Arithmetic Dynamics on Varieties of Dimension Greater Than 1
by
Benjamin A. Hutz
B. S., Duke University, 2000
Sc. M., Brown University, 2004
Submitted in partial fulfillment of the requirements
for the Degree of Doctor of Philosophy in the
Department of Mathematics at Brown University
Providence, Rhode Island
May 2007
c Copyright 2007 by Benjamin A. Hutz
This dissertation by Benjamin A. Hutz is accepted in its present form by
the Department of Mathematics as satisfying the dissertation requirement
for the degree of Doctor of Philosophy.
Date
Joseph Silverman, Director
Recommended to the Graduate Council
Date
Michael Rosen, Reader
Date
Dan Abramovich, Reader
Approved by the Graduate Council
Date
Sheila Bonde
Dean of the Graduate School
iii
Vita
March 27, 1978
Born in Pittsburgh, PA
June 1996
Graduated from Fox Chapel High School
1996-2000
Angier B. Duke Memorial Scholarship, Duke University
1998-1999
Visiting Student, Université de Paris VII
Summer 1999
Lord Rothermere Scholar, New College, Oxford
May 2000
B.S. cum laude in Mathematics with minors in French
and Computer Science, Duke University
2000-2002
Program Manager in Windows Security Group,
Microsoft Corporation
May 2004
Sc.M. in Mathematics, Brown University
Spring 2006
Brown University/Wheaton College Faculty Fellow
2006-2007
Brown University Mathematics Department
Outstanding Teaching Award
Spring 2007
Brown University Dissertation Fellowship
May 2007
Ph.D. in Mathematics, Brown University
iv
Acknowledgements
I would first like to thank my advisor Joseph Silverman, whose knowledge and keen insight
were invaluable to the completion of this work. Our weekly discussions never failed to
provide more ideas to ponder and a clearer understanding of the current problems. His
careful reading also kept many errors out of this work that I would have otherwise missed.
I would also like to thank my family for their support over the last few years. In
particular, I would like to thank my brother David for several programming suggestions, and
especially my mother Linda for proofreading two different versions of the entire manuscript.
Her ability to edit a document that is completely incomprehensible to her greatly helped
the clarity of this work.
I am also grateful to Natalie Johnson, Doreen Pappas, and Audrey Aguiar for their
support and their help with all of those pesky details that keep the mathematics department
functioning.
Particular thanks go to committee members Dan Abramovich and Michael Rosen, especially Dan Abramovich for many insightful suggestions. I would also like to thank the
Brown Center for Computation and Visualization for providing access to their computation
cluster.
I am also grateful to have a wonderful wife Catherine who provides unconditional love
and understanding regardless of what I am pursuing. Thanks also go to my two four-footed
helpers MacKenzie and Kazander for providing many hours of distraction and company
in addition to their excellent typing skills and ability to sleep on whatever material I was
trying to read at the time.
v
Contents
List of Tables
ix
List of Figures
x
1 Introduction
1
2 Good Reduction and Periodic Points
7
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2
Good Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.3
Point Moving and an Implicit Function Theorem . . . . . . . . . . . . . . .
18
2.4
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.5
A Computational Application . . . . . . . . . . . . . . . . . . . . . . . . . .
41
3 Dynatomic Cycles
43
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.2
Intersection Multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
3.3
Higher Tor Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.3.1
Some Facts from Serre’s Local Algebra [40] . . . . . . . . . . . . . .
46
Tor0 Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
3.4.1
Hironaka Standard Basis
. . . . . . . . . . . . . . . . . . . . . . . .
50
3.4.2
Proof of Main Result: a∗P (n) ≥ 0 for all n ≥ 1 . . . . . . . . . . . . .
58
Some Properties and Consequences . . . . . . . . . . . . . . . . . . . . . . .
84
3.4
3.5
4 Galois Theory of Periodic Points
89
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
4.2
Galois Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
4.3
Field Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
vi
5 Singularity Checking
99
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
5.2
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
101
5.2.1
Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
101
5.2.2
A-Resultants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
103
5.2.3
A-Discriminants . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
104
5.2.4
Triangulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
106
5.3
Bi-Degree of ∆A (L, Q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
107
5.4
Leading Monomials of ∆A (L, Q) . . . . . . . . . . . . . . . . . . . . . . . .
114
5.4.1
Coherent Triangulations and Newton Polytopes . . . . . . . . . . . .
114
5.4.2
Leading Monomials of S . . . . . . . . . . . . . . . . . . . . . . . . .
119
6 Periodic Points
6.1
6.2
121
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
121
6.1.1
Explicit Formulas for σx and σy . . . . . . . . . . . . . . . . . . . . .
