Rational points on curves

Henri Darmon

March 27, 2008

Contents

1 Preliminaries 9 1.1 Zero-dimensional varieties . . . . . . . . . . . . . . . . . . . . 9 1.2 Etale morphisms and the Chevalley–Weil theorem . . . . . . . 10

2 Faltings’ theorem 15 2.1 Prelude: the Shafarevich problem . . . . . . . . . . . . . . . . 15 2.2 First reduction: the Kodaira–Parshin trick . . . . . . . . . . . 17 2.3 Second reduction: passing to the jacobian . . . . . . . . . . . 19 2.4 Third reduction: passing to isogeny classes . . . . . . . . . . . 19 2.5 Fourth reduction: from isogeny classes to ℓ-adic representations 21 2.6 The isogeny conjecture . . . . . . . . . . . . . . . . . . . . . . 24 2.7 The finiteness principle for rational ℓ-adic representations … 26 2.8 A summary of Faltings’ proof . . . . . . . . . . . . . . . . . . 28

3 Modular curves and Mazur’s theorem 29 3.1 Modular curves . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Mazur’s criterion . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3 The jacobian J_{0}(p) . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4 The Birch and Swinnerton-Dyer conjecture . . . . . . . . . . . 36 3.5 Hecke theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.6 The winding quotient . . . . . . . . . . . . . . . . . . . . . . . 39 3.7 More results and questions . . . . . . . . . . . . . . . . . . . . 40 4 Fermat curves 42

4.1 Motivation for the strategy . . . . . . . . . . . . . . . . . . . . 43 4.2 Galois representations associated to Frey curves . . . . . . . . 45 4.3 Modular forms and Galois representations . . . . . . . . . . . 47 4.4 Serre’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.5 The Shimura–Taniyama conjecture . . . . . . . . . . . . . . . 49 4.6 A summary of Wiles’ proof . . . . . . . . . . . . . . . . . . . . 51

5 Elliptic curves 53 5.1 Modular parametrisations . . . . . . . . . . . . . . . . . . . . 54 5.2 Heegner points . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.3 Proof of Theorem 5.3 . . . . . . . . . . . . . . . . . . . . . . 58 5.4 Modularity of elliptic curves over totally real fields . . . . . . . 59 5.5 Shimura curves . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.6 Stark–Heegner points . . . . . . . . . . . . . . . . . . . . . . . 61

Introduction

Algebraic number theory is first and foremost the study of diophantine equations. Such a definition is arguably too narrow for a subject whose scope has expanded over the years to encompass an ever-growing list of fundamental notions: number fields and their class groups, abelian varieties, moduli spaces, Galois representations, p-divisible groups, modular forms, Shimura varieties, and L-functions, to name just a few. All of these subjects will be broached (sometimes too briefly, for reasons having less to do with their relative importance than with limitations of time, space, and the author’s grasp of the subject) in this survey, which is devoted to the first nontrivial class of diophantine equations: those associated to varieties of dimension one, or algebraic curves.

The term diophantine equation refers to a system of polynomial equations

X : \{ \begin{matrix}f_{1}(x_{1}, …, x_{n}) = 0 \\ ⋮ & (with f_{i} ∈ Z[x_{1}, …, x_{n}]). \\ f_{m}(x_{1}, …, x_{n}) = 0 \end{matrix}  (1)

Given such a system, one wishes to understand (and, if possible, determine completely) its set of integer or rational solutions.

Little of the essential features of the question are lost, and much flexibility is gained, if one replaces the base ring Z by a more general ring O. The prototypical examples are the ring of integers O_{K} of a number field K, or the ring O_{K,S} of its S-integers, for a suitable finite set S of primes of O_{K}.

Fix such a base ring O = O_{K,S} from now on, and assume that the polynomials in (1) have coefficients in O.

If R is any O-algebra, the set of solutions of (1) with coordinates in R is denoted X(R):

X(R) := {(x_{1}, …, x_{n}) ∈ R^{n} satisfying (1)} .

