [408] for arbitrary Kv. See [88, Section 5] for
the general dynamical case.
5.32. See [88] for a general construction of local canonical heights associated to
dynamical systems with eigendivisor classes.
5.45. This exercise, which appears in a paper of Rivera-Letelier, was suggested to
the author by Rob Benedetto.
Chapter 6. Dynamics Associated to Algebraic Groups
6.27. These examples are due to Noam Elkies [150].
Chapter 7. Dynamics in Dimension Greater Than One
7.4. deg¢n = 2l(n+ 2)/ 3J anddyndeg(¢) = ij2.
7.5. (c) This example is due to Hasselblatt and Propp [199].
7.12. This exercise was inspired by the work of Jogia, Roberts, and Vivaldi [219,
218, 383, 384], who prove results and state conjectures on how reversibility and
integrability affect the growth of Cp (¢) and related quantities.
7.14. This family of integrable maps was discovered by McMillan [293]. For an even
larger family of integrable maps called the QRT family, see [364, 365].
7.15. This result is due to Veselov [439].
7.16. This author thanks Shu Kawaguchi for providing a solution to this exercise
(private communication).
7.17-7.21. These exercises are due to Kawaguchi [230].
7.20. See [230, Proposition 4.2] and [413, remark following Theorem 3.1]).
7.24. See [233].
7.26. See [409, §5].
7.28. This exercise was suggested to the author by Shu Kawaguchi.
7.32. See [409, Corollary 2.3].
7.33. See [409, Corollary 2.4(b)].
7.36. See [409].
7.38. See [409].
7.40. (d) See [210, §11.8].
(f) The estimates (7.41) and (7.42) are due, respectively, to Cohn [108] and Zagier [451]. See also Baragar's articles [32, 33] for a higher-dimensional analogue in
which the counting function N(T) grows like (logT)' for an irrational exponent E.
444
Notes on Exercises
7.41-1.44. These exercises were inspired by the work of Baragar, Luijk, and Wang
[34, 3.5, 36, 37, 38, 39, 446], who study the arithmetic and dynamical properties of
these triple-involution K3-surfaces.
7.44. This example is due to Baragar [38, §4], who notes that ~13~12~13(Po) = Po
and ~12~13~12 (Po) = Po.
List of Notation
</Jn
<Po
nth iterate of the map </J, 1
the identity map, 1
orbit of o: by the map </J, 1
Oq,(o:)
set of periodic points of <PinS,
Per(</J, S)
PrePer( <P, S)
set ofpreperiodic points of <PinS,
torsion subgroup of the abelian group G, 2
Grors
automorphism group of the projective line, 10
Aut(lP' 1 )
projective linear group, 10
PGL2
general linear group, 10
GL2
<Pf
linear conjugation of <P by f, 11
chordal metric on lP' 1 ( IC), 11
p
ramification index of <Pat o:, 12
e,(</J)
>.q,(o:)
multiplier of <P at periodic point o:, 18
set of points of period n, 18
Pern(<P)
Per~· (<P)
set of points of exact period n, 18
space of differential one-forms, 19
n~
the Mandelbrot set, 26
M
multiplicative group, 29
IGm
the lfh Chebyshev polynomial, 29
Td
the additive group, 30
IGa
addition on an elliptic curve, 31
E9
Lattl~s map associated to multiplication by d, 32
</JE,d
Lattes map associated to an endomorphism u, 32
</JE,u
complex uniformization of an elliptic curve E, 33
'1/JE
Weierstrass p function, 34
f.J
residue fixed-point index of <P at o:, 38
L(<jJ,a)
the nth dynatomic polynomial, 39
4>~(z)
v-adic chordal metric on lP' 1 , 45
Pv
p
reduction of the point P modulo a prime, 48
¢
reduction of a rational map <P modulo a prime, 52
Res( A, B)
the resultant of A and B, 53
resultant of a rational map, 56
Res(<P)
""(P1, P2, Pa, P4) cross-ratio of P1, P2, P 3 , P4, 71
the filled Julia set of </J, 74
IC(<P)
H(P)
(multiplicative) height of a rational point, 82
set of standard absolute values on Q, 82
MQ
the usual absolute value on R, 82
/ · /oo
445
446
ordp(a)
I IP
0
MK
M'j(
M~
RK
Rs
nv
HK(P)
H(P)
k
K(P)
v'I
V(I)
I(V)
IPiv
lflv
Dv(m)
hK
h
0(1)
h<P
Av
~</>,v
Res( <P)
Per~* (<P)
Kn,</>
Gn,</>
Wreath(H, S)
G~<t>
o'
Kn,</>
J-l<i>
C(P/K)
Dp
J-lP
S¢
'P
J-l
Fn,Gn
<I>¢,n(X, Y)
<I>;,n(X, Y)
<I>n(z), 4>~(z)
Pern(¢)
Per~(¢)
Per~·(¢)
lld(n)
ap(n)
aj,(n)
List of Notation
the exponent of highest power of p dividing a, 82
the p-adic absolute value on Q, 82
the standard set of absolute values on K, 83
the archimedean absolute values on K, 83
the nonarchimedean absolute values on K, 83
the ring of integers of K, 83
the ring of S-integers of K, 83
local degree at an absolute value v, 83
(multiplicative) height of a K -rational point, 84
the absolute (multiplicative) height of P, 85
an algebraic closure ofthe field K, 85
field of definition of the point P, 86
the radical of the ideal J, 89
the algebraic set attached to the ideal I, 90
ideal attached to the algebraic set V, 90
maximum of absolute values of coordinates of P, 90
maximum absolute value of coefficients of a polynomial, 91
equal to m or 1 depending on whether v is archimedean, 91
logarithmic height, 93
absolute logarithmic height, 93
a bounded function, 93
canonical height associated to morphism ¢, 99
v-adic logarithmic distance function, 102
v-adic local canonical height, 102
resultant of a rational map, 112
periodic points of¢ of primitive period n, 122
dynatomic field generated by primitive n-periodic points, 123
Galois group of dynatomic field, 123
wreath product of HandS, 125
subgroup ofGn,</> leaving ¢-orbits invariant, 126
fixed field of G~, </>, 126
the canonical ¢-invariant probability measure on lP'N (C), 127
the set of Galois conjugates of P, 128
the Dirac measure supported at P, 128
discrete probability measure supported on Galois conjugates of P, 128
set of primes of bad reduction for¢, 132
Euler's totient function, 137
the Mobius J-t function, 148
coordinate functions of the nth iterate of¢, 149
then-period polynomial of¢, 149
that n 1h dynatomic polynomial of¢, 149
dehomogenized period and dynatomic polynomials, 149
the set of points of period n for ¢, 150
the set of points offormal period n for¢, 150
the set of points of primitive period n for¢, 150
number of points offormal period n for a map of degree d, 150
order of the period polynomial at the point P, 151
order of the dynatomic polynomial at the point P, 151
List of Notation
c/Jc(z)
lf>~(c,z)
Y1(n)
X1(n)
Formal(n)
Yo(n)
Xo(n)
e
Fa(X, Y)
[a,b]
Ratd
lP'~
Md
(. )
Q[RatdtGLz
812
PSL2
Ratd
Ratds
Md
Md
8
Fq,,n,G<j>,n
An(c/J)
A~(¢)
ain)(cj;)
~;n)(cj;)
O'd,N
•
O'd,N
A(¢)
M2
[¢]
[c/J]K
Aut(¢)
Twist(¢/K)
Twist(X/K)
9u
H 1 (r,A)
Gf
KJ
ordp(P)
FA,GA
tp ( cj;)
rytq,
SJlq,
aF,G
O.q,jK
Dn,q,(x)
447
the quadratic polynomial z 2 + c, 155
the nth dynatomic polynomial for z 2 + c, 157
(affine) dynatomic modular curve for z 2 + c, 157
(projective) dynatomic modular curve for z 2 + c, 157
PGL2-classes of quadratic with point offormal period n, 158
the quotient ofY1(n) by¢, 161
the quotient of xl (n) by¢, 161
conformal isomorphism to the complement of Mandelbrot set, 167
the homogeneous polynomial aoXd + a 1xd-ly + · · · + adYd, 169
the point [ao, ... , ad, bo, ... , bd] in lP'2 d+l, 169
the set of rational maps of degree d, 169
the projective line over V, 171
the moduli space of conjugacy classes of maps of degree d, 174
map from Ratd to Md, 174
the ring ofPGL2-invariant functions on Ratd., 175
the special linear group, 175
the projective special linear group, 175
set of stable rational maps, 178
set of semistable rational maps, 178
stable completion of Md, 178
semistable completion of Md, 178
coordinate functions of the nth iterate of¢, 181
the n-multiplier spectrum of¢, 182
the formal n-multiplier spectrum of¢, 182
symmetric polynomial of n-periodic multipliers of¢, 183
symmetric polynomial of formal n-periodic multipliers of¢, 183
map of Md using ain) with n:::; N, 187
map of Md using ~;n) with n:::; N, 187
the multiplier spectrum of¢, 187
map Ratz---> A 2 inducing an isomorphism M 2 ~ A 2 , 188
the completion M2 = M2s of M2, 194
set of rational maps k -equivalent to ¢, 195
set of rational maps K -equivalent to ¢, 195
the automorphism group of¢, 196
the set of twists of the rational map¢, 197
the set of twists of the object X, 199
the 1-cocycle associated to a twist, 201
cohomology set (group), 202
subgroup of Gal( k / K) associated to ¢, 207
field of moduli of¢, 207
minimum ordp of the coefficients of the polynomial P, 218
new coordinate functions for conjugate of [F, G], 218
exponent of the minimal resultant of¢, 220
the (global) minimal resultant of¢, 220
product of the primes ofbad reduction for¢, 221
ideal connecting minimal resultant to resultant of a model, 222
the Weierstrass class of¢ over K, 223
polynomial whose roots are multipliers of¢, 225
448
Fm,n(X, Y)
<I>;,,n(X, Y)
(S¢i)(z)
BiCritd
M~iCrit
Cp
R=RK
R=R'K
2)1
D(a,r)
D(a,r)
S(a,r)
D(Q,r)
D(Q,r)
orda( ¢!)
11¢lllr,a
v(z)
:F( ¢!)
.J(¢!)