122
Explicit Algorithm for Finding p−1
x (P ) . . . . . . . . . . . . . . . . . . . . .
124
6.2.1
Comparison to Checking All Points in
P2x
×
P2y
. . . . . . . . . . . .
130
6.3
Finding Periodic Points Over Q . . . . . . . . . . . . . . . . . . . . . . . . .
132
6.4
Point and Cycle Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . .
138
6.4.1
Average Number of Points . . . . . . . . . . . . . . . . . . . . . . . .
142
6.4.2
Average Length of Cycles . . . . . . . . . . . . . . . . . . . . . . . .
146
7 Point Counting and Zeta Functions
7.1
7.2
7.3
149
Point Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
149
7.1.1
Zeta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
150
7.1.2
Discussion of the Algorithm . . . . . . . . . . . . . . . . . . . . . . .
151
Zeta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
154
7.2.1
The Riemann Hypothesis . . . . . . . . . . . . . . . . . . . . . . . .
154
7.2.2
Some Equation Manipulation . . . . . . . . . . . . . . . . . . . . . .
156
7.2.3
The Newton-Girard Formulas . . . . . . . . . . . . . . . . . . . . . .
157
7.2.4
An Example Computation of the Nm s . . . . . . . . . . . . . . . . .
159
Exact Point Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
160
7.3.1
A Two-Parameter Family . . . . . . . . . . . . . . . . . . . . . . . .
160
7.3.2
A Second Two-Parameter Family . . . . . . . . . . . . . . . . . . . .
164
7.3.3
A Four-Parameter Family . . . . . . . . . . . . . . . . . . . . . . . .
168
vii
Appendix
A Periodic Point Search Code
175
A.1 Description of the Script . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
175
A.2 The Script . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
176
Bibliography
199
viii
List of Tables
5.1
Bi-Degree Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
113
6.1
Run Time Data for Finding All of the Rational Points . . . . . . . . . . . .
131
6.2
Run Time Data for Periodic Point Searching . . . . . . . . . . . . . . . . .
136
6.3
Primitive Periods 1-10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
137
6.4
Primitive Periods 11-16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
138
6.5
Counting Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
139
6.6
Distribution of the Number of Points . . . . . . . . . . . . . . . . . . . . . .
145
7.1
Estimated Point Counting Run Times: Modulus Variation . . . . . . . . . .
152
7.2
Time to Pre-Compute the Square Check . . . . . . . . . . . . . . . . . . . .
153
7.3
Estimated Point Counting Run Times: Square Check . . . . . . . . . . . . .
154
7.4
Actual Point Counting Run Times . . . . . . . . . . . . . . . . . . . . . . .
159
A.1 Detailed Run Time for Point Search Code. . . . . . . . . . . . . . . . . . . .
176
ix
List of Figures
5.1
The Convex Hull W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
108
5.2
Dimension 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
108
5.3
Dimension 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
109
5.4
Dimension 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
110
5.5
Dimension 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
111
5.6
Dimension 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
112
6.1
Average Number of Points versus p . . . . . . . . . . . . . . . . . . . . . . .
140
6.2
Extended Average Number of Points versus p . . . . . . . . . . . . . . . . .
140
6.3
Average Length of Cycles versus p . . . . . . . . . . . . . . . . . . . . . . .
141
6.4
Average Number of Cycles versus p . . . . . . . . . . . . . . . . . . . . . . .
141
6.5
Distribution of the Number of Points for p = 2 . . . . . . . . . . . . . . . .
144
6.6
Distribution of the Number of Points for p = 3 . . . . . . . . . . . . . . . .
146
x
Abstract of “Arithmetic Dynamics on Varieties of Dimension Greater Than 1”
by Benjamin A. Hutz, Ph.D., Brown University, May 2007.
This dissertation investigates properties of periodic points arising from iterating endomorphisms of projective varieties. It is divided into three main topics: first, an investigation
of good reduction properties of periodic points after reduction modulo the maximal ideal
of a local ring; secondly, an investigation of the zero-cycle of primitive periodic points of
a certain period including effectivity and associated Galois theoretic implications; finally,
a computational investigation of dynamical systems on a certain class of K3 surfaces first
introduced by J. Wehler (1988).
Chapter 1
Introduction
This dissertation investigates properties of periodic points arising from iterating endomorphisms of projective varieties. It is divided into three main topics: an investigation of good
reduction properties of periodic points after reduction modulo the maximal ideal of a local
ring (Chapter 2), an investigation of the zero-cycle of primitive periodic points of a certain
period including demonstrating that it is effective and associated Galois theoretic implications (Chapters 3 and 4), and a computational investigation of dynamical systems on a
certain class of K3 surfaces examined by J. Wehler [47] (Chapters 5, 6, and 7).