The functor R ↦ X(R) from the category of O-algebras to the category of sets is representable,

X(R) = Hom_{O}(A_{X}, R), where A_{X} = O[x_{1}, …, x_{n}]/(f_{1}, …, f_{m}). (2)

In this way the system (1) determines the affine scheme X := Spec(A_{X}) over Spec(O).

When the polynomials in (1) are homogenous, it is customary to view X as giving rise to a projective scheme over O. When R is a principal ideal domain, the set X(R) is a subset of the set ℙ_{n−1}(R) of n-tuples (x_{1}, …, x_{n}) ∈ R^{n} satisfying Rx_{1} + ⋯ + Rx_{n} = R, taken modulo the equivalence relation defined by

(x_{1}, …, x_{n}) ∼ (x′_{1}, …, x′_{n}) if x_{i}x′_{j} − x_{j}x′_{i} = 0, ∀ 1 ≤ i, j ≤ n.

Specifically,

X(R) := {(x_{1}, …, x_{n}) ∈ ℙ_{n−1}(R) satisfying (1)} .

In the projective setting, replacing the base ring O by its fraction field K, and X by its generic fiber X_{K}—a projective variety over K—does not change the diophantine problem. For instance, the natural map X(O) ⟶ X_{K}(K) is a bijection. So there is no distinction between the study of integral and rational points on a scheme whose generic fiber is a projective variety.

Here are some of the basic questions that can be asked about the behaviour of X(O).

Question 1. What is the cardinality of X(O)? Is it finite, or infinite?

Question 2. If X(O) is finite, can its cardinality be bounded by a quantity depending in a simple way on X and O?

Question 3. Can X(O) be effectively determined?

The arithmetic complexity of a point P ∈ X(O)—roughly speaking, the amount of space that would be required to store the coordinates of P on a computer—is measured by a (logarithmic) height function

h : X(O) ⟶ R.

The precise definitions and basic properties of heights are discussed elsewhere in this volume. Let us just mention that for any real B > 0, the number N(X; B) of P ∈ X(O) with h(P) ≤ B is finite, in any reasonable definition of h.

Question 4. When X(O) is infinite, what can be said about the asymptotics of the function N(X; B) as B ⟶ ∞?

A related question is concerned with the equidistribution properties of the points in X(O) (ordered by increasing height), relative to some natural measure on X(R) or X(C).

An algebraic curve over O is a scheme X (either affine or projective) of relative dimension one over Spec(O). If its generic fiber is smooth, the set X(C) (relative to a chosen embedding of O into C, through which C becomes an O-algebra) is a one-dimensional complex manifold. While a curve is often described by equations like (1), it is to be viewed up to isomorphism, as an equivalence class of such equations modulo suitable changes of variables. The main objects we will study are curves X over Spec(O), and the behaviour of the sets X(R) as R ranges over different O-algebras.

Remark. The term “integral points on elliptic curves” is often used (particularly by number theorists) to refer to to the integral solutions of an affine Weierstrass equation:

E_{0} : y^{2} = x^{3} + ax + b

which describes an affine curve over the base ring Z[a, b]. This is an abuse of terminology, since elliptic curves are always defined as projective varieties by passing to the projective equation

E : y^{2}z = x^{3} + axz^{2} + bz^{3},

resulting in the addition of the “point at infinity” O := (0, 1, 0) to E_{0}. This passage is crucial. Note, for instance, that E has the structure of an algebraic group, while E_{0} does not. It should be kept in mind that the common usage “integral points on E” refers to the integral points on the affine curve E_{0} = E − {O}, which is not an elliptic curve at all, and that, according to the definitions in standard usage, E(O) is equal to E(K) because E is projective.

The fundamental trichotomy for curves Suppose that the curve X is generically smooth, i.e., its generic fiber is a nonsingular curve over K, so that X(C) has the structure of a smooth Riemann surface. The set X(C) is (topologically and analytically) identified with

X(C) ≃ S − {P_{1}, …, P_{s}},

where S is a compact Riemann surface (of genus g, say) and P_{1}, …, P_{s} are distinct points. The invariants g and s, which completely determine the topological isomorphism class of X(C), can be packaged into the Euler characteristic

χ(X) = 2 − 2g − s.