:F(¢, K)
.J(¢,K)
A= A(¢)
Io,h
N
sN
1)
Dctosed
Dopen
TJC(¢)
7r
A;
ll(x,y)llv
Qq,(x, y)
~¢,E ([x, yJ)
jjB
~a,O
~a,r
~a,r
IIJIIR
jjB
jj~
']['R
La,r
La,r
U(f,B)
V(f,B)
ffi.a,r
/r\:,r
List of Notation
generalized period polynomial, 227
generalized dynatomic polynomial, 227
the Schwarzian derivative of¢!, 231
the space ofbicritical rational maps, 233
the moduli space ofbicritical rational maps, 233
the completion of the algebraic closure ofQp, 239
the ring of integers of a local field K, 242
the unit group of the ring of integers of a local field K, 242
the maximal ideal of the ring of integers of a local field K, 242
open disk of radius rat a, 242
closed disk of radius r at a, 242
circle of radius rat a, 242
open disk of radius r in IP' 1 ( K), 24 3
closed disk of radius r in IP' 1 ( K), 243
order of vanishing of a holomorphic function, 244
norm of¢! relative to the disk D(a, r), 24 7
valuation - logP Izl, 249
the Fatou set of¢!, 255
the Julia set of¢!, 255
the K -valued points of the Fatou set of¢!, 255
the K-valued points of the Julia set of¢!, 255
set of points with bounded i¢in(z) 1, 258
disks D(O, 1/p) and D(1, 1/p), 258
the set of natural numbers, 258
set of sequences chosen from the setS, 258
set of"disks" in a topological space, 277
standard collection of closed disks in 1P' 1 (Cp), 277
standard collection of open disks in IP' 1 (Cp), 278
set of disk components of the Fatou set, 283
projection map A 2 ---> IP' 1 , 287
the affine plane with (0, 0) removed, 287
sup norm of x andy, 288
Green function attached to <I> : A 2 ---> A 2 , 288
local canonical height associated to¢! and E, 291
the unit Berkovich disk, 295
Type-I point in Berkovich disk D5 , 295
Type-II or III point in Berkovich disk D5 , 295
Type-IV point in Berkovich disk jjB, 295
Gauss norm off E Cp[z], 296
the closed unit Berkovich disk, 297
the closed Berkovich disk of radius R, 297
the Tate algebra, 297
line segment in Berkovich disk, 298
line segment in Berkovich disk for Type-IV point, 298
open set in Gel'fond topology on Berkovich disk, 300
open set in Gel'fond topology on Berkovich disk, 300
closed branch ofBerkovich disk rooted at (a, r), 300
open branch ofBerkovich disk rooted at (a, r), 300
List of Notation
Gauss point ofBerkovich disk of radius R, 301
the Berkovich affine line, 301
JIDB
the Berkovich projective line, 302
Ann 5
annulus in Berkovich space, 302
branch at infinity ofBerkovich projective line, 302
k.:O
R(¢,a,r)
radius ofthe image disk¢( D( a, r) ), 305
.JB(</J)
the Berkovich Julia set of¢, 306
FB(¢)
the Berkovich Fatou set of¢, 306
canonical measure on Berkovich space associated to ¢, 306
P,q,
Resz=o: ( ¢( z) dz) residue of ¢(z) at z =a, 313
g<f.>
modified Green function, 318
local canonical height associated to a divisor, 320
5.q,,D
Dg
divisor of the rational function g, 320
path metric on the Berkovich disk, 323
K(~a,r, ~b,s)
the Hsia kernel on the Berkovich disk, 323
o(~a,r, ~b,s)
the power map zd, 325
Md
multiplicative group, 325
IGm
the group of nth roots ofunity, 326
J.Ln
the ~ Chebyshev polynomial, 329
Td
0
point at infinity on an elliptic curve, 336
the set of points of E defined over the field K, 337
E(K)
Div(E)
the group of divisors onE, 339
deg(D)
the degree of the divisor D, 339
sum
the summation map on the divisor group of E, 339
div(f)
the divisor associated to the rational function f, 339
ordp(f)
the order of the zero (pole) off at P, 339
Pic(E)
the Picard group (group of principal divisors on E), 339
the dual isogeny of 'lj!, 340
~
End(E)
the endomorphism ring of E, 340
Aut(E)
the automorphism group of E, 340
E
the reduction of E modulo p, 342
E['lj!]
kernel of the endomorphism 'lj!, 343
E[m]
kernel ofmultiplication-by-m map, 343
torsion subgroup of the elliptic curve E, 343
Etors
Te(E)
the Tate module of E, 344
the £-adic representation on an elliptic curve, 344
PE,e
the invariant differential onE, 344
WE
group of fractional ideals ofF, 350
LF
elliptic curves with CM by RF, 350
£U(RF)
ideal class group ofF, 350
CF
class number ofF, 350
hF
CritPtq,
set of critical points of¢, 353
CritValq,
set of critical values of¢, 353
PostCritq,
forward orbit of critical values of¢, 353
E(Z)
the set of points in E(Q) with integer coordinates, 372
v(E)
number of primes dividing the denominator of j(E), 372
p
homogenization of the polynomial F, 389
Z(¢)
locus of indeterminacy of the rational map¢, 389
~O,R
AB
449
450
q,
Div(V)
Ov,w
ordw(f)
Pic(V)
Dt "'D2
</>*D
f.D
</>*
deg
Base(D)
L(D)
£(D)
</>v
hv,v
SA,B
Pt,P2
Lt, L2
Lj,LY
Q'kt,Qfj
G'k, G%, Hij, Hij
A
A(P)
Dt,D2
h+,h-
h,+,h,h
p,(Q)
Cp(</>)
h+,h-,h
E+,ED
List of Notation
lift of the rational map ((>to AN +1, 389
the divisor group of V, 403
local ring of V at W, 403
order of vanishing off along W, 403
the Picard group of V, 403
linear equivalence of divisors, 404
pullback of the divisor D, 404
pushforward of the divisor D, 404
pullback homomorphism on Picard groups, 405
degree map on Div(IP'N) and Pic(IP'N), 405
base locus of the divisor D, 406
vector space associated to the divisor D, 406
dimension of the vector space L(D), 406
rational map associated to the divisor D, 406
height on the variety V associated to the divisor D, 407
the K3 surface in IP'2 x IP'2 determined by A and B, 410
projections of a K3 surface to IP'2 , 410
involutions on a K3 surface SA,B, 411
linear forms associated to a K3 surface, 412
quadratic forms associated to a K3 surface, 412
quartic forms associated to a K3 surface, 413
subgroup of Aut(SA,B) generated by Lt and t2, 418
the A-orbit of P on the K3 surface BA,B, 418
divisors on the K3 surface BA,B, 418
height functions on BA,B, 420
canonical heights on the K3 surface SA,B, 421
the sum h+ + h- on the K3 surface SA,B, 423
the order of the stabilizer of a point on a K3 surface, 425
number of distinct lFp-orbits of</>, 430
canonical heights on AN for a regular affine automorphism, 431
eigendivisors in Pic(SA,B) 181 JR, 435
subspace ofPic(Sc) generated by Dt, D2, and D3, 438
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Index
composition of elementary maps
and Henon maps, 402
composition of involutions, 429
degree, 394
degree of inverse, 394, 395
disjoint indeterminacy locus, 394
dynamical degree, 433
fixed point, 427
height inequality, 399, 431
height of periodic points, 402
Henon map, 390
indeterminacy locus, 389, 392
dimension of, 394
disjoint, 394
of inverse, 392
of iterate, 394
inverse may have different degree,
390
iterate may have wrong degree, 390
number of orbits over a finite field,
430
periodic points of, 394, 400
regular, 391, 394, 427
periodic points of, 428
reversible Henon map, 430
affine coordinate ring
of M2, 189
of Md, 176, 179
as Z-scheme, 186, 189
contains at), 183
contains ;~n), 183, 232
ofRatd, 169, 174
affine line, Berkovich, 30 1
affine minimal model of a rational map,
112,372,385
affine minimal rational map, 372
affine morphism, 375, 388
algebraically stable, 396, 433
degree of composition, 392
ABC conjecture, 372, 373
abelian group
preperiodic point equals torsion
point, 2, 41,326
torsion subgroup, 2
abelian variety, 409, 442
Bogomolov conjecture, 129
field of definition, 217
field of moduli, 217
Manin-Mumford conjecture, 127
Neron local height, 104
Raynaud's theorem, 127
absolute height, 85
logarithmic, 93
absolute value, 43
archimedean, 82, 83
completion of a field at an, 83
effect of polynomial map on, 288
equivalent, 44
extension formula, 83, 84
local degree, 83
nonarchimedean, 44, 82, 83, 242
of a point, 90, 288
of a polynomial, 91, 288
on function fields, 44
on Q, 82,312
p-adic, 44, 82
product formula, 83
standard set of(MK), 83
abstract dynamical system, 1
realizable sequence, 6
addition on an elliptic curve, 31
additive group, 30
additivity of height, 408, 421
affine automorphism, 390,427
algebraically reversible, 429
algebraically stable, 396, 433
canonical height, 431
473
474
affine morphism (continued)
dynamical degree, 397,428
finite quotient, 376
homogenization, 389
indeterminacy locus, 389, 392, 427
iterates, 376
jointly regular, 397, 429
lift of, 389
uniquely determines endomorphism
and translation, 376
affine plane
absolute value, 288
lift of rational map, 287
projection to projective line, 287
with origin removed, 287
affine space
absolute value, 90
height, 397
projection to projective space, 287
algebraic closure, 85
algebraic dynamics, 6
algebraic entropy, 397
algebraic family, 159, 230
field of definition, 159
induces map to dynatomic curve,
159
oflP' 1 's, 171
ofmorphisms, 171
of quadratic maps, 194
algebraic geometry, 402
algebraic group, 325
commutative, 375
conjugation is action on, 173
finite quotient, 376
iterates of affine morphism, 376
simple, 377
universal cover, 375
algebraic point in Julia set, 40
algebraic set, 90
attached to an ideal, 90
ideal of, 90
algebraic variety, 89, 147, 159
dynamically affine, 376
quotient by finite group, 161
quotient by infinite group, 161, 174
algebraically closed field, 26, 242
Cp is an, 239
algebraically integrable map, 430
Index
algebraically reversible affine
automorphism, 429
algebraically stable morphism, 396, 433
degree, 396
indeterminacy locus, 396
ample divisor, 406
bounded height is finite set, 408
analytic function, see holomorphic
function
annulus, 302
archimedean absolute value, 83
arithmetic complexity
height in Md, 221
minimal discriminant, 221
of orbit on K3 surface, 435
arithmetic function, 6
Arzela-Ascoli Theorem, 25
attracting basin of oo, 74, 138, 140, 141
attracting periodic point, 19, 47, 326, 362
accumlates on a critical point, 78
basin of attraction, 310
every periodic point is an, 78
immediate basin of attraction, 310
in Berkovich space, 310, 324
in Fatou set, 22, 40
limit is repelling, 274
of Type I, 310
reduction is critical, 78
automorphism
action on critical point, 234
affine, 390, 427
affine regular, 394
algebraically reversible, 429
canonical height, 431
finitely many periodic points, 435
height inequality, 399,431
height of periodic points, 402
induced by degree 2 map, 411,436,
437
is subgroup ofPGL2, 234
nondegenerate, 226
of order two, 96, 197, 235
rational, 430
automorphism group, 340, 341
containing (nz, 205, 234
contains p. 2, 235
cyclic, 197, 204
dihedral, 217, 234, 327
extension of cocycle to PGL2, 203
Index
automorphism group (continued)
involution, 410
is finite, 196
list of possible, 197
no twists if trivial, 198
noncommuting involutions, 410,
437
of Chebyshev polynomial, 332, 336
in characteristic p, 381
of conjugate, 196, 234
of elliptic curve, 365
of monomial map, 327
in characteristic p, 380
of power map, 327
in characteristic p, 380
of projective line, 10
of zd, 234
order two, 198, 200
points stabilized by, 425, 434
polynomial map with nontrivial,
234
rational map, 196
symmetric group, 197
trivial for most maps, 199, 234
{z± 1 },205,234
Backward orbit
dense for shift map, 314
dense in Berkovich Julia set, 311
dense in Julia set, 267
equidistributed, 315
equidistributed for shift map, 315
