Effective Chabauty for Symmetric Powers of Curves
by
Jennifer Mun Young Park
Submitted to the Department of Mathematics
in partial fulfillment of the requirements for the degree of
OASSACHU'NOI IN-C
OF TECHN~OLOGY
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
PRA RIES
June 2014
@ Massachusetts Institute of Technology 2014. All rights reserved.
Author .......
Signature redacted
Department of Mathematics
May 2, 2014
Certified by .
Signature redacted
1-r
Accepted by..
Bjorn Poonen
Professor
Thesis Supervisor
Signature redacted......................
Alexei Borodin
Chairman, Department Committee on Graduate Theses
2
Effective Chabauty for Symmetric Powers of Curves
by
Jennifer Mun Young Park
Submitted to the Department of Mathematics
on May 2, 2014, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
Faltings' theorem states that curves of genus g > 2 have finitely many rational points. Using
the ideas of Faltings, Mumford, Parshin and Raynaud, one obtains an upper bound on the
upper bound on the number of rational points [Szp85], XI, §2, but this bound is too large to
be used in any reasonable sense. In 1985, Coleman showed [Col85] that Chabauty's method,
which works when the Mordell-Weil rank of the Jacobian of the curve is smaller than g, can
be used to give a good effective bound on the number of rational points of curves of genus
g > 1. We draw ideas from nonarchimedean geometry to show that we can also give an
effective bound on the number of rational points outside of the special set of Symd X, where
X is a curve of genus g > d, when the Mordell-Weil rank of the Jacobian of the curve is at
most g - d.
Thesis Supervisor: Bjorn Poonen
Title: Professor
3
4
Acknowledgments
This thesis would not exist without the careful direction, patience and encouragement from
my advisor, Bjorn Poonen. Thank you, Bjorn, for sharing your excitement for math with
me. Thank you for being so generous with your time and with your ideas. Thank you for
being the mathematician and the person that I can look up to. Most of all, thank you for
being the person that I could always count on to stand by me. If I had to do grad school
one more time, I would choose to be your student again without a second thought!
I would like to thank Matt Baker, Jennifer S. Balakrishnan, Manjul Bhargava, Nils
Bruin, Frangois Charles, Henri Darmon, Kirsten Eisentraeger, Andrew Granville, Jochen
Koenigsmann, Barry Mazur, Gregory Minton, Joseph Rabinoff, Sug Woo Shin, Alexandra
Shlapentokh, Samir Siksek, Michael Stoll, Bernd Sturmfels, John Tate, and Bianca Viray.
I had many helpful conversations with these fantastic mathematicians; these interactions
shaped this thesis, as well as other projects that came into being while I was a graduate
student.
Jennifer S. Balakrishnan and Bianca Viray have been great mentors throughout graduate
school. Their friendship, guidance and support, as well as their mathematical expertise,
helped me in numerous ways throughout graduate school - thank you for everything! I
am also grateful to Andrew and Olena Blumberg, Sarah Chisholm, Henri Darmon, Andrew
Granville, Wei Ho, Sam Payne, Anthony Varilly-Alvarado, and Melanie Matchett Wood for
their advice and encouragement, especially during my final year of graduate school.
Thanks are due to my fellow graduate students at MIT and Harvard. I had a truly fun
five years in Cambridge because of you. In particular, I thank John Lesieutre, Tiankai Liu,
Gregory Minton, Roberto Svaldi, and Samuel Watson for many adventures. Thank you also
to my friends who are not mathematicians - especially Amy Forster, Clare Park and Sarah
Sun - for listening to me ramble on about math when it was probably really boring for you!
Finally, I thank my family - in particular, Kyunghun, Mikyung and Angela - for always
being there for me. Thank you for your love and support. I couldn't have done this without
you!
5
6
Contents
1
Introduction
1.1
2
3
4
9
H istorical D etails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
Chabauty-Coleman method
17
2.1
The logarithm map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.2
Chabauty-Coleman method
. . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.3
Explicit Coleman integration . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.3.1
p-adic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.3.2
Explicit parametrization of p-adic integrals on residue disks . . . . . .
22
2.3.3
Coleman's bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.4
Improvements to the Chabauty-Coleman method
. . . . . . . . . . . . . . .
25
2.5
Exam ple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.6
Chabauty for symmetric powers of curves . . . . . . . . . . . . . . . . . . . .
27
Chabauty-Coleman Method on Symd X
29
3.1
Chabauty on SymdX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.2
Explicit parametrization of points on residue disks . . . . . . . . . . . . . . .
35
3.2.1
The case of Sym 2 X . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.2.2
The case of Symd X . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
Comparison of the algebraic loci and analytic loci on Symd X
4.1
Rigid analytic geometry
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
41
42
4.2
Comparing algebraic and analytic loci in SymdX ................
5 p-adic geometry
6
46
49
5.1
Convex Geom etry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
5.2
Tropicalizing hypersurfaces convergent on a polyhedral domain . . . . . . . .
52
5.3
Tropical intersection theory and Newton polygons . . . . . . . . . . . . . . .
55
Continuity of roots
6.1
Deformation of power series via rigid analytic geometry and polynomial approxim ations
6.2
63
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
Explicit computation of the upper bound . . . . . . . . . . . . . . . . . . . .
67
7 Example
7.1
Hyperelliptic curves of genus3.
73
. . . . . . . . . . . . . . . . . . . . . . . . .
8
73
Chapter 1
Introduction
Throughout this thesis, we assume that X is a nice (smooth, projective and geometrically
integral) curve of genus g over Q that has a rational point 0 E X(Q).
The existence of
one rational point on X is not a strong assumption; everything that we prove in this thesis
can be modified, as long as we have a divisor of degree d to use in place of 0. We aim to
generalize the following theorem, which arises as a consequence of the statement of Coleman
[Col85] to the case of higher-dimensional varieties:
Theorem 1.0.1 ([Col85], Theorem 4). Let g > 1 and p a prime. Then there is an effectively
computable bound N(g,p) such that for every nice curve X of genus g over Q such that
g > rk(Jac(X))(Q) of good reduction at p , the inequality
#X(Q) < N(g, p)
holds.
Although this theorem is weaker than Faltings' theorem for curves, which states that any
nice curve of genus g > 2 has finitely many rational points, Coleman's bounds are sometimes
sharp [GG93]; so it is possible that Theorem 1.0.1 can be realistically used to find all rational
points of a given curve satisfying g > rk(Jac(X))(Q). Previously, all known bounds, such as
in [Szp85, XI §2], were too large to be practical.
9
Assuming that X has good reduction at p, the proof of Coleman's theorem splits up the
set of Qp-points of X into finitely many sets called the residue disks; the set of points on each
residue disk maps onto the same point in X(Fp) under the reduction mod p map. Further,
the points on a single residue disk are in bijection with pZ via a choice of a uniformizer,
chosen separately for each residue disk. A necessary condition for the Qp-points of X in
one of the residue disks (considered elements of pZp) to be rational points of X is given as
a power series equation in terms of the uniformizer; each rational point must be a solution
to the power series equation. The number of such solutions can be bounded above using
Newton polygons.
Furthermore, Coleman's method has been used to obtain a number of very interesting
consequences.
Definition 1.0.2. By an odd hyperelliptic curve, we mean a nice curve birational to an
affine curve given by the equation y 2 = f(x), where f(x)
c
Q[x] is a monic and separable
polynomial of degree 2g + 1. In this case, the genus of the corresponding curve is g, and this
curve has a rational point at infinity.
Chabauty's method aids in consequences such as:
Theorem 1.0.3.
(a)
[BG13],
Corollary 1.4) For any g > 2, a positive proportion (ordered by the naive
height just depending on the coefficients of the defining equations of the affine model)
of odd hyperelliptic curves of genus g have at most 3 rational points.
(b) ([BG13], Corollary 1.4) For any g > 3, a majority (i.e., a proportion of > 50%) of all
odd hyperelliptic curves of genus g have fewer than 20 rationalpoints.
(c) ([PS13], Theorem 10.3) For g > 3, a positive proportion of odd hyperelliptic curves
have just one rational point, namely the point at infinity.
(d) (PS13],
Theorem 10.6) For g > 1, the lower density of odd hyperelliptic curves having
just one rational point (namely, the point at infinity) is at least 1 - (12g + 20)2-9. In
10
particular,as g tends to infinity, the lower density of odd hyperelliptic curves having
just one rationalpoint tends to 1.
More generally, let Y be a nice variety over Q. In theory, using the Albanese varieties
as defined in [Ser58], it seems plausible that Chabauty's method could still apply, where
the Albanese variety Alb(Y) is substituted in place of the Jacobian. One then looks for all
rational points on the image of the Albanese map (assuming a choice of rational point on
j: Y
Y)
-+ Alb(Y); the necessary conditions for a Qp-point being a rational point would
be given by several multivariate power series. If, in addition, one can also understand the
rational points on the fibres of j, then one should be able to characterize all rational points
on Y. However, there exist several difficulties in generalizing the above theorem (often called
Chabauty's method) to arbitrary higher-dimensional varieties:
(1)
Alb(Y) may be trivial: since dim Alb(Y) = h 0 (Y, Q1 ), if h 0 (Y, Q1 ) = 0, then Chabauty's
method yields nothing.
For example, if Y is a rational surface, K3 surface or an
Enriques surface, then h 0 " = 0, so Chabauty's method does not apply.
(2) To understand Y(Q), we need to understand the rational points of the fibres of j as
well as the rational points of j(Y). Understanding the fibres may be complicated.
(3) The case of higher-dimensional varieties allows for the possibility that #Y(Q)
= oo,
even when the Albanese map is not trivial: for example, if Y = Sym 2 X for a hyperelliptic curve birational to the affine curve given by the equation Xaff : y 2 = f(x), then
{ (t,
1f(t)), (t, -
1f(t))} E (Sym 2 X)(Q) for all t
E Q, where the points on X are
identified with the points on Xaff via the birational map.
First, a definition:
Definition 1.0.4. Let X be a nice curve defined over Q.
We define Symd X
:-
Xd/Sd,
where Sd denotes the symmetric group on d elements.
It is known that Symd X is also a nice variety, and one may represent Q c (Symd X) (Q)
by a Gal(Q/Q)-stable multiset of d elements of X(Q).
11
We will consider the special case in which Y = Symd X for a nice curve X.
Then
Alb(Y) = Jac(X), so the Albanese variety is nontrivial, so (1) does not apply. Also, the
fibres of j are easy to understand, so (2) also is not really a problem. The case of (Symd X)(Q)
is of further interest because understanding the rational points on Symd X means that one
can find all degree-d points on X. To deal with the possibility of #((Symd X)(Q))
=
oo, we
recall from [Fal94]:
Theorem 1.0.5 ([Fal94]). Let A be an abelian variety over Q, and V C A be a closed
subvariety. Then there exist finitely many subvarieties Yi C V such that each Y is a coset
of an abelian subvariety of A and
V(Q)
=U
Yi(Q).
Apply this theorem with A = J and with V being the image of the map
j : (Symd X)(Q) -* J(Q)
{ P, i...,i Pd} - [P1 + --- + Pd - d - 0],
where J := Jac(X) = Alb(Symd(X)).
The fibres of this map are projective spaces, which
partially explains the existence of infinitely many rational points on Symd X. If dim Y > 0
for some i, this Y could also explain the existence of infinitely many rational points on
Symd X. The positive-dimensional Y have been studied in the literature, most notably in
the paper of Harris and Silverman [HS91], for the case d
=
2:
Theorem 1.0.6 ([HS91]). If Sym 2 X contains an elliptic curve, then X is either bielliptic
or hyperelliptic.
This suggests that in order to deal with the third difficulty, we may want to restrict our
attention to the rational points that are not contained in any positive-dimensional Y. Then,
by Theorem 1.0.5, it is already known that there are only finitely many such rational points,
so it makes sense to try to get an upper bound on the zero-dimensional Y that are "maximal"
12
in the sense that they are not contained in any positive-dimensional Y.
It turns out that
this notion is related to the following definition of Lang; the special set of a variety V,
denoted by S(V), is the Zariski closure of the union of the images of all nonconstant maps
f
: G -+ V of group varieties G defined over Q into V, which is defined over Q, where f is
defined over some finite extension of Q.
On the other hand, Chabauty's method concerns the the rigid analytic varieties defined
by the vanishing of the locally analytic functions (with power series representations on open
sets called residue disks; see Chapter 3 for details) of the form
rh : (Symd X)(CP)
Q
-+ Cp
/
fo (Q)
for 1 < i < d, with w E H0 (X(Q, Q') where the functions qj vanish on j-'(J(Q)) (for details,
see Chapter 2). Denote by (Symd X)y=0 the rigid analytic subvariety of (Symd X) a" defined
by the vanishing of the rj on the residue disks lying over the points in (Symd X)(F,).
It turns out that the set of rational points outside of the special set of Symd X is contained
in the set
{P E (Symd X)9=0 : P is the point in a 0-dimensional component in (Symd X)"=},
(this notion is made precise in Chapter 4), if one makes the following assumption:
Assumption 1.0.7. Retain the notation of the previous two paragraphs,as well as from the
beginning of this chapter. Assume that every positive-dimensional rigid analytic component
of (Symd X)7=0 is contained in S(Symd X)an.
The main result of this thesis is the following:
Theorem 1.0.8. Let d > 1, p a prime, and g
2. Then there exists a number N(p,d, g)
that can be computed effectively, such that for every nice curve X defined over Q of good
13
reduction at p with rk J < g - d satisfying Assumption 1.0.77,
#{Q E (Symd X)(Q) I Q does not
belong to the special set} < N(p, d, g).
If we impose extra hypotheses on X, we can get an even better bound on N(p, d, g). For
example:
Proposition 1.0.9. We can take N(2, 3,3) = 1539 for any degree 7 odd hyperelliptic curve
X such that rk J(Q) < 1 satisfying Assumption 1.0.7 and such that X has good reduction at
2.
Two main ideas are required to obtain an effective bound for the number of points outside
of the special set of Symd X. The first idea relies on the fact that as in the 1-dimensional
case, knowing the approximate shape of the Newton polygons coming from the multivariate
power series constraints on each residue disk U gives an upper bound on the number of
rational points of Symd X. In general, approximating the shape of the Newton polygons of
multivariate power series is hard, but in our case, we consider the following diagram:
Xd(CP)
. (Symd X)(Cp)
U(-> (Symd X)(Q)c
J
I
I
While Chabauty's method typically deals with residue disks U C (Symd X) (C,) lying
above a point P = {P 1 , ..
