Torsion Points on Modular Curves

by

Matthew Howard Baker

B.S. (University of Maryland at College Park) 1994

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

in Mathematics

in the

GRADUATE DIVISION

of the

UNIVERSITY of CALIFORNIA at BERKELEY

Committee in charge:

Professor Robert F. Coleman, Chair Professor Kenneth A. Ribet Professor Terence Speed

The dissertation of Matthew Howard Baker is approved:

Chair Date \overline{Date} \overline{Date}

University of California at Berkeley Torsion Points on Modular Curves

Copyright 1999 by Matthew Howard Baker

Abstract

Torsion Points on Modular Curves

by

Matthew Howard Baker Doctor of Philosophy in Mathematics

University of California at Berkeley

Professor Robert F. Coleman, Chair

Let N ≥ 23 be a prime number. In this thesis, we prove a conjecture of Coleman, Kaskel, and Ribet about the ℚ̄-valued points of the modular curve X_{0}(N) which map to torsion points on J_{0}(N) via the “usual” embedding. We also have results about the set of “torsion packets” on X_{0}(N), and on torsion points on other modular curves. In addition, we study some related questions. For example, we determine all modular curves X_{0}(N) which are trigonal, i.e., admit a degree three map to P^{1}. We do this by generalizing a method Ogg used to determine the hyperelliptic curves X_{0}(N). We also discuss more general questions about the gonality of modular curves, and discuss how these questions are related to the conjecture of Coleman, Kaskel, and Ribet. We also discuss Cartier points on curves, a concept motivated by Coleman’s paper “Ramified Torsion Points on Curves”. After proving some general results about Cartier points, we give some examples which yield information about torsion

points on certain modular curves.  Professor Robert F. Coleman Dissertation Committee Chair

To my parents, Michael and Fay Baker,

my brother, Darrin, and my (soon-to-be) wife Kate.

Contents

1 Introduction 1

2 Torsion Points on X_{0}(N) 6 2.1 The Coleman-Kaskel-Ribet Conjecture . . . . . . . . . . . . . . . . . . . . . 6 2.2 Proof of the Coleman-Kaskel-Ribet Conjecture . . . . . . . . . . . . . . . . 10 2.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Gonality of Modular Curves 22 3.1 A Generalization of Ogg’s Method . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Trigonal X_{0}(N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 Trigonal X_{0}^{+}(N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Cartier Points on Curves 36 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2 Duality, Linear Systems, and Ekedahl’s Theorem . . . . . . . . . . . . . . . 39 4.3 Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5 Additional Techniques 56 5.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.2 Useful Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.3 The Curve X_{0}(29) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.4 The Curve X_{0}(31) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.5 A Computational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.6 The Curve X_{0}(53) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6 Appendix 67

Bibliography 74

Acknowledgements

There are many people who provided valuable assistance with this work, and I

hope that their contributions are adequately recognized here.

First of all, I would like to thank Hendrik Lenstra, Arthur Ogus, Andrew Ogg,

and Bjorn Poonen, with whom I have engaged in many interesting conversations during my graduate career. Time and again they were able to clearly communicate mathematical ideas to me. For example, A. Ogus showed me the proofs of Lemmas 7 and 16, and B. Poonen helped out greatly with the ideas in Section 5.5 and the appendix.

Joe Harris, Robin Hartshorne, and Felipe Voloch also made useful remarks about

various parts of this thesis. In particular, J. Harris first told me about Prop. 11, which I needed to generalize Ekedahl’s theorem, and F. Voloch made a nice improvement to the proof of Prop. 14. I also am indebted to Voloch for suggesting I look at the paper [26], which helped inspire the main results of Chapter 3.

There are many fellow graduate students I should thank, too many to name them

all here. I would at least like to acknowledge here the many valuable ideas I learned in conversations with William Stein and Janos Csirik.

I would also like to thank Kevin Buzzard and Loic Merel for sharing many mathe-

matical ideas with me, and for the encouragement they gave me as a young, impressionable graduate student.

I would like to specially acknowledge at this point the crucial contributions of Ken

Ribet to Chapter 2 of this work. The second proof I give of the CKR conjecture is based on emails from and discussions with him. Ribet also came up with the idea of utilizing the interplay between torsion points on X_{0}(N) and X_{0}^{+}(N) while thinking about the specific case N = 389. In addition, he pointed out the elementary but very useful Lemma 1, and explained how Lemmas 4 and 17 could be used to simplify some of my arguments. In general, I am grateful for the many useful remarks Ribet made concerning modular curves and Galois representations.

Last but not least, I wish to thank Robert Coleman for suggesting that I work

on the Coleman-Kaskel-Ribet conjecture, and for his continuous help and encouragement. Many of the results in Chapter 5 also emerged from his suggestions.

Commutative diagrams in this thesis were designed using Paul Taylor’s Commutative Diagrams in TeX package. At various points during this research I made use of the software packages GAP, MAPLE, and PARI-GP.

This research was supported by an NDSEG Fellowship and a Sloan Dissertation

Fellowship.

Chapter 1

Introduction

The contents of this dissertation, while somewhat diverse, evolved from a common

source: a conjecture of Coleman, Kaskel, and Ribet (see [12]) concerning the way in which the modular curve X_{0}(N), with N prime, intersects the torsion subgroup of its Jacobian.

Let X be an algebraic curve of genus g ≥ 1 defined over a number field K. (For us,

the word curve used without further qualifications will always mean a complete, nonsingular, absolutely irreducible curve over a field.) Assume, furthermore, that X(K) is nonempty. Now choose an Albanese embedding defined over K of X into its Jacobian variety. In other words, choose a K-rational point P_{0} on X and define the map i : X ↪ J by sending P to the divisor class [(P) − (P_{0})]. (Recall that if L is an extension of K, the L-valued points of J correspond to linear equivalence classes of degree zero divisors on J defined over L.)

Now define the set T to be {P ∈ X(K̄) | i(X) ∈ J^{tor}}. In other words, T is the

set of K̄-valued points of X which map to torsion points on J.

If g = 1 (i.e., X is an elliptic curve), then i is an isomorphism, and so of course

T is infinite. But if g ≥ 2, the situation is entirely different. In fact, it is a theorem when g ≥ 2 that T is a finite set of points. This theorem, proved by M. Raynaud in 1983, is known as the Manin-Mumford conjecture. Many different proofs of this conjecture followed, including a 1987 proof by R. Coleman based on p-adic integration.

There is a striking analogy between the Manin-Mumford conjecture, on the one

hand, and the Mordell conjecture on the other. This analogy runs much deeper than the fact that both are theorems which are called conjectures. For example, Lang conjectured that i(X) ∩ Γ′ is finite whenever Γ is a finitely generated subgroup of J(K̄) and Γ′ is its division group, i.e., the set of points x in J(K̄) such nx ∈ Γ for some positive integer n. This is now a theorem, as are various generalizations to higher-dimensional varieties; see [44] for references and a summary of recent results in this direction. Note that Lang’s conjecture implies both the Manin-Mumford conjecture (taking Γ = 0) and the Mordell conjecture (taking Γ = J(K) and considering i(X) ∩ Γ ⊆ i(X) ∩ Γ′).

Determining the finite set of K-rational points on X (which we will refer to as

“Explicit Mordell”) for a “random” curve X is an extremely hard problem, to put it mildly. Faltings’ proof of the Mordell conjecture is ineffective, so even in principle this problem is difficult. Some of the most celebrated cases where X(ℚ) has been determined include the case where X is a Fermat curve (A. Wiles) and where X = X_{0}(N) is a modular curve (B. Mazur). There are also small industries devoted to solving this problem in the special case where X has genus 1 or 2.

Explicitly determining the set T of torsion points on X (“Explicit Manin-Mumford”)

is also, in general, a difficult one. Bjorn Poonen has pointed out, however, that this problem is, at least in principle, effective. In this setting as well, the appropriate test cases seem to be curves which either have small genus (see [6] for some examples when g = 2) or unusually rich arithmetic structure. For an example of the latter, see [14], in which the authors determine T when X is a Fermat curve embedded in J using a “cusp”.

In their joint paper [12], Coleman, Kaskel, and Ribet study the set of points on

the modular curve X_{0}(N) (here N ≥ 23 is a prime number) which map to torsion points of J_{0}(N) under the embedding i : P ↦ [(P) − (∞)]. (Here ∞ denotes one of the cusps on X_{0}(N).)

For the reader’s convenience, we recall a few definitions. X_{0}(N) is the (compacti-

fied) coarse moduli space for the set of (cyclic) isogenies E → E′ of degree N between elliptic curves. X_{0}(N) is an algebraic curve defined over ℚ, and the assumption that N ≥ 23 simply means that the genus of this curve is at least two. As a Riemann surface, X_{0}(N) can be thought of as the quotient of the complex upper half plane H by the action of the group Γ_{0}(N), at least once this quotient is suitably compactified by adding two “cusps”, which are ℚ-rational points that we call 0 and ∞. For additional background material on X_{0}(N) and its Jacobian J_{0}(N), as well as a number of important results we will use in what follows, see B. Mazur’s paper “Modular Curves and the Eisenstein Ideal” ([36]).

The curve X_{0}(N) has a natural involution w_{N}, the Atkin-Lehner involution, whose

moduli interpretation is that it takes an N-isogeny E → E′ to the dual isogeny E′ → E. The quotient of X_{0}(N) by w_{N} is denoted by X_{0}^{+}(N), which is also an algebraic curve defined over ℚ. We let g_{0}^{+}(N) (or simply g^{+}) be the genus of X_{0}^{+}(N).

When g^{+} happens to be zero (which happens, for N ≥ 23, if and only if N ∈

{23, 29, 31, 41, 47, 59, 71}), X_{0}(N) is forced to be a hyperelliptic curve (double cover of P^{1}). It is a theorem of Ogg [41] that the converse is almost true as well: X_{0}(N) is hyperelliptic if and only if g^{+} = 0 or N = 37. X_{0}(37) is unusual in that the hyperelliptic involution and Atkin-Lehner involution do not coincide.

We call the set T_{∞}(X_{0}(N)) of points Q on X_{0}(N) such that i(Q) ∈ J_{0}(N) has

finite order the cuspidal torsion packet on X_{0}(N). Certainly the two cusps ∞ and 0 are in this torsion packet; the image under i of latter point has order n = Num\frac{N−1}{12} in J_{0}(N) by a well-known theorem of Ogg.

Furthermore, it is not hard to show (see [12]) that when g^{+} = 0, the hyperelliptic

branch points on X_{0}(N) (the points which ramify in the degree two covering X_{0}(N) → P^{1}) are in the cuspidal torsion packet as well. When N = 37, however, the hyperelliptic branch points map to points of infinite order on J_{0}(37).

The authors of [12] make the following guess about the cuspidal torsion packet on

X_{0}(N), which we refer to as the Coleman-Kaskel-Ribet (CKR) Conjecture:

Conjecture 1. For all prime numbers N ≥ 23,

T_{∞}(X_{0}(N)) = \{ \begin{matrix}\{0, ∞\} if & g^{+} > 0 \\ \{0, ∞\} ∪ \{hyperelliptic branch points\} if & g^{+} = 0 \end{matrix}

They prove this result in the special case where N = 37 using results about J_{0}(37)

found in B. Kaskel’s thesis.

In chapter 2 of this thesis, we prove Conjecture 1 in full generality. In section

2.1, we summarize the work previously done on this problem, in particular the results of [12] and [47]. We also discuss a few technical results needed later on. In section 2.2, we give two proofs of the CKR conjecture. The first of these arguments relies on some of the material found in later chapters of the present work. For example, it uses our classification of the trigonal modular curves X_{0}(N) found in chapter 3. It also uses some of the results in chapters 4 and 5. The second proof, which came into being shortly after the first, is due mainly to Ribet and seems more ripe for generalization than the first proof. In particular, the trigonal modular curves no longer play an exceptional role in this approach.

We have recently learned that A. Tamagawa has independently proved Conjecture

1 by methods similar to Ribet’s.

We give proofs in section 2.3 of some generalizations of the Coleman-Kaskel-Ribet

conjecture. For example, we determine the set of torsion points on the modular curve X_{0}^{+}(N) for all primes N, where the embedding is via the unique cusp. We also study noncuspidal embeddings of these curves, and determine the complete set of torsion packets (see section 2.3 or the appendix) on X_{0}(N) and X_{0}^{+}(N) when N is sufficiently large.

More precisely, if X denotes either X_{0}(N) or X_{0}^{+}(N), we show that the torsion

packets on X are just the “expected ones” unless X admits a (non-constant) morphism of small degree to the projective line P^{1}. A theme which runs through section 2.2 is that to understand torsion points on X, it is useful to understand the gonality of X, i.e., the smallest degree of a morphism from X to P^{1}. So in chapter 3, we provide an extensive discussion of the gonality of modular curves. For example, we generalize Ogg’s classification of the hyperelliptic modular curves X_{0}(N) by determining all N such that X_{0}(N) is trigonal, i.e., admits a degree three map to P^{1}. As we mentioned above, this is used in our first proof of the Coleman-Kaskel-Ribet conjecture.

We remark that recently other authors (Nguyen-Saito, Hasegawa-Shimura) have

independently obtained results similar to those in chapter 3 – see [39], [29], and [30].

In chapter 4 we tackle a rather different topic, namely Cartier points on curves.

Without defining at the moment what these are, we simply note that their definition was motivated by work of Coleman (see [10]) concerning ramified torsion points on curves. Some of our early work on the Coleman-Kaskel-Ribet conjecture for small N hinged on this connection between Cartier points and ramified torsion points, and we give some concrete examples of this connection in section 4.4 and chapter 5. We also prove a theorem about Cartier points which generalizes and gives a new proof of a theorem of Ekedahl on superspecial curves in characteristic p.

Finally, in section 5 we present an assortment of techniques for dealing with torsion

points ramified at the special prime number 3. This material is largely based on suggestions CHAPTER 1. INTRODUCTION 5 of Robert Coleman. We hope that the techniques used in this section can also be used to study torsion points on other curves in other scenarios.

Chapter 2

Torsion Points on X_{0}(N)

2.1 The Coleman-Kaskel-Ribet Conjecture

We begin with a summary of [12], the paper in which the problem of determining

torsion points on X_{0}(N) was first posed.

The basic approach in [12] is to use the Chinese Remainder Theorem to decompose

a potential torsion point P on the modular curve X_{0}(N) (N prime) as a sum of its l-primary components, i_{∞}(P) = ∑ P_{l} (where P_{l} ∈ J_{0}(N)), and to try to show that P_{l} is in the cuspidal group for as many primes l as possible. The cuspidal group is the cyclic group C generated by the image in J_{0}(N) of the cusp 0, which Mazur proves in [36] is the full group of rational torsion points on J_{0}(N). Its order is n = num(\frac{N−1}{12}), and one knows that the intersection of C with the image of X_{0}(N) is just the set {0, ∞} of cusps (see [12]).

Let T = T_{0}(N) denote the full Hecke algebra for X_{0}(N); it is precisely the ring

of endomorphisms of J_{0}(N) (see [36, Prop. 9.5]). The main general result in [12] is the following theorem, which handles “most” l-primary components: Theorem 1. Let P be an element of T (N), and let l ≠ 2, 3 be a prime for which P_{l} does not belong to the cuspidal group C. Then at least one of the following holds: (i) l = N; (ii) l satisfies 5 ≤ l < 2g, X_{0}(N) does not have ordinary reduction at l, and l is ramified in T (in the sense that T/lT is not a product of fields).

The proof is based on Coleman’s theory of p-adic integration (see [12] and [10])

plus the following theorem [12, Thm. 2.2] proved using techniques of Mazur (see [36]): Theorem 2. Suppose l ≠ 2. Then P is unramified at l iff P_{l} ∈ C.

CHAPTER 2. TORSION POINTS ON X_{0}(N) 7

Ken Ribet’s papers [47],[48] suggest additional techniques for tackling the Coleman-

Kaskel-Ribet conjecture. Many of Ribet’s results involve a certain hypothesis (*), which we now explain. We recall that the Hecke algebra T = T_{0}(N) has the property that T ⊗ ℚ is a product of totally real number fields K_{i}, and T itself has finite index in its normalization T̃, which is the product of the maximal orders O_{i} of K_{i}. By the discriminant of T, we mean the product of the discriminants of the K_{i} multiplied by the square of the index of T in T̃. By definition, T is unramified at a prime l if l does not divide the discriminant of T; this is equivalent to saying that T/lT is a product of finite fields. Ribet’s auxiliary hypothesis is:

(*) The prime N is unramified in the Hecke algebra T.

William Stein has done computations (see [55]) which show that condition (*) is

satisfied by all N < 5000 except for N = 389.

One result which involves this hypothesis is:

Theorem 3. Let N and l ≠ N be prime numbers. Suppose P ∈ J_{0}(N)^{tor} is such that its l-primary component P_{l} is not contained in the cuspidal group C. Assume either that N does not divide the order of P or that N satisfies hypothesis (*). Then there is an element σ ∈ Gal(\overline{ℚ}/ℚ) such that σP − P has order l in J_{0}(N).

Remark. In the proof of this theorem given in [47], note that if P_{2} ∈ J_{0}(N)[ℑ] (where ℑ is the kernel of the Eisenstein ideal), the hypothesis that P_{2} ∉ C implies that the order 2^{a} of P_{2} is in fact divisible by 4, since C[2] = Σ[2]. So the extension ℚ(μ_{2^{a}})/ℚ(μ_{2^{a−1}}) really is ramified at 2.

As we remarked in the introduction, it is useful to work with the interplay between

torsion points on X_{0}(N) and its quotient curve X_{0}^{+}(N). The intuitive reason why X_{0}^{+}(N) is in many ways simpler than X_{0}(N) is that it is “non-Eisenstein”; we will make this more precise in a moment.

We discuss now some facts about X_{0}^{+}(N) that we will use in what follows.

Recall that X_{0}^{+}(N) is the quotient of X_{0}(N) by the Atkin-Lehner involution w =

w_{N}. There is a unique cusp at infinity on X_{0}^{+}(N), which we denote by ∞^{+}, or simply ∞ if no confusion is likely to arise. The fiber of the map π : X_{0}(N) → X_{0}^{+}(N) over ∞^{+} is just {0, ∞}. Let J_{0}^{+}(N) be the Picard (Jacobian) variety of X_{0}^{+}(N). The fact that J_{0}^{+}(N) is also the Albanese variety of X_{0}^{+}(N) implies there is a commutative diagram

[figure: X_{0}(N); i_{∞}; J_{0}(N); X_{0}^{+}(N); π; i_{∞}; J_{0}^{+}(N); π_{∗}]

where i_{∞} : X_{0}^{+}(N) → J_{0}^{+}(N) is the map which on closed points takes P to [(P) − (∞^{+})]. The map π_{∗} takes a point in J_{0}(N) represented by the degree zero divisor ∑ P_{i} − ∑ Q_{i} to the class of the divisor ∑ π(P_{i}) − ∑ π(Q_{i}), thought of as a point of J_{0}^{+}(N). Note that a point of X_{0}(N) mapping to a torsion point of J_{0}(N) is sent by π to a point of X_{0}^{+}(N) mapping to a torsion point of J_{0}^{+}(N).

The map π^{∗} : J_{0}^{+}(N) → J_{0}(N) induced by Picard functoriality is a closed immer-

sion, since w_{N} has fixed points, so identifies J_{0}^{+}(N) with an abelian subvariety of J_{0}(N). The composite map π^{∗} ∘ π_{∗} : J_{0}(N) → J_{0}(N) is easily seen to be the map 1 + w, so that J_{0}^{+}(N) is naturally identified with the subvariety J_{+} := (1 + w)J_{0}(N) of J_{0}(N) (see [36, II.10] for another discussion of this).

We also note that J_{−} := (1 − w)J_{0}(N) is naturally identified with the kernel of

multiplication by 1 + w on J_{0}(N). Indeed, J_{−} is certainly contained in this kernel; in fact, for dimension reasons J_{−} is the connected component of the identity in this kernel. But ker(1+w) is connected (this is equivalent to the fact that the map J_{+} → J_{0}(N) is injective).

For future reference, we define the Hecke algebra T^{+} to be the image of T in the

endomorphism ring of J_{0}^{+}(N) (thought of as the subvariety (1 + w)J_{0}(N) of J_{0}(N)).

Recall that C is the cuspidal subgroup of J_{0}(N), which is the cyclic subgroup

generated by the point c := i_{∞}(0) ∈ J_{0}(N), i.e., by the divisor (0) − (∞). Since w(c) = −c, 1 + w (and hence π_{∗}) annihilates C.

Furthermore, let ℑ be the Eisenstein ideal of T (see [36]). It is the ideal generated

by p+1−T_{p} for p not dividing N and by 1+w. The kernel J_{0}(N)[ℑ] of the Eisenstein ideal is a finite Galois module containing C. It, too, is annihilated by π_{∗}, since 1+w ∈ ℑ. It follows that if 𝔪 is any maximal ideal of T containing ℑ (i.e., 𝔪 is Eisenstein), π_{∗}(J_{0}(N)[𝔪]) = 0. This will be important in what follows. To give an idea as to why, recall that one of the results of [36] says that if 𝔪 is any maximal ideal of T of residue characteristic different from 2, then the Galois representation J_{0}(N)[𝔪] is two-dimensional, and is reducible if and only if 𝔪 is Eisenstein.

A stronger way of expressing the fact that X_{0}^{+}(N) is “non-Eisenstein” is to say

that J_{0}^{+}(N)[𝔪]=0 whenever 𝔪 is an Eisenstein prime. When the residue characteristic p of 𝔪 is different from 2, this is clear, since w acts as +1 on J_{0}^{+}(N) and −1 on J_{0}(N)[𝔪]. When p = 2 this is more subtle and is established in the proof of [36, II,Prop. 17.10].

Along similar lines, we have the following proposition.

Proposition 1. Let p be an odd prime. The Jordan-Holder factors (as a T^{+}[Gal(\overline{ℚ}/ℚ)]- module) of J_{0}^{+}(N)[p] are all two-dimensional, and are isomorphic to the representations J_{0}(N)[𝔪] for 𝔪 a maximal ideal of T containing 1 − w.