of rational map, 109, 142
bad reduction
becomes good reduction, 218
Berkovich Julia set of map with,
307,311
complex dynamics has, 294
composition of maps, 77
composition of maps can be good,
77
Green function of map with, 294,
318
of rj;(P) is not ¢(P), 77
product of primes of, 221
Baire category theorem, 268
Baker's theorem, 107, 155
Banach algebra, 298
Baragar's theorem, 443
475
baselocus,406,408
basin of attraction, 74, 138, 140, 141,310
immediate, 310, 311
Benedetto's theorem, 284-286
Berkovich affine line, 301
as ringed space, 304
contained in Berkovich projective
line, 303
contains Cp, 302, 304, 323
Gel'fond topology restricts to Cp
topology, 304, 323
has no Gauss point, 302
is Hausdorff, 304
is locally compact, 304
is tree, 302
is uniquely path connected, 304
no Type-IV fixed points, 321
polynomial map, 304
Berkovich disk, 295
annulus, 302
base of open sets, 300
big model, 323
branch, 300
closed branch, 300
closed of radius R, 297, 30 I
closed unit, 297
contains unit disk in Cp, 295, 301,
323
dynamics of z 2 , 307
Gauss point, 297, 301
Gel'fond topology
restricts to Cp topology, 301, 323
Gel'fond topology, 300, 322
good reduction polynomial map,
321,322
Hsia kernel, 323
is compact, 295, 300, 304
is connected, 295
is Hausdorff, 300, 304
is metric space, 295
is path-connected, 300
is set of bounded seminorms, 297
is set of seminorms, 296
is uniquely path connected, 304
Julia set of good reduction map, 322
line segment from Gauss point to
Type-IV point, 322
line segment in, 298
neutral fixed point, 322
476
Berkovich disk (continued)
of radius R, 297, 301
open branch, 300
potential theory, 323
radius of Type-IV point is positive,
296
seminorm associated to each type of
point, 297
small model, 323
topology on, 299
union of annulus and open branch,
302
visualizing, 298
Berkovich Julia set
backward orbit dense in, 311
equals closure of repelling periodic
points, 311
has empty interior, 311
is connected or infinitely many
components, 311
is perfect set, 307, 311
is uncountable, 307, 311
Berkovich projective line, 301, 302
as homogeneous two-variable
seminorms, 303, 305
as ringed space, 304
attracting fixed point, 324
basin of attraction, 31 0
branch at infinity, 302
canonical measure, 306
support of, 307
contains Berkovich affine line, 303
contains lP' 1 (Cp), 304,323
domain of quasiperiodicity, 310,
311
dynamics of z 2 , 307
Fatou set, 306, 311
contains classical, 306
Gel'fond topology restricts to Cp
topology, 304, 323
immediate basin of attraction, 310,
311
is compact, 304
is Hausdorff, 304
is uniquely path connected, 304
Julia set, 306
contains classical, 306
equals closure of repelling
periodic points, 311
Index
is nonempty, 307
of bad reduction map, 307, 311
of good reduction map, 307, 311,
322
Monte! theorem, 310, 311
rational map, 305
recurrent point, 31 0
Berkovich space, 294
attracting fixed point, 324
attracting periodic point, 31 0
basin of attraction, 31 0
canonical measure, 6, 306
support of, 307
connectivity of, 276
different centers yield same point,
295
domain of quasiperiodicity, 310,
311
dynamics of z 2 , 307
dynamics on, 304
Fatou set, 306, 311
fixed Gauss point, 321
Gauss point, 297
Gel'fond topology, 322
good reduction polynomial map,
321,322
Hsia kernel, 323
immediate basin of attraction, 310,
311
invariant measure, 104
is set of seminorms, 296
Julia set, 3 06
equals closure of repelling
periodic points, 311
is nonempty, 307
of bad reduction map, 307, 311
of good reduction map, 307, 311,
322
Laplacian of local canonical height,
104
line segment from Gauss point to
Type-IV point, 322
Monte! theorem, 310, 311
neutral fixed point, 322
path metric, 306, 323
potential theory, 323
radius of a point, 296
radius of Type-IV point is positive,
296
Index
Berkovich space (continued)
recurrent point, 31 0
repelling periodic point, 310
rigid analytic space, 298
seminorm associated to each type of
point, 297
topology on, 299
Type-I-IV points, 295
Bezivin's theorem, 274, 275
Bezout's theorem, 428
bicritical rational map, 233
bidegree,405,410
bifurcation point, 165
bifurcation polynomial, 165
is a power, 165, 226, 227
big model ofBerkovich disk, 323
big-Oh notation, 93
bihomogeneous polynomial, 405,410
binomial theorem, 206, 261
birational map on Markoff variety, 437
Bogomolov conjecture, 129
dynamical, 129
Borel probability measure, 127
bounded height
finitely many points of, 86, 407
is finite set, 408
Misiurewicz points have, 167
bounded seminorm
automatically nonarchimedean, 321
properties, 321
set of is Berkovich disk, 297
bounded seminorm on Cp[z], 296
branch at infinity in !P' 8 , 302
branch of Berkovich disk, 300
Brauer group, 204
Brauer-Siegel theorem, 368
Brolin measure, 307
bulb, 166
Canonical divisor, 214
canonical height, 97, 99, 100, 287
algorithm to compute, 99, 318
associated to an eigendivisor class,
104
commuting maps have same, 13 7
conjugation invariance, 137
finitely many points of bounded,
423
477
infinitely many points of bounded,
435
is sum of Green functions, 290, 318
Lehmer's conjecture, 101, 138
local, 102, 291, 320
local for a polynomial map, 140,
141
lower bound for, 101, 137, 138,221
normalization conditions, 422
normalized local, 141
of point in Fatou or Julia set, 138
on abelian variety, 409
on elliptic curve, 409
on K3 surface, 421, 423, 426, 435,
436
with three involutions, 438
product h+ h-, 435
quadratic map, 137, 138
regular affine automorphism, 431
sum of local canonical heights, 103,
293
Tate construction, 97
transformation formulas, 422, 432
uniqueness, 98
zero iff finite orbit, 423, 431
zero iff preperiodic point, 99
zeta function, 436
canonical measure, 127, 306
on Berkovich space, 306
support is Julia set, 307
supported at Gauss point, 307
cardiod, 166
Cauchy estimate for norm of derivative,
252,313
Cauchy residue theorem, 20, 248, 314
Cauchysequence,98,240,296
chain rule, 18, 360
calculate multiplier with, 19,47
change of variables, 11
chaos, 3,22
Chebyshev polynomial, 29, 30, 95, 329
algebraic properties, 41, 329
alternative normalization, 30, 331
automorphism group, 332, 336
in characteristic p, 381
binomial coefficient identity, 380
commutativity, 41, 329, 332, 378
defining equation, 30
derivative identity, 381
478
Chebyshev polynomial (continued)
derivative in characteristic 2, 381
differential equation characterizing,
334
dynamical properties, 41
even, 41, 329
explicit formula, 41, 329
fixed point, 332, 380
in characteristic p, 381
in characteristic 2, 381
in characteristic p, 380, 381
is monic of degree d, 329
Julia set, 30, 41, 336
multiplier of fixed point, 332, 380
in characteristic p, 381
multiplier summation formula, 380
not conjugate to monomial, 381
odd,41,329
periodic point, 41
recurrence relation, 41, 329
twist, 336
uniform boundedness of periodic
points, 137
value ofT2, T3, T4, 41
chordal metric, I 02
complex, 11, 144
dehomogenized form, 11
distance between closed disks, 268
effect of linear fractional
transformation, 76
equals distance on Riemann sphere,
11,35
invariant maps for, 35
inversion is isometry, 35
Lipschitz property satisfied, 11, 24,
36,56
on inverse image of rational map,
115, 144
related to Euclidean distance, 119
relation to cross-ratio, 72
resultant measures expansion, 56,
76
v-adic, 45, 46, 144, 243
value is between zero and one, 11
chordal sup norm, 269
circle, 242
is open and closed, 243
class field theory, 202, 204
class number, 350
Index
of quadratic imaginary field, 368
closed branch ofBerkovich disk, 300
closed disk, 242
closed under addition, 243
distance between, 268
equivalent nested sequences, 322
Gel'fond topology restricts to Cp
topology, 301, 323
image of D(O, 1) by PGL2, 317
image under inversion, 321
in lP' 1 , 243
is open, 243
is ring of integers, 243
nested sequence, 295, 322
radius of image by polynomial, 321
rational radius, 243, 278, 295
standard collection, 277
CM, see complex multiplication
coarse moduli space, 160
coboundary,201,202,209
trivial when extended, 203
cocycle, 201, 202
cohomologous, 202
determines field of moduli, 209
PGL2 gives twist oflP' 1 , 211
cocycle corresponds to twist?, 203
Cohn's theorem, 443
cohomology group, 202
maps to Brauer group, 204
twists map to, 202, 236
cohomology set, 202
commutative algebraic group, 375
universal cover, 375
commuting maps, 8
Chebyshev polynomial, 332
Fatou set, 378
have same canonical height, 13 7
have same preperiodic points, 38
higher-dimensional, 378
Julia set, 378
Lattes, 382
monomial, 325, 326, 380
polynomial, 378
power map, 326, 380
commuting rational map, 378
complete field, 240
circle, 242
closed disk, 242
Cp, 242
Index
complete field (continued)
extension, 241
holomorphic function, 244
open disk, 242
power series converges iff an ---+ 0,
244
projective space compact iff K
locally compact, 243
completely invariant set, 17, 258, 266
boundary of Julia set, 23
closed, 267
component at infinity, 40
contains Berkovich Julia set, 311
Fatou set, 23, 40, 266
finite, 16, 266
Julia set, 23, 40, 266
completion of a field, 83
valued, 241
complex chordal metric, 11, 144
complex multiplication, 32, 341, 348
by Z[p], 341
by Z[i], 341
degree of an endomorphism, 349
fractional ideal gives curve with,
350
ideal class group, 350
is quadratic imaginary, 349
complex orbifold, 176
complex projective line, see projective
line
complex torus, 33
fundamental domain, 33, 127
conformal map ofMandelbrot set
complement, 167
conjugacy class in space of rational maps,
174
conjugation, 11
canonical height is invariant, 13 7
linear, 11, 173
new coordinate functions, 218
connectivity
Berkovich space, 276
disk, 276
Julia set, 26, 165
Mandelbrot set, 167
rigid analytic, 276
continuity, 22
equi-, 254,264,265,271,306,313
uniform, 39, 254, 313
479
continuous function, 10, 253
on metric space, 22
convex hull, 249
coordinate ring, see affine coordinate ring
counting lenuna, 425
Cp, 239
bounded seminorm on on Cp [z],
296
Cauchysequence,296
compact Julia set example, 275
is algebraically closed, 239, 242
is complete, 242
is contained in Berkovich affine
line, 302
is totally disconnected, 239, 276,
294
not locally compact, 239, 294
projective space is bounded, 243
projective space not compact, 243
unit disk contained in Berkovich
disk, 295, 301, 323
valuation, 249
value group is IQ, 294
critical point, 12, 284, 353
bounded orbit, 26, 165
distinct modulo p, 237
image under automorphism, 234
Julia set with no, 285
number of for rational map, 14, 37
number of for separable map, 37
ofLattes projection, 384
orbit for quadratic map, 26, 165
periodic, 38
postcritical set, 280
ramification index, 12, 36
rational map with two, 233
reduction is if point is attracting, 78
repelling, 166
strictly preperiodic, 26, 166
tamely ramified, 37
totally ramified, 12
unbounded orbit, 26
whose reduction is periodic, 78
wildly ramified, 37
critical value, 353, 382
distinct modulo p, 237
cross-ratio, 71, 132, 348
j-invariant in terms of, 348
permutation invariant, 79
480
cross-ratio (continued)