.
, Pd} G (Symd X)(Fp), we will instead look at the preimage
of the residue disk U C (SymdX)(Cp)
in Xd(Cp).
The preimage of U decomposes into
a disjoint union of several of the products of d one-dimensional residue disks, where the
disjoint union indexed the different ordered multisets that lie above the unordered multiset
{Pi,..., Pd}. We fix one such product in the disjoint union, say HI
above Pi E X(Fp).
Ui, where Ui g X(Cp),
Considering the pullbacks of U to Hd_ Ui has the effect of change
of variables on the local coordinates of U into d uniformizing parameters of X(Cp) above
14
each of P 1 , . . . , Pd. We will see that we can write the multivariate power series constraints
as a sum of d single-variable power series in this way. From here, the Newton polygons
of the power series constraints are much easier to approximate, as we approximate onedimensional Newton polygon on each of the single-variable power series instead. Then the
one-dimensional approximations can be combined to give the multivariate approximation.
While the previous paragraph already gives an effective bound, we do something slightly
more complicated in this thesis: the minimal allowable valuation for the uniformizing parameters on Xd depends on the field of definition of P; by taking this information into account,
the effective bound becomes sharper.
The second main idea deals with the third difficulty mentioned above. Even with the
multivariate power series constraints, if j(Symd X) has infinitely many rational points,
Chabauty's method may not be useful. We can bypass some of this difficulty by looking
at Faltings' theorem for higher-dimensional varieties: if Symd X satisfies Assumption 1.0.7,
then rational points of Wd not contained in any cosets of a positive-dimensional abelian subvariety are contained in the stable intersections of the multivariate power series constraints.
Thus, we use deformation theory techniques coming from rigid analytic geometry, to deform
the power series to ensure finite intersection. This finite intersection number is an upper
bound on the points outside of the special set.
However, if Symd X does not satisfy Assumption 1.0.7, considering the multivariate power
series may not yield any useful information; this issue seems to be intrinsically embedded in
Chabauty's method.
This also means that if one could approximate the Newton polygon without resorting to
the change of variables method, the results outlined in this paper could generalize to other
classes of higher-dimensional varieties as well.
1.1
Historical Details
The problem of rational points on symmetric powers of curves have been studied by several
people in the past. One result is that of Debarre and Klassen [DK94], which studies the
15
Fermat curves, which are plane projective curves given by
XN + yN
=
ZN, N > 4.
While we already know that these curves only have finitely many K-points for any number
field K, and no nontrivial Q-points (Fermat's Last Theorem), [DK94] uses geometric methods
to prove the following theorem:
Theorem 1.1.1. [DK94] For N # 6, there are only finitely many number fields K with
degree d = [K : Q] < N - 2 such that FN(K) # FN(Q).
In [DK94], the question of applying Chabauty's method to symmetric powers of curves
is raised. Then in [Kla93], the first attempts to generalize [Col85] to symmetric powers of
curves:
Theorem 1.1.2. [Kla93, Theorem 13] Let 1 < d < y, and let X be a nice curve of genus
g > 2 and gonality -y, satisfying rk J(Q) < g - d. Then there exists a canonical divisor M
on (Sym X)Q, such that the complement Symd X\M has only finitely many rational points
(here, a canonical divisor is a divisor of a meromorphic d-form). Further,
#((Symd X) (Q) \ red-' (R(]F))) < # ((SyMd X) (IFP)\
(]F)),
where redp denotes the reduction modulo p map, and M denotes the reduction of M mod p.
Also, [Sik09] refines [Kla93 by removing the gonality from the hypothesis of the above
statement, and also giving a sufficient criterion for when a residue disk contains a single
rational point. Also, he developed a method that can be used to compute (Sym 2 X)(Q) for
some curves (in §6 of [Sik09], he works out two explicit examples).
16
Chapter 2
Chabauty-Coleman method
The Chabauty-Coleman method is a p-adic method that determines an explicit upper bound
on the number of rational points on a smooth, projective and geometrically integral curve
X of genus g > 2, subject to the hypothesis that g > r, where r denotes the Mordell-Weil
rank of the Jacobian. In this chapter, we discuss the classical Chabauty-Coleman method
on curves, its improvements, and an example of how it is used in practice to describe the set
of all rational points on curves.
2.1
The logarithm map
In this section, we explain the definition of the logarithm map of a Lie group to its Lie
algebra, which is essential in understanding Chabauty's method. Most of the exposition in
this section is taken from [Bou72]. We assume for this section that everything is defined over
a p-adic field; that is, a valued field of characteristic 0 satisfying the ultrametric inequality,
whose residue field has characteristic p.
Lemma 2.1.1 ([Bou72l, 111.7.6., Proposition 10). Let G be a finite-dimensional Lie group
over a p-adic field with e G G the identity element, and let g be the Lie algebra of G. We
define a subgroup Gf as the set of x G G such that there exists a strictly increasing sequence
of positive integers (ni) with limi,
0
x'i = e. Then
17
(i) Gf is open in G.
(ii) There exists a unique map
'i
: Gf -+ p satisfying:
(a) V)(x") = nrV(x)
(b)
There exists an open neighborhoodV of e inside Gf such that 0/|v is injective, and
the inverse of
4
on O(V), denoted ('V)- 1 , is an exponential map, as described
in [Bou72], §2, Proposition 3.
(c) The map 0 is analytic.
Definition 2.1.2. The local diffeomorphism 4 of Lemma 2.1.1 is called the logarithm map
of G, and is denoted logG, or log.
2.2
Chabauty-Coleman method
In this section, we give an overview of Chabauty's method, which finds an explicit upper
bound on the number of rational points of curves satisfying certain conditions. The exposition in this section follows [MP12].
Let X be a curve of genus at least 2 defined over the rational numbers Q, and let g be
the genus of the curve X. Let J = Pico
be the identity component of the Picard scheme
of X. Thus, we will view J as the moduli space of degree-0 line bundles on X. Let r denote
the rank of the finitely generated abelian group J(Q). Further, X(Q) denotes the set of
rational points on X.
Throughout the thesis, we will assume that X already has a rational point 0 C X(Q).
Then we can write the Q-points of J as degree-0 divisors of X (more generally, the Q-points
of J correspond to linear equivalence classes of degree 0 divisors on Xg). The existence of
0 E X(Q) is not a strong assumption; the rational point 0 may be replaced by any degree-I
divisor. Furthermore, if a rational point does not exist on X, there are other methods that
attempt to prove the non-existence of 0.
18
Using the base-point 0, we have the embedding of X into J:
(2.1)
X P-
[P -O].
Since 0 E X(Q), this embedding induces X(Q)
c
J(Q).
For a finite prime p, let Qp denote the field of p-adic numbers. Let XQ, and JQ, be the
base extension of the varieties X and J to the field Qp. We view J(Qp) as a Lie group over
Qp. The closure J(Q) of J(Q) under the p-adic topology is also an analytic subgroup of
J(Qp), and as in §2.1, there is the logarithm map log : J(Qp) -+ To(J(Qp))
!
Q.
The
logarithm map can be described in terms of "antiderivatives", as described in the following
paragraphs.
Suppose w G H0 (JQ,, Q1). Using the translation invariance of w, one can show that there
is a unique homomorphism
rj : J(QP) -Q
/Q
Q
jw0
such that there is an open subgroup U of J(Qp) such that if
Q
E U, then fww can be
computed by expanding w in power series in local coordinates, finding a formal antiderivative,
and evaluating the power series at the local coordinates of Q.
As J is a smooth abelian variety of dimension g over a field Q,, with TOJ the tangent
space at the identity element 0 E J, the logarithm map can be written by integrating the
basis {wi, . . . , wg} of the space of 1-forms H 0 (J,Q 1 ):
log : J(Qp) -+ To J E Qq
t
et
Its kernel is the torsion subgroup of J(Qp).
19
so
We denote r' := dim J(Q).
Lemma 2.2.1 (fMP12I, Lemma 4.2).
r' = dim J(Q) ; rk J(Q) = r.
Proof. Since log is a local diffeomorphism, we have
dim J(Q) = dim log J(Q) = dim log(J(Q)),
where the second equality follows since log is continuous and J(Q) is compact. Now we have:
r' = rkz, Z log(J(Q)) <_ rkz log(J(Q)) = rkz J(Q) = r.
0
Now, inside J(Qp), which is a g-dimensional p-adic Lie group, we have two objects of
interest: XQ,, which is a 1-dimensional p-adic manifold, and J(Q), which is at most an
r-dimensional p-adic Lie subgroup of J(Qp). If r < g, then X(Qp) and J(Q) are expected
to intersect at finitely many points, and this intersection further satisfies
X(Q) g_ X(Qp) n J(Q).
Chabauty's theorem (the statement of which was eventually subsumed in Faltings' theorem,
which does not require the hypothesis that r < g) shows that this intersection is in fact
finite:
Theorem 2.2.2 ([Cha4l]). Let X be a smooth, projective and geometrically integral curve
of genus g > 2 defined over Q. Let p be a prime, and suppose that r' = dim J(Q) < g. Then
X(Qp) n J(Q) is finite. In particular,this implies that X(Q) is finite.
20
2.3
Explicit Coleman integration
For a nice curve X of genus g > 2 defined over Q, Chabauty's theorem can be modified to
give an effective upper bound on #X(Q); while Faltings' theorem also leads to an upper
bound, the bounds are too large to be realistic. In this section, we sketch Coleman's theory
of integration, which leads to an effective upper bound on #X(Q).
Our goal is to obtain a
locally analytic function q on J(Qp) that vanishes on J(Q). Then we will bound the number
of zeros of lx((Q,). Since qlX(,Q)
bound on #X(Q).
2.3.1
vanishes on X(Q), the number of zeros will give an upper
We start by producing locally analytic functions on X(Qp).
p-adic integrals
Let p be a prime at which the curve X has good reduction. That is, there is a smooth proper
curve over Z, whose generic fibre is XQ,. We define p-adic integrals on J(Qp) by integrating
1-forms w E H0 (JQ,, Q1). Through the canonical isomorphism H0 (JQ,, Q1) 2 Ho(X(Qp Q'1)
we define the p-adic integral on X(Qp) by
1 : X(QP)
up
P [P-O]
where the integration on the right-hand side is defined on JQ,. To find an analytic function
that vanishes on J(Q), we look at the logarithm map:
log: J(Qp) -+ H0 (JQ,, Q1)* = To JQ,
where TOJQ, denotes the tangent space to JQ, at the origin 0. Since dimlog(J(Q)) < g,
there exists a hyperplane H C TOJQ, containing log(J(Q)). This hyperplane H is defined
by the vanishing of some Wj E HO(JQ,, Q1 ) - To J6,, and the restriction of Wj to XQ, can
be uniquely identified with an wx c H0 (XQ,, Q'). It remains to bound the zeros of i7 arising
from integrating Wx.
21
2.3.2
Explicit parametrization of p-adic integrals on residue disks
Regard X as a scheme over CP, and we continue to assume that X has good reduction at p.
Then there is a reduction map mod p:
redp : X(Cp) -* X(F).
of a point P E X(Fp) is called a
Definition 2.3.1. Let P E X(k). The preimage red-p(P)
1
residue disk over P . A uniformizer t is a rationalfunction on an open neighborhood
of P in the smooth OK-model of X, such that it vanishes to order 1 at P in the special fibre.
One can choose a uniformizer on each residue disk.
Let K be a finite extension of Q,, let OK be its ring of integers, and let k be its residue
field.
Definition 2.3.2. Let U be a residue disk of X. The K-points of U are denoted U(K), and
these are
U(K) := X(K) n u.
In this section, we are exclusively concerned about the cases where K
and k = FY.
=
Qp,
OK =
Zp,
However, in later sections, we will also consider the cases where K is an
algebraic extension of Q,. For now, assume that K = Q, (but everything in this section can
be generalized easily to the cases where K is a finite extension of Q,).
The uniformizer t provides a bijection
t : (red-'(P))(Qp) -+ pZp
to the P E X(Fp) to which it is associated, by an application of Hensel's lemma (in general,
replace Qp by K and pZ4 by {x c K: v(x) > 0}).
Since the points of the residue disk can be parametrized by pZp, one also shows that:
(1) Suppose that w E H 0 (X(Q,
Q1 ) can be normalized by an element of Q' so that it
22
reduces to a nonzero element in HO(XFp, Q1 ). Then w on the residue disk above can
be written as w(t)dt for w(t) E Z[[t]] with w(t)
$
0 modulo p.
(2) The function q, which is given by integrating w, is given on the residue disk by formally
integrating w(t)dt, so the coefficients of rj(t) are in Q,.
2.3.3
Coleman's bound
Let q : X(Qp) -+ Q, be the analytic function of §2.3.1 that vanishes on J(Q) n X(Qp),
obtained by integrating w E H0 (XQ, Q1 ). By an abuse of notation, we will also use
T
for the function on pZp, obtained by composing with the inverse of the bijection given by
a fixed uniformizer
tR
: redp 1 (P) -+ pZ, on each residue disk R. The statement that
X(Q) c X(Qp) n J(Q), combined with Chabauty's theorem 2.2.2, shows that
#X(Q) < #(X(Qp)
c X(Qp) :I(P)= 0}
n J(Q)) =
=
#{x E pZ, : q(x)
=
0},
(2.2)
R
where the sum in equation 2.2 runs over all residue disks R of X that are the preimages
of some P E X(Fp), with a fixed uniformizer tR, and
T(tR)
denotes the function 7 on the
parametrized residue disk R. We only count the solutions x of
q(tR)
with vp(x) > 1, since
by §2.3.2, the Qp-points on the residue disk R are parametrized by pZp via the uniformizer
tR-
In particular, equation 2.2 shows that the upper bound of #X(Q) can be computed by
considering each residue disk separately. Thus, we start by fixing a particular residue disk
R lying over a point P E X(Fp), with its uniformizer t. Since the number of residue disks is
finite and can be estimated by using the Weil conjectures, it remains to compute an upper
bound for the set
#{t C PZp : q(t) = 0}.