The field of complex dynamics has a long history; it was brought to prominence by
the work of Fatou and Julia around 1920 and is, in essence, the study of iterations on PN
C
with respect to the complex topology. Arithmetic dynamics is the study of iterations over
number fields with respect to the Zariski topology. The development of arithmetic dynamics
was initiated by Northcott [37] in his study of heights on PN , where he showed that the
set of rational preperiodic points for any endomorphism of PN of degree ≥ 2 is always
finite. In the 1990s, Call and Silverman [10] developed the modern theory of canonical
heights, generalizing the notion of Weil heights on PN and Néron-Tate heights on abelian
varieties. This allows many of the classical questions about abelian varieties, elliptic curves,
and multiplicative groups to be formulated for dynamical systems, such as the number of
rational and preperiodic points and their distributions.
Local Information
In Morton and Silverman [34], the authors explicitly describe the primitive period of a
periodic point in terms of the primitive period of the periodic point modulo a maximal
ideal for rational maps on P1 (and also automorphisms of PN ) with good reduction. Their
1
2
main theorem [34, Theorem 1.1] states
Theorem. (Morton-Silverman) Let (R, m) be a local ring with fraction field K and uniformizer π. Let φ : P1K → P1K be a rational function of degree d ≥ 2 defined over K with
good reduction. Let P ∈ P1K be a periodic point of φ. Denote P and φ the reductions of P
and φ modulo π respectively. Define the following quantities.
n = primitive period of P for φ.
m = primitive period of P for φ.
p = characteristic of k, where k = (R/m).
r = order of dφm P in GL1 (k), where dφm P is the map induced by φm on the cotangent
space of P1k at P .
Then n = m, n = mr, or n = mrpe for some e ≥ 1.
The main results of Chapter 2 investigate higher dimensional analogues of this theorem.
The case of morphisms on smooth, irreducible, projective varieties is studied. In particular,
Theorem 2.28 states
Theorem. Let X/K a non-singular, irreducible, projective variety defined over K and
φ : X → X be a morphism defined over K with good reduction. Let P ∈ X(K) be a periodic
point with primitive period n such that P is fixed by φ.
Then, the map induced on the cotangent space dφP has at least one eigenvalue whose
multiplicative order modulo π divides n.
This theorem is the best possible in some sense since Theorem 2.29 provides an infinite
class of examples where n = 2, m = 1, and r is any even positive integer. With some
additional hypotheses, it is possible to obtain the full result n = m, n = mr, or n = mrpe ,
as seen in Theorem 2.31.
The main tools of the proofs are a p-adic version of the Implicit Function Theorem,
information about the local ring of a scheme at a point, and good reduction information for
both the variety and the morphism.
Dynatomic Cycles
In the case of iteration of single-variable polynomial maps, φ(z) ∈ K[z], we can define
Φn (φ) = φn (z) − z.
3
This polynomial is often called the n-th division polynomial and its roots are periodic points
of period n. Going a step further, we can define the n-th dynatomic polynomial as
Y µ(n/d)
(φ),
Φ∗n (φ) =
Φd
d|n
where µ is the Möbius function. As shown by Morton [33, Theorem 2.5], this is in fact a
polynomial; and in the case where it has no multiple roots, its roots are exactly the points
of primitive period n. In the more general setting of morphisms on projective varieties,
these definitions are no longer suitable, so we turn to intersection theory for an appropriate
generalization. Let X be a projective variety and φ : X → X a morphism. Consider
P
the cycles in X × X: the graph of φn defined as Γn = x∈X (x, φn (x)) and the diagonal
P
∆ = x∈X (x, x). We say that φn is non-degenerate if ∆ and Γn intersect properly. Define
Φn (φ) by taking the intersection of Γn and ∆. If φn is non-degenerate, then Φn (φ) is a
zero-cycle whose points of positive multiplicity are the points of period n. We can then
proceed with the inclusion-exclusion sum as above, obtaining
X
X
Φ∗n (φ) =
µ(n/d)Φd (φ) =
a∗P (n)(P ),
P
d|n
where
a∗P (n)
is the multiplicity of the point P and the points of primitive period n have
a∗P (n) > 0. Define the n-th dynatomic cycle to be Φ∗n (φ).
In Morton and Silverman [35, Theorem 6.4], the authors construct the dynamic analogue
of cyclotomic (and elliptical) units. In the course of doing so, they prove that Φ∗n (φ) is an
effective zero-cycle for morphisms of curves (in particular, rational maps on P1 ) and also
for automorphisms of PN . In other words, they prove that a∗P (n) ≥ 0 for all points P on
the variety and all integers n ≥ 1 [35, Theorem 2.1, Theorem 3.1]. They also conjecture
that Φ∗n (φ) is effective for non-degenerate morphisms of non-singular projective varieties.