Proof. Let V be such a Jordan-Holder factor – its annihilator 𝔪′ is a maximal

ideal of T^{+} of characteristic p. Clearly 1 − w ∈ 𝔪′, and since p ≠ 2, 1 + w ∉ 𝔪′. The inverse image of 𝔪′ in T is a maximal ideal 𝔪 containing 1 − w but not 1 + w. We claim that J_{0}(N)[𝔪] = J_{0}^{+}(N)[𝔪] = V. Indeed, V is a subquotient of J_{0}^{+}(N)[𝔪], and hence of J_{0}(N)[𝔪], and it is stable under the action of T[Gal(\overline{ℚ}/ℚ)]. Since 𝔪 is not Eisenstein, J_{0}(N)[𝔪] is irreducible and two-dimensional. We must then have V = J_{0}^{+}(N)[𝔪] = J_{0}(N)[𝔪] as claimed.

■

For each maximal ideal 𝔪 of T^{+}, we can form the 𝔪-divisble group J_{0}^{+}(N)_{𝔪} :=

∪J_{0}^{+}(N)[𝔪^{i}]. If the residue characteristic p of 𝔪 is different from 2, then the 𝔪-adic Tate module Hom(ℚ_{p}/ℤ_{p}, J_{0}(N)_{𝔪}) is free of rank 2 over T_{𝔪}^{+}. This follows from replacing X_{0}(N) and J_{0}(N) by X_{0}^{+}(N) and J_{0}^{+}(N) in the proof of [36, II,Lemma 15.1].

We also have the following result about J_{0}^{+}(N) (compare with Theorem 3).

Theorem 4. Let N and l ≠ N be prime numbers. Suppose P ∈ J_{0}^{+}(N)^{tor} is such that its l-primary component P_{l} is nonzero. Assume either that N does not divide the order of P or that N is unramified in T^{+}. Then there is an element σ ∈ Gal(\overline{ℚ}/ℚ) such that σP − P has order l in J_{0}^{+}(N).

Proof. If N is unramified in T, this follows immediately from Theorem 3 from

the fact that J_{0}^{+}(N) can be thought of as a subvariety of J_{0}(N) whose intersection with J_{0}(N)[ℑ] is trivial. Otherwise, one simply notes that Ribet’s proof of Theorem 3 follows mutis mutandis for J_{0}^{+}(N). In fact, Theorem 4 is actually easier to prove than Theorem 3, because the complications arising from Eisenstein primes in the proof of the latter result do not occur here. ■

2.2 Proof of the Coleman-Kaskel-Ribet Conjecture

In this section we prove the Coleman-Kaskel-Ribet conjecture. In fact, we give

two proofs of this conjecture, with the hope that the ideas used in both proofs will be useful in other contexts. In the following section we apply similar arguments to determine torsion points on X_{0}^{+}(N) in the cuspidal embedding, and also to study arbitrary torsion packets on X_{0}(N) and X_{0}^{+}(N).

The key idea in the first proof of the CKR conjecture is to use π to project torsion

points on X_{0}(N) to torsion points on X_{0}^{+}(N), and then to use the fact that Gal(\overline{ℚ}/ℚ) acts in a particularly simple way on torsion points of J_{0}^{+}(N).

In what follows we will always use g^{+}(N) (or simply g^{+}) to denote the genus of

the modular curve X_{0}^{+}(N).

The proof of the following lifting lemma is reminiscent of the techniques of [49,

IV-23 Lemma 3].

Lemma 1. Let p be a prime, let A be a commutative ring with identity, and let R be a nilpotent ideal in A containing p. If H is a subgroup of GL(n, A) whose image \overline{H} in GL(n, A/R) contains the homothety −1, then H contains −1.

Proof. (Ribet) We are given that \overline{H} contains −1. This means that h ≡ −1 mod

R, i.e., there exist h ∈ H and r ∈ M(n, R) such that h = −1 + r. Since −1 and r commute, one can use the binomial theorem to see that h^{p} ≡ −1 mod R^{2}, and more generally h^{p^{i}} ≡ −1 mod R^{i+1}. Since R^{i} = 0 for i sufficiently large, it follows that the group H contains −1. ■

Though we originally conceived of the next result as an application of the main

theorem in [35], it follows from the much more elementary Lemma 1.

By an inertia group at p, we mean the inertia subgroup of Gal(\overline{ℚ}/ℚ) at some

prime lying over p. Also, by a Sylow subgroup of Gal(\overline{ℚ}/ℚ) we mean a projective limit of Sylow subgroups of finite quotients of Gal(K/ℚ).

Proposition 2. Let p ≥ 5 be a prime, and let T be either the completed Hecke algebra T ⊗ ℤ_{p} or its quotient T^{+} ⊗ ℤ_{p}. (Here T is associated to X_{0}(N) with N prime.) Let ρ = ∏ ρ_{i} : Gal(\overline{ℚ}/ℚ) → GL(2, T) = ∏ GL(2, T_{𝔪_{i}}) be the semistable representation coming from the action of Gal(\overline{ℚ}/ℚ) on the p-adic Tate module of J_{0}(N) (resp. J_{0}^{+}(N)). If T = T ⊗ ℤ_{p}, suppose that p does not divide N − 1. If p ≠ N, let X be the normal closure of an inertia group at p, and if p = N let X be the normal closure of an N-Sylow subgroup of an inertia group at N. Then the image ρ(X) ⊆ GL(2, T) contains the homothety −1.

Proof. For n ≥ 1, let A be the Artinian ring T/p^{n}T. If H denotes the image of

ρ(X) in GL(2, A), then it suffices to prove that −1 ∈ H for all n. Let R be the radical of A, i.e., the set of nilpotent elements in A. Then p ∈ R, and A/R is isomorphic to F := ∏ T/𝔪_{i}, a product of finite fields of characteristic p. By Lemma 1, it suffices to prove that the image \overline{H} of H inside GL(2, A/R) contains −1. In fact, \overline{H} contains all of SL(2, A/R). When T = T ⊗ ℤ_{p}, this follows from [48, Thm. 3.4, Prop. 6.3] when p ≠ N, and from [47, Prop. 6.4] when p = N. When T = T^{+} ⊗ ℤ_{p}, it follows from the same argument, noting that each ρ_{i} is irreducible (since there are no Eisenstein primes in T^{+}), and that F := ∏ T^{+}/𝔪_{i} is generated by the traces of elements ρ(σ) for σ ∈ Gal(\overline{ℚ}/ℚ), since the analogous statement is true for T. ■

Corollary 1. With the same notation and hypotheses as in the proposition, there exists a σ ∈ Gal(\overline{ℚ}/ℚ) such that σ acts as −1 on all torsion points of J_{0}(N) (resp. J_{0}^{+}(N)) of p-power order and as +1 on torsion points of order prime to p.

Proof. For p ≠ N, this follows from Prop. 2, together with the criterion of

Néron-Ogg-Shafarevich. At N this follows from the fact that elements of inertia groups at N act unipotently on prime-to-N torsion, so that the image ρ_{l}(I) of any wild inertia group at N under an l-adic representation with l ≠ N is both pro-l and pro-N, hence trivial. ■

We also need the following easy lemma.

Lemma 2. Let X be a curve of genus at least two mapping to its Jacobian via ϕ : P ↦ [(P) − (P_{0})] for some fixed P_{0} ∈ X. If there exists a point P ≠ P_{0} on X such that −ϕ(P) is in the image of X, then X is hyperelliptic and P_{0} is a hyperelliptic branch point.

Proof. We are given that (P_{0}) − (P) is linearly equivalent to (Q) − (P_{0}) for some

point Q. Therefore there is a rational function on X with divisor equal to (P)+(Q)−2(P_{0}), and since P ≠ P_{0} this forces X to be hyperelliptic. ■

We now give the first proof of the Coleman-Kaskel-Ribet conjecture, modulo a few

exceptional cases which follow either from the second proof below, or from results which will be established in chapters 3 and 4.

Theorem 1. Conjecture 1 is true for all N.

Proof. We first prove the conjecture under the hypothesis (*), which says that

N is unramified in the Hecke algebra T.

Suppose we have a point Q ∈ X_{0}(N)(ℚ̄) such that i_{∞}(Q) is a torsion point of

J_{0}(N). Write

P := i_{∞}(Q) = P_{2} + P_{3} + P_{N} + ∑_{l≠2,3,N} P_{l},

where P_{l} has l-power order in J_{0}(N) for all primes l.

If all P_{l} ∈ C, then P ∈ C, which implies that Q ∈ {0, ∞}. If P_{2} ∉ C, then by

Theorem 3 [since we are assuming (*)] there exists a σ ∈ Gal(\overline{ℚ}/ℚ) such that (σP) − (P) has order 2 in J_{0}(N). Since ∞ ∈ X_{0}(N)(ℚ), (σP) − (P) = (σQ) − (Q). Therefore X_{0}(N) is hyperelliptic and that Q is a hyperelliptic branch point. This possibility is already accounted for in the statement of the Coleman-Kaskel-Ribet conjecture. (Recall from [12] that the hyperelliptic branch points on X_{0}(37) are not torsion points.)

If P_{3} ∉ C, then there exists a σ ∈ Gal(\overline{ℚ}/ℚ) such that (σP) − (P) has order 3.

This means that X_{0}(N) is trigonal, i.e., admits a degree three morphism to P^{1}. We show in section 9 that this implies that N ∈ {23, 29, 31, 37, 43, 53, 61}. But in sections 4.4 and 5 we show that the CKR conjecture is true for these values of N.

So assume, then, that P_{2}, P_{3} ∈ C. Let Q^{+} = π(Q) ∈ X_{0}^{+}(N). Since the group C

is annihilated by π_{∗}, we see that

P^{+} = i_{∞}(Q^{+}) = ∑_{l≠2,3,N} P_{l}^{+} + P_{N}^{+},

where P_{l}^{+} = π_{∗}(P_{l}). By Corollary 1, we can find an element σ ∈ Gal(\overline{ℚ}/ℚ) such that σP^{+} = −P^{+}, so that −i_{∞}(Q^{+}) lies on the image of X_{0}^{+}(N). By Lemma 2, this implies that X_{0}^{+}(N) is sub-hyperelliptic, i.e., has genus 0 or 1 or is hyperelliptic. When g^{+} ≥ 2, it also implies that infinity is a hyperelliptic branch point on X_{0}^{+}(N), which it is not (see the proof of Lemma 5 below). Furthermore (as we discuss in section 3), it is proved in [28] that X_{0}^{+}(N) is sub-hyperelliptic if and only if the genus of X_{0}^{+}(N) is at most two. One can then explicitly check using Theorem 1 that the CKR conjecture is true whenever g^{+} ≤ 1 (the largest N for which X_{0}^{+}(N) has genus 0 or 1 is N = 131). Indeed, it suffices to show for these values of N that there are no primes p between 5 and 2g (g=genus of X_{0}(N)) which ramify in the Hecke algebra T and are simultaneously non-ordinary. That this is the case follows from the tables in [55].

This proves the CKR conjecture for all N satisfying hypothesis (*). We recall that

this hypothesis is satisfied for all primes N < 5000 except for N = 389.

For N = 389, we can use the fact that 389 is not ramified in T^{+} (see [55]).

Projecting a potential torsion point Q on X_{0}(389) right away to X_{0}^{+}(389), we see from the above arguments that Q is a cusp unless X_{0}^{+}(389) (which has genus 11) admits a map of degree 2 or 3 to P^{1}. But this is not the case – see chapter 3.

The key thing to notice in general when (*) is not necessarily satisfied is that we

can still apply Theorem 3 to a torsion point when the order of that point is not divisible by N.

Take a torsion point

P = i_{∞}(Q) = P_{2} + P_{3} + P_{N} + ∑_{l≠2,3,N} P_{l}

on the image of X_{0}(N) as before. We now project right away to X_{0}^{+}(N), so that we have P^{+} = i_{∞}(Q^{+}) = P_{2}^{+} + P_{3}^{+} + P_{N}^{+} + R^{+}, where

R^{+} = ∑_{l≠2,3,N} P_{l}^{+}

with the various P_{l}^{+} defined in the obvious way. By Corollary 1, there is a σ ∈ Gal(\overline{ℚ}/ℚ) such that σP^{+} = P_{2}^{+} + P_{3}^{+} − P_{N}^{+} − R. So P^{+} + σP^{+} = 2P_{2}^{+} + 2P_{3}^{+}.

Now this torsion point on J_{0}^{+}(N) has order prime to N, and so Theorem 4 applies

to it.

If 2P_{2}^{+} ≠ 0, we find that there exists a τ ∈ Gal(\overline{ℚ}/ℚ) such that P′ = (τP^{+} +

τσP^{+}) − (P^{+} + σP^{+}) has order 2 in J_{0}^{+}(N). We have P′ = τQ^{+} + τσQ^{+} − Q^{+} − σQ^{+}, so the divisor 2(τQ^{+}) + 2(τσQ^{+}) − 2(Q^{+}) − 2(σQ^{+}) is principal. This divisor is either identically zero, or else X_{0}^{+}(N) admits a map to P^{1} of degree at most four. The first case is impossible, because it implies that either Q^{+} = τQ^{+} and σQ^{+} = τσQ^{+} or Q^{+} = τσQ^{+} and σQ^{+} = τQ^{+}, but either way we would have P′ = 0 whereas we assumed P′ had order 2. So X_{0}^{+}(N) admits a map of degree at most four to P^{1}.

Similarly, if 2P_{3}^{+} ≠ 0, we find that X_{0}^{+}(N) admits a map of degree at most six

to P^{1}. But we show in chapter 3 that this implies that N ≤ 911. Since we have already established the CKR conjecture for primes less than 5000, we reduce to the case where P′ = 0. But then P^{+} = −σP^{+}, hence −i_{∞}(Q^{+}) is in the image of X_{0}^{+}(N) and by Lemma 2, X_{0}^{+}(N) is sub-hyperelliptic. But we have already dispensed of this case. So modulo some facts to be established in subsequent sections, this completes our first proof of the Coleman-Kaskel-Ribet conjecture. ■

Before discussing the second proof, we consider the special case X_{0}(N)∩J_{0}(N)[ℑ].

We recall that J_{0}(N)[ℑ] has order n^{2}, where n = Num\frac{N−1}{12}. Also, J_{0}(N)[ℑ] contains

both the cuspidal subgroup C and the Shimura subgroup Σ. When n is odd, J_{0}(N)[ℑ] is in fact equal to C +Σ. When n is even, C and Σ intersect in a Galois module isomorphic to μ_{2}, and consequently C+Σ has index 2 in J_{0}(N)[ℑ]. The Galois action of Gal(\overline{ℚ}/ℚ) on C is trivial and on Σ is given by the cyclotomic character χ : Gal(\overline{ℚ}/ℚ) ↠ Gal(ℚ(μ_{n})/ℚ)→^{∼}(ℤ/nℤ)^{∗}.

Lemma 3. Suppose X_{0}(N) is hyperelliptic and that Q is a hyperelliptic branch point. Then i_{∞}(Q) = [(Q) − (∞)] ∉ J_{0}(N)[ℑ].

Proof. When N = 37, the hyperelliptic branch points do not map to torsion

points of J_{0}(N) at all (see [12]). So we can assume that N ≠ 37. In this case, the hyperelliptic involution coincides with the Atkin-Lehner involution w.

One way to conclude is to note that a hyperelliptic branch point corresponds to an

elliptic curve with complex multiplication by an order in the ring of integers of ℚ(\sqrt{−N}). The field of definition of this point contains ℚ(\sqrt{−N}), which is ramified at N. Hence i_{∞}(Q) cannot be in J_{0}(N)[ℑ], which is unramified at N.

Here is an alternate argument. Let P = i_{∞}(Q) ∈ J_{0}(N). If P ∈ C then we’re done,

since C ∩i_{∞}(X_{0}(N)) consists only of the cusps (see [12]), which are not hyperelliptic branch points. (In fact, the cusps on X_{0}(N) are never Weierstrass points, see [43]). Otherwise Q is not rational, so there is some σ ∈ Gal(\overline{ℚ}/ℚ) such that Q′ = σQ ≠ Q is another hyperelliptic branch point. Then the divisor (Q) − (Q′) has order 2 in J_{0}(N). On the other hand, i_{∞}(Q) = [(Q) − (Q′)] + i_{∞}(Q′), so at least one of i_{∞}(Q), i_{∞}(Q′) has even order; since these points are conjugate, both have even order. Since w is the hyperelliptic involution on X_{0}(N), which acts on J_{0}(N) as −1, we have [(Q) − (∞)] = [(w∞) − (wQ)] = [(0) − (Q)] as elements of J_{0}(N). Adding [(Q) − (∞)] to both sides, we get 2P = [(0) − (∞)], which is a generator of the cyclic group C. This contradicts the fact that the order of P is even. ■

Lemma 4. If m is a positive integer not dividing 6, then there exist elements a, b ∈ (ℤ/mℤ)^{∗} with a + b = 2 and a ≠ 1, b ≠ 1.

Proof. By the Chinese Remainder Theorem, the result is true for m if it is true

for at least one prime power p^{t} exactly dividing m. By assumption we can choose such a p^{t} > 3. If p ≠ 3, then −1 and 3 work. If p = 3, use −2 and 4. ■

Proposition 3. Let N ≥ 23 be prime. The only points Q ∈ X_{0}(N)(ℚ̄) such that P = [(Q) − (∞)] lies in J_{0}(N)[ℑ] are 0 and ∞.

Proof. We provide two proofs of this result. First, we note that if P ∈ J_{0}(N)[ℑ],

then under the projection π_{∗}, P is sent to zero. Therefore, when the genus of X_{0}^{+}(N) is positive the map X_{0}^{+}(N) → J_{0}^{+}(N) is an embedding, and we have Q = 0 or Q = ∞ as desired. The genus of X_{0}^{+}(N) is zero exactly when X_{0}(N) is hyperelliptic and N ≠ 37, i.e., when N = 23, 29, 31, 41, 47, 59, or 71.

Suppose, then, that P ∈ J_{0}(N)[ℑ] (so its order divides n = Num\frac{N−1}{12}) and that

X_{0}(N) is hyperelliptic. Let g be the genus of X_{0}(N). One then checks using [55] that for each N in the above list that n is prime to 3, and that there are no primes between 5 and 2g which are simultaneously non-ordinary and ramified in the Hecke algebra. By Theorem 1 (which is based on p-adic integration techniques of [10]) and Theorem 2, it follows that P = P_{2} + P_{C}, with P_{2} of 2-power order and P_{C} ∈ C.

If P_{2} ∉ C, then by Theorem 3 there exists a σ in an inertia group for 2 in Gal(\overline{ℚ}/ℚ)

such that σP − P has order 2. This means that the divisor 2(σQ) − 2(Q) is principal, so Q is a hyperelliptic branch point. But the hyperelliptic branch points of X_{0}(N) do not map to J_{0}(N)[ℑ] by Lemma 3.

So we must have P_{2} ∈ C and therefore P ∈ C. But as explained in [12], the set of

points on X_{0}(N) mapping to C is always equal to {0, ∞}. This proves the result.

Here is another proof, which does not rely on any facts about ramified torsion

points on curves derived from the p-adic integration methods in [10].

We again may assume, after projecting to X_{0}^{+}(N), that X_{0}(N) is hyperelliptic

with N ≠ 37. And proceeding as above we see that P_{2} ∈ C, or else X_{0}(N) would be hyperelliptic and Q would be a hyperelliptic branch point of X_{0}(N), which is impossible. It follows that P ∈ C + Σ.

Since i^{−1}(C) = {0, ∞}, we may assume that P_{2} ∈ C but P ∉ C, and therefore we

may write P = P_{C} + P_{Σ}, where P_{C} ∈ C and P_{Σ} ∈ Σ is nonzero and of odd order m.

In fact, we may assume that m > 3, because n is prime to 3 for all N such that

g^{+} = 0, a fact we have already noticed above.

Since C has a trivial Galois action, it is easy to see that (σ − 1)P = (σ − 1)P_{Σ} for

all σ ∈ Gal(\overline{ℚ}/ℚ). Also, since Gal(\overline{ℚ}/ℚ) acts on Σ via the mod n cyclotomic character, it follows that for any μ ∈ (ℤ/mℤ)^{∗} we can find σ ∈ Gal(\overline{ℚ}/ℚ) such that σP_{Σ} = μP_{Σ}.

We conclude from Lemma 4 that there exist σ, τ ∈ Gal(\overline{ℚ}/ℚ) such that (σ−1)P_{Σ}+

(τ − 1)P_{Σ} = 0 but (σ − 1)P_{Σ} and (τ − 1)P_{Σ} are nonzero. It follows that (σQ) + (τQ) − 2(Q) is a nonzero principal divisor on X_{0}(N), and hence that Q is a hyperelliptic branch point. But this contradicts Lemma 3. ■

With this proposition in hand, we give the second proof of the Coleman-Kaskel-

Ribet conjecture.

Proof. Suppose Q ∈ X_{0}(N)(ℚ̄) maps to a torsion point P of J_{0}(N). If P ∈

J_{0}(N)[ℑ], then Q ∈ {0, ∞} by Prop. 3. So we can assume that P ∉ J_{0}(N)[ℑ], i.e., that P is ramified at N. We claim that there is an element σ ∈ Gal(\overline{ℚ}/ℚ) such that σP ≠ P but (σ − 1)^{2}P = 0. Given this, we see how to conclude: in terms of divisors this means that (σ^{2}Q) + (Q) − 2(σP) is linearly equivalent to zero. Hence X_{0}(N) is hyperelliptic and σP (and hence P, since the hyperelliptic involution is defined over ℚ) is a hyperelliptic branch point.

To prove the claim, we first assume that N is prime to the order of P. In this

case, we use the fact that P is ramified at N to find an inertia group I at N and an element σ ∈ I such that σP ≠ P. By Grothendieck’s semistable reduction theorem (see [47]), (σ − 1)^{2}P = 0 as desired.