For example, the following is an example of an upper bound; for the more general case, see
23
[Col85], Lemma 2.
Lemma 2.3.3 ([MP12, Lemma 5.1], rephrasing [Col85, Lemma 2]). Suppose that f(t) E
Qp[[t]] is such that f'(t) E Z,[[t]]. Let m = ordt=o(f'(t) mod p). If m < p - 2, then f has at
most m + I zeros in pZ..
Proof. We attempt to approximate the shape of the Newton polygon of f(t). Let f(t)
=
0antn. Recall that given a Newton polygon of f(t), if the slopes of the segments of
the Newton polygon (here, each segment is thought to have length 1, when projected to the
x-axis) are written as p 1 <
-- < t, then pi = -vp(si), where si E Qp satisfies f(si) = 0
with si having i-th smallest valuation.
In particular, we wish to find an upper bound for #{s
E pZ, : f(s) = 0}.
Since
f'(t) E Z,[[t]], we have v(am+) = 0 and v(ai) > -vp(i) > m + 1 - i for i > m + 1. So, to
the right of (m + 1, 0), the Newton polygon of f can have no slopes less than or equal to -1.
Thus,
f
D
has at most m + 1 zeros in pZ.
We now state a simplified version of Coleman's theorem, as well as its proof, as presented
in Theorem 5.3 of [MP12], since understanding its proof suggests the steps one can take in
generalizing Coleman's theorem to higher-dimensional varieties.
Theorem 2.3.4 ([Col85], Theorem 4). Let notation be as in the beginning of §2.3.3,
(a) Let q : X(Qp) -+ Qbe the analytic function of §2.3.1 given by integrating a nonzero
w C HO(JQI Q1 ). Scale w by an element of Qx so that it reduces to a nonzero 1-form
F
E H 0 (XF,
Q1 ). Suppose that Q G X(Fp), and let m = ord F. If m < p - 2, then
the number of points in X(Q) reducing to Q is at most m + 1.
(b) If p > 2g, then
#X(Q)
#X(Fp) + (2g - 2).
Proof.
(a) If there are no points in X(Q) reducing to
Q E
Q,
then we are done.
X(Q) reducing to Q. Then since foQW = fw w + fw w =
24
Otherwise, fix
fww,
one has a
power series expression I(t) of
w, for each Q' E X(Qp) reducing to Q. Applying
f
Lemma 2.3.3 to I(t) shows that 1(t) has at most m
+ 1 zeros, so there are at most
m + 1 rational points Q' in the residue class.
(b) For
Q
E X(Fp), let m
= ord,
D. By the Riemann-Roch theorem, the total number
of zeros of w in X(Fp) is 2g - 2. Thus EZEX(Fp) mn
< 2g
-
2. In particular, m, K
2g - 2 < p - 2 for each Q. Applying part (a) to each Q and summing yields
#X(Q)
E
(m + 1) = #X(Fp) +
E
m
#X(F) + (2g - 2).
E
QEX(Fp)
QEX(Fp)
Remark 2.3.5. The assumptions m < p - 2 of part (a) and p > 2g of part (b) can be
removed; in fact, in [Col85], Coleman develops a formula without these assumptions, although
it becomes more complicated. In this thesis, we will also not need these assumptions in
generalizing Coleman's results.
2.4
Improvements to the Chabauty-Coleman method
Since [Col85], there have been many generalizations and improvements to the original result
of Coleman in counting the upper bound on the rational points . In this section, we outline
some of these results. Although no analogue of these results in higher-dimensional variety
exists as of now, these could be one future direction in which this thesis could be taken.
Stoll, in [Sto06], noted that one can choose a different differential w E H0 (XQ,, Q1 ) for
each residue class, and that by choosing the 'best' differential, one can refine the bound of
Coleman, as written in Theorem 2.3.4.
Theorem 2.4.1 ([Sto06], Corollary 6.7). Suppose that X is a smooth projective and geometrically integral curve of genus g, and suppose further that r < g with p > 2g. Then
#X(Q) < #X(Fp) + (2g - 2),
25
where r = rk J(Q).
In another direction, one can also obtain the upper bound in the case where p is a prime
of bad reduction, as shown in the work of Lorenzini and Tucker [LT02]:
Theorem 2.4.2 ([LT02], Proposition 1.10). Suppose p > 2g and let X be a proper regular
model of X over Z, and let Xp'm be the smooth locus of the special fibre of X. Suppose that
r < g. Then
#X(Q) 5 #Xpsm (Fp) + 2r.
Recently, Katz and Zureick-Brown [KZB13] improved on these results by asking whether
one could improve the bounds on Theorem 2.4.2 by using the idea in 2.4.1 to choose a
different w for each residue class:
Theorem 2.4.3 ([KZB13]). Keeping the notation as before, suppose p > 2r + 2 is a prime.
Let X be a proper regular model of X over ZP, and suppose furthermore that r < g. Then
#X(Q) < #Xs m (Fp) + 2r.
Remark 2.4.4. All of these theorems can be generalized to the cases over number fields
without much difficulty.
2.5
Example
We give an example where Chabauty's method is used in practice. In particular, we illustrate
that Chabauty's bound can be sharp, which aids in the search for the complete set of rational
points on a curve. In this section, we consider the hyperelliptic curve X of genus 2 whose
affine model is given by the equation
y2 = x(x - 1)(x - 2)(x - 5)(x - 6),
whose Jacobian J has Mordell-Weil rank 1. This curve was first considered in [GG93], §4.
The curve X has good reduction at 7, and X(F7 ) = {(0, 0), (1, 0), (2, 0), (5, 0), (6, 0), (3, 6), (3, -6), oo}
26
By Theorem 2.3.4 (b), #X(Q) < 10. Furthermore, we can find ten rational points: the six
Weierstrass points, and also the points (3, ±6) and (10, ±120). Thus, #X(Q) = 10, and in
this case, Chabauty's bound is sharp.
It is known that the vector space H0 (X(Q, Q1 ) is generated by the two elements wo
and w, = xL. Thus, the locally analytic function
=
g
y
X(Q 7 )
--
Q7
j
P
1P
10
that vanishes on X(Q 7 ) n J(Q) arises from some w, which is a Q7 -linear combination of wo
and wi, say w
=
awo + bwi for a, b E Q7. In fact, one can determine
? explicitly
a and b; for example, one looks at the residue field over (3, 6), and solves for
by computing
f((36)120)
w=0.
For example, we can take
/(3,6)
Wi
(10,-120)
/(3,6)
b= -
2.6
f
0,)Wo.
S(10-120)
Chabauty for symmetric powers of curves
Our goal for the remainder of this thesis is to generalize this construction to the case of
higher-dimensional varieties. In particular, we will talk about applying Chabauty's method
to symmetric powers of curves of high enough genus (this will be made precise in the coming
chapters; the genus being high enough should be seen as an analogue of Chabauty's condition
saying g > r). This is a particularly interesting case, as
#
X(K)
X
[K:Q]=d
27
Furthermore, computations on Symd X are
can be bounded in terms of #(Symd X)(Q).
comparable in difficulty to computations on a curve, which makes the case of Symd X even
more appealing, although one expects that the methods outlined in this thesis should work
for any subvariety of an abelian variety.
As mentioned in the introduction, the original proof in the case of d = 1 requires significant modifications. One of the initial difficulties one encounters is the fact that Symd X can
have infinitely many rational points. In the next chapter, we start by proposing a way to
single out finitely many points on Symd X to be counted by a generalization of the ChabautyColeman method. Our starting point will be the generalized version of Faltings' theorem, as
in the case of curves.
28
Chapter 3
Chabauty-Coleman Method on Symd X
3.1
Chabauty on SymdX
Given Chabauty's method on curves, one can naturally ask, given a variety Y over Q of
dimension n with a choice of a rational point, whether Chabauty's method makes sense when
the Jacobian is replaced by the Albanese variety Alb(Y), using the associated Albanese map
j : Y -+ Alb(Y).
As remarked in the introduction,
j
is often of smaller dimension than Y. In the case when
is often not injective, and Alb(Y)
j
is not injective, we further need
information on the fibres of j.
In this section, we provide the setup for Chabauty's method when Y = Symd X, where
X is a nice curve of genus g defined over Q, with a rational point 0 E X(Q). Let Wd be
the subvariety j(Symd X) of J. In this setting, we have Alb(Y)
=
Jac(X) =: J. We will see
that when rk J < g - d, for each residue disk, we obtain d power series in d variables, whose
common zeros contain the points (Wd)(Qp) n J(Q).
In fact, one expects that similar techniques introduced in this thesis should apply for
other higher-dimensional varieties as well; however, these methods may be less feasible from
the computational point of view.
Even when we restrict to Y
of Y(Q) into Alb(Y)(Q).
=
Symd X, the Albanese map does not guarantee an injection
For example, suppose that X is an hyperelliptic curve with a
29
rational Weierstrass point; that is, an affine model of X is given by the equation y2
=
f(x)
for an odd-degree polynomial f E Q[x] of degree > 5. Then each of the points in the set
f(x)), (t, -
{{(t,
f(x))} I t E
Q}
C (Sym 2 X)(Q) maps into 0 E Alb(Sym 2 X)(Q). More
generally, the fibres of the Albanese map are well-understood:
Lemma 3.1.1. Let X be a nice curve over Q of genus g, with a rationalpoint 0 E X(),
and let
j : Symd X
-+
J
[P1+ -+
{Pl,..., Pd}
be the Albanese map. Suppose that Q E (Symd X)(Q).
Pd - dO]
Then the set of rationalpoints on the
fibre of j containing Q is isomorphic to the set of rationalpoints of a projective space IP' for
some n > 0.
Proof. Since J could be described as parametrizing the equivalence class of degree-0 divisors,
Q
can be identified with an effective divisor on X. By Theorem 11.5.19 of [Har77], the set
of points giving rise to the same divisor is isomorphic to a finite-dimensional vector space,
as required. In particular, if the fiber in question contains two distinct rational points, then
dimension of this vector space is at least 1.
0
However, the possible existence of infinitely many rational points on Symd X is not completely explained by the points on the fibre of the Albanese map; Faltings' theorem (Theorem
1.0.5) allows for the possibility that infinitely many rational points could occur in the image
of j as well.
Apply 1.0.5 to Wd C J. Then if
true for any such
QE
(Symd X)(Q), at least one of the following cases is
Q:
(1) #j - I(U(Q)) #' 1;
(2) there exists a coset Y of an abelian subvariety of J contained in Wd with dim Y > 0,
such that j(Q) E Y;
30
(3) #i(i(Q)) = 1, and there are no cosets Y of an abelian subvariety of J contained in
Wd
with j(Q) E Y, with dim Y > 0.
Case (1) is described by Lemma 3.1.1. Case (2) has also been studied in the literature,
most notably in
[HS91],
for the case of d = 2. As d grows, the geometry becomes more
complicated. Case (3) is disjoint from cases (1) and (2).
Theorem 3.1.2 ([HS91]). Let X be a curve over C. If Sym 2 X contains an elliptic curve,
then X is either bielliptic or hyperelliptic.
This is related to the following definition, as in [Lan9l], §1.3, page 16.
Definition 3.1.3. Let Y be a projective variety over Q. The special set of Y is the Zariski
closure of the union of all images of nonconstant rationalmaps f : G
-
Y of group varieties
G defined over Q into Y. We denote the special set of Y by S(Y). For a projective variety
Y over Q, let S(Y) be the closed subscheme of Y whose base extension S(Y)U is S(Y).
It turns out that the set of rational points satisfying (1) or (2) is contained in the set
of rational points in S(Symd X) (this point will be elaborated on, in the next chapter). As
remarked in the previous paragraph, the presence of rational points in the special set has
been studied in the literature, but few things are known about the rational points that are
not contained in the special set. In this thesis, we focus on
(S(Symd X))(Q)}.
#{Q
E (Symd X)(Q) I Q
By Theorem 1.0.5, this set is finite, so Chabauty's method has a hope of
making sense, if we just focus on this set. As remarked in the introduction, some work has
been done by [Sik09], [DK94], [Kla93].
Recall that X/Q is a nice curve of genus g, with a rational point 0 E X(Q). We further
fix d > 1, and assume that X satisfies r
rk J < g - d. Further, let p be a prime, and
assume that X have good reduction at p.
Chabauty's method for Symd X is based on the observation that
(SyMd X) (Q) g j- (Wd,(Qp) n J(Q).
31
Since J(Qp) and J(Q) are both p-adic Lie groups, with dim J(Qp) = g and dim J(Q) ; r =
rk J, there exist at least d independent locally analytic functions on J(Qp) vanishing on J(Q).
These locally analytic functions come from integrating certain 1-forms wi E H0 (JQ,, Q'), as
in the previous section. These functions can be locally expressed as power series; for example,
on residue disks (defined below), they have power series representations.
Definition 3.1.4. A residue disk U of Symd X is the preimage of a point in (Symd X) (PP)
under the reduction modulo-p map
redp : (Symd X)(CP) -- (Symd X)(FP).
If K is a finite extension of Q,, the set of K-points of the residue disk U are defined to be
the set U n (Symd X)(K), and these are denoted U(K).
Let U be the residue disk over 0 E (Symd X)(Fp). Then U fits into the exact sequence
0
-+ U -+ J(CP) = J(Oc') -+ J(Fp) -+ 0,
where the equality in the middle follows from the valuative criterion for properness. Then for
any finite extension K of Qp, we have U(K) = {P E (Symd X)(K) : redp(P) = {P 1 ,..
.
,Pd}}.
Thus, by a Hensel-type argument, there is a bijection
(ui,...
, ud)
: U(K)
-
(pK OKd
between the set of K-points of the residue disk mapping to {P
coordinates U 1 , ..
. , ud,
1 ,...,
Pd} via some local
where PK is the uniformizer of K.
In particular, for any w C H0 (JQ,, Q1 ), one can expand w in this residue disk to get
9
w = w(ui, ...
,ug)
=
wi(i,...,ug)du E Zp[[ui,..., ug]].