The answers to Questions 1—4 above depend on the sign of χ(X) in an essential way.

I. Positive Euler characteristic. If χ(X) > 0, then g = 0 and s = 0 or 1. Therefore X is isomorphic over K̄ either to the projective line ℙ_{1} or the affine line 𝔸^{1}. Forms of ℙ_{1} over K correspond to conics, for which one has the following basic result.

Theorem 5. Let X be a smooth conic over K. The following are equivalent.

(a) The curve X is isomorphic over K to ℙ_{1}.

(b) The set X(K) is nonempty.

(c) The set X(K_{v}) is nonempty, for all completions K_{v} of K.

The equivalence between (a) and (b) follows from the Riemann–Roch theorem: given a rational point ∞ ∈ X(K), there is a rational function with only a simple pole at ∞; such a function gives an isomorphism between X and ℙ_{1} over K. The equivalence between (a) and (c) is the Hasse–Minkowski theorem, one of the most basic instances of the so-called local-global principle which is discussed at greater length elsewhere in this volume.

Remark 6. The proof of the Hasse–Minkowski theorem, which relies on geometry of numbers, leads to an upper bound on the smallest height of a point on X(K), and thus is effective. Attempts to generalise Theorem 5 to higher dimensional varieties have led to a rich theory which forms the basis for some of the articles in this volume.

The case of positive Euler characteristic, for which the basic questions 1—4 are in some sense well-understood thanks to Theorem 5, will not be treated any further in these notes.

II. Euler characteristic zero. There are two types of curve with Euler characteristic zero:

• The affine case: g = 0 and s = 2.

• The projective case: g = 1 and s = 0.

The prototypical example of the affine case is when

X = ℙ_{1} − {0, ∞} = 𝔾_{m}.

The set X(O) = O^{×} is an abelian group under multiplication, and X is naturally equipped with the structure of a commutative group scheme over O. Something similar happens in the projective case: since X is a curve of genus one, it is isomorphic over K either to an elliptic curve, if X(K) ≠ ∅, or to a principal homogeneous space over such a curve. For the following theorem, suppose that X(O) ≠ ∅, and that X can be equipped with the structure of a group scheme over O.

Theorem 7. The group X(O) is finitely generated.

In the affine case, Theorem 7 is essentially Dirichlet’s S-unit theorem, while in the projective case it corresponds to the Mordell–Weil Theorem that the group of rational points on an elliptic curve over a number field is finitely generated.

III. Negative Euler characteristic. The theory of curves with negative Euler characteristic is dominated by the following basic finiteness result.

Theorem 8. If χ(X) < 0, then X(O) is finite.

In the affine case this is a theorem of Siegel proved in 1929, and can be reduced to the most interesting special case where X = ℙ_{1} − {0, 1, ∞}. The points in X(O) then correspond to solutions to the so-called S-unit equation

u + v = 1, with u, v ∈ O^{×}.

In the projective case Theorem 8 used to be known as the Mordell Conjecture. Its proof by Faltings in 1983 represents a significant achievement in the diophantine theory of curves.

We now describe the contents of these notes.

Chapter 1 recalls some preliminary results that are used heavily in later chapters: the main finiteness results of algebraic number theory, and the method of descent based on unramified coverings and the Chevalley–Weil theorem. Hugo Chapdelaine’s article [Chap] in these proceedings further develops these themes by describing a relatively elementary application of Falting’s theorem to a diophantine equation—the generalised Fermat equation x^{p} + y^{q} + z^{r} = 0—that appears to fall somewhat beyond the scope of the study of algebraic curves, but to which, it turns out, the “fundamental trichotomy” described in this introduction can still be applied.