If N divides the order of P, write P = P_{N} +P^{N} with P_{N} of N-power order and P^{N}

of order prime to N. We claim that there exists a σ ∈ Gal(\overline{ℚ}/ℚ) which fixes P^{N} but not P_{N} such that (σ − 1)^{2}P_{N} = 0. It then follows that (σ − 1)^{2}P = 0. We can find such a σ in X, the normal closure of a wild inertia group at N in Gal(\overline{ℚ}/ℚ). Elements of X fix prime-to-N torsion, and moreover it follows from [35] (or from [48] when N satisfies hypothesis (*)) that the image of X in Aut(T a_{N}(J_{0}(N))) ≅ GL(2, T ⊗ ℤ_{N}) contains SL(2, T ⊗ ℤ_{N}). In particular, there exist σ_{1}, σ_{2} ∈ X acting on an N-adic Tate module of J_{0}(N) as

σ_{1} = (\begin{matrix}1 & 0 \\ 1 & 1 \end{matrix}) , σ_{2} = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) .

It is easy to see that since P_{N} ≠ 0, one of σ_{1}, σ_{2} must act nontrivially on P_{N}. This

element σ of X also satisfies (σ − 1)^{2}P_{N} = 0, so we’re done. ■

We remark that we can modify this proof so that only the results of [48], and

not those of [35], are needed. According to Cor. 1, there exists τ ∈ Gal(\overline{ℚ}/ℚ) which acts trivially on prime-to-N torsion points of J_{0}(N) and as −1 on N-power torsion points. Therefore P′ = (1 + τ)P has order prime to N, and we find that there exists σ such that σP′ ≠ P′ but (σ − 1)^{2}P′ = 0 using the easier “order prime-to-N” case of Theorem 3. It is not hard to see that this implies that X_{0}(N) admits a morphism to P^{1} of degree at most 4. If such a morphism exists then N < 215 by the results of chapter 3. But we know from [55] that the smallest (and only?) prime for which (*) is not satisfied is 389. In particular, for prime numbers N less than 389 this condition is satisfied. So we are done by the argument above, which assumes (*).

2.3 Generalizations

We now present some generalizations of the Coleman-Kaskel-Ribet conjecture.

Let X/k be a curve. Follwing [10], we define an equivalence relation on X(k̄) by

saying that P is equivalent to Q if some multiple of the divisor (P) − (Q) is principal. The set of all points in a single equivalence class is called a torsion packet on X.

For example, we showed in the previous section that when g^{+}(N) > 0, the torsion

packet on X_{0}(N) containing ∞ is precisely the set of cusps. More generally we have the following.

Proposition 4. Let X be a modular curve covering some X_{0}(N) with g^{+}(N) > 0; for example, X could be X_{0}(NM) or X_{1}(NM) for any positive integer M. Then the set of cusps on X forms a complete torsion packet.

Proof. It follows from a theorem of Manin and Drinfeld (or from the work of

Kubert-Lang) that the cusps of X lie in a common torsion packet. Furthermore, the fiber of X → X_{0}(N) over a cusp of X_{0}(N) consists entirely of cusps. Suppose then that some multiple of the divisor (Q) − (∞) on X is principal. Let J be the Jacobian of X, and let i_{∞} : X ↪ J (resp. X_{0}(N) ↪ J_{0}(N)) be the Albanese embedding associated to the base point ∞. There is a commutative diagram

[figure: X; i_{∞}; J; X_{0}(N); i_{∞}; J_{0}(N)]

which shows that the image Q′ of Q in X_{0}(N) is a torsion point on J_{0}(N) via the mapping i_{∞}. Since g^{+}(N) > 0, we know that Q′ is a cusp. Therefore its preimage Q on X is also a cusp. ■

The following corollary follows directly by combining Prop. 4 with a theorem of

Mazur proved in [54, Thm. 0.4].

Corollary 2. Let X = X_{0}(NM) or X_{1}(NM) with g^{+}(N) > 0, let J be the Jacobian of X, and let i_{∞} : X → J be the embedding defined by Q ↦ [(Q) − (∞)]. Fix a noncuspidal point x ∈ X whose associated elliptic curve does not have CM. Let ℤT_{p}(x) be the ℤ-linear span in J of the p-Hecke points associated to x, i.e., if T_{p}(x) = ∑(y_{j}), then ℤT_{p}(x) is the subgroup of J generated by the p + 1 points i_{∞}(y_{j}). Then for all sufficiently large primes p, ℤT_{p}(x) has maximal rank p + 1.

Our next generalization concerns torsion points on X_{0}^{+}(N). Let N be a prime

number. When g^{+}(N) ≥ 1, let i_{∞} be the embedding of X_{0}^{+}(N) into J_{0}^{+}(N) defined by Q ↦ [(Q) − (∞)]. We will need the following proposition:

Proposition 5. If X_{0}^{+}(N) is hyperelliptic and Q is a hyperelliptic branch point, then i_{∞}(Q) is not a torsion point on J_{0}^{+}(N).

Proof. The hyperelliptic involution h operates on J_{0}^{+}(N) as −1, so that as

elements of J_{0}^{+}(N) we have

[(Q) − (∞)] = [(h∞) − (hQ)] = [(h∞) − (Q)]

and adding [(Q) − (∞)] to both sides of this equation,

2[(Q) − (∞)] = [(h∞) − (∞)].

Therefore i_{∞}(Q) is a torsion point if and only if i_{∞}(h∞) is a torsion point.

Now as we have seen, X_{0}^{+}(N) is hyperelliptic exactly when it has genus 2. Hasegawa

[31] found that for those N for which this is the case (namely N = 67, 73, 103, 107, 167, 191), the image of the cusp ∞ under the hyperelliptic involution h is a noncuspidal rational point. (Specifically, the image of ∞ is a Heegner point of class number one).

In other words, we have h∞ ≠ ∞. This can also be seen by looking at q-expansions

of weight-two cusp forms for Γ_{0}^{+}(N), since h∞ = ∞ iff ∞ is a Weierstrass point on X_{0}^{+}(N) iff there is a form f = a_{1}q+a_{2}q^{2}+a_{3}q^{3}+… in the two-dimensional space S_{2}(Γ_{0}^{+}(N), ℚ) such that a_{1} = a_{2} = 0. The result follows from scrutinizing the tables of [55]. (For somewhat larger prime values of N, however, it seems that ∞ usually is a Weierstrass point on X_{0}^{+}(N). See [19] for a discussion of this.)

In any case, it is a theorem of Mazur [36] that the Mordell-Weil group J_{0}^{+}(N)(ℚ)

is zero. So [(h∞) − (∞)], and therefore [(Q) − (∞)], has infinite order. ■

Theorem 5. When g^{+}(N) ≥ 2, ∞ is the only point Q ∈ X_{0}^{+}(N)(ℚ̄) such that i_{∞}(Q) ∈ J_{0}^{+}(N)^{tor}. In other words, the torsion packet on X_{0}^{+}(N) containing the cusp ∞ is trivial.

Proof. We emulate the second proof of the CKR conjecture. We know that on

J_{0}^{+}(N), every nonzero torsion point is ramified at N. So if Q ≠ ∞ maps to a torsion point P on J_{0}^{+}(N), then P is ramified at N, so there exists a σ in an inertia group for N such that σP − P is nontrivial and (σ − 1)^{2}P = 0. Hence X_{0}^{+}(N) is hyperelliptic and Q is a hyperelliptic branch point. But this is impossible by Proposition 5. ■

Our techniques extend in a rather straightforward manner to arbitrary torsion

packets on X_{0}(N) and X_{0}^{+}(N).

Theorem 6. If X_{0}^{+}(N) has a nontrivial torsion packet, then X_{0}^{+}(N) admits a map of degree at most 4 to P^{1}. In particular, if N > 479 then every torsion packet on X_{0}^{+}(N) is trivial.

Proof. By the results of chapter 3, the second assertion implies the first. So we

assume that P = [(Q_{1}) − (Q_{2})] ∈ J_{0}^{+}(N)^{tor} with Q_{1} ≠ Q_{2} and hope to deduce that X_{0}^{+}(N) admits a map of degree at most 4 to P^{1}.

The proof proceeds like our previous arguments. Since P is a nonzero torsion point

on J_{0}^{+}(N) it is ramified at N, and reasoning as above we can find a σ ∈ Gal(\overline{ℚ}/ℚ) such that σP ≠ P and (σ − 1)^{2}P = 0. Therefore (σ^{2}Q_{1}) + (Q_{1}) + 2(σQ_{2}) − (σ^{2}Q_{2}) − (Q_{2}) − 2(σQ_{1}) is principal. This implies that there is a rational function on X_{0}^{+}(N) of degree at most 4. For if not, we would have total cancellation in the above expression. But it is easy to see that this would contradict the fact that σP ≠ P. ■

For X_{0}(N), we have the following result.

Theorem 7. If X_{0}(N) has a nontrivial torsion packet other than the cuspidal packet {0, ∞}, then X_{0}(N) admits a map of degree at most 6 to P^{1}. In particular, if N > 479 then every noncuspidal torsion packet on X_{0}(N) is trivial.

Proof. Again it follows from chapter 3 that the second assertion implies the first.

So we assume that P = [(Q_{1}) − (Q_{2})] ∈ J_{0}(N)^{tor} with Q_{1} ≠ Q_{2} and hope to deduce that X_{0}(N) admits a map of degree at most 6 to P^{1}.

The proof proceeds like our previous arguments. As in the previous proof, if

P ∉ J_{0}(N)[ℑ] then it is ramified at N, and reasoning as before there exists a rational function on X_{0}(N) of degree at most 4.

It remains to consider the case where P ∈ J_{0}(N)[ℑ]. We may assume that P ∈

C + Σ; otherwise, as in the proof of Prop. 3, there exists σ ∈ Gal(\overline{ℚ}/ℚ) such that σP − P has order 2, which implies that X_{0}(N) admits a rational function of degree at most 4. Write P = P_{C} + P_{Σ} with P_{C} ∈ C and P_{Σ} in Σ. Let m be the order of P_{Σ}. If 3 divides m then there exists a σ ∈ Gal(\overline{ℚ}/ℚ) such that σP − P has order 3. It follows that X_{0}(N) admits a rational function of order at most 6. So we can assume that 3 does not divide m.

Notice that for all σ ∈ Gal(\overline{ℚ}/ℚ), (σ − 1)P = (σ − 1)P_{Σ}. If m > 2, then Lemma

4 implies that there exist σ, τ ∈ Gal(\overline{ℚ}/ℚ) such that (σ − 1)P + (τ − 1)P = 0 but (σ − 1)P and (τ − 1)P are nonzero. This easily implies that X_{0}(N) admits a rational function of degree at most 4.

So finally, without loss of generality we assume that m divides 2, i.e., that P ∈ C.

Then (σ − 1)P = 0 for all σ ∈ Gal(\overline{ℚ}/ℚ), so either X_{0}(N) is hyperelliptic and Q_{1} is a hyperelliptic branch point, or Q_{1} = σQ_{1} for all σ, i.e., Q_{1} is defined over the rational numbers. By results of Mazur [36], in the latter case Q_{1} (and similarly Q_{2}) is a cusp, unless N = 37, 43, 67, or 163. In each of these cases, there is a single noncuspidal rational point on X_{0}(N), which by uniqueness is fixed by the Atkin-Lehner involution w. But it is easy to see that Q_{1} = wQ_{2} whenever g^{+} > 0 using the fact that w acts on J_{0}(N)[ℑ] as −1. So in fact Q_{1} = Q_{2}, and hence P = 0, in each of these cases. ■

Corollary 3. For all prime numbers N > 479, there is no regular differential on X_{0}(N) vanishing to order 2g − 2 at a single point, where g denotes the genus of X_{0}(N).

Proof. Suppose, on the contrary, that some differential ω has divisor (2g − 2)(Q)

for some point Q ∈ X_{0}(N)(ℚ̄). This certainly implies that Q is a Weierstrass point on X_{0}(N). If Q is defined over ℚ, then results of Mazur show that Q must be a cusp, but according to [43] the cusps on X_{0}(N) are not Weierstrass points. Therefore there is some σ ∈ Gal(\overline{ℚ}/ℚ) such that (σω) = (2g − 2)(σQ) ≠ (2g − 2)(Q). The ratio of ω and σw is a rational function on X_{0}(N) with divisor (2g − 2)(Q) − (2g − 2)(σQ), so Q and σQ are in the same torsion packet on X_{0}(N). This is impossible by Theorem 7. ■

We also mention the following result, whose proof is nearly the same as the proofs

of Theorems 6 and 7.

Theorem 8. Let X_{0}(N)^{(d)} (resp. X_{0}^{+}(N)^{(d)}) map to J_{0}(N) (resp. J_{0}^{+}(N)) by the map i sending ∑ Q_{i} to [∑(Q_{i}) − Q′], where Q′ = ∑(Q′_{i}) is a ℚ-rational point. Then if Q ≠ Q′ in X_{0}(N)^{(d)}(ℚ̄) (resp. X_{0}^{+}(N)^{(d)}(ℚ̄)) maps to a torsion point via i, then X_{0}(N) (resp. X_{0}^{+}(N)) admits a map of degree at most 3d (resp. 2d) to P^{1}.

Chapter 3

Gonality of Modular Curves

3.1 A Generalization of Ogg’s Method

A curve X/k is hyperelliptic over k if it has genus g ≥ 2 and if admits a degree

two map to P^{1} defined over k. It is hyperelliptic if it is hyperelliptic over k̄. Similarly, we will say that X/k is trigonal if it is trigonal over k̄, which means that it has genus at least two and admits a degree three map to P^{1} defined over k̄.

In general, we say that the k-gonality of X is the degree of the smallest morphism

from X to P^{1} defined over k. The gonality of X (with no modifier) will mean its k̄-gonality. With our terminology, a curve of genus g ≥ 2 is hyperelliptic iff its gonality is two. It is well-known (and is easily deduced from Lemma 5) that X is trigonal iff g = 2 or the gonality of X is three.

In this section, unlike the previous ones, N is allowed to be a composite integer

unless otherwise specified.

In [41], A. Ogg proved that the modular curve X_{0}(N) (for N a positive integer)

is hyperelliptic exactly when N = 22, 23, 26, 28, 29, 30, 31, 33, 35, 37, 39, 41, 46, 47, 50, 59, 71.

His proof works by reducing modulo p and using the fact that if X_{0}(N) is hyper-

elliptic, then it is hyperelliptic over ℚ and the reduction mod p is hyperelliptic over F_{p} for p not dividing N. But then the number of points in X_{0}(N)(F_{q}) for q a power of p is bounded by twice the number of points of P^{1}(F_{q}), or 2(q + 1). However, when N is large X_{0}(N)(F_{p^{2}}) has many points, coming from supersingular elliptic curves in characteristic p. So one gets a bound on the values of N such that X_{0}(N) can be hyperelliptic, and studying Atkin-Lehner involutions and Weierstrass points gives the exact list mentioned above.

The main obstacle to a direct generalization of Ogg’s method to degree d maps

is that whereas a hyperelliptic curve over k possessing a k-rational point is automatically hyperelliptic over k, the analogous fact need not be true for d-gonal curves if d > 2. But under certain hypotheses it will be true when the genus is sufficiently large with respect to d.

In [1], Abramovich proves (using methods of Yau) that if X_{0}(N) admits a mor-

phism of degree d to P^{1} over the complex numbers, then N ≤ \frac{800d}{7}. For most d this lower bound for the gonality of X_{0}(N) is much better than we can obtain with our techniques, but for small d (such as d = 3), our methods give better bounds. We also note that our methods give information about the gonality of X_{0}(N)_{F_{p}}, the reduction modulo p of X_{0}(N).

In subsequent sections we give some concrete applications of our techniques. For

example, we determine all trigonal modular curves X_{0}(N), and we also determine when X_{0}^{+}(N) is trigonal in the case where N is a prime number.

We will have use for the following result, which we refer to as the Correspondence

Lemma, throughout this section. For some notation, if f_{1} : C → C_{1} and f_{2} : C → C_{2} are nonconstant morphisms between curves, we say that f_{1} and f_{2} are independent if C is birational to its image C′ under the induced map to C_{1} × C_{2}.

C
[figure: f_{2}; f_{1}]
C′  \overline{}C_{2}

C_{1}

Lemma 5. Let C_{1}, C_{2}, C be curves of genera g_{1}, g_{2}, and g, respectively, over the field k. Let f_{i} : C → C_{i} be morphisms of degree d_{i}, i = 1, 2. Assume that f_{1} and f_{2} are independent. Then

g ≤ (d_{1} − 1)(d_{2} − 1) + d_{1}g_{1} + d_{2}g_{2}.

Remark. The hypothesis of the lemma is satisfied, for example, if gcd(d_{1}, d_{2}) = 1. Proof. This is exercise VIII.C-1 in [4]. ■ We also give an easy lemma which allows us to recognize when maps are indepen-

dent.

Lemma 6. The morphisms f_{1} : C → C_{1} and f_{2} : C → C_{2} fail to be independent if and only if there exists a complete nonsingular curve D and morphisms h : C → D, h_{1} : D → C_{1}, and h_{2} : D → C_{2} such that f_{1} = h_{1} ∘ h, f_{2} = h_{2} ∘ h, and h has degree greater than 1.

Proof. Suppose the map f : C → C′ ⊂ C_{1} × C_{2} has degree d > 1. Then the

induced map h : C → D to the normalization D of C′ also has degree d > 1, and the maps f_{1} and f_{2} factor as required. Conversely, suppose there exists D as in the statement of the lemma. Then the map from C to its image C′ factors through the map from C to D, so does not have degree 1. ■

Using the Correspondence Lemma, we obtain a criterion for the existence of mor-

phisms of a given degree defined over a given base field.

Proposition 6. Suppose X, Y/k are curves of genera g, g′ over the perfect field k. Suppose X(k) is nonempty. Let f be a degree d morphism from X to Y defined over k̄, and suppose that f does not factor through any intermediate curve. If g > (d − 1)^{2} + 2dg′, then there exists a degree d morphism defined over k from X to a curve Y′ isomorphic over k̄ to Y.

Proof. (Compare with [26]) For each σ ∈ G_{k} := Gal(k̄/k), we get a degree d

morphism f^{σ} : X → Y. Since f does not factor through any intermediate curves, the degree of the map from X to its image under f × f^{σ} is either 1 or d. But the Correspondence Lemma shows that it cannot be 1, since g > (d − 1)^{2} + 2dg′. So f and f^{σ} must differ by an automorphism of Y, in other words for each σ ∈ G_{k}, there exists α_{σ} ∈ Aut(Y) such that f^{σ} = α_{σ} ∘ f. The collection {α_{σ}} forms a 1-cocycle in H^{1}(G_{k}, Aut(Y)), and by [53, X.2.2] corresponds to a twist of Y. In other words, there exists a curve Y′/k and an isomorphism λ : Y′ → Y defined over k̄ such that α_{σ} = λ_{σ} ∘ λ^{−1} for all σ ∈ G_{k}. The morphism we seek from X to Y′ is then λ^{−1} ∘ f. ■

Here is the simple case of this proposition which we will actually need.

Corollary 4. Let X be as in the proposition, and assume in addition that X has a krational point. Let l be a prime number, and suppose there exists a degree l morphism from X to P^{1} defined over k̄. If g > (l − 1)^{2} then there exists a degree l morphism from X to P^{1} defined over k.

Before we applying this result, we have a lemma which is an easier-to-prove special

case of the semicontinuity theorem [27, Theorem III.12.8]:

Lemma 7. Let R be a discrete valuation ring with field of fractions K, residue field k, and uniformizing parameter π. Let X be a one-dimensional regular scheme flat over Spec(R), with generic fiber X_{K} (a nonsingular curve over K) and special fiber X_{k} (a possibly singular curve over k). Then any line bundle L_{K} of degree d on X_{K} extends to a line bundle L on X, and the pullback L_{k} to X_{k} is a degree d line bundle with dim_{K} H^{0}(X_{K}, L) ≤ dim_{k}H^{0}(X_{k}, L_{k}).

Proof. With our hypothesis, it follows from [27, Prop. II.6.11], there is a natural

isomorphism between the divisor class group of X and the group Pic(X). The surjectivity of the natural map Pic(X) → Pic(X_{K}) then follows from [27, Prop. II.6.5(a)]. (If X/R is smooth then this map is also injective). Let L be a line bundle on X extending the degree d line bundle L_{K} on X_{K}. If we define the degree of a line bundle L to be χ(L) − χ(O_{X}), then this degree is constant on fibers because X/R is flat; by Riemann-Roch this degree must then be d.

We have dim_{K} H^{0}(X_{K}, L_{K}) = dim_{K} H^{0}(X, L) ⊗_{R} K because cohomology com-

mutes with the flat base extension R → K by [27, III.9.3].

The pullback (restriction) L_{k} of L is a line bundle of degree d on X_{k}, and it

remains to show that dim_{K} H^{0}(X, L) ⊗_{R} K ≤ dim_{k} H^{0}(X_{k}, L_{k}). This is a special case of the semicontinuity theorem [27, III.12.8], but for the reader’s convenience we give a self-contained and more elementary proof.

First of all, it is not hard to see that the “multiplication by π” map j : L → L

induces a short exact sequence of sheaves on X

0 → L → L → i_{∗}L_{k} → 0

where i : X_{k} ↪ X is the natural inclusion. For example, the map j is injective because L is locally free over O_{X} and hence R-torsion free. We then get a long exact sequence of cohomology

0 → H^{0}(X, L) ⊗_{R} K → H^{0}(X_{k}, L_{k}) → H^{1}(X, L)_{π} → 0.

In particular, the injection on the left yields the desired inequality. ■

Proposition 7. Let N be a positive integer, let ν(N) be the number of distinct primes dividing N, let μ_{N} = ∏_{p|N}(1 + \frac{1}{p}), and let g_{0}(N) be the genus of X_{0}(N). Let l, p be prime numbers with p not dividing N. If g_{0}(N) > (l − 1)^{2} and 2^{ν(N)} + \frac{p−1}{12} · μ_{N} > l(p^{2} + 1), then there does not exist (over ℚ̄) a degree l morphism from X_{0}(N) to P^{1}.