Using this, we use the following extended definition of 7
32
=
f w over CP, instead of the
usual definition over Q,
Definition 3.1.5. For any w E H 0 (JQ,, Q1), define the map q : J(Cp) -+ C, by first defining
it on the residue disk U over 0 E (Symd X)(Fp):
U (CP)
P
-
/Q
Q
JO
W(ui,..., lug),
where we integrate w formally. Then we extend q to J(Cp) by the following: Given
Q
E
J(Cp), find m > 1 such that mQ C U (such m exists since J(Fp) is a torsion abelian group),
and define q(Q) := -L(mQ). By an abuse of notation, call this map also 'q : J(Cp) -+ C,.
We note that this also shows that 71 has a power series expansion on each residue disk, not
just the residue disk above 0.
In order to use Chabauty's method to get an upper bound for the number of rational
points of Symd X not contained in (S(Symd X))(Q), we need to be able to pull back these
locally analytic functions on J(Qp) to (Symd X)(Qp). As in the case of classical Chabauty,
we use the p-adic integral defined on the Jacobian J of the curve X, which is assumed to be
of good reduction modulo p, via the natural morphisms of Xd and Symd X into J given by
Xd
-+
-+
SymdX
[P1+---+P--dO],
{P1, ... ,7Pd}
(P1,..Pd) H4
J
The p-adic integrals also satisfy:
Lemma 3.1.6. For Qi, Q2 E J((Q)
and w E H0 (JQ,,
Q'),
/Q1+Q2
Qi
O
1
Q2
0
Proof. We have
2
OQ 1+
JO
1
1+
W
JO
f1
33
2
1
2
W
0
where the first equality follows from linearity, and the second equality follows from the
translation-invariance of p-adic integrals.
El
Then we may define the integral on Symd X via the pullback of the integral on Wd C J,
which can be written using Lemma 3.1.6 as
]
[Pd-O
I
P1-O]
[Pl+...+Pd-d-O]
Therefore, if Wj E HO(JQ,, Q1 ), and if rj is its integral viewed as a locally analytic function
on J(Cp), then the corresponding locally analytic function on Symd X can be written as
n: (Symd X)(Qp) -+ Qp
{PI, P2 , ... ,P}
F) ?j([Pi +P2 +
+ Pd - do1)
P
P2
P1
where w E H0 (XQ,, Q1 ) is the differential corresponding to wj via the canonical isomorphism
HO(JQ,, Q1) -1+ H0 (XQ,, Q1 ), induced by the inclusion X(Q
--+ Ja, as in, for example,
[Mil86, Lemma 2.2].
Since dim J(Q) < g-d, there exist linearly independent
whose integrals vanish on J(Q). Let
obtained by integrating
M,
2,
...
, 7?
M,
2,2 . ..
, .q
(WJ)i, (WJ)2,..., (WJ)d,
(j)I,
(WJ)2, ...
,
(
E H(JQ,, Q 1 )
be the locally analytic functions on Symd X
respectively. We note that it is possible that
could have infinitely many common zeros in CP.
Remark 3.1.7. Let d = 2, and consider one residue disk U.
Then the locally analytic
functions ni have a power series representation on U. Call these qjju. If the zero sets over
Cp of the 77 iJu have infinite intersection, this intersectionmust have codimension 1, so the 7i u
must have a factor, say f, whose zeros correspond to the infinite intersection. Let fi := 1i ju.
We are interested in the zeros of the fi in (pZ7)
2
. This discussion leads to:
Theorem 3.1.8. Assume that dim J(Q) < g - 2. Then the number of points outside the
special set of Sym 2 X for a smooth, projective, and geometrically integral curve X of good
34
reduction at p is bounded above by the sum of the number of common zeros in (pZP) 2 of f,
and f2 over all residue disks, defined as in the above remark.
This makes the d = 2 case much easier than the d > 3 case. In fact, a simpler proof of
Theorem 1.0.8 bypassing much of the arguments in the following chapters exists when d = 2,
since factoring out f means that the deformation theory arguments are no longer necessary.
Further, the bound on the number of points outside of the special set would be improved.
3.2
Explicit parametrization of points on residue disks
As before, let X be a nice curve defined over Q. We view X to be over C, with good
reduction at p. Let K be a finite extension of Q,, with k its residue field, OK its ring of
integers. We have seen in the previous section that r has a power series expansion on each
residue disk.
In practice, this parametrization may be difficult to analyze: §3.1 suggests that we write
the higher-dimensional integrals in terms of several 1-dimensional integrals expanded around
various K-points P, but relating the local coordinates ui to the uniformizers tj of the points
Pi can be complicated. In this section, we explain how to do this.
3.2.1
The case of Sym 2 X
For example, consider the case when d = 2. And take a P E X(F), so that U(Qp) consists
of the Qp-points in Sym 2 X reducing to {P, P}
E (Sym 2 X)(F).
The completion of the local
ring of (X x X)(Q near any pair of points (Q1, Q2) E X x X reducing to {P,P} is given by
Qp[[ti, t 2 ]], where t1 and t 2 denote the uniformizers for the set of Qp-points of the residue
disks U1 around Q1 and U2 around Q2 in X, respectively. We further assume that ti and
t 2 vanish at
Q'
and Q', respectively. Take
Q
:= Q' = Q'. This means that we have two
bijections
ti : U1
~
pZP,
t2 : U2 -_-+Pp
35
Since tj(Q) = 0, and t 2 (Q) = 0. Then for any w E HO(JQ,, Q'), the Coleman integral
77: (Symd X)(QP)
-+
{Q1, Q2} M
Qp
/{01Q,Q2}
W
can be written as the following, in terms of the ti:
7({Q,
Q2}) =
/
1,2}
jO{ {
=
/Q /
i
W+
1i
2
+
where C0=2
f" w
(
2
+fQ W
i(Qi)
Jo
Q
ft2(Q2)
w(si)ds1 +
0
(s 2 )ds 2 ,
is a constant in Qp (that depends on the choice of
Q).
One can relate the t, to the original local coordinates of U(Qp). If r(Q1, Q2) = (P,P) in
2
(Symd X)(Fp), then the completion of the local ring at {Q1, Q2} E (Sym X)(Q,) is given by
Qp[[ti, t 2 ]]S2
=
where S 2 is the permutation group on the elements {ti, t 2 }. So one could choose the tj and
the ui to satisfy ui
=
t 1 +t
2
and u 2
=
tt
2
. This means that for each Qp-point {Qi, Q2} in the
residue disk, which corresponds bijectively to a unique element in pZ2 , there are generically
two pairs (ti, t 2 ) that correspond to it.
On the other hand, if Ub(Qp) were the Qp-points of the preimage of {P 1 , P 2 } c (Sym 2 X)(FP)
with P
#
P2 under the reduction by p map, the situation is simpler, as we have the following
description of U(Qp):
U(Qp) = {(Qi, Q2) : Qj E X(W(Fp2 )) reducing to Pi E X(Fp2 ), (Q1, Q2) E (Sym 2 X)(Q,)}
C X(W(IF P2))
X
X(W(lFp2)),
36
where W(F
2)
denotes the ring of Witt vectors over
(ti,
t2) : ll(Qp)
-~-+
Fp2.
Pp
X
Thus, there are the bijections
Ppi
Then, we can write for each w E H0 (XQ,, Q1 )
01
0,Q2}
rl({Ql,
= f0
Q2}) = 10
/JQ
=
+J Q1
2
+ J0
JQi W +
C + 1 ti(Qi) w(si)ds, +
0
1 t 2 (Q2 )
2
W
s 2 )ds 2 ,
0
where the w(si) express the w over the chosen residue disk, and C
=
W + f
9
w, is a
constant (that depends on the choice of Q' and Q'). In this case, the tj are parametrizations
of a degree-2 point on X, as given in §2.3.2.
In general, a class of multivariate power series that is particularly simple to deal with is
the following.
Definition 3.2.1. A power series f E K[[t 1 ,..., tj] is said to be pure if each of its terms
is of the form Ct7, with C E K and N E Z;>O. In particular, a pure power series does not
contain any term that is a product of more than one variable.
Our goal for the rest of the section is to find pure power series that are related to the
locally analytic functions in d variables that we obtain from Chabauty's method.
3.2.2
The case of Symd X
In this section, we generalize §3.2.1 to the case of Symd X. More concretely put, our aim
is to express each p-adic integral obtained from an w E H0 (XQ,, Q1 ) (see §3.1), which is
represented by a power series on each residue disk of (Symd X)(Qp), as a pure power series
over some extension field of
Q,
on the residue disk. We will show that this is possible by
doing a change of variables on the local coordinates of each residue disk. We fix a regular
37
differential w, from which we get one of the d power series vanishing on J(Q) n (Symd X) (Q,)
as in section §3.1.
We now consider the residue disk U given as the preimage of the point {P 1 , ... , Pd} E
(Symd X)(Fp) under the reduction map modulo p. Let u1 , ...
, uj
be its uniformizer, that
provides the bijection
(ui,... , Ud) :
(Qp)
*
(PZ)d.
Consider the locally analytic function obtained by p-adic integration of an W E H0 (JQP, Q1 )
restricted to U(Qp), given by
rq : (Symd X)(Qp) - Qp
{P1...
f~~~ P}
PsdW
E-4Zj.
1<i<dfOZ
The restriction of -q, to U(Qp) can be written as a power series using the parametrization
by (ui,... , Ud). However, in what follows, we explain how to find another parametrization
so that r7 can be written as a pure power series.
Given {P 1 ,..., Pd} E (Symd X)(F), choose one ordering of the P so that we have
(P 1 ,...
,
Pd) E Xd(PP). From the surjectivity of the following commutative diagram, each P
pulls back to a residue disk, call it Ul, in the i-th factor of Xd(Cp) over K, where K is the
p-adic field whose residue field is the compositum of the fields of definition Ki of the Pi.
Xd(CP)
I
X d((F)
(Symd X)(Cp)
I
(Symd X)(FP)
Pick a lift {Q,..., Qd} E (SymdX)(Cp) of {P1,..., Pd}, and let (Q1,... , Qd) E Xd(CP)
be a lift of (P 1 ,.. ., Pd) so that the above diagram commutes. Let tI,... , td be the uniformizers for each of these sets of Ki-points of residue disks Ub such that ti(Qj) = 0. Using these
38
uniformizers, we can re-write
1, 1u
as a sum of d 1-dimensional integrals as follows. If we let
77W'i : U,4(KI) -+ Kz C; K
Qi
/ti
-
(Qi)
o
(si)dsi
for 1 < i < d, then for each choice of lifts {Q1,..., Qd} and (Q1,..., Qd) of {P 1 ,...
P
that makes the above diagram commute, the following identity holds, as a consequence of
the discussion at the end of §3.2.1.
,i(Qi)
r7?({Q1,..., Qd})=
1<i<d
That is, each q, can be written as a pure power series. Furthermore, ti E 7r0K, where 7r is
a uniformizer of the p-adic field K, and OK is the ring of integers of K. Suppose that v is the
normalized valuation such that v(p) = 1. Then v(7r) ;> -, since {P 1 ,.
..
, P} E (Symd X)(Fp),
and so P is at most a degree-d point for each 1 < i < d.
Decompose the multiset {P 1 ,..., Pd} into disjoint multisets Si,...,S, each consisting
The above discussion leads to the following
of a single point with multiplicity si = #Si.
proposition:
Proposition 3.2.2.
(a) The power series q, = (Symd X)(Qp) -+ Q, obtained from p-adic integration of the
1-form w can be re-written via a change of variables as a pure power series, whose
coefficients are contained in some extension of Qp of degree at most d.
(b)
Suppose that one obtains d power series 71,...
,d
via Chabauty's method as outlined
in the previous section, and that one rewrites these power series as n',..., rk, where q'
are pure power series obtainedfrom part (a). Then there is a N-to-one correspondence
between the common zeros of the Ti and ', where N = fs
(si)!. Further,the solutions
to r7 that correspond to the points Symd X(Qp) have valuations of at least 1/d.
39
Now, it remains to associate Newton polygons to these power series, and apply arguments
analogous to [Col85].
40
Chapter 4
Comparison of the algebraic loci and
analytic loci on Symd X
In this section, we are interested in comparing different sets that contain (Symd X)
(Q) inside
(Symd X) (Cp). The different subsets of (Symd X) (Ce) that we consider are described below:
(i) Faltings' theorem says that given the natural embedding
j
: Symd X " J using the
basepoint 0 E X,
Wd(Q)
=
U
i(>Q)
finite
where the Y are cosets of abelian subvarieties of J with Y C Wd. The choice of the Y
is not necessarily unique; here, we fix a particular set of Y such that no 0-dimensional
Y is contained in a Z that is a coset of a positive-dimensional abelian subvariety of J
with Z C Wd. Then the set of points that we are interested in is the set of Cp-points
of the inverse image Uj
(Yi). It will be denoted F(Symd X)(Cp).
(ii) The set of Cp-points of the special set S(Symd X) of Symd X: recall that the special
set was defined in Definition 3.1.3. This set will be denoted S(Symd X)(Cp).
(iii) The set
{P E (Symd X)(Cp) : qj=(j(P)) = 0 for all 1 < i < d},
41
where the ri are d independent locally analytic functions on J(Cp) that vanish on J(Q)
arising from Chabauty's method. Since this definition depends on the choice of the 7m;
we will fix one such choice here. We denote this set by (Symd X)= 0 .
In this section, we relate these different sets.
If d = 1, g > 2 and rk J < g, then
all of the above sets are zero-dimensional, which makes the comparison simple: We have
0 = S(X)(CP) C F(X)(CP) C (X)= 0 .
For d > 1, we will see that these sets do not obey a linear containment relation; in particular, there does not seem to be any inclusion relation between S(Symd X) and (Symd X)1=0.
this necessitates an extra technical hypothesis to force such an inclusion. This seems to be
an intrinsic limitation of Chabauty's method on higher-dimensional varieties; a new idea
seems to be necessary to obtain more precise information on the rational points of Symd x.
In §4.1, we review some basics and terminology of rigid analytic geometry that will enable
the comparison of the sets above in §4.2.
4.1
Rigid analytic geometry
Rigid analytic geometry is used for two purposes in this thesis: one is to cut out and describe
a possible set of points in Symd X that contains all of the rational points via Chabauty's
method as mentioned in the previous chapters; the other is to use deformation theory. Even
with the generalizations of all the tools that are used in [Col85], we are still missing one
key ingredient: we need to make sense of the cases where #(Symd X) (Q)
=
oc. We do this
by deforming the analytic functions arising from Chabauty's method so that they only have
finitely many solutions, and interpreting the finitely many solutions as an upper bound on
the number of points outside of the special set. Rigid analytic geometry makes possible such
deformation theory techniques. In this section, we focus on the first purpose. In particular,
we review the basic terminology, and describe the irreducible components of a rigid analytic
variety. The exposition of this section is taken from [ConO8] and [Rab12].