relation to chordal metric, 72
cubes, sum of two, 141
curve
cubic, 30
function with given divisor, 214
genus, 15, 37
of genus at least two, I 07
quotient by finite group, 161
Riemann-Roch theorem, 214
twist, 200, 211, 215, 236
cuspidal cubic map, 193
cycle ofn-periodic points, 150,226
cyclic automorphism group, 204
cyclic group, 197
cyclotomic extension, maximal, 95
cyclotomic field, 69, 123
dynamical analog of, 123
Frobenius, 130
Galois group, 129
cyclotomic polynomial, 39, 148, 224
resultant with multiplier
polynomial, 226
cyclotomic unit, 69, 72, 133, 328
action of Galois, 133
Degenerate point for involution on K3
surface, 415
degenerate quadratic map, 194
degree
map on divisor class group, 405
map on divisor group, 405
map on function fields, 44
map on Picard group, 405
of algebraically stable morphism,
396
of composition of affine
morphisms, 392
of divisor, 339, 405
of endomorphism, 349
of family of morphisms, 171
of flexible Lattes map, 357
of inverse of affine automorphism,
390
of isogeny, 340
of iterate of affine automorphism,
390
ofLattes map, 34
of morphism, 389
Index
of polynomial, 389
ofrationalmap, 10,89,388,389
two map induces involution, 411
universal map of given, 171
degree 2 map induces involution, 436,
437
derivative
formal, 12, 37, 313
norm of, 252, 313
of Chebyshev polynomial, 381
in characteristic 2, 381
dessins d'enfant, 217
diagonal map, 150, 216, 226
differential equation characterizes
Chebyshev polynomial, 334
differential form used to define
multiplier, 19
dihedral group, 197,217,234,327
infinite, 419, 435
dimension of family ofK3 surfaces, 412,
438
Diophantine approximation, l 04
Baker's theorem, 107
Dirichlet's theorem, 104, 141
effective version, l 07
Roth's theorem, 104, 108
Diophantine equation, 4, 104, 202
Pell, 106
Thue's theorem, 105
Dirac measure, 128
Dirichlet's theorem on Diophantine
approximation, 104, 141
disconnected Julia set, 26
discrete dynamical system, I
discrete periodic point, 428
discrete valuation, 44
discriminant, 132, 134, 146, 368
elliptic curve, 30, 337, 343, 370,
385
minimal, 221, 342
of dynatomic polynomial, 226, 227
disk, 242
Berkovich, 295
closed, 242
closed under addition, 243
connected, 317
distance between, 268
image of unit disk by PG L2, 317
image under inversion, 321
Index
disk (continued)
in JID 1 , 243
in topological space, 277, 317
nested sequence, 295
open,242
polynomial sends to disk, 312
power series sends to disk, 252, 305
radius of image by polynomial, 321
rational radius, 243, 248, 252, 278,
295
Siegel, 28
standard collection, 277
sup norm of function on, 247
disk component, 2 77, 317
image may not be disk component,
283
is maximal disk connected set, 317
is open, 317
map on induced by rational map,
283
no wandering, 284
nonarchimedean has simple form,
278
of(z 2 - z)/p,283
open set is disjoint union of, 277,
317
periodic, 283
preperiodic, 283
same as path-connected component
over C, 277
set of, 283
wandering, 283
disk connected set, 276, 317
maximal is disk component, 31 7
disk domain, no wandering, 284, 285
distance
between closed disks, 268
between images via linear fractional
transformation, 269, 316
logarithmic, I 02
division polynomial, 362, 383
resultant, 383
divisor, 338, 403
action of involution, 418, 419, 438
ample, 406
associated to a rational function,
339
associated vector space L(D), 406
baselocus,406,408
481
degree of, 339, 405
effective, 406
positivity of height, 408
group of principal, 339
intersection on K3 surface, 434
is difference of effective, 406
linear equivalence, 404
linearly equivalent give same
height, 408
local ring, 403
of a rational function, 320, 403
of degree zero, 405
on K3 surface, 418
positive, see effective divisor
principal, 320, 403, 405
principal iff degree and sum are
zero, 339, 409
pullback,404
pullback of linearly equivalent, 405
pushforward, 404
rational map associated to, 406
Riemann-Roch theorem, 214
support, 403
very ample, 406
divisor class
ample, 406
base locus, 406
very ample, 406
divisor class group, 403
degree map, 405
exactsequence,404
height map on, 408
pullback homomorphism, 405
divisor group, 339, 403
degree map, 339,405
summation map, 339
domain of quasiperiodicity, 310, 311
domain, no wandering, 276, 284, 285
Douady-Hubbard theorem, 167
double cover, 410
of elliptic curve, 347
Drinfeld module, 6
dual isogeny, 340, 352
duplication map
elliptic curve, 32, 338, 351, 370
on Qj'll, 168, 229
dynamical Bogomolov conjecture, 129
dynamical degree, 397, 428, 433
dynamical Galois equidistribution, 128
482
dynamical Lehmer conjecture, 101, 138
dynamical Manin-Mumford conjecture,
127
dynamical system, I
abstract, I
algebraic, 6
discrete, I
integrable, 429, 430
iteration of arithmetic functions, 6
on Drinfeld modules, 6
on finite field, 2, 5
on finite set, I
on foliated spaces, 6
on function field, 5
on group, 2
on metric space, 3
on topological group, 8
on topological space, 3
reversible, 429
symbolic, 258, 314
dynamical unit, 69
action of Galois, 133
for power map, 328
generic monic polynomial, I46
generic rational map, 146
polynomial map, 71, 132
quadratic map, 134, 145, 146
rational map, 71, 72, 79, 132, 133
dynamically affine morphism, 376
dynatomic curve, 156, 157, 161
action of z 2 +con, 161
genus, 164,230
is coarse moduli space, I 60
is fine moduli space, 230
is irreducible, 164
is modular curve, 158
map X1(2)---+ X 0 (2), 162
map X1(3)---+ Xo(3), 230
points solve moduli problem, 159
quadratic polynomial, 156
quotient, 161
Xo(4),Xo(5),Xo(6),230
X1(l),X1(2),X1(3) are rational,
157
X1(4),X1(5),X1(6),230
dynatomic divisor, 150
dynatomlc field, 123
action of Galois, 124
Index
Galois group is subgroup of wreath
product, 125
quadratic polynomial, 123
ramification, 129, 131
unit, 129
dynatomic polynomial, 39, 148, 149, 181
associated multiplier polynomial,
225,227
dehomogenized, 149
discriminant, 226, 227
for z 2 + c, !56
generalized, 227, 229
homogeneous of degree vd(n), 181
is a polynomial, 151, 226
of z + 1/ z, 224
of z 2 + bz, 182
order at P (aj,(n)), 151,225,226
quadratic map, 39
reducible, 235
resultant of two, 165, 226, 227
dynatomic 0-cycle, 150, 226
is positive, 226
Effective divisor, 406
positivity of height, 408
eigenvalue, 421
involution, 438
elementary symmetric polynomial, 87,
180
elimination theory, 417
elliptic curve, 30, 336, 409
addition law, 31
automorphism group, 340, 341
explicit description, 342
canonical height, 409
complex multiplication, 32, 341
complex uniformization, 33, 127
critical values of quotient, 353
defined over K, 337
degree 2 map is even, 382
degree map on divisors, 339
degree of an endomorphism, 349
degree of an isogeny, 340
determined by critical values of
double cover, 347
discriminant, 30, 337, 383
division polynomial, 362, 383
divisor group of, 339
Index
elliptic curve (continued)
divisor is principal iff degree and
sum are zero, 339,409
divisor of a function, 339
divisor on, 338
dualisogeny,340,352
duplication formula, 338
duplication map, 32, 351, 370
endomorphism ring, 31, 340, 341,
378
field generated by torsion, 344, 363
flexible Lattes map, 355
fundamental domain, 33, 127
geometric points, 337
global minimal Weierstrass
equation, 343, 371
good reduction, 58, 62, 342, 343,
362,383
good reduction implies reduction is
homomorphism, 342
group law, 337
explicit formulas, 338
is algebraic, 338
Haar measure, 127
Heeke correspondence, 369
height on, 409
ideal class group maps to CM, 350
identity element, 337
integer point, 372
integral j-invariant, 370
invariant differential, 344, 359, 367
inverse of a point, 337
is a pair (E, 0), 337
is abelian group, 338
isogeny, 339
is homomorphism, 340
is unramified, 340, 353
isomorphic to Picard group, 339
isomorphism class, 337
j-invariant, 337
kernel of endomorphism, 343
Lattes map, 32, 97, 351
preperiodic points, 41, 352
level curve of integrable map, 430
£-series, 409
minimal Weierstrass equation, 222,
342
modified group law, 382
483
morphism is isogeny and
translation, 339
m-torsion subgroup, 343
multiplication map, 31, 112, 340,
383
effect on height, 409
kernel, 343
noncommutative endomorphism
ring,378
nonconjugate Lattes maps, 354
in characteristic 2, 383
nonsingular, 337
normalized embedding of
endomorphism ring, 345
point at infinity, 336
potential good reduction, 371
principal divisor, 339
has degree zero, 339
quasiminimal Weierstrass equation,
371,385
quotient by automorphism group,
346,365
rational point, 337
reduction modulo p, 342
reduction of torsion points, 62, 343,
364
subgroup of K -rational points, 338
summation map on divisors, 339
supersingular, 378
Szpiro conjecture, 221
Tate's algorithm, 342
torsion equidistributed, 127
torsion point, 41
torsion subgroup, 32, 343, 352
twist, 198, 341, 374
uniform bound for integer points,
372
uniform boundedness of torsion, 97,
369
universal cover, 346
Weierstrass !:J function, 34, 345
Weierstrass class, 237
Weierstrass equation, 30, 336, 409
elliptic modular curve, 163
endomorphism ring
elliptic curve, 340, 341, 378
kernel, 343
multiplicative group, 325, 378
noncommutative, 378
484
endomorphism ring (continued)
normalized embedding in C, 345
otder in quadratic imaginary field,
341
order in quatemion algebra, 341
entropy, 6
algebraic, 397
equicontinuity, 3, 22, 254, 264, 265, 271,
313
Fatou set is maximal open, 22
is not open condition, 40, 254
on a metric space, 22
topological, 306
equidistribution, 6
dynamical Galois, 128
Galois of a sequence, 128
ofpreperiodic points, 128
of small height points, 128
torsion on elliptic curve, 127
equi-Lipschitz, 313
equivalence relation from group action,
234
Eremenko's theorem, 378
ergodic theory, 6
essential singularity, 23
etale map, 353
Euclidean distance versus chordal metric,
119
Euler totient function, 137, 164, 182
iteration of, 6
exactperiod, 1, 18,38,150
condition for larger formal period,
151,165,226
rational map has points of different,
154, 155, 196
exceptional set, 16, 17, 266
exercise
hard(
textbf*), 7
unsolved (**), 7
extension formula, 83, 84
Faltings' theorem, 107,442
family of maps, 171, 194
Fatou domain, 240
Fatou set, 22, 255
all oflP' 1 , 26, 59, 239
Berkovich, 306, 311
canonical height of points in, 138
Index
classical contained in Berkovich,
306
classification of connected
components, 28
complement of Julia set, 22
component at infinity is completely
invariant, 40
contains attracting periodic points,
40
contains nonrepelling periodic
points, 256
empty, 26, 35, 361
Herman ring, 28
is completely invariant, 23, 40, 266
is nonempty, 256,285, 314
is open, 22
Julia set is boundary, 25
local canonical height of points in,
141
no wandering disk domains, 284,
285
no wandering domains, 28
nonarchimedean, 254
number of connected components,
27
of an iterate, 24, 255
of commuting maps, 378
of(z 2 - z)/p,283
of zd, 22
parabolic component, 28
polynomial cannot have Herman
ring, 28
recurrent critical point is periodic,
284,317
set of disk components, 283
Siegel disk, 28
Fermat's little theorem, 2, 252, 275