Proof. Suppose, to the contrary, that there exists a degree l morphism X_{0}(N) →

P^{1}. The hypothesis g_{0}(N) > (l − 1)^{2} means that there exists such a morphism defined over ℚ. By Lemma 7, there exists a degree l morphism to P^{1} from the reduction of X_{0}(N) mod p defined over F_{p}. In particular the number of F_{p^{2}}-valued points of X_{0}(N)_{F_{p}} is at most l(p^{2} +1). On the other hand, Ogg’s proof in [41], combined with the mass formula from [53, Ex. 5.9], shows that #X_{0}(N)(F_{p^{2}}) ≥ 2^{ν(N)} + \frac{p−1}{12} · μ(N) (see [57, Remark 4.1.55]). This is a contradiction. ■

We note that the Correspondence Lemma also gives the following result about the

gonality of X_{0}(N).

For N prime, define the function h_{0}(N) to be ϵ(N) · h(N), where h(N) is the

class number of ℚ(\sqrt{−N}) and ϵ(N) is \frac{1}{2} if N ≡ 1 (mod 4), 1 if N ≡ 7 (mod 8), and 2 if N ≡ 3 (mod 8). By [42, p. 20], h_{0}(N) is equal to half the number of fixed points of the Atkin-Lehner involution acting on X_{0}(N).

Proposition 8. Let d be a positive integer and let N be a prime such that h_{0}(N) > d. If X_{0}(N) admits a morphism f of degree d to P^{1}, then f factors through the quotient X_{0}^{+}(N) of X_{0}(N) by its Atkin-Lehner involution. In particular, if d is odd then X_{0}(N) does not admit a morphism to P^{1} of degree d.

Proof. Let g, g′ be the genera of X_{0}(N), X_{0}^{+}(N), respectively. Suppose X_{0}(N)

admits a morphism f of degree d to P^{1}. If this map does not factor through the map h from X_{0}(N) → X_{0}^{+}(N), then the maps f and h are independent. The Correspondence Lemmma then tells us that g ≤ d − 1 + 2g^{+}, i.e., g − 2g^{+} + 1 ≤ d. But g − 2g^{+} + 1 = h_{0}(N) by the Riemann-Hurwitz formula since h_{0}(N) is precisely half the number of ramification points of h. ■

Remark. Since h_{0}(N) grows roughly like \sqrt{N}, one can view this proposition as giving bounds similar to those of Prop. 7. In practice, a combination of these two propositions yields more information than either by itself.

Next, we show how to apply these methods in the case of certain degree d mor-

phisms to P^{1} with d composite. For similicity, we restrict to the case where d is a product of at most two primes, since this is the only case we need in Chapter 2.

First we have a couple of lemmas. The first is well-known.

Lemma 8. If there is a map of degree at most d from X to P^{1} and X covers Y, then there is a map of degree at most d from Y to P^{1} as well.

Proof. One can see this in terms of function fields by taking norms – see [26] for

a proof. ■

Lemma 9. Let f be a morphism of degree d = d_{1}d_{2} (d_{1}, d_{2} prime) from a curve X/k with X(k) nonempty to P^{1}. Then f satisfies one of the following three conditions:

(I) g(X) ≤ (d − 1)^{2}

(II) There exists a curve Y of genus at most max((d_{1} − 1)^{2}, (d_{2} − 1)^{2}) defined over

k and a k-rational morphism h : X → Y.

Proof. Suppose condition (I) does not hold. Then by the Correspondence

Lemma, f and f^{σ} are dependent for all σ ∈ Gal(\overline{k}/k). One possibility is that for each σ ∈ Gal(\overline{k}/k) there exists β_{σ} ∈ Aut(P^{1}) such that f^{σ} = β_{σ} ∘ f. In this case, the proof of Lemma 6 shows that there exists a morphism f′ of degree d defined over k from X to P^{1}. In particular, condition (II) is satisfied. By Lemma 6 and the Correspondence Lemma, the other possibility is that there exist σ ∈ Gal(\overline{ℚ}/ℚ), Y, f_{1} : X → Y and f_{2} : Y → P^{1} so that f = f_{2} ∘ f_{1}, where (switching d_{1} and d_{2} if necessary) f_{1} has degree d_{1} and f_{2} has degree d_{2}, and such that f and f^{σ} both factor through Y but do not differ by an automorphism of P^{1}. Since d_{2} is prime, f_{2} and f_{2}^{σ} are independent, and therefore g(Y) ≤ (d_{2} − 1)^{2}. Furthermore, since (I) does not hold, we also have g(X) > (d_{1} − 1)^{2} + 2d_{1}g(Y). Combining this fact with the Correspondence Lemma (and using the assumption that d_{1} is prime), we find that for all τ ∈ Gal(\overline{ℚ}/ℚ), there is an isomorphism α_{τ} : Y →^{∼}Y^{τ} such that f^{τ} = α_{τ} ∘ f. It follows as in the proof of Lemma 6 that f, Y can be chosen so they are defined over k. ■

Proposition 9. Fix a positive integer d = d_{1}d_{2} with d_{1} and d_{2} prime. Let N be an odd number and let X/ℚ be either X_{0}(N) or X_{0}^{+}(N). Let α_{4}(N) denote the number of F_{4}- rational points of the reduction of X mod 2, and let g be the genus of X. Assume that g > (d − 1)^{2} and

α_{4}(N) > max(5d, d_{1}(5 + 4(d_{2} − 1)^{2}), d_{2}(5 + 4(d_{1} − 1)^{2})).

Then there is no degree d morphism from X to P^{1}.

Proof. Suppose there is a degree d morphism from X to P^{1}. By Lemma 7, there

is such a morphism in characteristic 2 from X_{F_{2}} → P_{F_{2}}^{1}. The result follows from Lemma 9 using the Weil bound #Y (F_{4}) ≤ 5 + 4g(Y). ■ Remark. In the proof of Prop. 7, we saw that if X = X_{0}(N) then α_{4}(N) ≥ 2^{ν(N)} + \frac{N+1}{12}. Similarly, if X = X_{0}^{+}(N) then α_{4}(N) ≥ 2^{ν(N)−1} + \frac{N+1}{24}. This follows from the fact that there is a degree 2 map from X_{0}(N)_{F_{2}} → X_{0}^{+}(N)_{F_{2}} defined over F_{2}. Example 1. Maps of degree 4,5, or 6 from X to P^{1}.

For simplicity assume that X = X_{0}(N) or X_{0}^{+}(N) over ℚ with N prime. Suppose

there is a degree 4 map f from X to P^{1} defined over ℚ̄. Then either g(X) ≤ (4 − 1)^{2} = 9 or α_{4}(N) ≤ 4 · 5 = 20. Taken together, these conditions mean that N < 215 if X_{0}(N) admits a degree 4 map to P^{1}, or N ≤ 479 if X_{0}^{+}(N) does.

If we take d = 5, which is prime, we quickly find that either g(X) ≤ (5 − 1)^{2} = 16

or α_{4}(N) ≤ 25. This implies that N < 275 if X_{0}(N) admits a degree 5 map to P^{1}, and N ≤ 719 if X_{0}^{+}(N) does.

Finally, when d = 6 we have either g(X) ≤ (6 − 1)^{2} = 25 or α_{4}(N) ≤ 42. This

implies that N ≤ 479 if X_{0}(N) admits a degree 6 map to P^{1}, and N ≤ 911 if X_{0}^{+}(N) does.

3.2 Trigonal X_{0}(N)

We can now generalize Ogg’s results to determine for which N the modular curve

X_{0}(N) is trigonal.

In the statement of the following theorem, recall that g_{0}(N) denotes the genus of

X_{0}(N).

Theorem 9. X_{0}(N) is trigonal if and only if g_{0}(N) = 2 or g_{0}(N) is 3 or 4 and X_{0}(N) is not hyperelliptic.

Remark. When N is prime, this says that X_{0}(N) is trigonal if and only if

N ∈ {23, 29, 31, 37, 43, 53, 61}.

Proof. One direction is immediate: curves of genus 2 are trigonal by Riemann-

Roch, and nonhyperelliptic curves of genus 3 or 4 are trigonal by [27, Example 5.5.2]. Also, notice that a hyperelliptic curve of genus greater than 2 cannot be trigonal. This follows, for example, from the Correspondence Lemma.

For the converse, we first treat the case where N is prime, since it is simpler than

the general case but illustrates all the main ideas. Also, our application to the conjecture of Coleman-Kaskel-Ribet only concerns prime values of N.

It follows from Prop. 7 (taking l = 3 and p = 2) that for N prime, if g_{0}(N) > 4

and X_{0}(N) is trigonal then N ≤ 151. (The bound from [1] would only give N < 343.) In fact, we get something stronger, because for any specific N and p not dividing N we can find the exact value of #X_{0}(N)(F_{p^{2}}). A quick way to do this is to first calculate the characteristic polynomial F of the Hecke operator T_{p} acting on S_{2}(Γ_{0}(N)) using modular symbols, and then to compute the numerator f of the zeta function of X_{0}(N) over F_{p} using the identity F (x + p/x) = x^{−g}f(x) proved in [42, p. 23].

Prop. 7 implies that if X_{0}(N) is trigonal and g_{0}(N) > 4 then we have #X_{0}(N)(F_{4}) ≤

15 and #X_{0}(N)(F_{9}) ≤ 30. Computing these quantities exactly for N ≤ 151, we find that the set T of prime N such that g_{0}(N) > 4 and X_{0}(N) is trigonal is contained in the set {67, 73, 79, 83, 89, 101, 103, 131}.

Now note that a curve which is both bielliptic (admits a degree 2 map to an elliptic

curve) and trigonal has genus at most 4. This again follows from the Correspondence Lemma. Using the tables of [17], we see that T ⊆ {67, 73, 103}. Furthermore, X_{0}(103) cannot be trigonal because it violates the condition h_{0}(N) ≤ 3 given by Prop. 8. So we are left with the possibilities N = 67, 73, both of which have g_{0}(N) = 5. We use the criterion of [27, Example 5.5.3], which says that a curve of genus 5 is non-trigonal if and only if it is a complete intersection (of 3 quadric hypersurfaces) in its canonical embedding. Using the tables found in [20], we see that X_{0}(67), X_{0}(73) are both complete intersections in their canonical embeddings. So T = ∅ and we are done with the case where N is prime.

Now for the general case. In addition to the tools we have already used, we need

two more. The first is the fact that by Lemma 8, if X_{0}(MN) is trigonal and g_{0}(N) ≥ 2 then X_{0}(N) is either trigonal or hyperelliptic.

The second thing we need is a generalization of the criteria for trigonality given

above for curves of genus 5. This is afforded by Petri’s theorem (see [4, III.3]):

Theorem. Let C be a curve of genus g ≥ 4 canonically embedded in P^{g−1} by the homogeneous ideal I. C is neither trigonal nor a plane quintic if and only if I is generated by homogeneous quadratic relations. If C is trigonal or a plane quintic, then I is generated by homogeneous polynomials of degree 2 and 3.

The reason this is particularly convenient is that using a computer one can work

out generators for the canonical ideal I of X_{0}(N). In particular (supposing that X_{0}(N) is not hyperelliptic), we want to establish whether or not the vector space of cubic relations induced by the \frac{(g−2)(g−3)}{2} quadratic ones spans the (\frac{g(g+7)(g−4)}{6} + 5)-dimensional vector space of all cubic relations. The dimension counts just given follow from Max Noether’s theorem ([4, III.2]) that the natural maps Sym^{n}H^{0}(C, K) → H^{0}(C, K^{n}) are surjective for n ≥ 1, where K^{n} is the sheaf of n-fold differentials. By “induced cubic relations” we mean the image of H^{0}(C, K) ⊗ I_{2} inside the kernel of the map Sym^{3}H^{0}(C, K) → H^{0}(C, K^{3}), where I_{2} is the vector space of quadratic relations thought of as the kernel of the map Sym^{2}H^{0}(C, K) → H^{0}(C, K^{2}). In the case of the modular curves X_{0}(N), we may think of elements of Sym^{n}H^{0}(C, K) as cusp forms of weight k = 2n for Γ_{0}(N), and by [56] a cusp form of weight k is determined by the first μ_{N}k/12 coefficients of its q-expansion, where μ_{N} = ∏_{p|N}(1 + \frac{1}{p}). So the question of determining whether or not a given curve is trigonal reduces to linear algebra over ℚ, provided we can compute enough terms in the q-expansions for a basis of S_{2}(Γ_{0}(N); ℚ). This can be done using, for example, modular symbols – see the tables in [55].

In theory, this gives a recipe for determining trigonality for any given value of N.

In practice the linear algebra outlined above can become computationally difficult, so it is better to use other arguments when possible. Counting points over F_{p^{2}} for various primes p not dividing N, as in the case where N is prime, is a very helpful in this regard, because the computations are much faster.

Combining these techniques, we first compute the N for which X_{0}(N) is trigonal

and one of 2,3, or 5 does not divide N. This is done the same way as the case where N is prime, and we only have to compute the dimension of induced cubic relations a handful of times because counting points usually does the job. We are now done by combining Lemma 8 with the fact that by Petri’s theorem, X_{0}(N) is not trigonal for N = 2·3^{2} ·5, 3·5^{2}, 2^{2} ·3·5. This proves theorem 9. ■

Remark. Combining Theorem 9 with Petri’s Theorem actually provides a shortcut for finding equations for the canonical embedding of X_{0}(N). For when g_{0}(N) > 4 and g_{0}(N) ≠ 6, we see that the canonical ideal is generated by degree 2 relations. Using this fact, one needs only to determine the quadratic relations among differentials (cusp forms); it is not necessary to look for any higher relations.

3.3 Trigonal X_{0}^{+}(N)

In this section we make some further remarks about the gonality of modular curves,

and in particular study when X_{0}^{+}(N) is trigonal for N a prime number.

Notice a pattern one observes in looking at hyperelliptic and trigonal X_{0}(N).

In the latter case, one may rephrase Theorem 9 in a loose way by saying that X_{0}(N) is trigonal if and only if “it has to be”, in the sense that any (nonhyperelliptic) curve of that genus is trigonal. And similarly, Ogg’s result can be phrased as saying that X_{0}(N) is only hyperelliptic when it has to be, meaning that it has genus 2 (which explains X_{0}(37) being hyperelliptic, for example), or a quotient by some Atkin-Lehner involution has genus zero. So one is tempted to speculate that the gonality of X_{0}(N) is governed by only two factors: the gonality of a general curve of genus g, and the presence of Atkin-Lehner involutions. We remark that the general curve of genus g has gonality equal to [\frac{g+3}{2}] (see [4]).

Let us restrict ourselves now to the case when N is prime. It seems plausible that

when X_{0}^{+}(N) has genus at least 3 it will have no automorphisms. (This seems to be an open question.) As a special case of this statement, it is a theorem that X_{0}^{+}(N) is hyperelliptic precisely when it has genus 2. This is proved in [28], where the reader can in fact find a more general statement which applies when N is square-free. Is it possible, more generally, that X_{0}^{+}(N) has gonality equal to that of the general curve of genus g^{+}, namely [\frac{g^{+}+3}{2}]? (The gonality of X_{0}^{+}(N) is, in any case, a linear function of g^{+} by the results of [1]). More generally, is the gonality of X_{0}(N), N prime, simply the minimum of [\frac{g+3}{2}] and 2 · [\frac{g^{+}+3}{2}]?

We now show that the answer to these questions in general is no. More precisely:

Theorem 10. Let N be a prime number. Then X_{0}^{+}(N) is trigonal iff 2 ≤ g^{+} ≤ 4 or g^{+} = 5 and N is either 181 or 227.

Remark. The exact list of primes N satisfying the conditions of the theorem is as follows: g^{+} = 2 ⇔ N ∈ {67, 73, 103, 107, 167, 191} g^{+} = 3 ⇔ N ∈ {97, 109, 113, 127, 139, 149, 151, 179, 239} g^{+} = 4 ⇔ N ∈ {137, 173, 199, 251, 311} To get an upper bound for the set of N such that g^{+}(N) = α, one can use the

Weil bound: if g^{+}(N) = α then X_{0}(N)_{F_{2}} maps over F_{2} to a curve of genus α defined over F_{2}, so X_{0}(N)(F_{4}) can have at most 2(5 + 4α) elements. On the other hand, we know that X_{0}(N)(F_{4}) has at least 2 + \frac{N+1}{12} elements. It follows that N < 96(α + 1).

Proof. The theorem follows from the same arguments as in the proof of Theorem

9. If X_{0}(N) is trigonal and g^{+} ≥ 5, then from the Correspondence Lemma X_{0}^{+}(N) is actually trigonal over ℚ, and reducing modulo 2 and estimating points over F_{4} gives a bound N < 335. Actually counting points on X_{0}(N) over F_{4} and F_{9} shows that in fact N ≤ 277. For the remaining values of N, we decide trigonality by computing generators for the canonical ideal and applying Petri’s Theorem. ■

We remark that the degree three coverings X_{0}^{+}(181) → P^{1} and X_{0}^{+}(227) → P^{1}

are not Galois. This follows from the following lemma:

Lemma 10. The order of any automorphism of X_{0}^{+}(N), N a prime number, is a power of 2.

Proof. The group Aut(X_{0}^{+}(N)) injects into the multiplicative group of the ring

End(J_{0}^{+}(N)). The latter is known to be a subring of ∏ K_{i}, where the K_{i} are totally real number fields – this follows from the fact that T ⊗ ℚ = End^{0}(J_{0}(N)) = End^{0}(J_{0}^{+}(N)) × End^{0}(J_{0}^{−}(N)), where End^{0}(J) means End(J)⊗ℚ, since the simple factors of J are mutually non-isogenous. Therefore Aut(X_{0}^{+}(N)) injects into the group ∏ μ_{i}, where μ_{i} is the group of roots of unity in K_{i}. As K_{i} is totally real, we have μ_{i} = {±1}, which proves the lemma.

■

We also remark that before having computed generators for the canonical ideal of

any of the curves X_{0}^{+}(N), we had already found evidence that X_{0}^{+}(181) and X_{0}^{+}(227) might be trigonal by an entirely different method. Since this method could be useful in analyzing the gonality of X_{0}^{+}(N) in general, we describe it here. The reader should consider the rest of this section as work in progress, and the details are correspondingly sketchy!

For the rest of this section, let N be a prime number, let R denote the ring ℤ_{N},

and let R^{ur} be the ring of integers of the completion of the maximal unramified extension of ℚ_{N} in a fixed algebraic closure \overline{ℚ}_{N}.

Let X_{0}^{+}(N)_{R} be the Deligne-Rapoport model for X_{0}^{+}(N), and let X_{0}^{+}(N)_{F_{N}} be

its special fiber. This special fiber is a copy of the j-line P_{F_{N}}^{1} which intersects itself transeversely at the g^{+} points corresponding to conjugate pairs of supersingular elliptic curves defined over F_{N^{2}} but not F_{N} (see the appendix to [28]). Denote by { \atop X_{0}^{+}(N)} the normalization of X_{0}^{+}(N)_{F_{N}}, which as we just saw is isomorphic to P_{F_{N}}^{1} , and let {e_{i}, e′_{i}} be the conjugate pairs of j-values of supersingular elliptic curves which are identified in the map

{ \atop X_{0}^{+}(N)} → X_{0}^{+}(N)_{F_{N}} , (i = 1, …, g^{+}).

Because the elliptic curves with “extra automorphisms” (i.e., those with j-invariant

0 or 1728) are defined over F_{N}, X_{0}^{+}(N)_{R} should in fact be regular over R. (We have not worked out the details of this.)

Assuming X_{0}^{+}(N)_{R} is regular, we can use Lemma 7 to define the reduction mod

N of a rational function f on X_{0}^{+}(N) defined over R^{ur}.

In particular, if there is a rational function on X_{0}^{+}(N) of degree d defined over R

(resp. R^{ur}), then there is also a rational function on X_{0}^{+}(N)_{F_{N}} of degree d defined over F_{N} (resp. \overline{F}_{N}). Such an object is simply a rational function \overline{f} of degree d on the normalization

{ \atop X_{0}^{+}(N)} such that \overline{f}(e_{i}) = \overline{f}(e′_{i}) for i = 1, …, g^{+}.

With the above discussion as motivation, we specialize now to the following con-

crete problem.

Let F be an algebraically closed field, let d, t be positive integers, and let x_{i}, y_{i} (1 ≤

i ≤ t) be distinct elements of F^{∗} thought of as points in P_{F}^{1}. Suppose h is a rational function of degree d on P_{F}^{1}, and say that h is admissible if h(x_{i}) = h(y_{i}) for 1 ≤ i ≤ t. What can we say about the existence or nonexistence of admissible functions h?

Suppose for example that d = 1. Then h(x_{1}) = h(y_{1}) is then impossible, so no

admissible functions h exist when t ≥ 1.

If d = 2 and t ≥ 2, then without loss of generality we have h(x_{1}) = h(y_{1}) = 0

and h(x_{2}) = h(y_{2}) = ∞. The divisor of h is then (h) = (x_{1}) + (y_{1}) − (x_{2}) − (y_{2}), so h is determined up to scalars. For generic choices of x_{3}, y_{3}, then, we will not have h(x_{3}) = h(y_{3}). In other words, for generic (x_{i}, y_{i}) we expect an admissible h to exists iff t ≤ 2.

Consider now general d.

A rational function h of degree d on P^{1} can be thought of as a ratio

h = \frac{a_{d}z^{d} + a_{d−1}z^{d−1} + ⋯ + a_{1}z + a_{0}}{b_{d}z^{d} + b_{d−1}z^{d−1} + ⋯ + b_{1}z + b_{0}}

of polynomials of degree d. The conditions h(x_{k}) = h(y_{k}) for 1 ≤ k ≤ t then give t equations of the form

L_{k} := ∑_{0≤j<i≤d} c_{i,j,k}(a_{i}b_{j} − a_{j}b_{i}) = 0.

for various c_{i,j,k} ∈ F. If we think of each z_{ij} := a_{i}b_{j} − a_{j}b_{i} as a separate variable, then each L_{k} is a linear equation in r := \binom{d+1}{2} variables. Moreover, there are certain quadratic relations between the z_{ij}, namely (fixing α and β) we have

Q_{ij} := z_{ij}z_{βα} − z_{iα}z_{βj} + z_{jα}z_{βi} = 0

and these give nontrivial independent relations whenever i > j and neither i, j is α or β.