As before, let K be a field complete with respect to a nontrivial non-archimedean absolute
42
value I- 1, with corresponding valuation v.
Definition 4.1.1. The Tate algebra in n variables is the K-algebra
K(xi,..., xn) :=
auxu E K[[x 1 ,..., xn]] : laul -+ 0 as |jull -+ oo
where f|(u1,...,un)|| = E_ 1 ui for u E ZO. Then the Tate algebra K(x1,... ,xn) is an
integral domain, that is Noetherian, regular and a UFD. A K-affinoid algebra is a Kalgebra that is isomorphic to a quotient of a Tate algebra. The maximal spectrum of an
affinoid algebra A, denoted MaxSpec A, is the set of maximal ideals of A.
in [Con08, Exercise 1.2.3], MaxSpec(A) is
functorial
As remarked
via pullback; that is, given a map
f : A -+ B of K-affinoid algebras, the prime ideal f-1 (m)
C
B is a maximal ideal of B for
every maximal ideal m of A. So we may talk about the map MaxSpec(f) : MaxSpec(B)
-+
MaxSpec(A).
Remark 4.1.2. We note in particularthat K(x) is a Dedekind domain.
If K is algebraically closed, then MaxSpec(K(x1,... , xn)) is the unit ball B7n
xil
=
{(x 1,...
,)
1, 1 < i < n}, which plays the same role as affine space An in algebraic geometry.
Definition 4.1.3. Let A be a K-affinoid algebra. A subset U C MaxSpec(A) is an affinoid
subdomain if there exists a map i: A -+ A' of K-affinoids such that the image of the
induced map MaxSpec(i): MaxSpec(A') --+ MaxSpec(A) lands in U and is universal for this
condition in the following sense: for any map of K-affinoid algebras q: A -* B, there is a
commutative diagram
A
A'
B
if and only if MaxSpec(#) carries MaxSpec(B) into U, in which case such a diagram is
unique. In fact, by Yoneda's lemma, such A' is unique up to unique K-isomorphism, so we
write Au = A', and call it the coordinate ring of U.
43
Some examples of affinoid subdomains are the following:
Example 4.1.4. Let A be an affinoid algebra.
(i) A Laurent domain is a subset of MaxSpec(A) of the form
D(f, g- 1 ) = { C MaxSpec(A) : f1( ) 1,.-, fn( )j < 1, jgi(), ..., 9M()1 > 1}
for some f = (fi,.
..
, fm), g = (gi, -- ,gM) with fi, gi E A. If m = 0, we call D(f) a
Weierstrass domain. The coordinate ring for the above Laurent domain is given by
A(f, g 1 ) := A(x 1,... , Xn, Y, ... , Ym)/(Xi - fi, ... , Xn - fm, y1g1
-
1,--.
, Ymgm -
1).
(ii) A rational domain is a subset of MaxSpec(A) of the form
{{ E MaxSpec(A) : jfi(x)I,..., Ifm (x)l < Ig(x)I}
for f1,...
, fm,
g c A with no common zeros.
The coordinate ring for this rational
domain is
A( fi
g
f) :=A(x,...,xn)/(gx -
9
fi,...,gXm - fm).
In fact, affinoid subdomains admit a simple description in terms of the rational domains,
by the following theorem of Gerritzen-Grauert:
Proposition 4.1.5. Let A be a k-affinoid algebra. Every affinoid subdomain U C MaxSpec(A)
is a finite union of rational domains.
In order to define a suitable topology on affinoid domains, we need a few more definitions:
Definition 4.1.6. A subset U C MaxSpec(A) is an admissible open subset if it has a settheoretic covering {Ui} by affinoid subdomains such that for any map of affinoids f : A -+ B
44
with (MaxSpec(f))(MaxSpec(B)) C U, the cover (MaxSpec(f))- 1 (Ui) of MaxSpec(B) has
a finite subcover.
A cover {Ui} of an admissible open subset U is an admissible cover provided that for
any map of affinoids f : A -+ B such that f*(MaxSpec(B)) C U, the covering {(f*) -(Ui)}
of MaxSpec(B) has a refinement consisting of finitely many affinoid subdomains.
Then we impose the Tate topology on MaxSpec(A), which is a Grothendieck topology
that has as objects the admissible open subsets and as coverings the admissible open coverings. The Tate acyclicity theorem lets one define a sheaf of rings on the topological space
MaxSpec(A).
Theorem 4.1.7. Let A be a K-affinoid algebra. The assignment U
H->
Au of the coordinate
ring to every affinoid subdomain of MaxSpec(A) uniquely extends to a sheaf OA with respect
to the Tate topology on MaxSpec(A). In particular, if {Ui} is a finite collection of affinoid
subdomains with U
=
U also an affinoid subdomain of MaxSpec(A) then the sequence
0
+
Au
-+
7 Au,
-+
f
Auynu3
is exact.
Then the locally ringed topological space (MaxSpec(A),
OA)
is called an affinoid space
and is denoted Sp(A).
To globalize this notion, we use the notion of G-topologized spaces, which is a set X
equipped with a set of subsets (meant to generalize the notion of open sets), with a structure
sheaf. As we will only need affinoid spaces in this thesis, we do not go into the details of the
construction.
Definition 4.1.8. A rigid-analytic space is a topological space (satisfying some technical
hypotheses) which admits an admissible cover by affinoid spaces, and a morphism of rigid
analytic spaces is a morphism in the category of locally ringed G-topological spaces.
To conclude, we show that there is a notion of irreducible components on rigid analytic
45
spaces. The theory of irreducible components was first suggested in [CM98], and simplified
in [Con99]. We quickly summarize [Con99]:
Definition 4.1.9. A rigid analytic space X is disconnected if there exists an admissible
open covering {U, V} of X with U n V
=
0, where U, V
#
0. Otherwise, X is said to be
connected.
Definition 4.1.10. Let X be a rigid analytic space that admits a cover of affinoid spaces
{Sp A}EA. A morphism 7r : X -+ X is said to be a normalization if it is isomorphic to
the morphism obtained by gluing Sp(AA) -+ Sp A,\, where AA denotes the normalization of A
in the usual sense.
It is known that for any rigid analytic space X, we can find a normalization 7r X
-+
X;
for example, see [Con99, Theorem 1.2.21.
Definition 4.1.11 ([Con99], Definition 2.2.2). The irreducible components of a rigid
analytic space X are the images of the connected components Xi of the normalization X
under the normalization map ir : X -> X.
Remark 4.1.12. When X = Sp(A) is affinoid, the irreducible components of X are the
analytic sets Sp(A/p) for the finitely many minimal prime ideals p of the noetherian ring A.
4.2
Comparing algebraic and analytic loci in Symd X
Let X be a smooth projective curve with the choice of a rational point 0 E X(Q), with
good reduction at p. The goal of this section is to determine the containment relations of
the three sets mentioned at the beginning of this chapter; namely, .F(Symd X), S(Symd X),
and (Symd X)p=o.
Lemma 4.2.1. For any smooth projective curve X with the choice of a rational point 0 E
X(Q) with good reduction at p, one has
(Symd X)(Q) C (Symd X)?= .
46
Proof. Recall that each 7i mentioned in part (iii) of the introduction of this chapter satisfies
,i(P)
=
0 for all P E J(Q). In particular, (Symd X)(Q)
C
j-(J(Q)) g (Symd X)(Cp)"=O.
El
Lemma 4.2.2. We keep the notation of Y and S(Symd X)(Cp) from the beginning of this
chapter. Let Y = Y for some i such that dimY > 0. Then j
1
(Y(C,)) C S(Symd X)(Cp).
Proof. We consider two cases: if the generic point of Y has a positive-dimensional preimage,
then each Q E Y(Cp) is P for some n > 0, so each fibre is contained in S(SymdX)(Cp).
On the other hand, if the preimage of the generic point of Y is 0-dimensional, then any
irreducible component of j 1 (Y) is either covered by positive-dimensional projective spaces,
or is birational to Y via the restriction of j. All of these irreducible components are then in
M
the special set.
Finally, it remains to relate S(Symd X) (Cp) and (Symd X)y= 0 . It seems likely that in general, neither set is contained in the other; however, with Assumption 1.0.7, we immediately
get:
Proposition 4.2.3. Let R 1 ,..., R, be the irreducible components of the rigid analytic space
(Symd X)n=0 . Further, suppose that Symd X satisfies Assumption 1.0.7. Then for each Ri
with dim Ri > 1,
Ri(Cp) C S(SymdX)(Cp).
Then taking complements of the relation obtained in Proposition 4.2.3 inside Symd X
and looking at the Q-points, one obtains:
Corollary 4.2.4. Under the hypothesis of Proposition 4.2.3, one has
{Q-points of Symd X\S(Symd X)} C UJo-dimensional Ri}.
Thus, under the hypothesis of Proposition 4.2.3, one is still able to interpret the results
given from Chabauty's method for higher-dimensional varieties, as Chabauty's method gives
47
an upper bound on U(O-dimensional Ri).
For the rest of the thesis, we assume that the
conditions of Proposition 4.2.3 hold for Symd X.
48
Chapter 5
p-adic geometry
The goal of this section is to associate a "generalized Newton polygon" to each multivariate
power series, and to state an approximation theorem for the number of roots of a system of
equations given by d power series in d variables in general position - that is, having finitely
may common zeros - using these Newton polygons. The classical case of d = 1 is well-known
in the literature. To define the Newton polygons for multivariate power series, we review
the language necessary to define tropical objects, and state the results in tropical geometry.
For a more detailed treatment of tropical geometry, see [MS13] and [Rab12].
5.1
Convex Geometry
This preliminary section establishes the language necessary to define the generalized Newton
polygon. The notation follows [Rab12].
Notation 5.1.1. We fix for the rest of the thesis the following notation:
49
N 7Zd
a lattice
M = Homz(N, Z)
the dual lattice of N
NR = N Oz R 2-- R d
d-dimensional vector space
M=
N;
(-, -)
M
the dual vector space of NR
x
NR -+ R
canonical pairing
FcR
nonzero additive subgroup
Nr= N OZ r
the subgroup of F-rationalpoints of NR
Mr.= M oz
likewise for MR.
The following is standard in convex geometry:
Definition 5.1.2. Let L be a d-dimensional lattice, and Lv be its dual lattice. That is, we
may take L = N or L = M. As before, LR and L' denote the d-dimensional vector spaces
over R.
(a) An affine half-space (or just half-space) in LR is of the form
H = {w E LR I (u,w) < a}
for some u E Lv\{O} and a C R. It is an integral (affine) half-space if u C Lv,
and it is integral IF-affine if u C Lv and a E IF.
(b) A polyhedron in LR is a nonempty intersection of finitely many half-spaces. if each
half-space in the intersection is integral (resp. integral 1-affine), the polyhedron is also
said to be integral (resp. integral 1-affine). A bounded polyhedron is called a polytope.
(c) Given a polyhedron P C LR, and u E L', we define
faceu(P) = {w E P I(u, w) > (u, w') for all w' E P}.
Each nonempty such subset is called a face of P. A face consisting of a single point
is called a vertex of P. Further,P0 denotes the interior of P.
50
(d) A polyhedral complex is a finite collection U of polyhedra in LR called the faces or
cells of H, satisfying
then PnP' is a face of P and P';
(i) If P,P' E H and PnP'#0
(ii) If P C U and if F is a face of P, then F e H.
The support of H is the set |ul
=
UPEH P.
(e) A cone a- is an intersection o of finitely many half-spaces in LR each defined by {w E
LR :
(u,w)
We say that a cone - is pointed if 0 is a
0} for some u E Lv\{0}.
vertex of -. A fan is a polyhedral complex A whose cells are cones. A fan is complete
if its underlying set
(f) Let - =
|J
is LR, and pointed if all cells of A are pointed cones.
2 R;>owi be a cone in L. The dual cone of o- is the cone
r
or = {u E L
: (u, w)
<
0 for all w E a} =
n{u
E Lv: (u, wi)
0}.
i=1
In particular,avv =
(g) For each face F of a polyhedron P C LR, define
Af(P, F) := {u E Lv : F c faceu(P)}.
The normal fan to P is the fan X(P) in Lv whose cells are the cones Af(P, F)
for each face F of P.
Example 5.1.3. Let m =
mi for 1 < i < d}.
(Mi,
E Qd,
... ,id)
and let Pm := {(xi,.
Let ej : N -+ Z be the i-th coordinate map.
Hi = {w E NR : (-ej,w) < -mi}.
Then
d
Pm =
51
H.
..
,xn) E NR : xi
Then -ei
>
E M. Let
If -mi E F for all i, then P, is an integral F-affine polyhedron.
5.2
Tropicalizing hypersurfaces convergent on a polyhedral domain
In this section, we will discuss the tropicalization of hypersurfaces defined by power series
over CP, convergent on some open neighborhood of 0 in (C
)d.
The tropicalization should be
seen as the dual of a Newton polygon; this notion will be made precise in this section. More
generally, everything in this section works for a nontrivially valued field K that is complete
with respect to the nontrivial valuation. We further assume that K is algebraically closed.
Tropical geometry will generalize the theory of Newton polygons of single-variable power
series to apply to power series of several variables. Most of the exposition from this section
is taken from [Rab12].
Let R:= R U {-oo}, which is endowed with the topology that restricts to the standard
topology on R, and whose neighborhood basis of -oc is given by the sets of the form [-oo, a)
for a C R. Then R>o acts on R by continuously extending the action of JR>O on R.
Definition 5.2.1. Let a be a cone in NR.
The partial compactification of NR with
respect to a is the space Na-) := HonR,>(9V,K) of monoid homomorphisms respecting
multiplication by the elements of R>O; hence, NR(-) is equipped with the topology of pointwise
convergence. Its underlying set is in bijection with
U NR/ Span(-'),
where a' ranges over the faces of a. Roughly speaking, Na-) compactifies NR in the direction
of the faces of a.