Fermat's sum of two squares theorem,
200,236
field
complete, 240
completion, 241
extension of complete, 241
finite, 2
of characteristic p, 37
separable extension, 37
valued, 44, 240
field of definition, 206
abelian variety, 217
Index
field of definition (continued)
cocycle is coboundary, 209
contains field of moduli, 208
covering map, 217
intersection of all is field of moduli,
236
not equal to field of moduli, 208,
213,236
of algebraic family, 159
of morphism, 89
of point, 86, 122, 145
of quotient variety, 161
field of moduli, 177, 207
abelian variety, 217
cocycle, 209
contained in field of definition, 208
covering map, 217
equals field of definition for
polynomials, 215
equals field of definition if degree is
even, 215
from image in Md, 207
is intersection of all fields of
definition, 236
not equal to field of definition, 208,
213,236
of a point, 145
filled Julia set, 74, 140
Green function, 140
fine moduli space, 160, 172, 230
finite field
dynamical system on, 2, 5
function field over, 5
finite invariant set, 16, 266
finite subgroup ofPGL2, 197
fixed point, 18
attracting in Berkovich space, 324
flexible Lattes map, 382
Galois conjugates, 180, 184
Galois group, 427
index summation formula, 20, 38,
255,314,380,382
Lattes map, 34
multiplier, 18, 47, 180
at infinity, 38
neutral, 322
no Type-IV, 321
nonrepelling, 255, 314
in Fatou set, 256
485
of affine automorphism, 427
of Chebyshev polynomial, 332, 380
in characteristic p, 381
of Type I, II, III, IV, 322
rational, 427
rational map, 149
real, 427
repelling in Julia set, 256
residue index, 38
rigid Lattes map, 366, 384
summation formula, 20, 38,255,
314,380,382
symmetric polynomial of
multipliers, 180
totally ramified, 16, 17,37
flexible Lattes map, 187, 355
composition of, 382
degree, 357
fixed points, 382, 384
good reduction, 362, 383
multiplier, 358, 366, 382, 384
spectrum, 356
multiplier at oo, 383
periodic points, 358, 382, 384
FOD, see field of definition
foliated space, 6
FOM, see field of moduli
formal derivative, 12, 37,313
formal group, 67
formal multiplier spectrum, 182, 183
of z 2 + bz, 182
of zd, 182, 183
formal period, 18, 39, 150
algebraic family induces map to
dynatomic curve, 159
algebraic family of points of given,
159
condition for smaller primitive
period, 151, 165,226
moduli problem, 158
vd(n) points of formal period n,
150,224
formal power series, 5
periodic points over ring of, 78
forward orbit, 1, 108, 280, 353
fractional ideal, 350
exactsequence,404
height formula, 136
486
fractional linear transformation respects
reduction, 50
free product of cyclic groups, 419, 435
Frobenius map, 351
cyclotomic field, 130
periodic points are attracting, 78
function
analytic, 244
equicontinuous family, 254, 264,
265,271,306,313
equi-Lipschitz family, 313
holomorphic, 244
meromorphic, 244
uniformly continuous family, 39,
254,313
uniformly Lipschitz family, 39, 254,
256,264,265,271,313,315
function field, 5
absolute values on, 44
dynamics, 5
functoriality ofheight, 407, 421
fundamental domain, 33, 127
Galois cohomology, 205
Galois conjugate, measure supported on,
128
Galois equidistributed sequence, 128
Galois equidistribution, 128
Galois group
action on dynatomic field, 124
action on PGL2, 203
action on projective space, 122
action on rational map, 207, 211
cyclotomic field, 129
of dynatomic field, 123, 125
of field generated by periodic
points, 122
of fixed point, 427
orbit of conjugate point, 137
profinite topology, 211
subgroup associated to rational
map, 207
subgroup of wreath product, 125
Galois invariance of height, 85
Galois-invariant orbit on K3 surface, 436
Galois theory, 122
Gauss norm, 296
associated Gauss point, 297
is sup norm, 296
Index
on ICp[z], 296
on Tate algebra, 298
Gauss point, 297
fixed by polynomial, 321
is Julia set of good reduction map,
307,322
line segment to Type-IV point, 322
ofBerkovich disk of radius R, 301
Gel'fond metric, 306
Gel'fond topology, 300, 322
base of open sets, 300
not induced by path metric, 306,
323
restricts to ICp topology, 301, 304,
323
general linear group, 10
generalized dynatomic polynomial, 227
for quadratic map, 229
generalized Weierstrass equation, 336
generating function for degree of rational
map, 397, 428
generic fiber, 77
generic rational map
dynamical unit, 146
not very highly ramified, 231
genus, 15,37
of Xo(n), 164
of X1 (4), X1 (5), X1 (6), 230
of X1(n), 164
geometric invariant theory, 176
geometric point, 337
geometric quotient, I 79
GL2, 10
global minimal model, 222
implies trivial Weierstrass class,
223
overQ, 236
global minimal Weierstrass equation,
343,371
difference from quasiminimal, 385
global S-minimal model, 237
good reduction, 43, 58, 218
abelian variety, 442
attracting periodic points, 64
Berkovich Julia set of map with,
307,311,322
composition of maps, 59, 77
degree of reduction is same, 58
elliptic curve, 342, 362, 383
Index
good reduction (continued)
finiteness of periodic points, 66
from bad reduction, 218
Green function of map with, 289,
294,318
iff resultant is unit, 58, 218
implies reduction is
homomorphism, 342
is everywhere nonexpanding, 59
Julia set is empty, 59, 239
local canonical height for, 103, 141
map on Berkovich disk, 321, 322
nonrepelling periodic points, 64
of ¢>(P) is ¢(P), 59
of a rational map, 58
over power series ring of
characteristic p, 78
period of reduced point, 62, 66, 78
periodic points reduce to periodic
points, 61
potential, 371
scheme-theoretic characterization,
59,61, 77
separable, 64
twists with outsideS, 237
v-adic distance between periodic
points, 69
graph, 150,216,226
Green function, 287, 288
algorithm to compute, 318
complex, 140
defining properties, 289
homogeneity properties, 289
is continuous, 289
is Holder continuous, 318
modified, 318
of bad reduction map, 294, 318
of good reduction map, 289, 294,
318
relation to local canonical height,
291
sum is canonical height, 290, 318
group, 2
action, 234
additive, 30
algebraic, 325
commutative algebraic, 375
dynamical system on, 2
law on elliptic curve, 338
487
multiplicative, 29
multiplicative scheme, 29
periodic points form subgroup, 8
variety PGL2, 170
wreath product, 125
H1 ,202,236
Haar measure, 127
half a loaf, 133
hard problem (
textbf*), 7
harmonic function, local canonical height
is, 140
Hausdorff space, Berkovich disk is, 300
Heeke correspondence, 369
height, 4, 81
absolute, 85
absolute logarithmic, 93
additivity, 408, 421
bounded for ample divisor is finite
set, 408
canonical, 97, 99, 100, 287,409
on K3 surface, 421, 426,435,
436
effect of morphism of degree d, 90,
136,398,407,409
effect of rational map, 388
effectivity of constants, 408
elliptic curve, 409
effect of multiplication map, 409
equals one iff root of unity, 88, 100
field extension formula, 84
finitely many points of bounded, 86,
407,423
for linearly equivalent divisors, 408
fractional ideal formula, 136
functoriality, 407, 421
Galois invariance of, 85
h([u, a, b]) ::; h([u, a])+ h([u, b]),
398,428
in affine space, 397
inequality for jointly regular affine
morphisms, 397, 429
inequality for nonregular affine
automorphism, 429
inequality for regular affine
automorphisms, 399, 431
infinitely many points of bounded,
435
488
height (continued)
is at least one, 84
is well-defined, 84
Lehmer's conjecture, 100, 101, 138
logarithmic, 93
lower bound for, 100, 101, 13 7,
138, 221
machine, 407
map on Picard group, 408
Misiurewicz points have bounded,
167
number of points of bounded, 135,
136,441
on K3 surface, 425
ofarationalmap, 143
of a rational point, 82
of an algebraic point, 84
of isolated periodic points is
bounded,402
of polynomial and its roots, 136,
138
of wandering point for regular
affine automorphism, 432
on K3 surface, 420
onMd, 237
on JP'N X JP'M, 409, 420
on projective space, 407
positivity, 408
product h+ h- of canonical, 435
transformation rules on K3 surface,
420
uniqueness, 408
Wei!, 89, 407
zeta function, 436
height machine, 407
Henon map, 390, 402
number of orbits over a finite field,
430
reversible, 430
Hensel's lemma, 48
Herman ring, 28
Hilbert basis theorem, 90
Hilbert Nullstellensatz, 90, 92
Hilbert theorem 90, 199, 205, 207
Holder continuity of Green function, 318
holomorphic function, 244
coefficients uniquely determined,
312
distance between image disks, 268
Index
dynamics of, 23
equicontinuous family, 254, 264,
313
equi-Lipschitz family, 313
family omitting one point, 264, 315
Fatou set, 24, 255
is continuous, 253
is open, 253
Julia set, 24, 255
Lipschitz bound, 248, 254, 272
maximum modulus principle, 250
Monte! theorem, 264,271,315
Newton polygon, 249
norm,247
norm of derivative, 252, 313
order of, 244
product is holomorphic, 312
reciprocal is holomorphic, 313
roots determined by Newton
polygon, 249
sends disks to disks, 252, 305
uniformly continuous family, 254,
313
uniformly Lipschitz, 254, 264, 265,
271,313,315
zeros are isolated, 244
homogeneous coordinates, 10
homogeneous ideal, 89
homogeneous polynomial, 10
representation of integer by, 105
homogeneous seminorm, 303, 305
homogeneousspace,6
homogenization, 389
Hsia kernel, 323
Hsia's theorem, 264, 265, 271,273
Hurwitz formula, 13,217
hyperbolic component ofMandelbrot set,
166
hyperbolic map, 317
nonarchimedean, 279, 285, 317
lcosohedral group, 197
ideal
algebraic set attached to, 90
attached to an algebraic set, 90
Hilbert basis theorem, 90
homogeneous, 89
radical, 89
ideal class group, 350
Index
ideal class group (continued)
exactsequence,404
map to CM elliptic curves, 350
identity element of an elliptic curve, 337
imaginary field, quadratic, 341
immediate basin of attraction, 310, 311
inclusion-exclusion, 148, 224
indeterminacy locus, 389, 427
dimension, 394
disjoint, 394
of involution on K3 surface, 413
of iterate, 394
indifferent periodic point, 19, 4 7, 326
one implies infinitely many, 314
induced rational map, 389
inertia group, 344
infinite dihedral group, 419,435
integer point
counting on Markoff variety, 437
cutoff for in orbits, 375
in orbit, 3, 108, 109, 112, 142, 143,
145
in orbit of z + 1/ z, 8
on lP' 1 minus 3 points, I 06, 108
on elliptic curve, 3 72
on Markoff variety, 437
orbits with many, 110, Ill, 143,
375
uniform bound in orbits, 112, 372,
385
value ofrational function, 143
integrable map, 429,430
level curve is elliptic curve, 430
level set, 4 31
McMillan family, 443
QRT family, 443
intersection theory, 434
invariant differential, 344, 359, 367
is holomorphic, 345
is translation-invariant, 345
transformation for [m], 345
invariant measure, 307
invariant set, 16
completely, 17, 266
contains Berkovich Julia set, 311
Fatou and Julia sets are completely,
23
finite, 16, 266
inverse function theorem, 115, 144
489
over Cp, 312
inverse of point on elliptic curve, 337
inversion maps disk toj, 321
involution, 410
action on divisor, 418, 419, 438
as linear transform on Picard group,
420
composition is reversible affine
automorphism, 429
degenerate point, 415
eigenvalue of, 421,438
formulas to compute, 433