If we think of the z_{ij} as coordinates of a projective space P^{r−1}, then each of the

r − (2d − 1) forms Q_{ij} is a hypersurface in P^{r−1}. For there to be a_{0}, a_{1}, …, a_{d}, b_{0}, …, b_{d} as above, we must have all Q_{ij} = 0 and L_{k} = 0 for k = 1, …, t. If the points x_{i}, y_{i} are chosen generically, then the dimension of this intersection of hyperplanes and quadric hypersurfaces in P^{r−1} will be (r − 1) − (r − 2d + 1) − t = (2d − 2) − t. So for generic x_{i}, y_{i}, one expects this intersection to be empty when t ≥ 2d − 1. The converse is true as well; in other words, for generic values of x_{i}, y_{i}, admissible functions of degree d exist iff t ≤ (2d − 2).

For a rigorous treatment of the calculation just sketched, and an exposition of how

all this relates to Grassmannians, Schubert calculus, and Brill-Noether theory, among other things, see [22] and [25, Sec. 5A].

For example, if we let {e_{i}, e′_{i}} be the g^{+} conjugate pairs of supersingular j-values

not in F_{N}, then putting x_{i} = e_{i} and y_{i} = e′_{i} we expect (assuming the e_{i}, e′_{i} are in sufficiently general position on the j-line) to find an admissible h of degree d iff g^{+} ≤ (2d − 2). This is equivalent to saying that the smallest d such that there exists an admissible function of degree d is [\frac{g^{+}+3}{2}], which is exactly what we might hope it would be.

However, the assumption that the e_{i}, e′_{i} are in “general position” is not always

satisfied.

Here is a concrete example with d = 3. An admissible function of degree 3 exists on

the reduction of X_{0}^{+}(N) if and only if the closed subvariety of P^{5} defined by the equations

Q_{21} = z_{21}z_{30} − z_{20}z_{31} + z_{10}z_{32}

and

L_{k} = ∑_{0≤j<i≤3}(x_{k}^{i}y_{k}^{j} − x_{k}^{j}y_{k}^{i})z_{ij} = 0

for k = 1, … g^{+} is nonempty.

We carried out this computation for a few values of N such that g^{+} = 5. When

N = 157 it turns out that there is no admissible function of degree 3, and this corroborates the fact that X_{0}^{+}(157) is not trigonal. When N = 181, we were surprised to find that there is an admissible function of degree 3, namely

h(z) = \frac{105z^{3} + 51z^{2} + z}{93z^{2} + 134z + 1} .

As we mentioned, it turned out that in fact X_{0}^{+}(181) is trigonal. Similarly, the

linear algebra described above shows that the reduction of the trigonal curve X_{0}^{+}(227) has an admissible function of degree 3.

In addition to this technique of reducing mod N and thinking about rational

functions on the reduced scheme, one could also try to study morphisms of degree d > 3 from X_{0}(N) and X_{0}^{+}(N) to P^{1} by replacing the systematic use of Petri’s theorem by an application of Green’s conjectures [21]. These conjectures say roughly that one should be able to see the Clifford index of a curve (and so gather some information about its gonality) by looking at higher syzygies of the canonical ideal.

Chapter 4

Cartier Points on Curves

4.1 Introduction

Throughout this chapter Y will denote a complete, nonsingular, and irreducible

algebraic curve of genus g > 0 over the algebraically closed field k of characteristic p. Eventually we will assume that g ≥ 2. Denote by W the g-dimensional k-vector space of regular differentials of Y/k,

which is just the vector space H^{0}(Y, Ω_{Y/k}^{1}). Also let K(Y) denote the function field of Y, and let Ω_{K(Y)/k}^{1} be the k-vector space of meromorphic differentials on Y.

Recall (see, for example, [34][A2]) that there is a map C from Ω_{K(Y)/k}^{1} to itself

called the Cartier operator. The existence of this map only uses the fact that k is perfect. It preserves the space W of regular differentials. The Cartier operator is additive and also satisfies the identity

C(z^{p}ω) = z · C(ω)

for all z ∈ K(Y) and ω ∈ Ω_{K(Y)/k}^{1}.

In particular, the action of C on the k-vector space W is σ^{−1}-linear, meaning that

C(σ(α)ω) = α · C(ω), where α ∈ k and σ is the Frobenius (p-th power) map on k.

The action of the Cartier operator on W is dual to the action of the Frobenius

map on H^{1}(Y, O_{Y}). So the action of C on W is intimately related to the structure of the p-torsion on the Jacobian of Y/k. A curve is called ordinary if the action of C on W is an isomorphism, and superspecial if C is the zero operator on W. The following is a nontrivial fact about superspecial curves (see [40]):

Theorem 2. Y is superspecial if and only if the Jacobian of Y is isomorphic to a product of supersingular elliptic curves.

In section 4.2, we will give a new proof (and generalization) of the following theo-

rem of Ekedahl (see [18]):

Theorem 3. Let Y be a curve of genus g over an algebraically closed field k of characteristic p whose Jacobian is isomorphic to a product of supersingular elliptic curves. Then g ≤ \frac{p(p−1)}{2}. If Y is hyperelliptic then g ≤ \frac{(p−1)}{2}.

Remark. Ekedahl shows in [18], considering Fermat curves of degree p + 1 and hyperelliptic curves of the form y^{2} = x^{p} − x that these bounds are in fact sharp.

Our proof of Ekedahl’s theorem is based on the hypothesis that the Cartier oper-

ator annihilates W, and ultimately relies on general facts about linear systems on curves. Ekedahl’s proof, by contrast, has as its direct input the fact that Jac(Y) is isomorphic to a product of supersingular elliptic curves. He shows that this implies that Y descends down to F_{p^{2}} with Frobenius acting on H_{et}^{1}(Y_{F_{p^{2}}} , ℤ_{l}) as multiplication by ±p, which cannot happen (by an observation of Serre) if g > \frac{p(p−1)}{2}.

Now we come to the main object of study of this chapter.

Definition. A closed point P of Y is said to be a Cartier point if the hyperplane W (P) of regular differentials vanishing at P is stable under the Cartier operator. Example 2. On an elliptic curve every point is a Cartier point, since W (P) is always zero. On a superspecial curve as well, every point is a Cartier point, since any subspace of a vector space is stable under the zero operator!

Notice that the Cartier operator C induces a σ-linear map C^{∗} on the dual space

W^{∗} via the formula

(C^{∗}λ, ω) = (λ, Cω)^{σ}.

There is a natural identification of C^{∗} with the p-linear Frobenius map acting on W^{∗} ≅ H^{1}(Y, O_{Y}) (see [52]).

If Y is ordinary then there is also an induced map P(C) on the projective space

P(W), which is the set of hyperplanes in W.

For a fixed choice of a differential ν ∈ W, let λ_{P} be the element of W^{∗} sending

ω ↦ \frac{ω}{ν} (P). The image j(P) in P(W) of λ_{P} depends only on P, not on the differential ν.

The map j : Y → P(W) is just the canonical map of Y to projective space. The following lemma can be proved by unwinding definitons (or see the proof of Prop. 15).

Lemma 11. Suppose Y is ordinary. A point P ∈ Y (k) is a Cartier point if and only if P(C)(j(P)) = j(P).

In other words, when Y is ordinary a Cartier point can be interpreted geometrically

as a point on Y whose image via the canonical embedding is a fixed point of the action of Frobenius on the ambient projective space.

Our original motivation for our study of Cartier points was the following theorem

of Coleman:

Theorem 4. Suppose X is a (smooth proper) curve of genus g ≥ 2 defined over the ring of integers R in an unramified finite extension K of ℚ_{p}, p ≥ 3. Let X̃ denote the special fiber of X, and suppose that X̃ is ordinary. If P ∈ X(K̄) is a torsion point with respect to some embedding of X in Jac(X) defined over K, and if P is ramified (i.e., the extension K(P)/K is ramified), then the reduction P̃ of P is a Cartier point on X̃.

Proof. Using Lemma 11, we see that this theorem is proved in parts (ii) and

(iii) of [12, Prop. 3.6]. ■

In fact, Coleman also proves in [10] that under the hypoetheses of the above

theorem, there can be no torsion points at all ramified at p if p ≥ 5. [He also reaches this conclusion for p > 2g without the hypothesis that X̃ is ordinary.] So really we have found Theorem 4 useful only when p = 3. Nonetheless, this is an interesting case. For instance, in section 4.4 and chapter 5 we apply Theorem 4 to some concrete examples of the Coleman-Kaskel-Ribet conjecture.

As another application of Theorem 4, consider the following theorem of Buium

(see [7]):

Theorem 5. Let X → J be an Albanese embedding defined over a number field K of a smooth proper curve of genus g ≥ 2 into its Jacobian. Let 𝔭 be a prime of K such that K/ℚ is unramified at 𝔭, X/K has good reduction at 𝔭, and no point of X(K̄) ramified at 𝔭 maps to a torsion point of J. Suppose p = char(𝔭) is odd. Then

#(X(K̄) ∩ J(K̄)_{tors}) ≤ p^{4g} · 3^{g} · [p(2g − 2) + 6g] · g!.

If p = 2, then the same formula holds with p^{4g} replaced by 64^{g}.

Buium applies Coleman’s results to conclude that p in the above theorem can be

taken to be any prime bigger than 2g, or any ordinary prime at least 5. If X is ordinary at some prime lying over 3, then one can take p equal to 3 in Buium’s bound if one can show that the reduction X̃ of X has no Cartier points. (Even if there are Cartier points on X̃, one can sometimes successfully show that they do not lift to torsion points on X. See chapter 5 for examples of this.)

We also mention that Cartier points arise naturally when studying the moduli

space of hyperelliptic curves in characteristic p. This is illustrated in the work of B. Chisala (see [8]).

4.2 Duality, Linear Systems, and Ekedahl’s Theorem

In preperation for our extension of Ekedahl’s theorem, we review some of the

duality theory for curves and its relation to the Cartier operator in characteristic p. A good reference for this is [51].

Let k be a field and C/k a (complete nonsingular) algebraic curve with function

field K(C). Assume the genus g of C is positive. A repartition ξ is a family {ξ_{P} }_{P ∈C} of elements of K(C) such that ξ_{P} ∈ O_{P} for almost all P ∈ C. Let R be the k-algebra of repartitions of K(C). K(C) is embedded diagonally in R, allowing one to view K(C) as a subring of R.

If D is a divisor, write R(D) for the vector subspace of R consisting of all ξ = {ξ_{P} }

such that v_{P} (ξ_{P}) ≥ −v_{P} (D). We set L(D) equal to the vector space consisting of 0 and all rational functions f such that (f) ≥ −D. We have L(D) = H^{0}(C, L(D)), where L(D) is the subsheaf of the constant sheaf K(C) whose stalk L(D)_{P} at a point P ∈ C(k̄) is the set of rational functions f such that v_{P} (F) ≥ −v_{P} (D).

For any divisor D, I(D) = H^{1}(C, L(D)) is canonically isomorphic to R/(R(D) +

K(C)).

Let Ω_{C/k}^{1} be the sheaf of differentials of C, and let Ω_{K(C)/k}^{1} be the k-vector space

of meromorphic differentials on C. There is a pairing between elements ω of Ω_{K(C)/k}^{1} and repartitions ξ ∈ R via

⟨ω, ξ⟩ = ∑_{P ∈C} Res_{P} (ξ_{P} ω).

Let Ω(D) be the k-vector space formed by 0 and the elements ω ∈ Ω_{K(C)/k}^{1} such that (ω) ≥ D. Then by [51, II.8 Thm. 2], for every divisor D the scalar product ⟨ω, ξ⟩ puts the vector spaces Ω(D) and I(D) in duality.

Now suppose that C = Y is a curve in characteristic p > 0. Then the Cartier

operator C on the meromorphic differentials of Y satisfies

⟨Cω, ξ⟩^{p} = ⟨ω, ξ^{p}⟩.

Using this duality theory, we have the following proposition giving an alternate

characterization of Cartier points.

Proposition 10. A point P ∈ Y (k) is a Cartier point if and only if there exists a rational function f on Y and a local parameter T at P such that f is regular outside P, and at P has the Laurent expansion

f = T^{−p} + cT^{−1} + holomorphic terms,

where c ∈ k.

Proof. By the duality between differentials and repartitions, we see that C

preserves W (P) ⊂ W if and only if the p-th power map on R preserves the subspace R(P)+K(Y). Since R(0)+K(Y) has codimension one in R(P)+K(Y), and R(0)+K(Y) is stable under p-th powers, this occurs if and only if the repartition (0, 0, …, (T^{−1})^{p}, 0, 0, …) is in R(P)+K(Y), where the nonzero component corresponds to the place P. This is easily seen to be equivalent to the existence of an f as in the statement of the proposition. ■

This proposition shows that every Cartier point P on Y, together with a choice

of a local parameter T at P, gives rise to degree p rational function f_{P,T} on Y. If f′_{P,T} is another degree p rational function regular outside P and with Laurent expansion T^{−p} + c′T^{−1} + holomorphic terms, then f′_{P,T} = f_{P,T} + a for some constant a ∈ k by the residue theorem. So we get a uniquely determined base-point free g_{p}^{1} on Y, depending only on P and T. It is interesting to consider when different Cartier points can give rise to equivalent g_{p}^{1}’s.

Lemma 12. With notation as above, suppose P, Q are distinct Cartier points on Y and choose local parameters T and T′ at P and Q, respectively. If the corresponding g_{p}^{1}’s are equivalent, then [(P) − (Q)] is a p-torsion point of Jac(Y).

Proof. We can normalize f_{P,T} so that it vanishes at Q. The hypothesis that the

g_{p}^{1}’s are equivalent means that there is an automorphism α of P^{1} such that f_{Q,T′} = α ∘f_{P,T}. It follows that α maps 0 to ∞, and that the inverse image of 0 under f_{P,T} consists of the point Q with multiplicity p. Hence the divisor of f_{P,T} is p(Q) − p(P), which proves what we want. ■

We will use this in conjuction with a result concerning base-point free pencils on

curves. We will say that two base-point free g_{d}^{1}’s on C are independent if the corresponding map from C to its image C′ in P^{1} × P^{1} has degree one. Thinking about the geometry of P^{1} × P^{1}, we see that a g_{e}^{1} and g_{f}^{1} are independent iff there exists a Zariski-dense open set U ⊆ C such that for all P ∈ C, there are divisors E ∈ g_{e}^{1} and F ∈ g_{f}^{1} with E · F = {P }. Indeed, if the g_{e}^{1} and g_{f}^{1} are dependent, then E · F will always have degree at least 2. On the other hand, if the g_{e}^{1} and g_{f}^{1} are independent, we can take U to be the inverse image of the complement of the singular locus on C′.

The proof of the following result is very similar to results found in [15] and [3].

Unlike [15, Prop. 3], we do not assume the g_{d}^{1}’s in question are complete, nor do we assume that the ground field has characteristic zero. We include a slight variant of the proof in [15] here to convince the reader that these extra assumptions are not necessary.

Proposition 11. Let C be a (smooth) curve of genus g defined over an algebraically closed field k. If C has at least d mutually independent base-point free g_{d}^{1}’s, then g ≤ d(d − 1)/2.

Remark. This result is sharp, since a plane curve of degree d + 2 with d − 1 nodes has d − 1 independent g_{d}^{1}’s and genus equal to d(d − 1)/2 − 1. Also, the independence hypothesis is necessary, since there exist double covers of elliptic curves of arbitrary genus and these have infinitely many g_{4}^{1}’s.

Before giving the proof, we need two lemmas. Notation will be as follows: if

A ⊂ P^{r}, then ⟨A⟩ will denote the linear span of A. If ϕ : C → P^{r} is a morphism and D is an effective divisor on C, then ⟨ϕ(D)⟩ is the intersection of all hyperplanes H in P^{r} such that either ϕ(C) ⊂ H or ϕ^{∗}(H) ≥ D. If ϕ is an embedding we often write simply ⟨D⟩ for ⟨ϕ(D)⟩.

The geometric Riemann-Roch theorem ([4, p. 12]) says that for any effective

divisor D of degree d on C,

l(D) = 1 + dim(|D|) = d − dim(⟨j(D)⟩)

where j is the canonical map to P^{g−1}.

Lemma 13. Let C be a non-hyperelliptic smooth curve of genus g and let g_{d}^{1} be a special base-point free linear system on C. Let g_{e}^{1} and g_{f}^{1} be independent base-point free linear systems on C. If dim(|g_{d}^{1}|)=b, dim(|g_{d}^{1} + g_{e}^{1}|)=a_{1}, and dim(|g_{d}^{1} + g_{f}^{1}|)=a_{2}, then dim(|g_{d}^{1} + g_{e}^{1} + g_{f}^{1}|) ≥ a_{1} + a_{2} + 1 − b. Also, if r denotes the dimension of |g_{e}^{1}| and s denotes the dimension of |g_{f}^{1}|, then dim(|g_{e}^{1} + g_{f}^{1}|) ≥ r + s + 1.

Proof. Assume that C is canonically embedded. For any x ∈ C, denote by

E_{x}, F_{x} the unique divisors in g_{e}^{1}, g_{f}^{1}, respectively, which contain x. As we saw previously, the assumption that g_{e}^{1} and g_{f}^{1} are independent is equivalent to the assertion that for almost all x ∈ C we have E_{x} · F_{x} = {x}. Let D be a divisor in the linear system g_{d}^{1}. The fact that D is special means that ⟨D⟩ is not all of P^{g−1}. So for a point x ∈ C to lie outside ⟨D⟩ is a nonempty Zariski-open condition. Also the condition that x ∈ C is such that E_{x} · D = F_{x} · D = ∅ is non-empty and open. We conclude that there exists a point x_{0} ∈ C and divisors E, F in g_{e}^{1}, g_{f}^{1}, respectively, such that E · D = F · D = ∅, E · F = x_{0}, and x_{0} ∉ ⟨D⟩.

Then we must have ⟨D + E + F − x_{0}⟩ = ⟨⟨D + E⟩ ∪ ⟨F ⟩⟩ since D + E and F − x_{0}

are disjoint, and so

dim(⟨D + E + F − x_{0}⟩) = dim(⟨D + E⟩) + dim(⟨F ⟩) − dim(⟨D + E⟩ ∩ ⟨F ⟩).

By the geometric Riemann-Roch theorem, we have

dim(⟨D⟩) = d − b − 1,
dim(⟨D + E⟩) = d + e − a_{1} − 1,
dim(⟨D + F ⟩) = d + f − a_{2} − 1,
dim(⟨F ⟩) = f − 1 − s.

So we have dim(⟨D⟩ ∩ ⟨F ⟩) = a_{2} − b − 1 − s. And as

⟨F ⟩ ∩ ⟨D + E⟩ ⊃ ⟨(⟨D⟩ ∩ ⟨F ⟩) ∪ {x_{0}}⟩,

and x_{0} ∉ ⟨D⟩, we have dim(⟨F ⟩ ∩ ⟨D + E⟩) ≥ a_{2} − b − s. Therefore

dim(⟨D + E + F − x_{0}⟩) ≤ d + e + f − a_{1} − a_{2} + b − 2.

So by geometric Riemann-Roch, we get

dim(|D + E + F − x_{0}|) ≥ a_{1} + a_{2} − b.

Since x_{0} is not a base point of |D + E + F |, we see that

dim(|D + E + F |) ≥ a_{1} + a_{2} − b + 1.

as desired.

Finally, we have

dim(⟨E⟩) = e − 1 − r

and

dim(⟨F − x_{0}⟩) = f − 2 − (s − 1),

so

dim(⟨E + F − x_{0}⟩) = dim(⟨E⟩) + dim(⟨F − x_{0}⟩) − dim(⟨E⟩ ∩ ⟨F − x_{0}⟩)

≤ e + f − r − s − 2.

Therefore by geometric Riemann-Roch,

dim(|E + F − x_{0}|) ≥ e + f − 2 − (e + f − r − s − 2) = r + s.

Since x_{0} is not a base point for |E + F |, it follows that dim(|E + F |) ≥ r + s + 1

as claimed. ■

Lemma 14. Let C/k be hyperelliptic. Then any special g_{d}^{1} factors through the (unique) g_{2}^{1}. In other words, the map f : C → P^{1} coming from the g_{d}^{1} factors as f = g ∘ h, where h : C → P^{1} has degree 2.

Proof. It suffices to show that f(P) = f(Q) whenever P + Q ∈ g_{2}^{1}, since then f

factors through the quotient of C by the hyperelliptic involution w. So if D is an effective divisor in g_{d}^{1}, we have to show for all P ∈ C that Q = w(P) is a base point of |D − P |. This is equivalent to the statement that l(D − P − Q) = l(D − P), which by Riemann-Roch is equivalent to l(E + P) − l(E + P + Q) = 1, where E is a residual effective divisor to D, i.e., D + E = K. The desired equality follows immediately from the geometric Riemann-Roch theorem. ■

Corollary 5. Suppose C is a hyperelliptic curve of genus g defined over an algebraically closed field k. Then any rational function of degree at most g on C must have even degree.

We now give the promised proof of Proposition 11.

Proof. Assume g > \frac{d(d−1)}{2}. and d ≥ 3. Without loss of generality we may assume

that d ≥ 3, because for d = 1 the Proposition is trivial and for d = 2 it is equivalent to the fact that the g_{2}^{1} on a hyperelliptic curve of genus at least 2 is unique, which is proved in [27, Prop. 5.3]. Let D_{1}, … D_{d} be effective divisors on C corresponding to the d independent g_{d}^{1}’s. As d ≥ 3, it follows that g ≥ d, and in particular each D_{i} is special by Riemann-Roch. Thus C is not hyperelliptic, since on a hyperelliptic curve no two special g_{d}^{1}’s can be independent by Lemma 14.

Using Lemma 13 and induction on k, we see that if k ≥ 3 and if |D_{1} + … + D_{k−2}|

is special then

dim(|D_{1} + … + D_{k}|) ≥ \binom{k + 1}{2}.

By the induction hypothesis, we have

dim(|D_{1} + … + D_{k−2}|) ≥ \binom{k − 1}{2}

and so |D_{1} + … D_{k−2}| is special if g > k(k − 2) − \binom{k−1}{2} = \frac{(k+1)(k−2)}{2}. Therefore we find that

dim(|D_{1} + … + D_{d}|) ≥ \frac{d(d + 1)}{2}.