Definition 5.2.2. Let P be a polyhedron. The cone of unbounded directions of P,
denoted -p or a (when there is no ambiguity), is given by the dual of the cone A(P). The
compactification P of P is the closure of P in NR(-).
52
Example 5.2.3. Let NR = R2 and let P c R 2 be the polyhedron
P = {(x, y) E R2 : X
1, y > 1, x + y > 3}.
Then its cone of unbounded directions Up is the first quadrant. We then get
P = P U ({oo} x [1, oc)) U ([1, o) x {o}) Li {(oo, oo)}.
Definition 5.2.4. For the rest of the thesis, we work over a non-archimedean field K,
endowed with a nontrivial valuation v : K -+ R U {oo}, that is algebraically closed and
complete with respect to the nontrivial non-archimedean absolute value |
=
exp(-v(.)). We
will let F denote the corresponding value group of v, and assume that F is dense in R. Then
we define the tropicalization map to be
trop: Kd
-+
(V)
(v(
((1, ....,
Let P be a polyhedron in Na where N
U
(R U {oo})d
1),
. . . , V(
) .
Zd, and define Up := trop- 1 (P) C Kd. Further, let
-
be the cone of unbounded directions of P, and let S, := -n M. We define
K(Up) :=
E auxu : au E K, v(au) + (u, w) -+ oc for all w E P
where the convergence v(au) + (u, w) -+ oc holds as u ranges over the elements of S, in any
order. This can be interpreted as the set of power series that converge on the points in Kn
whose valuations belong to P.
Example 5.2.5. Let 0 = (0,0,... ,0)
C
Zd,
and let P = Po, as defined in Example 5.1.3.
Then
Up
=
{(X1 ,...
, Xd)
:v(xi) > 0 for all i},
53
and
K(Up) = K(x1,.
. .,x)
Remark 5.2.6. More generally, it is known that K(Up) is a K-affinoid algebra (Rab12,
Lemma 6.9(i)]), a Cohen-Macaulay ring ([Rabl2, Lemma 6.9(v)]), and that Up = Sp K(Up).
Definition 5.2.7. Let P be a polyhedron, and let f1,..., f, E K(Up).
Let (f1,...,fn) be
the ideal in K(Up) generated by f..... , fn. Then
V(f 1 , ...
Then V(f 1 ,.
.
, fn)
:= Sp K(Up)/(fi,...
, fn).
, fn) is an affinoid subspace of Sp K(Up).
In our case, each the power series
f
E K[[x1,...
, Xd]]
that arises from Chabauty's method
on SymdX converges when v(xi) > 0 for 1 < i < d. Thus,
f
E K(Up) for any P = Pm,
where Pm is defined as in Example 5.1.3, where m E Qd.
Now fix an integral l-affine pointed polyhedron P, and let
fine Trop(f), the tropical variety corresponding to
f,
f
E K(Up).
We will de-
and then outline the procedure for
computing Trop(f) in § 5.3.
Definition 5.2.8. Let P be an integral F-affine pointed polyhedron. For f C K(Up),
Trop(f) := trop(V(f))
=
{(V(1), .. . ,V(d)) : f()
=
0,
C
E Up},
where V(f) is the affinoid subspace defined by the ideal a = (f) C K(Up).
Here, we take the
topological closure in P.
Often, the easiest way to compute Trop(f) is by using Lemma 5.2.10, which requires
these definitions:
Definition 5.2.9. Let P be an integral F-affine pointed polyhedron, and let a be the cone of
unbounded directionsfor P. Let T be a face of o. For 0 # f E K(Up), write f = ZCs' auxl.
54
Let w E P. That is, w E NR/ Span T for some face
T
of the cone of directions of P. The
height graph of f with respect to r is
H(f, r) ={(u,v(au)) : u E S, O r', au
0 } C (S, fL)
n
x
R.
We also define
mf (w) = m(w) = min{(-w, 1) - H(f, T)},
where - denotes the usual dot product, and
vertw(f)
=
{(u, v(au)) E H(f, T) : (-w, 1) - (u, v(au)) = m(w)} 9 H(f, T).
Intuitively, m(w) denotes the minimum valuation achieved assuming that v(x) = w, among
the terms of f.
Then vertw(f) records the corresponding terms of f with the minimum
valuation, again assuming that v(x) = w.
The following is the power-series analogue of a well-known result proved for polynomials;
the original result for polynomials is first recorded in an unpublished manuscript by Kapranov, and a proof of this lemma for power series can be found in [Rab12], Lemma 8.4; also
see, for example [MS13], Theorem 3.1.3. This gives a useful method to computing Trop(f).
Lemma 5.2.10.
Trop(f) = {v E P: # vert(f) > 1}.
5.3
Tropical intersection theory and Newton polygons
In this section, we assume that we have a nontrivially valued field K that is complete with
respect to the nontrivial valuation. We further assume that K is algebraically closed. We
also assume that P is an integral I'-affine polyhedron of dimension d, and that we have d
power series in d variables in K(Up) that have finitely many common zeros. We will explain
that in order to bound the number of common zeros of d power series in d variables, it
55
suffices to know their tropicalizations and their Newton polygons. Since the tropicalizations
and the Newton polygons depend only on finitely many terms of the power series, this section shows that one can approximate a power series of several variables by a polynomial for
the purposes of intersection theory. In a sense, this is a stronger approximation than what
Weierstrass approximation can tell us; Weierstrass approximation for multivariate power series approximates f E K[[ti,... , td]] by f' E K[t 1 ][[t 2 , . .. , td]], whereas here, we approximate
f by f" C K[ti, ...,i
td].
Let P be an integral F-affine polyhedron, and let
vertp(f) :=
U
f
vert
E K(Up). Write f = E auxu. Define
>(f.
wEP
[Rab12, Lemma 8.2 proves vertp(f) is finite via the following:
Let F be the union of the bounded faces of P, and let o- be the cone of unbounded
directions of P. Then P
=
Fb + -. Let a C K[S,] be the ideal generated by {xU : a,
Since K[S,] is noetherian by Gordan's lemma, there exist U1 , ...
,
0}.
Ur E S, such that a =
(XU1,... I,Xu). Let
a = max{v(a,,) + (ui, w)
:i=1,...,r, and w C F}.
Let v 1 ,... , v, be the vertices of P. Let
f = T = {uE S, : v(a) + (u, v) < a for some j = 1, ... ,s}.
Then T is finite since v(au) + (u, vj) -+ oc as a ranges over S,, for each j.
Further, let 7r : M x F -+ M be the projection map that forgets the last coordinate.
Then:
Lemma 5.3.1 ([Rabl2], Lemma 8.2). Let P c NR be an integral F-affine polyhedron and
let f c K(Up) be nonzero. Then the set ,r(vertp(f)) is contained in 'I; hence, vertp(f) is
finite.
56
This lemma, along with Lemma 5.2.10, tells us that Trop(f) determined by only finitely
many terms of
f.
Now, the following lemma shows that if the coefficients of a power series f are perturbed
in a way so that their v-adic valuations do not change, and so that vertp(f) does not change,
then the tropicalization also stays the same.
Lemma 5.3.2. Let P be an integral F7-affine polyhedron, and let f, f' E K(Up), with f
Z auxu and f'
=
Z a'ex.
Suppose that vertp(f)
=
=
vertp(f'). Then Trop(f) = Trop(f').
Proof. Fix w E P. We claim that vert.(f) = verte(f') for each such v. Choose uo E S,
minimizing v(au) + (u, w). This means
mf (w)
=
Thus, (uo,v(auO)) E verte(f) C vertp(f)
v(auo) + (uo, w).
=
vertp(f'). So (uo,v(auO)) E vert"'(f) for some
w' E P. In particular, (uo, v(au0 )) = (uo, v(a')). Thus,
min (v(au) + (u, w))
uES,
=
v(auo) + (u, w)
=
v(a'u) + (uo, w) > min(v(a' ) + (u, w)).
UEes
The symmetric argument proves the inequality in the other direction, showing that vert" (f) =
vertw(f') for each w E P. Then by Lemma 5.2.10, Trop(f) = Trop(f').
l
Given f E K(Up), we can find a Laurent polynomial g E K(Up) such that vertp f =
vertp g, in which case Lemma 5.3.2 implies that Trop(f)
=
Trop(g):
Corollary 5.3.3. Let P be an integral F-affine polyhedron, and let f E K(Up), with f =
auxu. By an abuse of notation, we also denote the projection map tensored with R given
by MR x R - MR by 7r. Let S C M be a finite set containing ,(vertp(f)). Define the
ZU
auxiliary polynomial of the power series f with respect to S by
gs = Z auxu E K(Up).
Ues
57
Then Trop(f) = Trop(gs).
Proof. Since vertp(f) = vertp(gs), and since Trop(f) only depends on vertp(f), the conclusion follows.
D
Now, we look at the dual situation, which is described by the Newton polygons. As
in the case of the theory of Newton polygons in the classical case of power series with
one variable, the tropicalization Trop(f) of
f,
which can be considered to be the dual of
the generalized Newton polygon (to be defined in the next paragraph), contains all the
information about the valuations of zeros of
f
counting multiplicity. We will now define the
Newton polygon, explain that it too depends on only finitely many terms of
f, and
conclude
that the information about the intersection theory of the power series also depends on finitely
many terms of
f,
for any power series f that converges on some polyhedral domain P.
If S is a finite set of points in a lattice N, its convex hull in NR is denoted conv(S).
Definition 5.3.4. Let P C NR be an integral 1-affine polyhedron, let a be its cone of
unbounded directions, let f E K(Up), and let T be a face of -. For each w c P n N,,, define
its associated Newton polytope
- (f) = y
=
7r(conv(vert (f))).
In this thesis, we will be especially concerned with the case where
T =
{0}.
Remark 5.3.5. We note that the gs from Corollary 5.3.3 are chosen so that the 7"(f) =
-yw(gs) as well.
Now, we outline some theorems from [Rab12] that show that the information about the
common roots of
fi
E K(Up) having a specified valuation w is encoded in Trop(fi) and my.
For the rest of this chapter, we assume that P = Pw, where Pw is a translate by a w E Q
0
of the first quadrant of dimension d, as in Example 5.1.3.
Definition 5.3.6. If fi,..., fd E K(Up) C K[[xi,... , xd]1, Y = V(fi),
and Y =
Assume that Y is 0-dimensional. Then the intersection multiplicity of Y 1 ,...,Yd
58
i Yi.
at
w E Nr n P is defined as
i(w;Y 1 -. Yd) = dimK H0 (Y n U{w, Oynu,}),
where we view {w} as a 0-dimensional polytope. In simpler terms, this intersection multiplicity at w is the number of common zeros of the fi that have the same coordinate-wise
valuation as w, counting with multiplicity.
The intersection number of Y,..
.,
Y is
i(Y, ... , Yd) := dim H (Y Oy).
Definition 5.3.7. Let P 1 ,..., Pd be bounded polytopes. Define a function
Vp ,...,pd(,
...
., Ad) := vol(AiP1 + - - - + AdPd)
where + denotes the Minkowski sum. The mixed volume of the Pi, denoted MV(P 1 ,... , Pd),
is defined as then coefficient of the A -- -Ad-term of Vp,...,pd(Al,....,
Ad).
In fact, as long as ni V(fi) is the union of finitely many isolated points, the intersection
number can be deduced from just the information present in Trop(f) and New(fi), as stated
in the following theorem:
Theorem 5.3.8. [Rab12, Theorem 11.7] Let P
=
PMc
NR with m E
J'-affine pointed polyhedron as in Example 5.1.3, let fi,..
common zeros, and let w
E niL Trop(f)
fd
QiO be an integral
E K(Up) have finitely many
be an isolated point in P'. For i = 1,...,d let
Yi = V(fi) and let -i = w( fi). Then
i(W, Y - . -Yd) = MV(-hi, . .. ,).
Remark 5.3.9. In fact, as long as the fi have finite intersection number, even if there
were a positive-dimensional connected component C of
59
% Trop(fi), then there exist e > 0
and vectors v 1 ,...
intersection fl
1
such that one can shift Trop(fi) by the small vector Evi so that the
,v,
(Top(fi) +Evi) only has isolated points. Then Theorem 5.3.8 can be applied
to each of these points to compute the intersection multiplicities of the fi.
In particular, Theorem 5.3.8 implies that we may approximate power series by polynomials, if we want to compute the intersection number.
Theorem 5.3.10. Let P = Pm g NR as before, where NR is an d-dimensional vector space.
Suppose that f1,... , fd C K(Up), and let gi be the auxiliary polynomials of the fi with respect
to some finite set S C M containing all u such that (u, v(au)) E vertp(f ). Then
: i(W, V(f1) ...V (Ma)
=
: i (* V (1)
..-.
V(gd)),
wEP 0
wEPo
if both are finite.
Proof. By the choice of the gi, Trop(fi) = Trop(g ) for each 1 < i < d, and -yw(fi) = -yw(gj)
for each w C P. We note from Theorem 5.3.8 that Trop(fi), Trop(gi), /,(fi) and -y(gi) are
the only information required in computing the intersection multiplicities.
F
The following results for polynomials are useful in estimating the number of zeros of
power series. First, a definition:
Definition 5.3.11. Let f =
uCA auxu be a polynomial, where A C Zd is a finite set. Then
the Newton polygon of f is given by
New(f)
=
conv({u : u E A, au / 0}) C R.
We recall Bernstein's theorem:
Theorem 5.3.12 ([Ber75]). Let fi,... , fdE Kfxi,... , Xd] be polynomials with finitely many
common zeros. Then the number of common zeros with multiplicity of the fi in (KX
given by
MV(New(fi),..., New(fd)).
60
)d
is
Further, suppose that the
fi have finitely many common zeros whose valuations belong to
P, and also suppose that the gi have finitely many common zeros in Kd. By Theorem 5.3.10,
number of common zeros of the
fi
with valuations in P
= number of common zeros of the Ps(fi) with valuations in P
< number of common zeros of the Ps(fi) in (K )d
5 MV(New(gi),
...
, New(gd)),
where the last inequality follows by Bernstein's theorem.
Thus, we conclude:
Theorem 5.3.13. Let P = P, C NR as before, where NR is an d-dimensional vector space.