indeterminacy locus, 413
induced by degree 2 map, 411, 436,
437
is morphism on most K3 surfaces,
417,434,438
noncommuting, 410
not a morphism, 436, 437
on Markoff variety, 436
quotient by, 410
relation between noncommuting,
419,435
surface in lP'N x lP'N with two, 436
three noncommuting, 437
irrational number approximated by
rational number, 104
irrationally neutral periodic point, 19, 47
isogenous elliptic curves, 339
isogeny, 339
degree, 340
dual, 340, 352
is homomorphism, 340
is unramified, 340, 353
isolated point, 402
Julia set contains no, 25
of bounded height, 402
periodic, 402
preperiodic, 8
isospectral family, 186, 188, 356, 362
iterate, coordinate functions of, 149, 226
iteration commutes with conjugation, II
itinerary, 258
as map on sequence space, 260
left shift, 260
Jacobian variety, 369
]-invariant, 221,337
in terms of cross-ratio, 348
490
jointly regular affine morphisms, 397,
429
height inequality, 397, 429
Julia set, 22, 255, 284
algebraic points, 40
all of!P'1 , 26, 35, 361
backward orbit is dense, 25
Berkovich, 306
backward orbit dense in, 311
for bad reduction map, 307, 311
for good reduction map, 307,
311,322
has empty interior, 311
is connected or infinitely many
components, 311
boundary is completely invariant,
23
boundary ofFatou set, 25
canonical height of points in, 138
chaotic behavior, 22
Chebyshev polynomial, 30, 41
classical contained in Berkovich,
306
closure of repelling periodic points,
41
compact, 275
complement ofFatou set, 22
connected, 26, 165
contained in closure of periodic
points, 273
contains repelling periodic points,
40,256
disjoint from postcritical set
closure, 280
effect of strictly preperiodic critical
points, 26
equal to Zp, 275
equals closure of repelling periodic
points, 27, 27 4, 311
filled, 74, 140
Green function for filled, 140
has empty interior, 267
in two disks, 257
is closed, 22
is closure of backward orbit, 267
is completely invariant, 23, 40, 74,
266
is empty for good reduction map,
26,59,239
Index
is nonempty, 25
is perfect set, 23, 25, 267, 307, 311
is uncountable, 267, 307, 311
local canonical height of points in,
141
no critical points, 285
nonarchimedean, 254
nonempty interior, 26
of an iterate, 24, 255
of Chebyshev polynomial, 336
of commuting maps, 378
ofLattes map, 35, 361
of(z 2 - z)/p,262,283,3l5
ofz 2 - 2,40
of zd, 22
of z · · · (z- d + 1)/p, 315
open set orbit omits at most one
point, 266
orbit of open subset, 27
periodic points dense, 23,263,315
smallest closed completely invariant
set, 267
strictly expanding map on, 279, 317
support of canonical measure is,
307
topologically transitive map, 263,
315
totally disconnected, 23, 26
with no critical points, 279
K3 surface, 410, 412
action of involution on divisors,
418,419,438
A orbit, 418
aritlunetic complexity of orbit, 435
canonical height, 421, 423, 426,
435,436
infinitely many points of
bounded,435
normalization conditions, 422
product h+ h-, 435
transformation formulas, 422
zero iff finite orbit, 423
degenerate point for involution, 415
dimension offamily of, 412,438
divisors on, 418
eigenvalue of involution on
divisors, 421, 438
finitely many periodic points, 435
Index
491
K3 surface (continued)
finitely many points of bounded
canonical height, 423
Galois-invariant orbit, 436
H 1 (S, 0 5 ) = 0, 412
height functions on, 420
height transformation rules, 420
height zeta function, 436
homogeneous forms associated to,
412,433
in JP' 1 X JP' 1 X JP'i, 437
indeterminacy locus of involution,
413
intersection theory, 434
involution, 411
as linear transform on Picard
group, 420
example, 411
formulas, 433
is morphism on most, 417, 434,
438
Kodaira dimension, 412
nonsingular, 438
number of points of bounded
height, 425
orbit is Zariski dense, 435
Picard group of general, 418
projections to lP' 2 , 410
relation between noncommuting
involutions, 419, 435
stabilizer of point on, 425, 434
with three involutions, 437,438
Kawaguchi's theorem, 399, 431
Kodaira dimension, 412
Kronecker's theorem, 88, 100
Krylov-Bogolubov theorem, 127
Kummer sequence, 205
£-adic representation, 344
Lang's height lower bound conjecture,
101, 221
Lang's integer points conjecture, 112
Lattes diagram, 365
Lattes map, 32, 97, 186, 187,351
affine minimal, 372, 385
commuting, 378
composition of, 382
critical point of 1r, 384
critical values, 353
cutoff for integral points in orbit,
375
defining equation, 32, 351
degree, 34
fixed points, 34, 366, 382, 384
flexible, 355
for multiplication-by-(1 + i), 32
for multiplication-by-2, 32, 351,
370
on j = 0 curve, 351,381
on j = 1728 curve, 351
good reduction, 362, 383
higher-dimensional, 377
in characteristic p, 366
integral j-invariant, 370
isospectral family, 362
iterate, 382
multiplier, 34, 186, 358, 366, 382,
384
at oo, 383
of fixed point, 382
spectrum, 362
summation formula, 382
nonconjugate, 354
in characteristic 2, 383
to polynomial, 381
periodic points, 186, 358, 366, 382,
384
all attracting, 362
all nonrepelling, 362
expanding, 35
in Julia set, 35, 361
padically
p-adically, 361
postcritical set, 353
postcritically finite, 353
preperiodic points, 32, 41, 352
projection toE ;r, 365
reduced diagram, 365
rigid, 364
set of is monoid, 382
torsion points map to preperiodic
points, 41
translation fixed by r, 365
uniform bound for integral points in
orbit, 372, 385
uniform boundedness of periodic
points, 137
lattice, 33
492
Laurent series, 245
coefficients uniquely determined by
function, 312
of Schwarzian derivative, 232
product of, 312
least period, see exact period
Lefschetz principle, 20, 365
left shift map, 259, 314
as itinerary map, 260
backward orbit dense, 314
backward orbit equidistributed, 315
continuous, 259
is topologically transitive, 259
Lipschitz, 259
periodic points, 259
are dense, 314
properties of, 259
topologically transitive, 314
uniformly expanding, 259
Lehmer's conjecture, 100, 138
dynamical, 101, 138
level curve of integrable map, 430
level set of an integrable map, 431
Lie group, 6
lift, 287
from lFp to Zp, 48
of affine morphism, 389
of rational map, 287, 389
line bundle, metrized, 41 0
line segment
from Gauss point to Type-IV point,
322
in Berkovich disk, 298
of Newton polygon, 249
linear conjugation, 11, 173
commutes with iteration, 11
linear equivalence, 403, 404
divisors give same height, 408
pullback preserves, 405
linear fractional transformation, 10
action of Galois, 203
chordal sup norm, 269
distance between images, 269, 316
effect on chordal metric, 76
image of unit disk, 317
Lipschitz constant, 36
move three points, 36
on IP'N, 226
resultant of, 76
Index
linear group
general, 10
projective special, 175
special, 17 5
Lipschitz, 11, 24, 36, 56
equi-, 313
holomorphic function, 248, 254,
272
shift map, 259
uniform family, 39, 254, 256, 264,
265,271,313,315
local canonical height, 102, 291
associated to a divisor, 320
associated to an eigendivisor class,
104
computation of, 103
existence proof, 291
for a polynomial map, 103, 140,
141
good reduction case, 103, 141
Green function and, 291
is harmonic, 140
Laplacian is invariant measure, 104
normalized, 141
of point in Fatou or Julia set, 141
properties of, 102, 291, 320
sum to global canonical height, 103,
293
transformation law, 320
local degree, 83
local ring, 67, 79
at a divisor, 403
normalized valuation, 403
locally compact, 74, 239, 243, 304
Cp is not, 239, 268, 294, 295
locally constant function, 318
locus of indeterminacy, 389
affine automorphism, 392
affine morphism, 392
algebraically stable morphism, 396
jointly regular affine morphisms,
397,429
logarithmic distance, 102
logarithmic height, 93
absolute, 93
logarithmic singularity, 140
long exact sequence, 205
Luroth's theorem, 158
Lyubich measure, 307
Index
Mahler measure, 138
Mandelbrot set, 26, 165
analytic description ofMisiurewicz
points, 229
bulb, 166
complement conformal to unit disk
exterior, 167, 168
component of interior, 166
in disk of radius 2, 166
is connected, 167
Misiurewicz point, 166
analytic description, 168
dense in, 166
in boundary, 166
root of hyperbolic component, 166
spider algorithm, 168
uniformization, 167, 168
Manes's theorem, 235
Manin-Mumford conjecture, 127
dynamical, 127
Markoff equation, 436
integer points, 437
Markoff triple, 437
unicity conjecture, 437
variety is rational, 437
Masser-Oesterle conjecture, 372, 373
maximum modulus principle, 248, 250,
252,253,268,296,298,305,
321
false if K not algebraically closed,
252
for rational functions, 250
Mazur's theorem, 97
Mazur-Kamienny-Merel theorem, 369
McMillan family of integrable maps, 443
McMullen's theorem, 187, 367
M2, 188
completion equals lP' 2 , 194
explicit formula for a1, a2, 189
is a rational variety, 232
is a scheme over Z, 189
isomorphic to A2 , 188
locus of polynomial maps, 232
point with Aut( cjJ) = S3, 234
subvariety with Aut = J.L 2 , 235
Md, 174
affine coordinate ring, 176
493
a;
coordinate ring contains n) and
;;.(n) 183 232
' ' '
field of moduli, 207
generic map not very highly
ramified, 231
has dimension 2d - 2, 176
height, 221,237
is a rational variety?, 232
is a scheme over Z, 179, 186
is a unirational variety, 232
is a variety, 17 5
is affine, 17 6
is complex orbifold, 176
is connected, 176
is integral, 176
is nonsingular?, 193
M2 ~ A2 , 189
M3 embedded by a;n>?, 232
map from Ratd, 174, 175
multiplier 1 is Zariski closed subset,
230
not embedded by a;n), 186, 187,
356,367
rational points, 177
semistable completion, 178
stable completion, 178
subvariety with nontrivial
automorphisms, 199,234
twists have same image, 198, 221
M~, 178
has natural quotient property, 179
is a scheme over Z, 179
is geometric quotient, 179
is quasiprojective, 178
isomorphic to M~s iff dis even,
178
M~·. 178
is a scheme over Z, 179
is categorical quotient, 179
is projective, 178
isomorphic toM~ iff dis even, 178
measure, 6, 36
Brolin, 307
canonical on Berkovich space, 129,
306
Dirac, 128
Haar, 127
invariant, 104, 127, 307
Lyubich, 307
494
measure (continued)
spherical, 36
support on Galois conjugates of
point, 128
weak convergence, 128
Merel's theorem, 97, 369
meromorphic function, 244
coefficients uniquely determined,
312
dynamics of, 23
equicontinuous family, 254, 265,
313
equi-Lipschitz family, 313
family omitting two points, 265,
315
Fatou set, 24, 255
is continuous, 253
is open, 253
Julia set, 24, 255
Laurent series, 245
Montel theorem, 265,315
order of, 244
pole, 244
product is meromorphic, 312
uniformly continuous family, 254,
313
uniformly Lipschitz, 254, 313
zero, 244
metric space, 3
chaotic map, 3
continuous function, 22
equicontinuous collection of
functions, 22
equicontinuous map, 3
Fatou set, 22
Julia set, 22
normal family of maps, 25
ofsequences,258,314
path in, 306, 323
SN is a, 258, 314
uniform continuity, 39
uniform Lipschitz, 39
metrized line bundle, 410
minimal discriminant, 221
minimal model, 220, 222
global S-, 237
minimal resultant, 220, 221, 237
exponent at p, 220
is PGL2(K)-invariant, 221
Index
is resultant times power, 222
of conjugate, 219