Clifford’s theorem implies that |D_{1} + … + D_{d}| is nonspecial, so by Riemann-Roch

we find that g ≤ \frac{d(d−1)}{2}, a contradiction. ■

Using Cor. 5, we deduce the following proposition, which generalizes results of

Coleman (see [9, 5.5]).

Proposition 12. Let Y be an algebraic curve of genus g ≥ 2 over an algebraically closed field k of characteristic p.

1. If p = 2, then P ∈ Y (k) is a Cartier point if and only if Y is hyperelliptic and P is a hyperelliptic branch point.

2. If Y/k is hyperelliptic and p ≠ 2, then:

(a) If p ≤ g, then there are no Cartier points on Y.

(b) If p < 2g, the hyperelliptic branch points on Y are not Cartier points.

Proof. The first part of the proposition follows immediately from Prop. 10. Now

consider the second part. If P is a Cartier point on Y then there exists a rational function f on Y regular outside P and with a pole of order p at P. As p is odd, the statement in part (a) follows from Cor. 5, and the statement in part (b) follows from the fact that the Weierstrass gap sequence at a hyperelliptic branch point P is 1, 3, …, 2g − 1, so in particular there is no rational function regular outside P and with a pole of order p < 2g at P. ■

We now state our generalization of Ekedahl’s results.

Theorem 6. Let Y be a curve of genus g over an algebraically closed field of characteristic p.

1. If Y has at least p distinct Cartier points, no two of which differ by a p-torsion point on Jac(Y), then g ≤ \frac{p(p−1)}{2}.

2. If Y is hyperelliptic of genus g, p is odd, and some hyperelliptic branch point of Y is a Cartier point, then g ≤ \frac{p−1}{2}.

Proof. The first assertion follows from Prop. 12 and Prop. 11 after noting that

any two nonequivalent g_{d}^{1}’s are independant if d is a prime number. The second follows immediately from Prop. 12(b). ■

Corollary 6 (Ekedahl). Let Y be a superspecial curve of genus g over an algebraically closed field of characteristic p. Then g ≤ \frac{p(p−1)}{2}. If Y is hyperelliptic then g ≤ \frac{p−1}{2} or p = 2 and g = 1.

Proof. If Y is superspecial then C = 0 on W, so every point on Y is a Cartier

point. Also the Jacobian of Y has no p-torsion, so each Cartier point gives rise to a distinct g_{p}^{1} on Y by Lemma 12. The conclusions of the corollary now follow from the proposition.

■

Remark. If the results of [16] are valid in arbitrary characteristic (they should be, but we have not checked this), then it follows from our proof that if Y is superspecial and g = \frac{p(p−1)}{2}, then Y is a nonsingular plane curve of degree p + 1. In [58] it is proved that if Y is superspecial and hyperelliptic with g = \frac{p−1}{2}, then Y is defined by an equation y^{2} = x^{p}−x.

4.3 Bounds

We keep the same general notations as before. In particular Y will denote a curve

of genus at least 2 over the algebraically closed field k of characteristic p, though we may repeat this information for emphasis.

It is clear from Prop. 15 that an ordinary curve can only possess finitely many

Cartier points. On the other hand, every point on a superspecial curve is a Cartier point. What is the general picture? In this section we show that a curve has infinitely many Cartier points if and only if it is superspecial. We also give some bounds for the number of Cartier points a non-superspecial curve can possess.

Before we begin, we need a few facts from σ-linear algebra which can be found in

[52, Sec. 9] and [5].

Let k be an algebraically closed field of characteristic p and let σ be the Frobenius

endomorphism of k. Let T be a σ or σ^{−1}-linear operator on the k-vector space V of dimension g. Then the kernel of T is a linear subspace of V, and the rank s of T is well-defined as g minus the dimension of the kernel of T. The vector space V has a unique direct-sum decomposition V = V_{1}⊕V_{2}, where the action of T is bijective on V_{1} and nilpotent on V_{2}. The dimension r of V_{1} is called the Hasse-Witt invariant of T. Furthermore, there exists a basis {v_{1}, … v_{r}} of V_{1} such that T v_{i} = v_{i} for i = 1, 2, …, r. The solutions x ∈ V to T x = x are exactly the p^{r} elements of the F_{p}-vector space generated by v_{1}, …, v_{r}. The induced operator T^{∗} on V^{∗} has the same invariants r and s as T.

By convention, we say that the Hasse-Witt invariant of the curve Y is the Hasse-

Witt invariant of the Cartier operator acting on W := H^{0}(Y, Ω_{Y/k}^{1}), or equivalently of the Frobenius operator acting on W^{∗} = H^{1}(Y, O_{Y}).

We make the following definition.

Definition. Suppose Y is ordinary. A set of canonical coordinates for Y is a choice of a basis for H^{1}(Y, O_{Y}) consisting of fixed points of C^{∗}. Equivalently, we say that the map from Y to P^{g−1} given by sending P to [ω_{1}(P) : ω_{2}(P) : ⋯: ω_{g}(P)] is given in canonical coordinates if ω_{1}, …, ω_{g} is a basis for W consisting of differentials fixed by C (whose existence is guaranteed by our facts from σ-linear algebra).

The following fact follows immediately from Lemma 11.

Lemma 15. Assume Y is ordinary. A point P on Y is a Cartier point if and only if it maps to an F_{p}-rational point with respect to some canonical coordinates for Y.

If Y is a curve of genus g ≥ 2, we will define δ_{Y} to be the degree of the canonical

map Y → P^{g−1}, namely 2 if Y is hyperelliptic and 1 otherwise.

Proposition 13. Suppose Y/k is ordinary. Then the number of Cartier points on Y is bounded above by

δ_{Y} · Max(p + 1 + 2g\sqrt{p}, (\frac{2g − 2}{δ_{Y}})^{2}).

Proof. (See Prop. 5.1 of [9]). Map Y to P^{g−1} using canonical coordinates and

call the image curve E, so that the Cartier points on Y are just those points of Y which map to F_{p}-rational points of E by Lemma 15. Now if E is defined over F_{p}, then the Riemann hypothesis gives a bound of p+1+2g\sqrt{p} for the number of rational points on E. Otherwise, E^{σ} ≠ E, where σ is the Frobenius map on P^{g−1}. In this case, by projecting E down to a suitable two-dimensional plane we can apply Bezout’s theorem to estimate the number of points of E fixed by the Frobenius of P^{g−1}. The degree of both E and E^{σ} is \frac{2g−2}{δ_{Y}}, so the cardinality of E ∩ E^{σ} is bounded by (\frac{2g−2}{δ_{Y}})^{2}. ■

Remark. We do not know of any examples of E as above which are defined over the prime field F_{p}. It would be interesting to find such a curve or to prove that none exists. To get a flavor for this remark, see example 5 in the next section.

When p is small relative to g, which is the case in our applications to torsion points

on curves (where p = 3), the following bound is better.

Proposition 14. Suppose Y/k is ordinary. Then there are at most (g−2)+(p+1)g Cartier points on Y.

Proof. Let F := F_{p}. Y ordinary implies that W^{C} ⊗_{F} k ≅ W, where W^{C} denotes

the space of logarithmic differentials, i.e., the elements of W := H^{0}(Y, Ω_{Y/k}^{1}) fixed by the Cartier operator. We have that P is a Cartier point iff W (P)^{C} ⊗_{F} k ≅ W (P).

If there are fewer than g − 2 Cartier points on Y then we are done. Otherwise,

let Q_{1}, …, Q_{g−2} be distinct Cartier points. The k-vector space of differentials vanishing at each of the Q_{i} has dimension at least 2 by Riemann-Roch, and has a basis {ω_{i}} of logarithmic differentials.

Pick any two such logarithmic ω_{i}; they generate a 2-dimensional F-vector space

V. Let {v_{1}, …, v_{p+1}} be elements of V representing distinct elements of P^{1}(V).

Now let P be a Cartier point different from Q_{1}, …, Q_{g−2}. As dim_{F}W (P)^{C} +

dim_{F}V = g + 1, we have W (P)^{C} ∩ V ≠ (0), so some nonzero element of V is in W (P). But then one of v_{1}, …, v_{p+1} ∈ W (P). So

P ∈ ⋃_{i=1}^{p+1}{zeros of v_{i}}.

Now each of the v_{i} can vanish at most at g points different from the Q_{i}. So we

find that a Cartier point on Y must be one of the g − 2 points Q_{1}, …, Q_{g−2}, or else one of at most (p + 1)g other points where some v_{i} has a zero. This gives the desired bound. ■

A curve Y is called extraordinary if it is neither ordinary nor superspecial (see

[10]).

Proposition 15. Let Y be an extraordinary curve with Hasse-Witt invariant r. Then the number of Cartier points on Y is bounded by

Min(4g − 4, 2g − 2 + δ_{Y} · \frac{p^{r} − 1}{p − 1}).

Furthermore, let s be the rank of C acting on W. If s = 1 then there is at least one Cartier point on Y.

Proof. Given a hyperplane H in W, let f_{H} ∈ W^{∗} denote any nonzero element of

the 1-dimensional subspace of W^{∗} which annihilates H under the natural pairing. Also let f̄_{H} denote the image of f_{H} in P(W). It follows from the definition of C^{∗} that H is stable under C if and only if C^{∗}f_{H} = c · f_{H} for some c ∈ k.

If c ≠ 0 then the solutions of C^{∗}y = cy correspond bijectively to the solutions

of C^{∗}x = x by setting x = c^{\frac{−1}{p−1}} y for some fixed (p − 1)-st root of c. So the set S of hyperplanes H ⊂ W such that C^{∗}f_{H} = c · f_{H} for some c ≠ 0 corresponds bijectively to the \frac{p^{r}−1}{p−1} values of f̄_{H} such that C^{∗}f_{H} = f_{H}, where r is the Hasse-Witt invariant for C^{∗}. Now pick a basis ω_{1}, ω_{2}, …, ω_{g} for W such that ω_{1}, …, ω_{r} are fixed by the Cartier operator, and map Y to its image E in P^{g−1} using these coordinates. Then the points of S which lie on E are just the points of E with coordinates (x_{1} : ⋯: x_{g}) such that x_{1}, …, x_{r} ∈ F_{p} and x_{r+1} = ⋯ = x_{g} = 0. In particular, since r < g, the intersection S ∩ E lies in the intersection of E with the hyperplane x_{g} = 0. Since the degree of E in P^{g−1} is \frac{2g−2}{δ_{Y}}, we have #(S ∩ E) ≤ Min(\frac{p^{r}−1}{p−1}, \frac{2g−2}{δ_{Y}}).

On the other hand, the set of H such that C^{∗}f_{H} = 0 is exactly the set of H such

that f_{H} ∈ K, where K is the kernel of C^{∗}, a vector space of dimension g − s. This set forms a projective space L inside P(W) of dimension g − s − 1. Note that the intersection L ∩ E is empty, since every point in L has x_{1} = ⋯ x_{r} = 0 (notation as above). By hypothesis the dimension of L is between 0 and g − 2, inclusive. Since the image E of Y inside P(W) does not lie in any hyperplane, the intersection L ∩ E is finite, and its cardinality is bounded by the degree of E, which is \frac{2g−2}{δ_{Y}}.

We are now done, because the Cartier points on Y are exactly those points which

map to points in E ∩ (S ∪ L). If s = 1 then L is a hyperplane in P(W) and therefore intersects E nontrivially. ■

Corollary 7. The set of Cartier points on Y is Zariski-closed, and is infinite if and only if Y is superspecial. Remark. We mention an application to modular forms. If we fix p it follows from the above proposition that the rank s of the Hecke operator T_{p} mod p on weight two cusp forms for Γ_{0}(N) is at least two for all prime numbers N sufficiently large with respect to p. For T_{p} mod p is the Cartier operator on H^{0}(X_{0}(N), Ω_{X_{0}(N)/F̄_{p}}^{1}) for p ≠ N by the Eichler-Shimura relation. So if s ≤ 1 then there exist Cartier points on X_{0}(N)F̄_{p}, hence X_{0}(N)F̄_{p} admits a degree p morphism to P^{1}. But one can show using the techniques of Chapter 3 that this is not the case for N >> p.

4.4 Examples

In this section we provide some examples of Cartier points on curves, along with

some techniques for calculating them. We already know from Proposition 12 that if p ≠ 2 and p ≤ g, there are no Cartier points on a hyperelliptic curve, and that when p = 2 the Cartier points are just the hyperelliptic branch points. By explicit calculations one can determine exactly what the Cartier points on a given hyperelliptic curve are. Example 3. Cartier points on ordinary hyperelliptic curves.

Let Y be an ordinary hyperelliptic curve over the (algebraically closed) field k of

characteristic p ≠ 2. Suppose f is given by the equation y^{2} = f(x), with f of degree 2g + 2 and having distinct roots. Write f^{\frac{p−1}{2}} = ∑_{i=0}^{p−1} g_{i}^{p}x^{p−1−i}, where the g_{i} are elements of k[x]. Let ω = \frac{dx}{2y}, so that \{ω, xω, …, x^{g−1}ω\} is a basis for the regular differentials on Y. It is wellknown that C(x^{pk+i}ω) = x^{k}g_{i}ω for 0 ≤ i ≤ p − 1 (see [60]). Since V = {ω, xω, …, x^{g−2}ω} is a basis for the differentials which vanish at ∞ (and the same for ∞′), the two points at infinity are Cartier points if and only if V is stable under C if and only if deg(x^{k}g_{i}) ≤ g − 2 whenever 0 ≤ pk + i ≤ g − 2. For example, when p = 3 and g = 2, the points at infinity are Cartier points if and only if g_{0} is a constant, i.e. deg(f′(x)) ≤ 4.

Now let P = (x(P), y(P)) be a point on Y with x(P), y(P) ∈ k. Let h_{P} ∈ W^{∗} be

the function given by the rule h_{P} (ν) = \frac{ν}{ω} (P). Then

C^{∗}(h_{P})(x^{pk+i}ω) = h_{P} (C(x^{pk+i}ω))^{p} = (h_{P} (x^{k}g_{i}ω))^{p} = (x^{k}g_{i})(P)^{p}

for 0 ≤ pk + i ≤ g − 1. Using Lemma 11, we see that P is a Cartier point if and only if there exists a constant c ∈ k^{∗} such that x(P)^{pk+i} = c · x(P)^{pk}g^{i}(P)^{p} for 0 ≤ pk + i ≤ g − 1. This happens if and only if there exists c ∈ k^{∗} such that x(P)^{i} = c · g_{i}(P)^{p} for all i = 1, 2, …, m := min(p − 1, g − 1) if and only if g_{0}(P) ≠ 0 and the polynomials h_{i} := xg_{i}^{p} − g_{i+1}^{p} vanish at P for i = 0, 1, …, m − 1. Remark. One can adapt these techniques to the non-ordinary case. One can also use the calculations given in this example to give a computational proof of part 2 of Prop. 12.

As a special case of our computations, we get the following proposition, which

slightly generalizes [9, Theorem 5.5] and is relevant to the study of torsion points on genus 2 curves.

Proposition 16. Let Y be an ordinary genus 2 curve over a field k of characteristic 3 given by the equation y^{2} = f(x), with f of degree 6 and having distinct roots. Then the Cartier points on Y are the points whose x-coordinates are roots of f′(x), plus the two points at infinity when deg(f′(x)) ≤ 4.

Proof. The only things to note in addition to what we have already done are

that f′(x) = g_{1}^{3} − xg_{0}^{3} and that g_{0} and g_{1} cannot simultaneously vanish at any point when Y is ordinary. ■

We now give some examples of Cartier points on nonhyperelliptic ordinary curves.

We need the following lemma on plane curves.

Lemma 16. Let Y/k be a plane curve given by the degree d homogeneous polynomial f. Then the following diagram commutes:

H^{1}(Y, O_{Y})  ⟶^{a↦a^{p}}  H^{1}(Y, O_{Y}^{p})  ⟶^{i}  H^{1}(Y, O_{Y})
·f↑  ↑^{·f}
H^{2}(P^{2}, O_{P^{2}}(−d)) ⟶^{g↦g^{p}} H^{2}(P^{2}, O_{P^{2}}(−dp)) ⟶^{·f^{p−1}} H^{2}(P^{2}, O_{P^{2}}(−d))

Proof. From the exact sequence of sheaves

0 → I_{Y} → O_{P^{2}} → O_{Y} → 0

we see that

H^{1}(Y, O_{Y})→^{∼}H^{2}(P^{2}, I_{Y}).

As O_{P^{2}}(−d)→^{∼}I_{Y} (the map is multiplication by f), we have

H^{2}(P^{2}, I_{Y}) ≅ H^{2}(P^{2}, O_{P^{2}}(−d)).

The remaining details are explained in the proof of [27, Prop. IV.4.21]. ■

Remark. Note that the map F : O_{Y} → O_{Y} is the p-th power map O_{Y} → O_{Y}^{p} followed by the inclusion map O_{Y}^{p} ↪ O_{Y}. So the top row of the above exact sequence gives the action of F on H^{1}(Y, O_{Y}), which is dual to the action of C on H^{0}(Y, Ω_{Y}). The composite map in the bottom row of the diagram can be calculated quite explicitly using the basis described in [27, Section III.5] for the cohomology of projective space.

Assume that Y is ordinary, and choose a set of coordinates for P(W), in other

words fix a basis B for H^{1}(Y, O_{Y}). We can then consider the matrix A = A_{B} giving the action of C^{∗} with respect to the basis B. A is called the Hasse-Witt matrix of Y with respect to the basis B. It is easy to see from σ-linear algebra that there will be exactly \frac{p^{g}−1}{p−1} projective solutions to the equation Av^{(p)} = v, where v ∈ H^{1}(Y, O_{Y}) is expressed in terms of the basis B, and v^{(p)} means the vector obtained from v by raising all coordinates to the p-th power (See section 4.3 for a more general statement). Using Lemma 11, we have the following proposition.

Proposition 17. Assume Y is ordinary. With notation as in the preceding paragraph, a point P ∈ Y (k) is a Cartier point if and only if its coordinate vector v = j(P) with respect to the basis B satisfies the equation Av^{(p)} = v.

Remark. This proposition is useful for calculations because given A one can algorithmically find the vectors v such that Av^{(p)} = v. This is discussed in [32, Proof of Satz 12] and [60, Proof of Thm. 2.1]. Example 4. The points (0:0:1) and (1:0:0) are the Cartier points in characteristic 3 on the genus three plane curve Y given by the equation

f(x, y, z) = y^{3}z + xy^{2}z + x^{3}y + 2x^{2}yz + xz^{3} = 0.

Proof. It is straightforward to check that the given curve is nonsingular.

We can compute the action of Frobenius on H^{1}(Y, O_{Y}) with respect to the basis

{x^{−2}y^{−1}z^{−1}, x^{−1}y^{−2}z^{−1}, x^{−1}y^{−1}z^{−2}} for H^{2}(P^{2}, O_{P^{2}}(−4))) using Lemma 16. The discussion in [27, Section III.5] shows that this basis corresponds to the canonical embedding given by f with respect to the coordinates x, y, z for P^{2}.

With respect to this basis, Frobenius acts on H^{1}(Y, O_{Y}) via the matrix A equal

to

(\begin{matrix}1 & 2 & 0 \\ 0 & 2 & 0 \\ 1 & 0 & 2 \end{matrix}) .

Thus Y is ordinary. By Prop. 17, the Cartier points on Y are just the points

with projective coordinates (a : b : c) such that f(a, b, c) = 0 and Av = v^{(p)}, where v is the column vector (a, b, c)^{t}. Using the algorithm from [60], or simply computing by hand, one finds that the solutions to Av^{(p)} = v correspond to the 13 projective points

{(0 : 0 : 1), (1 : 0 : 0), (1 : 0 : i), (1 : 0 : −i), (α : i : γ)}

where i is a fixed square root of −1 and α^{3} − α = −i, γ^{3} + γ = α. A computation now shows that of these points, only (0 : 0 : 1) and (1 : 0 : 0) lie on the curve.

We verify from another point of view that these two points are Cartier points. To

see that P = (0 : 0 : 1) is a Cartier point, we think of P as the origin on the affine plane curve given by the equation y^{3}+xy^{2}+x^{3}y−x^{2}y+x = 0 and apply Lemma 10. A calculation shows that the divisors of the degree three affine coordinate functions x and y are

(x) = 3(0 : 0 : 1) − (1 : 0 : 0) − 2(0 : 1 : 0)
(y) = (0 : 0 : 1) + 2(1 : 0 : 0) − 3(0 : 1 : 0).

So y is a uniformizing parameter at P, and since

\frac{y^{3}}{x} = −1 − y^{2} + y(x − x^{2}) = −1 − y^{2} + y^{4}(unit),

we have

\frac{−1}{x} = y^{−3} + y^{−1} + holomorphic

is the Laurent expansion of \frac{−1}{x} in terms of y, and this function has a pole only at P. Thus P is a Cartier point.

One can check in a similar way that \frac{y}{z} is a uniformizer at Q = (1 : 0 : 0), that \frac{z−1}{y^{2}}

is regular outside Q, and that at Q it has a Laurent expansion

\frac{z − 1}{y^{2}} = (\frac{y}{z})^{−3} + (\frac{y}{z})^{−1} + holomorphic

so that Q is also a Cartier point. ■

Example 5. The plane curve Y in characteristic 3 given by the equation

y^{4} − z^{4} + x^{2}y^{2} − x^{2}z^{2} + α · xy^{3} = 0.

with α ∉ F_{3} defines a nonsingular curve of genus 3 possessing exactly 5 Cartier points.

Proof. One easily checks that the given curve is nonsingular; this would be false

if we allowed α to lie in F_{3}.

We can compute the action of Frobenius on H^{1}(Y, O_{Y}) with respect to the ordered

basis {x^{−2}y^{−1}z^{−1}, x^{−1}y^{−2}z^{−1}, x^{−1}y^{−1}z^{−2}} for H^{2}(P^{2}, O_{P^{2}}(−4))) using Lemma 16. The discussion in [27, Section III.5] shows that this is exactly the ordered basis of H^{1}(Y, O_{Y}) determined by choosing the coordinates {x, y, z} for P(W) ≅ P^{2}.