Suppose that f1,... , fd E K(Up), and let Ps(fi) be the associated polynomials of the fi with
respect to some finite set S C M containing all u such that (u, v(au)) E vertp(f). Suppose
further that ri V(f )
#
< oc and
= I V(Ps(fi))
/
d
(Kx )d n
V(fi)
< oc. Then
MV(New(fi),.. .,New(fd)).
Proof. Again, the proof follows from the fact that by the choice of the gi, we have Trop(fi)
=
Trop(gi) and -y,(fi) = 7,(gj) for each 1< i < d and w C P.
El
Remark 5.3.14. Note that the power series obtained from Chabauty's method are allowed
to have solutions of the form (x 1 , ...
, Xd),
where some of the xi may be 0. This means that
to get an upper bound on all of the solutions of the power series, one needs to apply Theorem
5.3.13 multiple times, while setting some of the xi = 0.
61
62
Chapter 6
Continuity of roots
Throughout Chapter 6, K is a complete, algebraically closed valued field with respect to a
nontrivial, nonarchimedean valuation v: K
-+
Q.
In [Rabl2], one studies the intersection
theory of power series in K(Up) that have finite intersection. Our goal in this chapter is to
analyze what happens when the power series in K(Up) have possibly infinite intersection;
we will show that these power series have "small" deformations that have finite intersection,
and that they preserve information about the 0-dimensional components of the original
intersection. In this chapter, we make this notion precise, and we obtain an upper bound on
the number of 0-dimensional components (counting multiplicity) of the original intersection,
using the new power series that were obtained via small p-adic deformations.
6.1
Deformation of power series via rigid analytic geometry and polynomial approximations
We first state some theorems from [Rabl2] showing that small deformations do not affect
the multiplicity of 0-dimensional components of intersections in rigid analytic spaces.
Theorem 6.1.1 ([Rabl2, Theorem 10.2], Local continuity of roots). Let A be a K-affinoid
algebra that is a Dedekind domain and let S = Sp(A). Let X = Sp(B) be a Cohen-Macaulay
affinoid space of dimension d + 1, let fl, . . . , fd E B, and let Y C X be the subspace defined
63
by the ideal a = (fi,... , fd). Suppose that we are given a morphism a : X
t G
|S|
-4
S and a point
such that the fibre Y = a-(t) n Y has dimension zero. Then there is an affinoid
subdomain U C S containing t such that a-1 (U) -+ U is finite and flat.
The following is immediate from the above theorem, and is applicable to our situation
arising from Chabauty's method:
Corollary 6.1.2 ([Rab12, Example 10.3]). Let X = BK x Bk and S = Bk, with a: X -+ S
the projection onto the second factor. Let fi,..., fd
fi,o, ...
,
G
K(x 1 ,...
, xd,
t). If the specializations
fd,o at t = 0 have only finitely many zeros in Bd then there exists E > 0 such that f
|s| < E, then f1,,..., fd,, have the same number of zeros (counted with multiplicity) in BdPt(s)
as fi,o, ...
, fd,o.
Proof. From Remark 4.1.2, Tate algebra in one variable is a Dedekind domain, and by
El
Remark 5.2.6, X is Cohen-Macaulay.
Definition 6.1.3. Let P C NR be an integral F-affine pointed polyhedron, where
d-dimensional vector space. Suppose that fi,..., fd
G
K(Up)
NR
is an
are power series such that
V(f,... , fd) is possibly infinite. Define
d
No(fi,...,
fd)
If Y := V(f 1 ,...
:= number of 0-dimensional components of
, fd)
is
V(fi), counting multiplicity.
finite, then let
N(fi, .. , fd) :=dim H(Y Oy).
Definition 6.1.4. Let f = Z
auxu e K(Up), and define
M(f) := the set of monomials appearing in
S{x"'
: a,, # 0}.
64
f
Call f nondegenerate if for every i, some power of xi appears with a nonzero coefficient
in f.
Remark 6.1.5. Any pure power series arising from Chabauty's method involves every variable, and hence is nondegenerate.
We will need to impose the nondegeneracy conditions in all power series f in order to be
able to carry out deformations. However, this is not a strong condition, as setting xi = 0
will lead to a lower-dimensional problem. Now we prove a series of deformation results for
non-stable intersections.
Further, we will assume from now on that P C NR is of a particular form; namely, we
assume that P = [
,10)d.
Lemma 6.1.6. Let P C NR be as above. Let f c K(Up) be a nondegenerate power series,
and suppose that P 1 , P 2 ,.
, P, E Up such that Pi $ 0 in Kd. Then there exists a polynomial
h such that h does not vanish on any of P1 ,..., Pe and M(h) C M(f).
Proof. We will prove by induction on f that there exists g such that M(g) C M(f) and
g(Pi), ... , g(Pj) $ 0. The statement is clear when f = 0, so assume that there is a polynomial
g with M(g) C M(f) satisfying g(P 1 ),.. . ,g(Pt) - 0 and g(P+1) = ''' = g(Pn) = 0, after
possibly reordering the P. If f = n, then we are done, so assume otherwise. We will show
that there exists another polynomial g' E M(f) such that g' does not vanish on at least f + 1
of the points P.
Choose a monomial m E M(f) such that m(Pf+i) $ 0; such m exists due to the nondegeneracy condition on f.
We may choose c E K' such that v(cm(P)) > v(g(P)) for
1< i < f.
Then g'
g + cm satisfies the property that g'(P) # 0,..., g'(Pe) # 0. Also, g'(Pt+1) =
cm(P+i) = 0. The lemma then follows from an inductive argument on f.
E
Proposition 6.1.7.
(a) Let P C NR be as before. Suppose that f,
series. Then there exist nondegenerate 91,
65
...
, fd C K(Up) are nondegenerate power
. .,
gd
K(Up) with Trop(fj) = Trop(gi)
and -yw(fi ) = -yw (gi), for 1 < i < d and w E P, with
No(f, ....
,fd) < N(gi,...,gd).
(b) Moreover, if the fi are polynomials, then the gi may be chosen to be polynomials.
Proof. We will deform the
fi
to gi one by one. Specifically, we will prove by induction on r
that there exist gi, . . . , gr such that Trop(fi) = Trop(gi) and ye (fi) = -y(gi) for i E {1, ..., r}
and w E P, satisfying codim F)> V(gi) > r for each r E {1, ...
, d},
and
No(f, . ...,fd) < No(gi,..., gr, fr+,.. - , fd)-
When r = 1, the statement above is clear, by taking fi = gi. Now we prove the statement
for r + 1. Let C1, . . . , Cj be the codimension r irreducible components of n)> V(gi). Choose
points Pi C Ci, such that Pi
#
0 in Kd. We will deform fr+1 to g,+1 so that gr+1(P) # 0 for
each i, while keeping Trop(fr+i) = Trop(gr+i) and
that codim
(nit7
V(gr+1))
> r+
'yw(fr+i)
= Ywq(r+1). This will guarantee
1.
From the nondegeneracy assumption of fr,+, we have that V(M(fr+i)) = 0 (if M(fr+i)
contains 1) or V(M(fr+i)) = {0} (if M(fr+i) does not contain 1).
may adjust only the constant term to get the desired g,+1.
In the first case, we
Thus, we may assume that
we are in the second case. Then by Lemma 6.1.6, we may pick a polynomial h such that
M(h) C M(fr+i) such that h that does not vanish on any P. For small enough nonzero e,
the deformation fr+1 4 fr+1 + 6h =: gr+ does not vanish on any of the P; in this case, the
intersection q
V(gi) has codimension r + 1, as required. Further, since h E M(fr+i), after
possibly making e even smaller, both the tropicalization and the 'yo of fr+1 are identical to
those of gr+1Now we will prove that No(gi,...
, d)
No(fi,...
, fd).
It suffices to show that
No (gi, ....
, gr+1, fr+2, . .. , fd) > NO (g1, . .. , 9r, fr+li ..
- fd)
66
for r
E {o,...
,d
- 1}; if r = 0, then the previous inequality will be interpreted as
No (gi, f2, . .7
fd)
> No(fi, ._..I
fd).
Let I be the ideal that cut out the dimension > 1 components of V(g 1,
...
, gr,
fr+i,--- , fc)
in K(Up) and let p1,... , pe denote the maximal ideals corresponding to the 0-dimensional
components of V(g,. . .,
gr, fr+1-i-...
, f).
Choose a
f
E I such that
f
pi for 1 < i < e.
Such choice is possible by the prime avoidance theorem, see for example [AM69, Proposition 1.11]. Now we apply Theorem 6.1.1 on Sp B, where B = K(Up)f, which states that a
small deformation of V(gi, ...
, gr,
fr+1, -. . , fd) preserves all 0-dimensional components away
from the positive-dimensional locus. This proves the inequality at the beginning of this
paragraph, and consequently part (a) of the proposition.
Part (b) of the proposition follows, since we deform the the
fi by monomials
appear in fi.
6.2
that already
E
Explicit computation of the upper bound
Let X be a nice curve over
Q,
p a prime, and d > 1. Fix a residue disk U C (Symd X)(Cp)
whose points reduce to a given {P 1 ,... , P4} E (Symd X)(Fp). Recall from Proposition 3.2.2
that Chabauty's method on U yields d pure power series fi, . . . , fd in d variables, whose
common zeros in Cd with valuations at least 1/d correspond to a set containing the points
in j-1 (W(Qp)) n J(Q)). Using the results of the previous section, we will obtain an explicit
upper bound on the number of common zeros of the
fi in this
section by estimating New(fi).
The methods used in this section are reminiscent of [Col85].
Definition 6.2.1. Let K and v be as in the beginning of this chapter. Fore E (0, ), k E Z>o
and d, f G Z;> with d > f, let
e(k, v, f) := max N E Z>o : v(k + N) > ( -e)N + v(k)}.
67
Remark 6.2.2. We note that 6e(k,v, f) is well-defined; v(k + N) = O(log N) as N -+ oo,
while ( - E)(N + 1) increases linearly with N.
Notation 6.2.3. Given f c W(Fq)[[t]], we mean by f the image of f under the natural
reduction map of the coefficients W(Fq)[[t]]
lFq[[tll. We will denote ordo(f) := ordt o(f),
-+
the exponent of the first term that does not vanish under the reduction map.
Lemma 6.2.4. For any rational number e satisfying 0 < E < I, the following holds: Let
f E W(Fq)
[1] [[t]]
be such that its derivative f' is in W(IFq)[[t]], and ordo
some k > 1. Let
F(t...
, tj)
= f(ti) + -
Z
+ f (t) =
f'
=
k - 1 for
aut,
uezg 0
where tu denotes tu1 -- t"'.
Let w c P
[>,oo)e. If u
=
ui > k+6,(k,vj) for some 1 <i < , then (u,v(au))
(u1,...,u1) E Z% satisfies
=
vert.(F).
Proof. Fix w E [&, oo)'. Since F is pure, it suffices to consider u E ZtO such that ui >
k + 6e6(k, v, f) and U2
=
U3 =
=
ud
=
0. We will show that there exists u' E Zt such that
v(au,) + (u',w) < v(au) + (u, w).
Then by the definition of vert, (F), the conclusion would follow.
Write f'(t) =
furthermore, v(ckl)
j>O cit', so f(t) =
=
Zj
1
0 ci t'+ . Then ci
C
Z since f' E
0, with v(cj) > 0 for 1 < j < k - 1, since ordo f'
Then autu = 'i t"+1 , where m > k+6e(k, v, f). We claim that u'
For any w E
[E,
=
oo), consider
m(w) := min {v(asu) + (u", w)}
< V (Ck1)
=v
Ck_1
+ ((k, , ... , 0),
+ kw1
68
(W1, W2, -
W,))
=
[[t]], and
k - 1.
(k, 0, . .. , 0) suffices.
Since m > k + J, (k, v, f), we have
v(m+1) < (m + 1 - k)w + v(k),
which rearranges to
-v(k) + kwi < -v(m + 1) + (m + 1)wi.
Using
V(Ckl)
=
0 and v(cm) > 0, this inequality becomes
v
( Ck-1)
k /
+ kw1 <V (m C
+ (m + 1)wI.
m+I
3
That is, (u, v(a,,)) Z vert (F), as required.
Remark 6.2.5. Lemma 6.2.4 shows that any pure power series as in the statement of the
lemma can be approximated by polynomials whose terms are pure, and whose degree is less
than k + J6 (k, v, f). This, in turn, means that the Newton polygons of these polynomials are
{(k + 66(kv,))e: 1 < i < f}, where
at worst the convex hull of the points {(0,...,0)}
the ej denotes the i-th standard vector. Thus, the Newton polygon can be approximated by a
simplex.
Definition 6.2.6. Let A = (a21 ) be a d x d matrix. The permanent of A is
d
Per(A) =
1(aig()).
JCSd i=1
Lemma 6.2.7. Let A = (aij) be a d x d matrix of positive real numbers, and define the
polytopes Xi C Rd for 1 < i < d by the following:
Xi = conv(0, a, 1e,... , ai,ded)
Then
MV(X 1 ,. . . ,)Xd)
69
1
=
Per(A).
Proof. The mixed volume is
coefficient of A1
...
Ad of vol (conv(O, (Alai + --- + Adad1)e, ...
= coefficient of A,--
Ad
of
1
I
(Alai +---
+Adadl)
...
(Aia
+-
,
(A ai +-- - -+ Adadd)ed))
+
Adadd)
dd
OSd i=1
=
Per(A).
Lemma 6.2.8. Let K be as in the beginning of Chapter 6. Let R be the ring of integers of
K. Let
fi E K[[ti]] for 1 < i < d. Suppose further that f4(ti) = Z'o cijtj E R[[t1]] for all i,
and that the fi converge when v(ti) ;>
. Suppose also that for each i there exists ki E Z>0
such that the coefficients cij satisfy v(cij) > 0 for j < ki and v(cik) = 0. From these data,
define a multivariate pure power series
F(ti,... , tn) := f(t)
+ -
+ fd(tn).
Then the Newton polygon of the pure power series F with respect to the polyhedral domain
Pd (where w = (w1,...
, wn))
is contained in the d-dimensional simplex defined by the convex
hull of the vectors
(ki + 6(ki, v, wi))ei,
where ei is the i-th standard vector.
Proof. Straightforward application of Lemma 6.2.4 and Remark 6.2.5 to each
fi
that show
-
up in the pure power series.
Now, let X be a nice curve with good reduction at p. Let P = {P1, ... , Pd} c (Symd X)(F).