valuation, 219
minimal Weierstrass equation, 222, 342
global, 343, 371, 385
quasi, 371, 385
Tate's algorithm, 342
Misiurewicz point, 166
analytic description, 168, 229
dense in Mandelbrot set, 166
has bounded height, 167
in boundary ofMandelbrot set, 166
minimal polynomial, 230
no neutral cycles, 166
oftype (m,n), 166
repelling critical orbit, 166
spider algorithm, 168
subhyperbolic, 166
Mobius inversion formula, 148, 224
Mobius 1-L function, 39, 148, 149, 164,
224
modified Green function, 318
modular curve
dynatomic, 157, 158, 161
elliptic, 163, 369
Heeke correspondence, 369
modular Jacobian, 369
moduli space, 4, 26, 147
coarse, 160
degree of symmetric polynomial
map, 187,232,367,384
fine, 160,172,230
isospectral family, 188, 356
M2 ~A?, 188
M2 ~ 1P'2 , 194
map defined by symmetric
polynomials of multipliers,
187,232,367,384
of rational maps of degree d, 174
scheme over Z, 179, 186, 189
moduli, field of, 207
Mobius transformation, see linear
fractional transformation
monoid ofLattes maps, 382
monomial map, 29, 95, 325, 428
automorphism group, 234, 327
in characteristic p, 380
commuting maps, 326,380
dynamical unit, 328
Index
monomial map (continued)
in characteristic p, 380
multiplier of periodic point, 326
periodic point, 326
preperiodic point, 326
twist, 206, 328, 380
monomial, number of given degree, 136
Monte! theorem, 25
nonarchimedean, 264, 265, 271,
281,311,315
ofHsia, 265, 271
ofRivera-Letelier, 311
on Berkovich space, 310, 311
with moving targets, 271
morphism, 89, 388
affine, 375, 388
algebraic family of, 171
bijective =fo- isomorphism, 193
Bogomolov conjecture, 129
canonical height associated to, 99,
287
canonical measure, 127
defined over K, 89
degree of, 389
degree of composition, 392
dynamical degree, 397, 428
dynamically affine, 376
effect on absolute value, 288
effect on height, 90, 136, 398, 407,
409
functoriality of height for, 407, 421
Green function, 288
induced on family, 171
invariant measure, 127
involution on most K3 surfaces is,
417,434,438
is open map, 433
jointly regular, 397, 429
local canonical height, 102
over a variety, 171
pullback of divisor, 404
pullback of linearly equivalent
divisors, 405
pushforward of divisor, 404
Morton-Silverman Conjecture, 96, 135,
368
for zd, 137
for Lattes map, 370
moving targets, 271
495
m-torsion subgroup of elliptic curve, 343
multiplication map
by 2 on QjZ, 168, 229
on elliptic curve, 31, 340, 383
effect on height, 409
kernel of, 343
multiplicative group, 29, 30
endomorphism, 29, 325
endomorphism ring, 378
preperiodic points, 326
quotient by { ±1 }, 29, 328
torsion, 326
multiplicative group scheme, 29
multiplier, 18, 19, 47, 180
at infinity, 38
Lattes map, 383
chain rule used to calculate, 19, 47
Chebyshev polynomial, 41
condition for unequal formal and
primitive periods, 151, 226
conjugacy invariant, 19
differential one-form definition, 19
equal to one, 225, 230
flexible Lattes map, 186, 358, 382
Galois conjugates, 180, 184
integral over Q[Ratd], 180, 185
is PGL 2 invariant, 18, 180, 185
minimal polynomial, 225, 227
not equal to one, 41
of a Lattes map, 34
of critical periodic point, 38
of degree two rational map, 190,
233
of fixed point of Chebyshev
polynomial, 332, 380
in characteristic p, 381
of fixed point ofLattes map, 382
of z 2 + bz, 182
rigid Lattes map, 366, 384
root of unity, 151,226
symmetric polynomial of, 180, 183,
187,232,367
multiplier polynomial, 225, 227
resultant with cyclotomic
polynomial, 226
multiplier spectrum, 182, 187
flexible Lattes map, 356
formal, 182
isospectral family, 188, 356, 362
496
multiplier spectrum (continued)
of Lattes map, 186
of z 2 + bz, 182
of zd, 182
Natural numbers, 258
nth-dynatomic polynomial, 149, 181
homogeneous of degree vd(n), 181
Neron local height, 104
nested sequence of closed disks, 295, 322
associated seminorm, 297, 322
equivalent, 322
neutral fixed point, 322
neutral periodic point, 19, 47
Newton polygon, 248, 249, 252, 275, 276
determines roots, 249
line segment, 249
n-multiplier spectrum, 182
formal, 182
of zd, 182
no wandering domains
disk conjecture, 284
disk theorem, 284, 285
theorem, 28, 276
nonarchimedean absolute value, 44, 83,
242
morphism is open map, 433
nonarchimedean analysis, 242
nonarchimedean Cauchy residue formula,
314
nonarchimedean disk component
has simple form, 278
nonarchimedean hyperbolic map, 279,
285,317
nonarchimedean inverse function
theorem, 312
nonarchimedean chordal metric, see
v-adic chordal metric
noncommutative monoid of Lattes maps,
382
noncommuting involutions, 410
three, 437
nondegenerate automorphism, 226
nonrepelling cycles, number of, 27
nonrepelling fixed point, 255, 314
nonrepelling periodic point, 362
in Fatou set, 256
nonsingular elliptic curve, 337
nonsingular K3 surface, 438
Index
norm,247,288
Gauss on Cv[z], 296
of a holomorphic function, 247
of derivative, 252, 313
semi-, 296
sup,288
normal family, 25
normal forms lemma, 190,233
normalized coordinates, 49
normalized Markoff triple, 437
normalized rational map, 51, 75
normalized valuation, 44, 249
Northcott's theorem, 94, 407,423
false for rational maps, 136
notation, 7
n-period polynomial, 149, 181
homogeneous of degree dn + 1, 181
Nullstellensatz, 90, 92, 408
number field
algebraic closure, 85
completion of, 83
extension formula for absolute
values, 83, 84
product formula, 83
ring of integers, 83
ring of 8-integers, 83
numerical criterion for (semi)stability,
178
0(1), 93
octahedral group, 197
one coboundary, 20 I, 202, 209
trivial when extended, 203
one cocycle, 20 I, 202
corresponds to twist?, 203
determines field of moduli, 209
extension from Aut(¢) to PGL2,
203
of twist, 201
open branch of Berkovich disk, 300
open disk, 242
closed under addition, 243
image of D(O, 1) by PGL2, 317
in IP' 1 , 243
is closed, 243
is maximal ideal, 243
rational radius, 243, 278
standard collection, 278
open map, 10, 253, 312
497
Index
open map (continued)
morphism is, 433
rational map is, 10, 24
open set
base offor Gel 'fond topology, 300
disjoint union of disk components,
277,317
disk connected, 317
iterates omit at most one point, 266
orbit, 1
backward is dense in Julia set, 267
finite iff canonical height zero, 423,
431
Galois-invariant on K3 surface, 436
height zeta function, 436
integral points, 3, 108, 109, 142,
143
construction of many, ll 0, 111,
143,375
cutoff for, 375
uniform bound for, 112, 372, 385
integrality estimate for points in,
ll2, 145
number over a finite field, 430
of critical points, 280
is bounded, 26, 165
is unbounded, 26
of Galois conjugate, 137
of open set omits at most one point,
266
product of multipliers, 19,47
S-integral points in, 143
two sided, 432
under collection of rational maps,
143
Zariski dense on K3 surface, 435
orbit-counting function, 432
orda, 12
order
in quadratic imaginary field, 341
in quaternion algebra, 341
of a holomorphic function, 244
of a meromorphic function, 244
ordp, 82
of a polynomial, 218
Ostrowski's theorem, 44, 312
p, Weierstrass function, 34, 345
p-adic absolute value, 44
p-adic Cauchy residue formula, 314
p-adic hyperbolic map, 279, 285, 317
p-adic inverse function theorem, 312
parabolic component, 28
path-connected, 304
Berkovich disk is, 300
same as disk-connected over C, 277
path metric, 306, 323
does not induce Gel'fond topology,
323
Pell equation, 106
perfect set
Berkovich Julia set is, 307, 311
Julia set is, 267
period
exact, 150
formal, 150
of a point, 1, 18
primitive, 150
period polynomial, 149, 181
dehomogenized, 149
for z 2 + c, 156
generalized, 227
homogeneous of degree dn + 1, 181
multiple root, 230
order at P (ap(n)), 151, 226
periodic disk component, 283
periodic domain, 28
periodic point, 1
algebraic family induces map to
dynatomic curve, 159
algebraic family of with given
formal period, 159
algebraic properties, 3
attracting, 19, 47, 326, 362
for good separable reduction, 64
iff reduction is critical, 78
in Berkovich space, 31 0
in Fatou set, 22, 40
basin of attraction, 31 0
canonical height zero, 99,431
Chebyshev polynomial, 41
closure contains Julia set, 273
closure of repelling equals Julia set,
274,311
condition for unequal formal and
primitive periods, 151, 165,
226
critical, 38
498
Index
periodic point (continued)
critical point with reduction a, 78
cycle of, 150, 226
cyclotomic, 95
dense for shift map, 314
differential one-form definition of
multiplier, 19
discrete, 428
dynamical unit, 69, 71, 72, 79, 132,
133
dynamics on finite set, 7
dynatomic field, 123
equation of, 122
equidistribution, 6
every is attracting, 78
exactperiod, 18,38,150
field generated by, 122
finitely many on K3 surface, 435
finitely many rational, 8
finiteness for regular affine
automorphism, 400
finiteness over number field, 66, 94,
407
finiteness over Q, 3
flexible Lattes map, 358, 382
form subgroup, 8
formalperiod, 18,39,150
moduli problem, 158
Galois theory of field generated by,
122
generalized dynatomic polynomial,
227,229
generalized period polynomial, 227
immediate basin of attraction, 310
index summation formula, 20, 38,
255,314,380,382
indifferent, 19, 47, 326
irrationally neutral, 19, 47
isolated, 402
Julia set is closure of nonrepelling,
27
Lattes map, 186
least period, 18
multiplier, 18, 19,47
at infinity, 38
Lattes map, 35, 186
of critical, 38
power map, 326
nonrepelling, 362
for good reduction, 64
in Fatou set, 256
number of, 39
given period, 122
nonrepelling cycles, 27
of commuting maps, 8, 378
of power map, 326
of quadratic polynomial map, 96
of regular affine automorphism,
394,428
of(z 2 - z)/p,263
of z 2 + 1, 8
of z 2 -1,8
of zd + a over finite field, 8
of z · · · (z- d + 1)/p, 315
on subvariety, 127
one indifferent implies infinitely
many, 314
p-adic, 361
prime period, 18
primitive, 122
field generated by, 123
primitive period, 18, 150
product of multipliers in orbit, 19,
47
rational map can have infinitely
many rational, 136
rational map has points of different
periods, 154, 155, 196
rationally neutral, 19, 47
in hyperbolic component, 28
reduction of, 61, 62, 66, 78
regular affine automorphism, 400
relative abelian extension, 126
repelling, 19, 47, 326, 361
in Berkovich space, 310
in Julia set, 22, 40
limit of attracting, 27 4
repelling in Julia set, 256
residual degree, 310
rigid Lattes map, 366, 384
set of is Galois-invariant, 123
shift map, 259
summation formula, 20, 38, 255,
314,380,382
superattracting, 19, 47
Type-1, 310
Type-II, 310
Type-IV, 321
Index
periodic point (continued)
uniform boundedness conjecture,
96, 135,368
forzd, 137
for Lattf$ map, 370
v-adic distance, 69
periodic subvariety, 127
permutation group, 124
permutation polynomial, 5
Pern(¢), 18
PGL2, 10
action of Galois, 203
automorphism group is subgroup,
234
cocycle gives twist of lP' 1, 211, 215
conjugation is algebraic group
action, 173
equivalence, 195,233
extension of cocycle to Aut(¢),
203
finite subgroups, 197
invariant functions on Ratd, 175
is group variety, 170
is Rat1, 170
map from PSL 2, 175, 231
minimal resultant is invariant, 221