We find that Frobenius acts on the vector space H^{1}(Y, O_{Y}) via the 3 × 3 identity

matrix. Thus not only is Y is ordinary, but in fact it is embedded in P^{2} via canonical coordinates. So by Lemma 15, the Cartier points on Y are just the points whose coordinates lie in F_{3}. The only such points are {(0 : 0 : 1), (0 : 1 : 0), (0 : 1 : 1), (0 : 1 : −1), (1 : 0 : 0)}.

■

The following example is motivated by the Coleman-Kaskel-Ribet conjecture (see

chapter 2). Example 6. There are no Cartier points on the modular curves X_{0}(43) or X_{0}(61) in characteristic 3.

Proof. From [20], we find models for the canonical embeddings of these two

curves in characteristic 3. The curve Y = X_{0}(43) in characteristic 3 is given by the homogeneous quartic equation

f(x, y, z) = x^{4} + 2x^{3}y + 2x^{2}y^{2} + 2x^{2}yz + x^{2}z^{2} + xy^{3} + 2xy^{2}z + xyz^{2} + y^{3}z + 2y^{2}z^{2} + z^{4} = 0.

With respect to the basis {x^{−2}y^{−1}z^{−1}, x^{−1}y^{−2}z^{−1}, x^{−1}y^{−1}z^{−2}} for H^{2}(P^{2}, O_{P^{2}}(−4))

we find that Frobenius acts on H^{1}(Y, O_{Y}) via the matrix A equal to

(\begin{matrix}1 & −1 & 1 \\ 0 & 0 & 1 \\ 0 & −1 & 0 \end{matrix}) .

In particular, Y is ordinary. It follows from Lemma 11 that the Cartier points

on Y are just the points on Y whose projective coordinates (a, b, c) satisfy the equation Av^{(p)} = v, where v = (a, b, c)^{t}. The algorithm given in [32] (which can also be found in [60]), which is easily implemented on a computer, allows one to calculate the \frac{3^{3}−1}{3−1} = 13 projective points v such that Av^{(p)} = v. Having done this, one discovers that none of these points v = (a, b, c)^{t} satisfy f(a, b, c)=0. In other words, there are no Cartier points on Y.

The curve Y := X_{0}(61) in characteristic 3 is a complete intersection in P^{3} of the

two hypersurfaces:

g(x, y, z, w) = w^{2} + 2x^{2} + 2xy + z^{2} = 0,

xdv

h(x, y, z, w) = x^{2}z + xy^{2} + xyz + 2xz^{2} + y^{2}z + 2yz^{2} = 0.

An argument similar to the proof of Lemma 16 shows that the action of Frobenius

on H^{1}(Y, O_{Y}) with respect to the basis B which corresponds to the given coordinates x, y, z, w for P(W) ≅ P^{3} can be computed as follows.

Taking long exact sequences of cohomology shows that H^{1}(Y, O_{Y}) is isomorphic

to H^{3}(P^{3}, O_{P^{3}}(−5)), and under this isomorphism B corresponds to the basis

B′ = {x^{−2}y^{−1}z^{−1}w^{−1}, …, x^{−1}y^{−1}z^{−1}w^{−2}}

for the cohomology of projective space. The Frobenius map induces the map on H^{3}(P^{3}, O_{P^{3}}(−5)) which takes b ∈ B′ to the class of b^{p}(gh)^{p−1}, where in this example p = 3. Recall that monomials of the form x^{a}y^{b}z^{c}w^{d} are zero in H^{3}(P^{3}, O_{P^{3}}(−5)) whenever any of a, b, c, d are nonnegative.

Using the basis B′, one computes the matrix A of F on H^{1}(Y, O_{Y}) as

(\begin{matrix}0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ −1 & −1 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) .

Again, a computer-aided computation shows that none of the solutions to Av^{(p)} =

v correspond to points on the curve. ■

Chapter 5

Additional Techniques

5.1 Purpose

In this section we present additional methods for studying torsion points on curves.

The purpose of this is two-fold: first, we expect that these techniques may be of some general use in trying to explicitly find torsion points on curves. In addition, the results of this section finish up our first proof of Theorem 1 by determining the full cuspidal torsion packet T (N) on the trigonal modular curves X_{0}(N) with N = 29, 31, 43, 53, 61. In each case we show that T (N) is indeed the conjectured set S(N), namely {0, ∞} if N = 43, 53, 61 and {0, ∞}∪{hyperelliptic branch points} if N = 29, 31. The trigonal curve X_{0}(37) is dealt with in [12], and a determination of T (23) is made in [13] by methods similar to those used here for N = 29 and N = 31.

5.2 Useful Facts

For each prime p, fix a prime of \overline{ℚ} lying over p, and a corresponding decomposition

group D_{p} of Gal(\overline{ℚ}/ℚ).

We will have use for the following two facts in the arguments that follow. The first

we learned in conversation with Ken Ribet.

Lemma 17. Suppose N is unramified in the Hecke algebra for X_{0}(N), and that P is a torsion point on J_{0}(N) of order divisible by N. Then P has at least N^{2}−1 Galois conjugates.

Proof. Let T_{N} := T ⊗ ℤ_{N}, and T̃_{N} := T_{N}/NT_{N}. Without loss of generality we may replace P by a multiple Q of P having order N, and we choose a basis for the N-torsion of J_{0}(N) as a 2-dimensional T[Gal(\overline{ℚ}/ℚ)]-module.

By [48], the image of Gal(\overline{ℚ}/ℚ) in GL(2, T̃_{N}) contains \{m ∈ GL(2, T̃_{N})|det(m) ∈

(ℤ/Nℤ)^{∗}}. It is then easy to see that the lemma is true if Q is of the form (u, 0) with respect to our given basis.

So it suffices to show that any vector (u, v) with coefficients in T̃_{N} can be trans-

formed into something with second component 0 by a matrix in the image of Gal(\overline{ℚ}/ℚ). Since any nonzero (u, v) is a multiple of a vector with one of u, v a unit in T̃_{N}, we may assume the latter property holds.

Now if u is a unit, then ( \begin{matrix}1 & 0 \\ −vu^{−1} & 1 \end{matrix} ) ( \begin{matrix}u \\ v \end{matrix}) = ( \begin{matrix}u \\ 0 \end{matrix} ).

And if v is a unit, then we have ( \begin{matrix}0 & 1 \\ 1 & −uv^{−1} \end{matrix} ) ( \begin{matrix}u \\ v \end{matrix}) = ( \begin{matrix}v \\ 0 \end{matrix} ). ■

For a proof of the following theorem, see [23].

Theorem 11. Suppose f = ∑ a_{n}q^{n} is a normalized weight 2 cusp form for Γ_{0}(N) with coefficients in a field E of characteristic p ≠ N. Assume that a_{p} ≠ 0, and let D_{p} be a decomposition group at p in Gal(\overline{ℚ}/ℚ). Then the restriction of the 2-dimensional Galois representation ρ_{f} associated to f to D_{p} has the form (after picking an appropriate basis)

(\begin{matrix}χλ(\frac{1}{a_{p}}) & ∗ \\ 0 & λ(a_{p}) \end{matrix}) .

Here χ is the mod p cyclotomic character and λ(x) is the unramified character

which takes a Frobenius element at p to x.

Remark. If p ≠ 2 and p is unramified in the Hecke algebra T, then the representation ρ is the direct sum of the representations ρ_{f_{i}} above, where the f_{i} are representatives for each conjugacy class of weight two cusp forms for Γ_{0}(N).

5.3 The Curve X_{0}(29)

The curve X := X_{0}(29) is hyperelliptic of genus 2. Embed X in its Jacobian

via the map i sending x to the class of (x) − (∞), and let Q be a point of X such that P = i(Q) is a torsion point in this embedding. We can decompose P into a sum of l-primary components for various primes l,

P = ∑ P_{l},

where the P_{l} are points in the Jacobian of l-power order.

Proposition 18. Suppose that P_{3} = 0. Then we have T (29) = S(29).

Proof. Suppose Q ∈ T (29), Q ∉ S(29). By Theorem 1, we have P = P_{2} +

P_{3} + P_{29} + P_{c}, with P_{c} in the cuspidal group, and by hypothesis P_{3} = 0. Also, since the characteristic polynomial of T_{2} has discriminant 8, all primes other than 2 (and in particular 29) are unramified in T. So Theorem 3 shows that P_{2} = 0. Thus P = P_{29} + P_{c}.

We can now see that P_{29} = 0 as follows. By [48], there is a σ ∈ Gal(\overline{ℚ}/ℚ) which

acts on P_{29} as the homothety −6, since 6^{2} − 1 is prime to 29. This σ fixes the rational point P_{c} of order dividing 7, hence acts as multiplication by −6 on P itself. By the spirit-of-Lang estimate [12, Lemma 4.1], P can have at most g · 6^{2} = 72 conjugates. But an element of order divisible by 29 in Jac(X) has at least 29^{2} − 1 conjugates by Lemma 17, so P_{29} = 0. Thus P ∈ C. As explained in [12], it now follows that Q is one of the cusps, contradicting Q ∉ S(29). ■

Theorem 12. T (29) = S(29).

Proof. Suppose P ∈ T (29), P ∉ S(29). By Prop. 18, we may assume that

P_{3} ≠ 0. Then P_{3} is in fact ramified at 3 by Theorem 2, since the order n = 7 of the cuspidal subgroup is not divisible by 3.

To obtain a contradiction, we use explicit data pertaining to the curve X. First,

we need some data concerning the Hecke algebra T, which we gather using the tables of [55]. T is isomorphic to ℤ[\sqrt{2}], so the prime 3 is inert; also, the characteristic polynomial of T_{3} is x^{2} − 2x − 1, which shows that X mod 3 is ordinary (since ordinariness is equivalent to the constant term of this characteristic polynomial being a 3-adic unit).

In addition, we use an explicit equation for X, provided in [20] as

y^{2} = x^{6} − 4x^{5} − 12x^{4} + 2x^{3} + 8x^{2} + 8x − 7.

Let f(x) denote the right-hand side of this equation and f̃(x) its reduction mod 3. Since f̃ is irreducible over F_{3}, this equation provides a model for X over ℤ_{3}. By Thm. 4, the ramified torsion point P on the ordinary curve X must reduce to a Cartier point mod 3. By Prop. 16, the x-coordinate of the reduction of P must then be a root of f′(x) mod 3. One sees easily that f̃′(x) = x^{4} + x + 2 is irreducible over F_{3}. If α is a root of this polynomial defined over F_{81}, then f(α) is a square in F_{81} (A MAPLE computation shows that its norm to F_{3} is a square, which is equivalent). This shows that all torsion points on X ramified at 3 have their reduction defined over F_{81}.

Now according to Theorem 11, the action of a decomposition group at 3 on 3-

torsion points of Jac(X) is given matricially by

(\begin{matrix}χλ(\frac{1}{a_{3}} ) & ∗ \\ 0 & λ(a_{3}) \end{matrix}) .

Here χ is the mod 3 cyclotomic character and λ(x) is the unramified character

which takes a Frobenius element at 3 to x. The order of λ is thus the order of x in the multiplicative group of \overline{F}_{3}. In this case, a_{3} is a root of char(T_{3}) = x^{2} − 2x − 1, so has order 8. The action of Galois on the 3-torsion in the kernel of reduction is given by the character χλ(\frac{1}{a_{3}} ), and the action of Galois on the reduction of the 3-torsion by the character λ(a_{3}) (see [23]).

Now we may assume that P_{3} is not in the kernel of reduction. This is because

some conjugate of P_{3} under Gal(\overline{ℚ}/ℚ) must lie outside the kernel of reduction or else the representation J[3] would not be irreducible. This representation is irreducible since n is not divisible by 3, i.e., 3 is not an Eisenstein prime (see [36]).

An alternative argument shows in fact that P_{3} cannot be in the kernel of reduction

at all. First, we note that P_{N} = 0. For otherwise some multiple P′ of P having order N has at least N^{2} − 1 global Galois conjugates by Lemma 17. These conjugates all reduce to different points mod 3, since reduction is injective on prime-to-3 torsion. Thus P̃ itself would have at least N^{2} − 1 = 840 conjugates, which contradicts the fact that P̃ is defined over F_{81}. So P = P_{3} +P_{c}, and if P_{3} were in the kernel of reduction, it would have a multiple of 8 conjugates under a decomposition group at 3, and hence P = P_{3} + P_{c} would have a multiple of 8 conjugates in a single residue class (namely that of P_{c}). This violates Cor. 3.7 of [12].

At any rate, without loss of generality we now suppose that 3 divides the order

of the reduction of P_{3}; P̃_{3} thus has a multiple of 8 conjugates, i.e. it is minimally defined over an extension of F_{3} of degree a multiple of 8. But then so is the reduction of P, a contradiction (P̃ is defined over F_{81}).

This proves that P_{3} = 0 as desired. ■

5.4 The Curve X_{0}(31)

One would hope to treat the curve X_{0}(31) in a similar manner as we did X_{0}(29)

in the previous section. Indeed, most of the arguments go through again.

The Hecke algebra T = ℤ[\frac{1+\sqrt{5}}{2}] is ramified only at 5, and the order of the cuspidal

group is also 5. The reasoning used to prove Prop. 18 shows that it suffices to prove P_{3} = 0 for an arbitrary torsion point Q on the curve.

We have an equation

y^{2} = g(x) := x^{6} − 8x^{5} + 6x^{4} + 18x^{3} − 11x^{2} − 14x − 3

for X := X_{0}(31). This model has good reduction mod 3, and g′(x) mod 3 is congruent to 2x^{4} + 2x + 1, which is irreducible. But a root β of this irreducible polynomial has g(β) not a square in F_{81}, so we can say only that the reduction of P is defined over an extension of F_{3} of degree 8, not 4. The contradiction in the previous argument for X_{0}(29) vanishes due to this mercurial change! (Otherwise, life would be easy: 3 is still inert in T, and the characteristic polynomial of T_{3} is x^{2} + 2x − 4, so not only is X mod 3 ordinary, but a_{3} still has order 8 in the multiplicative group). So another argument is needed to show that P_{3} is zero. Theorem 13. T (31) = S(31).

Proof. Let Q be a torsion point on X_{0}(31) ramified at 3, and let P = i(Q). We

know we can write P = P_{3} + P_{31} + P_{c} by Theorems 1 and 3. (Note that the discriminant of the Hecke algebra is prime to 31). Here P_{c} is a point in the cuspidal subgroup, which has order 5. Also, Q must be a Cartier point mod 3, and we have seen that this implies that the reduction of Q mod 3 is defined over F_{3^{8}}.

Using the characteristic polynomial of T_{3}, we can compute the zeta function of

X mod 3, and in particular |J_{0}(31)(F_{3^{8}})|. Using, for example, [41], we see that the zeta function of X/F_{3} is

\frac{T^{4} + 2T^{3} + 2T^{2} + 6T + 9}{(1 − T)(1 − 3T)},

and then one sees from standard properties of the zeta function that

|J_{0}(31)(F_{3^{8}})| = ∏_{ζ} f(ζ),

where f is the numerator of the zeta function over F_{3} and ζ runs through all 8th roots of unity. Indeed, such a formula holds whenever 8 is replaced by a power of 2; e.g. |J_{0}(31)(F_{3})| = f(1).

In particular, we find that |J_{0}(31)(F_{3^{8}})| = 2^{12} · 3^{2} · 5^{2} · 7^{2}. The mod 3 reduction

of P thus has order prime to 31, and since the reduction map is injective on prime-to-3 torsion, we find that in fact P_{31} = 0. The order of P mod 3, call it m, is then 3^{a} · 5^{b}, where b = 0 or 1, and a = 1 or 2. (If a = 0 then P_{3} is in the kernel of reduction, but as with X_{0}(29) we may assume that this is not the case.)

We now claim that a = 1. For since 3 is inert in the Hecke algebra, T/3T ≅ F_{9} and

consequently J_{0}(31)[3] is an F_{9}-vector space; it is then easy to see that a = 2 is impossible.

Thus m = 3 or 15, so 15P̃ = 0. What we do now is the following: we have an

explicit hyperelliptic model for X (in which the cusps corresponds to the two points at infinity) and we know the x-coordinate of P̃ explicitly, so we should be able to compute the order of P̃ in the Jacobian of X. In particular, if 15P̃ ≠ 0 then we get the desired contradiction.

We have done exactly this computation, and in fact 15P̃ ≠ 0 on the Jacobian of

X mod 3. This gives us P_{3} = 0, as we set out to prove. ■

For lack of a suitable reference, we briefly sketch our method for carrying out the

above computation that 15P̃ ≠ 0.

5.5 A Computational Method

The general setup for this section is as follows: Let C be a genus 2 hyperelliptic

curve given by the equation y^{2} = g(x), with g of degree 6 having coefficients in a field of characteristic p ≠ 2. [The restriction to genus 2 is merely for the sake of concreteness in notation, it is not essential to our method.] For simplicity, we assume that g(x) is monic, so we can write

g(x) = x^{6} + a_{5}x^{5} + a_{4}x^{4} + a_{3}x^{3} + a_{2}x^{2} + a_{1}x + a_{0}.

Label the two (non-Weierstrass) points at infinity on this curve ∞, ∞′. Let P be a non-Weierstrass point on the affine part of C with x-coordinate x_{0}, and let m ≥ 3 be an integer. We want to know if mP = 0 in Jac(C) with respect to an Albanese embedding of C into Jac(C) via one of the points at infinity.

To do this, we find an explicit basis for the vector space L(m∞), which has di-

mension m − 1 by Riemann-Roch (and our assumption m ≥ 3). Now a basis for the 2m − 1-dimensional space L(m∞ + m∞′) is easy to find, using the fact that (x) = (0) + (0′) − (∞) − (∞′) and (y) = ∑_{i=1}^{6}(P_{i}) − 3(∞) − 3(∞′). Such a basis is

{1, x, x^{2}, …, x^{m}, y, xy, x^{2}y, …, x^{m−3}y}.

To distinguish between the two points at infinity comes down to choosing a branch

of the square root function, in the following sense. Since g(x) is monic, we can write g(x) = x^{6}(1 + G(1/x)), with G a polynomial having no constant term. Using the binomial theorem, we can then expand y as a Laurent series in 1/x, obtaining

y = ±x^{3}(1 + \frac{1}{2}G(\frac{1}{x}) − \frac{1}{8}G(\frac{1}{x})^{2} + ⋯)
= ±(g_{1}(x) + ∑_{i=1}^{∞} b_{i}x^{−i})

Here g_{1}(x) is a monic polynomial of degree 3 in x and b_{i} ∈ k. Note that g_{1}(x) is

the polar part of y as an expansion in \frac{1}{x}, which is a local parameter at ∞ (and also at ∞′). Without loss of generality, suppose that y − g_{1}(x) (as opposed to y + g_{1}(x)) is holomorphic (i.e. regular) at ∞′. This is equivalent to choosing a branch of the square root. If for 2 ≤ n ≤ m − 2 we let

g_{n}(x) = polar part of x^{n−1}(g_{1}(x) + ∑_{i=0}^{∞} b_{i}x^{−i})

then the function x^{n−1}y − g_{n}(x) is also regular at ∞′, and the set

{1, y − g_{1}(x), xy − g_{2}(x), x^{2}y − g_{3}(x), …, x^{m−3}y − g_{m−2}(x)}

is easily seen to form a basis for L(m∞).

We are trying to find out if there exists a function f ∈ L(m∞) which vanishes to

order m at our point P. Since by assumption P is not a Weierstrass point, w = x − x_{0} is a local parameter at P. Changing variables, let Y (w) = +\sqrt{g(w + x_{0})} be a power series in w representing the expansion of the regular function y around the parameter w. Note that this involves taking a square root of g(x_{0}) in k (and g(x_{0}) ≠ 0 by hypothesis). Let h_{0}(w) = 1, and for 1 ≤ n ≤ m − 2 let

h_{n}(w) = (w + x_{0})^{n−1}Y (w) − g_{n}(w + x_{0})

be power series in w representing the expansions of our basis elements for L(m∞) around w.

Without loss of generality, we are trying to see if there exists a nontrivial linear

combination

∑_{i=0}^{m−2} c_{i}h_{i}(w) = w^{m} + d_{m+1}w^{m+1} + ⋯

as power series in w, where the d_{n} for n ≥ m + 1 are in k. This is simply a linear algebra problem: writing

h_{i}(w) = ∑_{j=0}^{m−1} a_{ij}w^{j}

for 0 ≤ i ≤ m − 2, we want to determine whether or not the m × (m − 1) matrix (a_{ji}) has a nontrivial kernel.

Even when k is a finite field (over which the curve C and also P are defined),

this entire computation can be effectively carried out on a computer. For example, we implemented this algorithm using the software package GAP. In the case where C is the hyperelliptic model

y^{2} = g(x) = x^{6} + x^{5} + x^{2} + x

for X_{0}(31) in characteristic 3 (see the previous section) and P is a point on C with xcoordinate x_{0} equal to a root of g′(x), the program assures us that the divisors 15(P)−15(∞) and 15(P) − 15(∞′) are not principal. As explained in the last section, this proves that T (31) = S(31).

5.6 The Curve X_{0}(53)

The technique we use to study X_{0}(53) is quite different from those employed thus

far. Since X_{0}(53) does not have ordinary reduction mod 3, we cannot use Theorem 4.

Instead, we make some technical refinements of the proof of [12, Prop. 3.12] and exploit the existence of an elliptic curve defined over ℚ with conductor 53 and supersingular reduction mod 3. It should be noted that this method makes no use of an explicit equation for the modular curve X_{0}(53).

Let Q ∈ T (53) be a torsion point on X = X_{0}(53). The discriminant of the

characteristic polynomial of T_{2} is 2^{4} · 37; in particular, the Hecke algebra T is unramified at 3,5, and 53. By Theorems 1 and 3, we have P = P_{3} + P_{53} + P_{c}, where P_{c} has order dividing 13. Also, the same trick as before shows that to prove T (53) = S(53), it is enough to prove that P_{3} = 0. For in that case there exists an element σ ∈ Gal(\overline{ℚ}/ℚ) which acts as the homothety −12 on P, so P has at most 12^{2} · g = 576 conjugates. But if P_{53} ≠ 0, then P_{53} (and hence P) must have at least 53^{2} − 1 > 576 conjugates.