The main goal of Chapter 2 is to give a survey of Faltings’ proof of the Mordell Conjecture. In many ways, this chapter forms the heart of these notes. The ideas in Chapter 2 are used to motivate the startlingly diverse array of techniques that arise in the diophantine study of curves. These techniques are deployed in subsequent chapters to study several important and illustrative classes of algebraic curves—specifically, modular curves, Fermat curves, and elliptic curves.

Chapter 3 focusses on what may appear at first glance to be a rather special collection of algebraic curves, the so-called modular curves over Q classifying isomorphism classes of elliptic curves with extra level structure. Singling out modular curves for careful study can be justified on (at least) two grounds.

1. They are the simplest examples of moduli spaces. Classifying the rational points on modular curves translates into “uniform boundedness” statements for the size of torsion subgroups of elliptic curves over Q, and therefore leads to nontrivial results concerning rational points on curves of genus one.

2. Modular curves are also the simplest examples of Shimura varieties, and their jacobians and ℓ-adic cohomology are closely tied to spaces of modular forms. (It is from this connection that they derive their name.) This makes it feasible to address finer questions about the rational points on modular curves, following a line of attack that was initiated by Mazur [Ma1] in his landmark paper on the Eisenstein ideal.

Chapter 3 attempts to convey some of the flavour of Mazur’s approach by describing a simple but illustrative special case of his general results: namely, his proof of the conjecture, originally due to Ogg, that the size of the torsion subgroup of elliptic curves over Q is uniformly bounded, by 14. The approach we describe incorporates an important strengthening due to Merel exploiting progress on the Birch and Swinnerton-Dyer conjecture that grew out of later work of Gross–Zagier and Kolyvagin–Logachev. Marusia Rebolledo’s article [Reb] in these proceedings takes this development one step further by describing Merel’s proof of the strong uniform boundedness conjecture over number fields: given d ≥ 1, the modular curves Y_{1}(p) contain no points of degree d when p is large enough (relative to d).

Chapter 4 describes the approach initiated by Frey, Serre, and Ribet for reducing Fermat’s Last Theorem to deep questions about the relationship between elliptic curves and modular forms. This subject is only lightly touched upon in these notes. Pierre Charollois’s article in this volume [Char] describes a technique of Halberstadt and Kraus that strengthens the “modular approach” to prove a result on the generalised Fermat equation ax^{p}+by^{p}+cz^{p} = 0 that is notable for its generality. This result also suggests that it might be profitable to view the modular approach as part of a general method, rather than just a serendipitous “trick” for proving Fermat’s Last Theorem.

Chapter 5 gives a rapid summary of the author’s second week of lectures at the Göttingen summer school, devoted largely to curves of genus 1, particularly elliptic curves. This chapter is less detailed than the others, partly because it covers topics that have already been treated elsewhere, notably in [Da2]. The main topics that are touched upon (albeit briefly) in Chapter 5 are:

1. The collection of Heegner points on a modular elliptic curve, and Kolyvagin’s use of them to prove essentially all of the Birch and Swinnerton-Dyer conjecture for elliptic curves with analytic rank ≤ 1. Kolyvagin’s techniques also supply a crucial ingredient in Merel’s proof of the uniform boundedness conjecture, further justifying its inclusion as a topic in the present notes. The article by Samit Dasgupta and John Voight [DV] in these proceedings describes an application of the theory of Heegner points to Sylvester’s conjecture on the primes that can be expressed as a sum of two rational cubes.

2. Variants of the modular parametrisation which can be used to produce more general systems of algebraic points on elliptic curves over Q. Such systems are likely to continue to play an important role in further progress on the Birch and Swinnerton-Dyer conjecture. A key example is the fact that many elliptic curves defined over totally real fields are expected to occur as factors of the jacobians of Shimura curves attached to certain quaternion algebras. The articles by John Voight [Vo] and Matthew Greenberg [Gre1] in these proceedings discuss the problem of calculating with Shimura curves and their associated parametrisations from two different angles: from the point of view of producing explicit equations in [Vo], and relying on the Cerednik–Drinfeld p-adic uniformisation in [Gre1].