PGL 2(K) equivalence, 195, 233
quotient of Ratd by, 17 4
PGLn, 226
map from PSLn, 231
Picard group, 339, 403
action of involution, 418,419,438
degree map, 405
eigenvalue of involution, 421, 438
exactsequence,404
generated by hyperplane, 405
height map on, 408
intersection theory for K3 surface,
434
isomorphic to elliptic curve, 339
linear transform induced by
involution, 420
of general K3 surface, 418
oflP' 2 x lP' 2 , 418
ofJP'N X JP'M, 405,406,409
pullback homomorphism, 405
point
absolute value, 90, 288
at infinity, 10, 11
499
on elliptic curve, 336
field of definition, 86, 122
polynomial action on Berkovich,
305
radius of in Berkovich space, 296
reduction modulo a prime, 48
sup norm, 90, 288
pole, 244, 403
polynomial
absolute value of, 91,288
bidegree, 405, 410
bihomogeneous,405,410
bounded seminorm on ring of, 296
cyclotomic, 148, 224
degree of, 389
dynatomic, 148, 149, 181
elementary symmetric, 87
Gauss norm, 296
height of roots, 136, 138
homogeneous, 10
homogeneous ideal, 89
local canonical height, 140
normalized, 51
number of monomials, 136
ordp, 218
permutation, 5
resultant of two, 53
sup norm, 91, 288
trihomogeneous,437
polynomial map, 17
cannot have Herman ring, 28
canonical measure on Berkovich
space, 306
Chebyshe~29,95,329
in characteristic p, 3 81
commuting, 378
complete invariance of component
at infinity, 40
condition for unequal formal and
primitive periods, 165
cyclotomic preperiodic points, 95
dynamical unit, 71, 132
of generic monic, 146
effect on absolute value, 288
filled Julia set, 74, 140
finitely many rational periodic
points, 8
fixes Gauss point, 321
FOM = FOD, 215
500
polynomial map (continued)
good reduction, 321, 322
has primitive periodic points of
different periods, 155
iff 0"1 ( ¢) = 2, 232
iterate is a, 17, 37, 431
iterates of open set omit at most one
point, 266
LaW$ map not conjugate to, 381
local canonical height, 103, 140,
141
monomial, 29, 95, 325
multipliers not equal to one, 41
nontrivial automorphism group, 234
number of nonrepelling periodic
cycles, 27
on Berkovich space, 304
power, 95, 325
power map, 29, 95, 234, 321, 325
in characteristic p, 380
on Berkovich space, 321
radius of image disk, 321
sends disks to disks, 305, 312
wandering disk domains exist over
Cp, 286
polynomial type rational map, 143
positive divisor, see effective divisor
positivity of height, 408
postcritical set, 280, 353
potential good reduction, 371
potential theory on Berkovich space, 323
Hsia kernel, 323
power map, 29, 95, 325
automorphism group, 234, 327
in characteristic p, 380
commuting maps, 326, 378, 380
dynamical unit, 328
in characteristic p, 380
multiplier of periodic point, 326
on Berkovich space, 321
periodic point, 326
preperiodic point, 326
twist, 206, 328, 380
power series, 244
coefficients uniquely determined by
function, 312
converges iff an -----> 0, 244
distance between image disks, 268
family omitting one point, 264, 315
Index
formal, 5
maximum modulus principle, 250
Monte! theorem, 264,271,315
Newton polygon, 249
norm of derivative, 252, 313
periodic points in ring of, 78
product of, 312
sends disks to disks, 252, 305
Tate algebra, 297
preperiodic disk component, 283
preperiodic disk domain, 284, 285
preperiodic domain, 28, 276
preperiodic point, 1
canonical height zero, 99
critical, 26, 166
cyclotomic, 95
dynamics on finite set, 8
finitely many, 38
finiteness over number field, 94,
407
Galois equidistribution, 128
generalized dynatomic polynomial,
227,229
generalized period polynomial, 227
isolated, 8
large ramification index implies is,
37
Lattes map, 32, 352
of abelian group is torsion, 2, 41,
326
of commuting maps, 8, 38, 378
of multiplicative group, 326
of power map, 326
on subvariety, 127
reduction of, 61
Type-IV, 321
uniform boundedness conjecture,
96,135,368
for zd, 137
for Lattes map, 370
preperiodic subvariety, 127, 128
prerequisites, 7
prime divisor, 403
degree, 405
local ring, 403
prime is sum of two squares, 200, 236
prime period, see exact period
primitive period, see exact period
primitive periodic point, 122
Index
primitive periodic point (continued)
dynatomic field, 123
field generated by, 123
set of is Galois-invariant, 123
principal divisor, 320, 339, 403
has degree zero, 339
probability measure, 127
Borel, 127
canonical, 127
canonical on Berkovich space, 306
invariant, 127
product formula, 83, 84, 290
profinite topology, 211
projection map, 287
of K3 surface to IP' 2 , 41 0
projective general linear group, 10
projective line
automorphism group, I 0
Berkovich, 302
change of variables, 11
chordal metric, II, 35, 45, 144, 243
construction of, 10
integer points on, I 06, 108
inversion is isometry, 35
linear fractional transformation, 10
morphism of over a variety, 171
moving three points, I 0, 36, 51
not locally compact over Cp, 294
over a variety, 171
point at infinity, 10, 11
rational map, 10
Riemann sphere, 10
Riemann-Hurwitz formula, 13, 217
spherical measure, 36
twist, 200,211, 236
twist trivial iff C (K) # 0, 215
universal map of degree d, 171
projective space
absolute value of a point, 90
action of Galois group, 122
compact iff K locally compact, 243
cross-ratio, 71, 79, 132, 348
degree map on divisors, 405
field of definition of point, 86, 122
finitely many points of bounded
height, 86, 407
height, 82, 84, 407
hyperplane generates Picard group,
405
501
monomial map, 428
morphism, 89, 388
is open map, 433
moving N + 2 points, 75
nondegenerate automorphism, 226
normalized coordinates, 49
number of points ofbounded
height, 135, 136,441
probability measure, 127
productoftwo,405,406,409,418
K3 surface in, 41 0
surface with involutions in, 436
projection map to, 287
rational map, 89, 388
Segre embedding, 406, 409
projective special linear group, see PSL
PSL, 175
map to PGL, 175,231
stable locus, 176
Puiseux series, 5
pullback divisor, 404
linearly equivalent, 405
pushforward divisor, 404
QRT family of integrable maps, 443
quadratic formula, 261
quadratic imaginary field, 341
class number, 350, 368
fractional ideals in, 350
ideal class group, 350
quadratic map
algebraic family
induces map to dynatomic curve,
159
algebraic family of, 159, 194, 230
associated multiplier polynomial,
227
automorphism group contains J1 2 ,
235
automorphism of order two, 96, 235
bifurcation point, 165
bound for periodic points, 96
canonical height, 137, 138
conjugation to z 2 + c, 156
degenerate, 194
dynamical unit, 134, 145, 146
dynatomic curve, 156, 157, 161
dynatomic field, 123
ramification, 131
502
quadratic map (continued)
dynatomic polynomial, 39
explicit formula for a1(¢), 181
generalized dynatomic polynomial,
229
local canonical height, 141
Mandelbrot set, 26, 165
Misiurewicz point, 166
minimal polynomial, 230
orbit of critical point, 26, 165
parameter space has dimension
three, 168
polynomial iff a1 (¢) = 2, 232
rational periodic points, 137
reducible dynatomic polynomial,
235
space of conjugacy classes has
dimension one, 168
with preperiodic critical point, 166
quadratic twist, 198
rational map, 200
quasiminimal Weierstrass equation, 371
difference from global minimal, 385
quaternion algebra, 341, 378
quotient curve, 161
critical values, 353
dynatomic, 161
elliptic, 346
quotient variety, 410
by finite group, 161
by infinite group, 161, 174
categorical, 179
field of definition, 161
geometric, 179
Ratd modulo PSL2, 175
rational points on, 177
semistable, 179
stable, 179
Radical of an ideal, 89
radius
of Berkovich disk, 297, 30 I
of Berkovich point, 296
ofType-IV point is positive, 296
ramification
field generated by torsion on elliptic
curve,363
in dynatomic field, 129, 131
Index
in field generated by torsion on
elliptic curve, 344
ramification index, 12, 36, 340
automorphism invariant, 234
divisibility by p, 37
effect on chordal metric, 115, 144
large implies preperiodic, 37
of iterate, 117
ofLattes projection, 384
sum of over inverse image of a
point, 15
ramification point, see critical point
Rat1 is PGL2, 170
Rat2, 170, 174
normal forms lemma, 190,233
subvariety with Aut = JL 2, 235
Ratd, 169
affine coordinate ring, 169, 174
coordinate ring contains a~ n) and
~~n), 183, 232
generic map not very highly
ramified, 231
geometry of boundary, 170
in PSL 2-stable locus, 176
is a rational variety, 232
is subset oflP'2d+ 1, 169
map induced by Schwarzian
derivative, 232
maptoMd, 174,175
multiplier 1 is Zariski closed subset,
230
PGL2-invariant functions on, 175
quotient by PGL2, 174
quotient by PSL 2 is a variety, 175
semistable rational maps, 178
stable rational maps, 178
subvariety with nontrivial
automorphisms, 199,234
universal map over, 171
Ratd, 178, 179
Ratd•, 178, 179
rational automorphism, 430
rational function, 9
divisor, 320, 403
finitely many nonzero residues, 314
formal derivative, 12, 313
induces rational map, 10
integer values of, 106, 108
Index
rational function (continued)
maximum modulus principle, 250
pole, 403
principal divisor, 320, 403
residue, 313
Schwarzian derivative, 231
space of associated to a divisor, 406
Taylor series, 12
zero, 403
rational map, 10, 89, 388
action of automorphism on critical
point, 234
action of Galois, 207, 211
actsonX1(n), 161
affine minimal, 112, 372, 385
algebraic entropy, 397
algebraically integrable
automorphism, 430
Arzela-Ascoli Theorem, 25
associated field extension, 207
associated Galois subgroup, 207
automorphism group, 196
contains 11 2 , 235
is finite, 196
of conjugate, 196, 234
order two, 235
S3, 197
{z±l}, 205, 234
automorphism of order two, 96,
197,198,200
az + bjz, 96,235
backward orbit, 109, 142
bad reduction, 77, 239
bicritical, 233
Bogomolov conjecture, 129
can have infinitely many rational
periodic points, 136
canonical height, 99, 287
lower bound, 221
canonical measure, 127
on Berkovich space, 306
cocycle gives twist?, 203
coefficients well-defined up to
homogeneity, 169
commuting, 378
have same canonical height, 13 7
have same preperiodic points, 38
with polynomial, 378
completely invariant set, 17, 266
503
contains Berkovich Julia set, 311
condition for unequal formal and
primitive periods, 151, 165,
226
conjugation is algebraic group
action, 173
coordinate functions of iterate, 149,
226
critical point, 12, 284, 353
distinct modulo p, 237
periodic, 38
critical value, 353, 382
cutoff for integral points in orbit,
375
cyclic automorphism group, 197,
204
defined at P, 89
degenerate quadratic, 194
degree, 10,89,388,389
of composition, 392
of regular, 394
degree generating function, 397,
428
determined by 2d + 2 coefficients,
169
domain of quasiperiodicity, 310,
311
dynamical degree, 397,428
dynamical unit, 71, 72, 79, 132, 133
of generic, 146
dynamically affine, 376
dynatomic field, 123
dynatomic polynomial, 39, 148,
149, 181 .
is a polynomial, 151, 226
effect on absolute value, 288
effect on height, 90, 388, 398, 407,
409
equicontinuous, 22
equicontinuous family, 254, 265,
313
equi-Lipschitz family, 313
etale, 353
even degree implies FOM = FOD,
215
expanding on average, 36
family omitting two p