It remains to prove that P_{3} = 0. Suppose not, so that (since n = 13 is not divisible

by 3) Q is a torsion point ramified at 3.

Exploiting the tables of [55], one sees for p = 3 (using the notation of [12]) that

dim W_{1} = 3 and dim W_{2} = 1. This means that W := H^{0}(X, Ω_{X/ℤ_{3}}) has a 3-dimensional ordinary part and a 1-dimensional non-ordinary part. Since 3 is unramified in T (or simply because dim W_{2} = 1 and C is nilpotent on W̃_{2}), H_{2} (and by abuse of notation W_{2}) is in fact superspecial. (See [12, Prop. 3.10])

Pick ω_{0} ∈ W_{2} (it is unique up to a constant). The following proposition will be

important for our upcoming analysis.

Proposition 19. The differential ω_{0} ∈ W_{2} has only simple zeros mod 3. These zeros occur exactly at the branch points of the modular parametrization map from X to the unique rational elliptic curve of conductor 53.

Proof. According to the tables of [55], the Jacobian J_{0}(53) is isogenous to a

product A × E of a simple 3-dimensional abelian vareity A with ordinary reduction mod 3 and an elliptic curve E with supersingular reduction mod 3. By [17], the degree d of the modular parametrization map ϕ : X_{0}(53) ⟶ E is 2. Up to constants, ω_{0} is the pullback ϕ^{∗}ν of an invariant differential on E. Since ν has no zeros mod 3, the zeros of ω_{0} mod 3 occur exactly at the ramification points of ϕ, and (since the ramification must be tame given d < 3) have multiplicity d − 1 = 1. ■

Setting k_{ω}(x) = ordx̃(ω̃) + 1 for ω ∈ W, we see that k_{ω_{0}}(x) ≤ 2 for all x ∈ X,

where the tilde denotes reduction mod 3.

Let Q′ be a point unramified at 3 in the same mod 3 residue class as Q, and let

T be a parameter at Q′ which gives an analytic isomorphism, defined over ℚ_{p}^{nr}, from U(Q′) (the residue class of Q′) to the open unit ball in ℂ_{p}. Let s = v(T (P)), where v is the normalized valuation of ℂ_{p} taking p to 1.

By Riemann-Roch, Q̃ is not a base point for one of W̃_{1}, W̃_{2}. Our argument for

proving that P_{3} = 0 will fall into two cases accordingly.

Case 1: Q̃ is not a base point for W_{2}, i.e. k_{ω_{0}}(Q) = 1.
By [12, Lemma 3.11], M_{P} (H_{2}) = 0. So by [12, Prop. 3.6],
s = \frac{1}{p^{2n}(p^{2} − 1)} = \frac{1}{8 · 3^{2n}}  (5.1)

for some integer n ≥ 0.

Now suppose, momentarily, that k_{ν}(Q) ≥ p = 3 for all ν ∈ W_{1}. Since it is ordinary,

W̃_{1} has a basis of differentials fixed by the Cartier operator C. For these differentials, k_{ν}(Q) ≥ 3 implies that in fact k_{ν}(Q) ≥ 4 by [12, Lemma 11]. Thus in fact k_{ν}(Q) ≥ 4 for all ν ∈ W_{1}. As no differential in W̃_{2} vanishes at Q̃, we see that every differential of W̃ that vanishes at Q̃ vanishes to order at least 3. This is impossible by Riemann-Roch. (See [9, Proof of Thm. 5.5(a)]).

So some element ν of W_{1} has k_{ν}(Q) ≤ p − 1 = 2, i.e. k_{Q}(H_{1}) ≤ p − 1 in the

notation of [12]. By [12, Prop. 3.5], then, we see that s is either of the form \frac{r}{t} with t ≤ 2, \frac{1}{kp^{n}} = \frac{1}{k·3^{n}}, or \frac{1}{kp^{n}(p−1)} = \frac{1}{2k·3^{n}} with n ≥ 0 and k equal to 1 or 2. But these possibilities are all clearly incompatible with (5.1), so this case does not occur.

Case 2: Q̃ is not a base point for W_{1}, so k_{Q}(H_{1}) = 1 and (by Prop. 19 and Case

1) k_{Q}(H_{2}) = 2.

Then by [12, Prop. 3.5,3.6] applied to H_{1}, we have that s is either of the form \frac{1}{(p−1)p^{n}} = \frac{1}{2·3^{n}} or \frac{1}{p^{M}} = \frac{1}{3^{M}} with n ≥ 0 and M > 0.

Applying the same proposion to H_{2} (and noting again that M_{Q}(H_{2}) = 0), we see

that s must be of the form \frac{1}{k(p^{2}−1)p^{n}} = \frac{1}{16·3^{n}} with n ≥ 0 or \frac{r}{t} with t ≤ k = 2. The only conceivable overlap with the results derived from analyzing H_{1} is that s = \frac{1}{2}.

So we will have shown that P_{3} = 0, and thus that T (53) = S(53), if we can manage

to rule out s = \frac{1}{2}. We remark that our analysis shows, using [12, Lemma 14], that there are at most two torsion points ramified at 3 in any single residue class of X. Since there are at most 2g − 2 = 6 residue classes where ω̃_{0}(Q̃) = 0, there can be at most 12 torsion points on X ramified at 3. This shows in particular that P_{3} ramified at 3 implies P_{53} = 0, since otherwise by Lemma 17, P would have at least 53^{2} − 1 Galois conjugates, all ramified at 3.

Now we know that J_{0}(53) is isogenous to A × E with A ordinary mod 3 and E

supersingular. Write the image of P_{3} in A × E componentwise as (P_{A}, P_{E}). Now since E is a supersingular elliptic curve and P_{E} a point of 3-power order on E, the residue class mod 3 of all conjugates of P_{E} is the same (namely 0). Now according to [50, Prop. 2], the image of an inertia group at 3 in Aut(E[3]) is a non-split Cartan subgroup, hence cyclic of order 8. So if P_{A} = 0, then P = P_{E} + P_{c} has at least 8 distinct conjugates in the same residue class. This is impossible by our above remark.

Now suppose P_{A} ≠ 0. Then we can exploit our knowledge of the Galois represen-

tation on 3-torsion of the ordinary part of J_{0}(53). As the characteristic polynomial of T_{3} acting on the ordinary cusp forms is x^{3} − 3x^{2} − x + 1, a MAPLE computation shows (using the shape of the Galois representation in the ordinary case, see Thm. 11) that a nontrivial 3-torsion point in the ordinary part of J_{0}(53) is either in the kernel of reduction and has at least 26 conjugates in one residue class, or else its reduction has at least 26 conjugates. So our point P, or its reduction mod 3, must have at least 26 Galois conjugates, which contradicts our observation that there are at most 12 torsion points on X ramified at 3.

Thus P_{A} = P_{E} = 0, i.e., P_{3} is in the kernel of the isogeny f : J_{0}(53) → A × E.

But the degree of this isogeny can only be divisible by primes which ramify in the Hecke algebra. In particular, the degree of f is prime to 3. So P_{3}, which is of 3-power order, cannot be in the kernel of f unless it is zero.

We have thus proved the following theorem:

Theorem 14. T (53) = S(53).

Chapter 6

Appendix

In this appendix, we give proofs of some results concerning torsion packets on

curves. Some of these results seem to be known to the experts, but to our knowledge the proofs we present have not appeared in the literature.

Let us briefly set up some notation and recall a few definitions. Let X be an

algebraic curve defined over a field K of characteristic zero. As usual, when we say that X is a curve we require it to be smooth, proper, and geometrically irreducible over K. We will denote by J the Jacobian variety of X, and by g the genus of X.

Let K̄ be an algebraic closure of K. We define an equivalence relation on X(K̄)

by defining P ∼ Q iff the divisor m(P) − m(Q) on X is principal for some positive integer m. We call an equivalence class under ∼ a torsion packet on X. Clearly the torsion packet containing P ∈ X(K̄) is nothing more than the set of points of X(K̄) which map to torsion points of J via the Albanese map from X to J sending Q ∈ X(K̄) to the class of the divisor (Q) − (P). A torsion packet is said to be trivial if it has only one element.

Recall that the Manin-Mumford conjecture (proved by M. Raynaud in [45]) states

that if g ≥ 2, then every torsion packet on X is finite. In fact, Raynaud [46] proved the following more general result about torsion points on subvarieties of abelian varieties:

Theorem 15. Suppose A is an abelian variety over the algebraically closed field L, and that V is a geometrically irreducible closed K-subvariety of A. If the torsion points of A are Zariski-dense in V, then X is the translate, by a torsion point, of an abelian subvariety of A.

Remark. Raynaud’s theorem is equivalent to the following assertion: With hypotheses as in Theorem 15, the set of torsion points of A lying on X is contained in a finite union ∪Z_{j}, where each Z_{j} ⊆ V and is a translate of a (possibly zero-dimensional) abelian subvariety of A by a torsion point.

We show how Raynaud’s theorem can be used to prove some interesting results

about the size of torsion packets on a curve X.

First of all, we have the following result, which says that if g ≥ 2 then there is a

uniform bound on the size of torsion packets on X. This result is stated without proof in [54]. The result seems to be nearly proved in the literature – Raynaud [46, Prop. 9.1] gives a proof when g > 2 – but for completeness we provide a proof due to Bjorn Poonen. Theorem 16. Suppose g ≥ 2. Then there exists a constant M depending only on the curve X such that every torsion packet on X has size at most M.

Proof. Assume that there are torsion packets on X of arbitrarily large size.

Fix a positive integer n ≥ 3, and consider the map α_{n} : X^{n} → J^{n−1} given by sending (P_{1}, …, P_{n}) to ([(P_{1}) − (P_{2})], [(P_{2}) − (P_{3})], …, [(P_{n−1}) − (P_{n})]). The map α_{n} is just the composition of the Albanese map X^{n} → J^{n} given by (P_{1}, …, P_{n}) ↦ ([(P_{1}) − (P_{0})], [(P_{2}) − (P_{0})], …, [(P_{n}) − (P_{0})]) for some fixed point P_{0} on X with the map J^{n} ↠ J^{n−1} obtained by quotienting J^{n} by J, embedded diagonally. Since the image of X^{n} under the Albanese map generates J^{n} as a group, the image of X^{n} under α_{n} generates J^{n−1}.

So X^{n} cannot be the translate by a torsion point of an abelian subvariety of J^{n−1},

because the former variety has dimension n while the latter has dimension (n−1)g > n. By Theorem 15, it is therefore enough to show that the torsion points of J^{n−1} are Zariski-dense in X^{n}.

Notice that a point (P_{1}, …, P_{n}) on X^{n} maps to a torsion point on J^{n−1} iff

P_{1}, …, P_{n} all lie in a common torsion packet.

For each positive integer i, let T_{i} be the set of n-tuples (P_{1}, …, P_{n}) such that the

P_{i} all lie in a common torsion packet having at least N elements.

The Zariski closures of the sets T_{i} in X^{n} form a descending chain

T̄_{1} ⊇ T̄_{2} ⊇ ⋯ ⊇ T̄_{i} ⊇ ⋯

and since X^{n} is Noetherian this chain stabilizes, starting with say T_{M}, so that T̄_{N} = T̄_{N+1} = ⋯

Let V be the Zariski closure of T_{M} in X^{n}. We claim that V = X^{n}. As previously

noted, this claim implies the theorem. In any case, our initial assumption shows that V has dimension at least one.

To prove the claim, suppose that V is contained in but not equal to X^{n}. Let

0 < k < n be its dimension. A consideration of tangent spaces shows that there is some coordinate projection X^{n} → X^{k} so that the induced map π from V to X_{k} is dominant. To see this, choose a nonsingular point P of V; by linear algebra there is some projection π_{0} : X^{n} → X^{k} such that the corresponding map π : V → X^{k} has π_{∗} : T_{P} (V) → T_{π(P)}(X^{k}) an isomorphism. The map π_{∗} is then an isomorphism on tangent spaces in a whole open neighborhood U_{0} of P. If V′ denotes the scheme-theoretic image of V in X_{k} under π, it follows that there is some nonsingular point Q in V′ whose tangent space has dimension at least k. Therefore V′ = X^{k}, i.e., π is dominant.

It follows that π is generically finite of degree d. This means that there exists a

Zariski open set U ⊆ X^{k} such that the induced map π : U′ := π^{−1}(U) → U has finite fibers all of size d.

Since T_{i} is dense in V for i ≥ M, for each i there is some element P^{i} of T_{i} in U′.

Also we must have P′ ∈ U′ for all P′ such that π(P′) = π(P^{i}), since U′ = π^{−1}(U). So if i > n − k + d, then there are at least d + 1 elements in T_{i} ∩ U′ lying in the fiber of π over π(P^{i}). This contradiction establishes the claim, and hence the theorem. ■

Notice that Theorem 16 does not directly address the issue of how many nontrivial

torsion packets a curve can have. With this question the picture is slightly more complicated. For example, since the subtraction map ϕ : X × X → J given by (P, Q) ↦ [(P) − (Q)] is surjective when g = 2, it follows easily that every curve of genus 2 has infinitely many nontrivial torsion packets.

In [11, Example (iv)], one finds that statement that it follows from [46] that for

g ≥ 3, there are only finitely many nontrivial torsion packets. This conclusion is false, as B. Poonen pointed out: we claim that if X has genus three and is both hyperelliptic and bielliptic (for example, X could be the projective curve corresponding to the equation y^{2} = x^{8} + 1), then there are infinitely many nontrivial torsion packets on X.

To see this, let h be the hyperelliptic involution on X, and choose a Weierstrass

point S on X, and note that 2(S) is linearly equivalent to (P) + (hP) for all points P on X. Let f : X → E be a degree 2 map. Then the image of the homomorphism E → J given by P ↦ [f^{∗}(P) − 2(S)] is contained in the image of ϕ, because if f^{∗}(P) = (Q) + (Q′), then [f^{∗}(P) − 2(S)] = [(Q) − (hQ′)]. The claim follows.

In fact, we can show that Poonen’s example is the only way a curve of genus greater

than 2 can have infinitely many nontrivial torsion packets.

Theorem 17. Let X be a curve of genus g ≥ 2. If X has infinitely many nontrivial torsion packets, then either g = 2, or g = 3 and X is both hyperelliptic and bielliptic.

Proof. We continue to denote by ϕ the subtraction morphism X × X → J.

Suppose X is not hyperelliptic; then ϕ is birational to its image W. Indeed, it is easy to see that W is two-dimensional, so that ϕ is generically finite; on the other hand, the generic degree of ϕ is clearly 1, so ϕ must be a birational isomorphism.

More precisely, we claim that ϕ induces an isomorphism between the open subva-

rieties U = X × X − Δ and U′ = W − 0, where Δ ⊂ X × X is the diagonal. To see this, we use the fact (see [4, Exer. VI.A-1]) that if K = K̄ and ψ : X → P(W) denotes the canonical embedding, then the projectivized image of the differential map

ϕ_{∗} : T_{(P,Q)}(X × X) → T_{[(P)−(Q)]}J→^{∼}W^{∗}

is the linear subspace of P(W) spanned by ψ(P) and ψ(Q). (Here P(W) denotes the space of hyperplanes in W, which is isomorphic to the space of lines in W^{∗}.)

Using this description, we see easily that ϕ_{∗} is an isomorphism on tangent spaces

when restricted to U. Since ϕ : U → U′ is clearly bijective when X is not hyperelliptic, our claim follows from Lemma 14.8 and Theorem 14.9 of [24] (“inverse function theorem” for varieties).

Now suppose that W contains infinitely many torsion points of J. If g > 2, then

W is not an abelian surface, because it generates J, which has dimension at least 3, as an abelian group. (This follows from the stronger fact that any Albanese embedding of X generates J). Therefore, by Raynaud’s theorem, W must contain the translate of an elliptic curve E by a torsion point of J. Since W contains the origin of J, we can in fact assume that E is an abelian subvariety of J.

If we assume that X is not hyperelliptic, then there is a birational map E ↪ W →

X^{2} ↠ X, where the map X^{2} ↠ X is projection onto the second factor. Since E and X are nonsingular curves, this map extends to a morphism E → X. Our assumption that X has infinitely many nontrivial torsion packets implies that this morphism is not constant, which contradicts the fact that g > 1.

Therefore X is hyperelliptic with involution h. If S is a hyperelliptic branch point

on X, then the composite map

X^{2} → X^{2} → X^{(2)} → J

given by

(P, Q) ↦ (P, hQ) ↦ P + hQ ↦ [(P) + (hQ) − 2(S)]

is just the difference map ϕ. Therefore E is contained in the image of the symmetric square map X^{(2)} → J. It follows from [2, Thm. 3] that X is bielliptic. An application of the Correspondence Lemma (Lemma 5) shows in this case that g ≤ 3. ■

We conclude with an intriguing open problem concerning torsion packets on curves,

a conjecture of Coleman first put forth in his paper [10].

Conjecture 2. Suppose K is a number field, and that X is a curve of genus g ≥ 2 defined over K. Let 𝔭 be a prime of K of characteristic p not dividing 6 for which K/ℚ is unramified and X has good reduction. Let T be a torsion packet in X(K̄) stable under Gal(K̄/K). Then the extension K(T)/K is unramified above 𝔭.

A good discussion of what is known about this conjecture can be found in [11].

For example, Coleman proved in [10] that the conjecture is true for p > 2g, and he proved it for p < 2g as well when the reduction of X at 𝔭 is either ordinary or superspecial. Another bit of evidence for the conjecture is given by [11, Prop. 5].

Without the condition that K/ℚ is unramified at 𝔭, the conjecture is false, as is

explained in example (2) at the end of [10].

We now give the details of an example, also due to Coleman, which shows that

the other hypotheses of the conjecture are necessary as well. Specifically, we show that the cuspidal torsion packet on X_{1}(13) (which has at least 22 elements) is stable under Gal(\overline{ℚ}/ℚ) and is ramified above the primes 2, 3, and 13.

For a number of details concerning the structure of X_{1}(13), we refer the reader to

the excellent paper [37].

X_{1}(13) (which we refer to from now on as X) is a curve of genus 2 defined over ℚ

which has good reduction everywhere except 13. It has 12 cusps, 6 of them rational over ℚ and the other 6 rational over K, the maximal totally real subfield of ℚ(μ_{13}). The group Δ, which is dihedral of order 12, acts on X as a group of automorphisms. Δ consists of the six diamond bracket operators ⟨d⟩ with d ∈ (ℤ/13ℤ)^{∗}/ ± 1 and the six Atkin-Lehner involutions w_{ζ} for ζ ∈ μ_{13}, where w_{ζ} = w_{ζ^{−1}}.

Explicitly, if (E, P) corresponds to a “point” of X, where E is an elliptic curve

and P a point of order 13 on E, then ⟨d⟩ : (E, P) ↦ (E, dP) and w_{ζ} : (E, P) ↦ (E′, ϕ(Q)), where ϕ : E → E′ is the isogeny with cyclic kernel generated by P and Q is a point on E whose Weil pairing with P is ζ.

It is not hard to see that the hyperelliptic involution on X coincides with the

automorphism ⟨5⟩. Indeed, it follows from [37, p. 44] (where ⟨d⟩ is called γ_{d}) that the involution ⟨5⟩ = ⟨2⟩^{3} acts on J = J_{1}(13) as −1, so it must be the hyperelliptic involution. Similarly, the quotient of X by the cubic automorphism ⟨3⟩ has genus zero. One way to see this is that by the Riemann-Hurwitz formula, ⟨3⟩ either has four fixed points and X/⟨3⟩ has genus zero, or else ⟨3⟩ has 1 fixed point and X/⟨3⟩ has genus one. The latter is impossible, as can be seen by applying Riemann-Hurwitz to the degree 6 Galois covering X → X/⟨2⟩. (Alternatively, we will construct momentarily 4 fixed points of ⟨3⟩). A similar application of Riemann-Hurwitz shows that the 6 hyperelliptic branch points P_{i} and the four fixed points Q_{i} of ⟨3⟩ are all distinct.

The 6 rational cusps generate a subgroup of J(ℚ) of order 19 (in fact, this is the

full Mordell-Weil group of J). Since Δ acts freely on the set of cusps, all 12 cusps C_{i} lie in a common torsion packet, which we refer to as the cuspidal torsion packet T. We also see from the freeness of this action that the 22 points C_{i}, P_{i}, Q_{i} are all distinct. In addition, they all lie in the torsion packet T. This follows directly from the very useful Lemma 6.1 of [9]. Clearly T is stable under Gal(\overline{ℚ}/ℚ).

It remains to show that the torsion packet T is ramified above the primes 2,3, and

13. It is ramified at 13, for example, because six of the cusps are defined only over K, which is totally ramified at 13. We can see that T is ramified at 2 from the fact that the divisor classes [(P_{i}) − (P_{j})] generate the entire group J[2] of 2-torsion points on J, which is certainly ramified at 2.

It suffices, then, to show that the points Q_{i} are ramified at 3. To see this, we identify these points quite explicitly. Let E/ℚ have complex multiplication by ℤ[ω], where ω is a cube root of unity (so that j(E) = 0). The prime ideal 13 splits in ℤ[ω] as a product (13) = (π)(π′) of principal ideals. Since (ω − 3)(ω − 9) ≡ 0 (mod π), we have that either ω ≡ 3 or ω ≡ 9 (mod π). Without loss of generality, let us assume that ω ≡ 3 (mod π). Then if we choose a point P ∈ E[π], it follows that [ω] induces an isomorphism between the pairs (E, P) and (E, 3P). So (E, P) corresponds to a fixed point of ⟨3⟩. (Conversely, it is not hard to see that all four fixed points of ⟨3⟩ arise in this way, either from P ∈ E[π] or P ∈ E[π′].)

We claim that a field of definition for a pair (E, P) as above must contain ℚ(\sqrt{−3}),

hence is ramified at 3. In other words, we want to know that every element of Gal(\overline{ℚ}/ℚ) which fixes P also fixes ω. This is clear, because if τ ∈ Gal(\overline{ℚ}/ℚ) fixes P but not ω, then τπ = π′ and τP = P. This implies that P ∈ E[π] and τP = P ∈ E[π′], but the intersection of E[π] and E[π′] is trivial. This proves the claim and finishes the proof that T is ramified at 2, 3, and 13.

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