Let U be the residue disk of (Symd X)(C)
reducing to {P 1 ,... , Pd}. Decompose the mul-
tiset {P 1 ,... , Pd} into disjoint multisets S,... ,S, each consisting of a single point with
multiplicity sj = #Sj.
70
Let Lj be the degree sj unramified extension of the field of definition K of the points in
Si, and let Rf be the ring of integers of the Lj. For 1 < i < d, let fij E Lj[[tj]] be the power
series obtained from Chabauty's method, applied to the residue disk in (Sym'sj X)(Kj) above
the point Si, such that their derivatives fj', are in Rj[[tj]]. Let
F(t,..., t)
=
fA, 1 (ti) +
- -+ fi,d(td),
and let kij = ordo(fij).
Then define the d x d matrix Ap = (aij) by aij = kij + J(ki,, v, si) for each residue disk.
Theorem 6.2.9. Keep the notation from the previous paragraph. Then the F satisfy
No(F 1,... ,Fd) < +Per(A).
Proof. By Proposition 6.1.7, we may as well assume that the power series that we get from
Chabauty's method have finitely many common zeros (that is, a deformation of the power
series exists, such that the tropicalizations and the
'Yw
stay constant). This means that, by
Theorem 5.3.13, that the number of solutions can be written as the mixed volume of Newton
D
polygons. Now combine Lemma 6.2.7 and Lemma 6.2.8.
Now, we recall that NO(F 1 ,... , Fd) counts the 0-dimensional components of the common
zeros of the F in (C, )d. Thus, we need to count the solutions in which some of the coordinates are 0 separately. For example, if we wish to count the solutions that are of the form
(Cx)(d-1) x {0}, it suffices to consider
NO (F1 (ti, t2,
which is bounded above by (d1')'
td-1, 0), . .. , Fd-1(ti, t2, .
td-1, 0)),
Per(B), where B is a (d - 1) x (d - 1) minor of A that
takes the first (d - 1) rows and columns. Thus, let
Per(A)' :=1Per(Aij),
Oi
djEAi
71
where Aij denotes the i x i minor of A that takes the first i columns (and any i rows), and
A00 is the 0 x 0 matrix whose permanent is understood to be 1 (since if (0,
... ,
0) were a
solution to the F, it would contribute at most 1 to No(F 1 , . . . , Fd)).
Theorem 6.2.10. Suppose X is a nice curve over Q with good reduction at p satisfying
Assumption 1.0.7, and let w 1 ,...
vanish on J(Q) such that 69i
#
,
E H0 (XQ,, Q1 ) be independent differential forms that
0. Then keeping the notation as above, with the ki, corre-
sponding to the order of vanishing of wi at the point P1,
the number of points outside of the
special set of (Symd X) (Q,) is at most
Per(Ap)'
PE (Symd X)(F,)
Proof. Apply the above theorem to each residue disk of (Symd X)(Qp), and use Corollary
4.2.4. The IN accounts for the ordering 6f the solutions, since the order of the points does
E
not matter in Symd X.
The above theorem shows that there is an upper bound on the number of points outside of
the special set, depending only on the choice of g, d and p. If we could bound #(Symd X) (Fp)
in terms of g, d and p, then this would complete the proof of Theorem 1.0.8. The following
proposition does that.
Proposition 6.2.11. Let X be a nice curve of genus d with good reduction at p, and let
d > 1. Then
#((Symd X)(Fp)) < (1 + 2gp/2 + pd)d.
Proof. We use the Hasse-Weil bound on X, along with the fact that if {P 1 ,..., Pd} E
(Symd X)(Fp), then Pi E
X(Fpd)
for 1
i < d.
El
Then the proof of Theorem 1.0.8 follows by combining the statements of Proposition and
Theorem 6.2.10.
72
Chapter 7
Example
7.1
Hyperelliptic curves of genus 3
In this chapter, we apply the methods that we have developed to find an upper bound on the
number of points outside of the special set for nice hyperelliptic curves of genus 3 satisfying
Chabauty's hypothesis and also the Assumption 1.0.7. The strategy is exactly as outlined
before; we bound the total number of residue disks, and also the number of possible points
outside of the special set of Sym 2 X on each residue disk.
Lemma 7.1.1. Let X be a smooth projective odd hyperelliptic curve of genus 3 that has good
reduction at 2. Then #(Sym2 X)(F 2 ) < 19.
Proof. We first note that a mod-2 reduction of an odd hyperelliptic curve of genus 3 corresponds to an equation of the form y2 + g(x)y = h(x), with g(x), h(x) E F 2 [X], with
deg g < 3, deg h = 7. Let P E (Sym 2 X)(F 2 ). It can be viewed as a multiset of two points
P = {P1, P 2 }. We denote by x(P) and y(P) the x- and y-coordinates of P, respectively, for
i = 1, 2. We have two cases:
Case 1: When P 1, P 2 E X(F 2 ). If P E X(F 2 ), then x(P), y(P) E F2 or x(P) = 00, so
in particular, one must have x(P) E {0, 1, oo}. There are at most two points above each
F 2-point in the map X -+ P1, so there are at most 5 points in X(F 2 ). Let #X(F 2 ) = a.
Then there are
(")
+ a points P E (Sym 2 X)(F 2 ) of the form {P 1 , P2 } with P E X(F 2 ); the
73
first term counts {P 1, P2 } with P, and P 2 distinct, and the second terms counts {P1, P 2 }
with P = P2 .
Case 2: When P1, P 2 E X(F 4 )\X(F2 ) are Galois conjugates. In this case, there are at
most 4 points of X(F 4 ) - X(F 2 ) above P'(F 4 ) - P(F 2 ). But there could also be points of
X(F 4 ) - X(F 2 ) above P1 (F 2 ); the number of these is 5 - a, since all of the 5 F2 -points of X
above Pl'(F
2)
are either F 2 -points or F 4 -points. So there are at most 9 - a such points.
Clearly, the choice of 1 < a < 5 that maximizes
are at most 19 points in (Sym
2
(")
+ 9 is a = 5, which means that there
E
X)(F 2 ).
Now we focus on a single residue disk of (Sym 2 X)(F 2 ) and compute the possible number
of points on each residue disk.
Since g = 3, the degree of 6j is 2g - 2 = 4. We start by computing 6,(k, v, f) in Definition
6.2.1 for when k = 1, 2,3, 4. We take E E (0, 1) as small as possible, as that minimizes
6,(k, v, f). Then we have
&(4, 2, 2) = 0,
6e(3, 2, 2) = 5,
6e(2, 2, 2) = 2,
6(1, 2, 2) = 3.
Thus, for a residue disk over P, the largest value of Per Ap is given from the 2 x 2 matrix
whose entries are all k + 6,(k, v, f) with k = 3. That is, the maximal value for Per Ap is 128.
Now, there are two 1 x 1 minors that we need to compute, from the definition of Per(A)'
in the previous chapter. Again, the maximal values for these are 8, obtained when k = 3.
This gives Per(A)' < 1 - 128
8+8 + 1 = 81.
Now, we apply Theorem 6.2.10 on the 19 residue disks with N
> 1 and Per(Ap)' < 81.
This gives the upper bound of 81 x 19 = 1539. This completes the proof of Proposition 1.0.9.
74
Bibliography
[AM69] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR0242802 (39 #4129) T6.1
[Ber75] D. N. Bernstein, The number of roots of a system of equations, Funkcional. Anal. i Prilozen. 9
(1975), no. 3, 1-4 (Russian). MR0435072 (55 #8034) T5.3.12
[BG13] Manjul Bhargava and Benedict H. Gross, The average size of the 2-Selmer group of Jacobians
of hyperelliptic curves having a rational Weierstrass point, arXiv preprint 1208.1007v2 (2013).
T1.0.3
[BGR84] S. Bosch, U. Giintzer, and R. Remmert, Non-Archimedean analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 261, Springer-Verlag,
Berlin, 1984. A systematic approach to rigid analytic geometry. MR746961 (86b:32031) T
[Bou72] N. Bourbaki, Elements de mathematique. Fasc. XXXVII. Groupes et algebres de Lie. Chapitre II:
Algebres de Lie libres. ChapitreIII: Groupes de Lie, Hermann, Paris, 1972. Actualites Scientifiques
et Industrielles, No. 1349. MR0573068 (58 #28083a) 12.1, 2.1.1
[Cha4l] Claude Chabauty, Sur les points rationnels des courbes alg6briques de genre supirieur e l'uniti, C.
R. Acad. Sci. Paris 212 (1941), 882-885 (French). MR0004484 (3,14d) T2.2.2
[Col85] Robert F. Coleman, Effective Chabauty, Duke Math. J. 52 (1985), no. 3, 765-770, DOI
10.1215/S0012-7094-85-05240-8. MR808103 (87f:11043) T(document), 1, 1.0.1, 1.1, 2.3.3, 2.3.3,
2.3.4, 2.3.5, 2.4, 3.2.2, 4.1, 6.2
[CM98] R. Coleman and B. Mazur, The eigencurve, Galois representations in arithmetic algebraic geometry
(Durham, 1996), London Math. Soc. Lecture Note Ser., vol. 254, Cambridge Univ. Press, Cambridge, 1998, pp. 1-113, DOI 10.1017/CB09780511662010.003, (to appear in print). MR1696469
(2000m:11039) T4.1
[Con99] Brian Conrad, Irreducible components of rigid spaces, Ann. Inst. Fourier (Grenoble) 49 (1999),
no. 2, 473-541 (English, with English and French summaries). MR1697371 (2001c:14045) T4.1,
4.1, 4.1.11
[Con08]
, Several approaches to non-Archimedean geometry, p-adic geometry, Univ. Lecture Ser.,
vol. 45, Amer. Math. Soc., Providence, RI, 2008, pp. 9-63. MR2482345 (2011a:14047) T4.1, 4.1.1
[DK94] Olivier Debarre and Matthew J. Klassen, Points of low degree on smooth plane curves, J. Reine
Angew. Math. 446 (1994), 81-87. MR1256148 (95f:14052) T1.1, 1.1.1, 1.1, 3.1
[Fal94] Gerd Faltings, The general case of S. Lang's conjecture, Barsotti Symposium in Algebraic Geometry
(Abano Terme, 1991), Perspect. Math., vol. 15, Academic Press, San Diego, CA, 1994, pp. 175-182.
MR1307396 (95m:11061) T1, 1.0.5
[GG93] Daniel M. Gordon and David Grant, Computing the Mordell- Weil rank of Jacobians of curves of
genus two, Trans. Amer. Math. Soc. 337 (1993), no. 2, 807-824, DOI 10.2307/2154244. MR1094558
(93h:11057) T1, 2.5
75
[Har77] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52. MR0463157 (57 #3116) T3.1
[HS91] Joe Harris and Joe Silverman, Bielliptic curves and symmetric products, Proc. Amer. Math. Soc.
112 (1991), no. 2, 347-356, DOI 10.2307/2048726. MR1055774 (91i:11067) T1, 1.0.6, 3.1, 3.1.2
[KZB13] Eric Katz and David Zureick-Brown, The Chabauty-Coleman bound at a prime of bad reduction
and Clifford bounds for geometric rank functions, Compos. Math. 149 (2013), no. 11, 1818-1838,
DOI 10.1112/50010437X13007410. MR3133294 12.4, 2.4.3
[Kla93] Matthew James Klassen, Algebraic points of low degree on curves of low rank, ProQuest LLC, Ann
Arbor, MI, 1993. Thesis (Ph.D.)-The University of Arizona. MR2690239 T1.1, 1.1.2, 1.1, 3.1
[Lan9l] Serge Lang, Number theory. III, Encyclopaedia of Mathematical Sciences, vol. 60, Springer-Verlag,
Berlin, 1991. Diophantine geometry. MR1112552 (93a:11048) T3.1
[LT02] Dino Lorenzini and Thomas J. Tucker, Thue equations and the method of Chabauty-Coleman,
Invent. Math. 148 (2002), no. 1, 47-77, DOI 10.1007/s002220100186. MR1892843 (2003d:11088)
T2.4, 2.4.2
[MS13] Diane Maclagan and Bernd Sturmfels, Introduction to tropical geometry, preprint (2013).
15, 5.2
[MP12] William McCallum and Bjorn Poonen, The method of Chabauty and Coleman, Explicit Methods
in Number Theory: Rational Points and Diophantine Equations, Panoramas et Syntheses, vol. 36,
Societe Mathematique de France, Paris, 2012, pp. 99-117. 12.2, 2.2.1, 2.3.3
[Mil86] J. S. Milne, Abelian varieties, Arithmetic geometry (Storrs, Conn., 1984), Springer, New York,
1986, pp. 103-150. MR861974 T3.1
[PS13] Bjorn Poonen and Michael Stoll, Most odd degree hyperelliptic curves have only one rationalpoint,
arXiv preprint 1302.0061v2 (2013). T1.0.3
[Rab12] Joseph Rabinoff, Tropical analytic geometry, Newton polygons, and tropical intersections, Adv.
Math. 229 (2012), no. 6, 3192-3255, DOI 10.1016/j.aim.2012.02.003. MR2900439 14.1, 5, 5.1, 5.2,
5.2.6, 5.2, 5.3, 5.3.1, 5.3, 5.3.8, 6, 6.1, 6.1.1, 6.1.2
[Ser58] Jean-Pierre Serre, Morphismes universels et variiti d'Albanese, Seminaire Claude Chevalley 4
(1958), 1-22. T1
[Sik09] Samir Siksek, Chabauty for symmetric powers of curves, Algebra Number Theory 3 (2009), no. 2,
209-236, DOI 10.2140/ant.2009.3.209. MR2491943 (2010b:11069) T1.1, 3.1
[Sto06] Michael Stoll, Independence of rational points on twists of a given curve, Compos. Math. 142
(2006), no. 5, 1201-1214, DOI 10.1112/S0010437X06002168.
2.4.1
MR2264661 (2007m:14025) T2.4,
[Szp85] Lucien Szpiro (ed.), Seminaire sur les pinceaux arithmitiques: la conjecture de Mordell, Societe
Mathematique de France, Paris, 1985. Papers from the seminar held at the Ecole Normale
Superieure, Paris, 1983-84; Asterisque No. 127 (1985). MR801916 (87h:14017) T(document), 1
76