Theorem 3.56 in Chapter 3 resolves this conjecture in the affirmative.
Theorem. Let X be a non-singular, irreducible, projective variety defined over a field K.
Let φ : X → X be a morphism defined over K and fix a point P ∈ X(K). Define integers
m = the exact φ-period of P (set m = ∞ if P 6∈ Per(φ)).
p = the characteristic of K.
∗
ri = the multiplicative period of λm
i in K (set ri = ∞ if λi is not a root of unity).
m
Let dφm
P be the map induced by φ on the cotangent space of X at P . Let λ1 , . . . , λl be the
distinct eigenvalues of dφm
P.
Assume that φn is non-degenerate. Then
4
(a) a∗P (n) ≥ 0 for all n ≥ 1.
(b) Let n ≥ 1. If a∗P (n) ≥ 1, then n has one of the following forms
(i) n = m.
(ii) n = m lcm(ri1 , . . . , rik ) for 1 ≤ k ≤ l.
(iii) n = m lcm(ri1 , . . . , rik )pe for 1 ≤ k ≤ l and some e ≥ 1.
As in the one-dimensional case, the proof is carried out by carefully examining when the
multiplicity of a point P in Φn (φ) is greater than the multiplicity of P in Φ1 (φ). However,
several new ideas and a lot of additional work are needed in the higher dimensional case.
Some of the difficulties encountered are taking into account the higher Tor modules in
the intersection theory, which turn out to all be identically 0 (Theorem 3.20), using the
theory of standard bases to obtain information about the multiplicity of a point in Φn (φ)
(Proposition 3.52), and iteration of local power series representations of the morphism.
Φn (φ) and Φ∗n (φ) occur with great frequency in the literature, under a variety of notations and with a number of results stemming from the fact they are effective, see for
example [34], [35], [30], [31], [32], [33], [36], [46]. In particular, [33] and [46] contain Galois
theoretic results in the single-variable polynomial case where Φ∗n (φ) has no multiple roots.
Along with proving some general properties of Φ∗n (φ) in Chapter 3, Chapter 4 contains
generalizations of several of the Galois theoretic results of [33] and [46].
Wehler’s K3 Surfaces
Let S ⊂ P2 × P2 be a smooth surface given by the intersection of a (1,1) and a (2,2)
effective divisor. Wehler [47, Theorem 2.9] shows that this is a K3 surface with an infinite
automorphism group. Silverman [41, Theorem 1.1] constructs a height function on these
surfaces and shows that the set of preperiodic points is a set of bounded height. Call and
Silverman [11, Appendix] give explicit algorithms for computing the automorphisms and
the height function. Combining their work with the results above allows for an explicit
investigation of the rational points on these surfaces.
In particular, an algorithm developed in Chapter 6 searches for periodic points over Q
using information from Theorem 2.12 and Corollary 2.14 in Chapter 2. The key fact is that
if there is a periodic point, then the primitive period of the reduction of a primitive periodic
point for a prime of good reduction must divide the primitive period of the point. While this
may return false positives due to failure of the Hasse principle (i.e., there may be a solution
modulo every prime but no rational solution), it provides an effective way to search quickly.
5
This algorithm is used successfully in Pari/gp to find surfaces with Q-rational points of
primitive period 1, . . . , 16.
In Chapter 7, this algorithm is used to obtain information on the number of points,
number of cycles, and cycle lengths for S non-degenerate over Fp . Experimentally, S has
approximately p2 + 2p + 1 Fp -rational points on average divided into p cycles of average
length p. However, for a random permutation of p2 points, we would expect to have ln(p2 )
cycles on average. A suitable explanation for this is still open.
Also in Chapter 7, the algorithm is optimized for counting rational points using the fact
that projection to each component is a double cover and checking whether there are 0, 1,
or 2 points over each point in P2 (Fp ). It is then shown how this algorithm can be used to
determine the zeta function of such a surface using the Weil conjectures. However, for this
calculation to be successful, one must know when the intersection of a given (1, 1) and (2, 2)
divisor results in a smooth surface. In this direction, Chapter 5 uses methods from GKZ
[16], as in Grier et al. [19], to determine information about the homogeneous polynomial
which vanishes if and only if the surface is singular, i.e., the A-discriminant. In particular,
we determine the bi-degree of the A-discriminant (Theorem 5.21) and illustrate a method
for determining the leading monomials.
Note that there are several similarly constructed classes of surfaces in Baragar [3], Billard
[7], and McMullen [29] which may allow the same type of explicit investigation.