3. The theory of Stark–Heegner points which is meant to generalise classical Heegner points. Matthew Greenberg’s second article [Gre2] in these proceedings discusses Stark–Heegner points attached to elliptic curves over imaginary quadratic fields. Proving the existence and basic algebraicity properties of the points that Greenberg describes how to calculate numerically would lead to significant progress on the Birch and Swinnerton-Dyer conjecture—at present, there is no elliptic curve that is “genuinely” defined over a quadratic imaginary field for which this conjecture is proved in even its weakest form.

1 Preliminaries

1.1 Zero-dimensional varieties

In order to get a good understanding of algebraic varieties of dimension d+1, it is useful to understand the totality of algebraic varieties of dimension d. Such a principle is hardly surprising, since a (d + 1)-dimensional variety can be expressed as a family of d-dimensional varieties, parametrized by a one-dimensional base. Any discussion of the diophantine properties of curves must therefore necessarily begin with a mention of the zero-dimensional case.

A zero-dimensional variety (of finite type) over a field K is an affine scheme of the form X = Spec(R), where R is a finite-dimensional commutative K-algebra without nilpotent elements. Let

n := #X(K̄) = #Hom(R, K̄) = dim_{K}(R),

where K̄ denotes as usual an algebraic closure of the field K. Finding the rational points on X amounts to solving a degree n polynomial in one variable over K.

An integral model of X over O is an affine scheme of the form Spec(R_{O}), where R_{O} ⊂ R is an O-algebra satisfying R_{O} ⊗_{O} K = R. Such a model is said to be smooth if R_{O} is finitely generated as an O-module and R_{O}/𝔭 is a ring without nilpotent elements for all 𝔭 ∈ Spec(O). The reader can check that X has a smooth model over Spec(O) if and only if R = ∏_{i} L_{i} is a product of field extensions L_{i}/K which are unramified outside of S.

It is of interest to consider the collection of zero-dimensional varieties of fixed cardinality n which possesses a smooth model over Spec(O). The following classical finiteness result is extremely useful in the study of curves.

Theorem 1.1 (Hermite–Minkowski). Given n and O = O_{K,S}, there are finitely many isomorphism classes of varieties of cardinality n over K which possess a smooth model over Spec(O). Equivalently, there are finitely many field extensions of K of degree at most n which are unramified outside of S.

The proof is explained, for example, in [Sz], p. 91. In the simplest special case where K = Q and S = ∅, we mention the following more precise statement:

Theorem 1.2 (Minkowski). Any zero–dimensional variety over Q which has a smooth model over Spec(Z) is isomorphic to Spec(Q^{n}) for some n ≥ 1. Equivalently, there are no nontrivial everywhere unramified field extensions of Q.

1.2 Etale morphisms and the Chevalley–Weil theorem

If π : X ⟶ Y is a nonconstant, finite morphism of projective curves defined over K (or of affine curves over O = O_{K,S}), then π induces finite-to-one maps π_{K} : X(K) ⟶ Y (K) and π_{O} : X(O) ⟶ Y (O). In particular, if Y (K) is finite, then so is X(K). This simple principle reduces the study of rational points on a curve X to the often simpler study of points on the image curve Y. (For instance, the genus of Y is less than or equal to the genus of X, by the Riemann–Hurwitz formula.) As an historical illustration, Fermat proved that the equation x^{4} + y^{4} = z^{4} (which corresponds to a projective curve of genus 3 over Q) has no nontrivial rational points by studying the integer solutions of the auxiliary equation x^{4} + y^{4} = z^{2} which are primitive in the sense of [Chap]. These primitive solutions correspond to rational points on a curve of genus one (in line with the principles explained in [Chap]), and Fermat is able to dispose of these rational points by his method of descent.

In contrast, the finiteness of X(K) does not imply the finiteness of Y (K) in general, because the maps π_{K} or π_{O} need not be surjective, and in fact are usually far from being so. The following weakening of the notion of surjectivity is frequently useful in practice.

Definition 1.3. The map π : X ⟶ Y of curves over Spec(O_{K,S}) is said to be almost surjective if there is a finite extension L of K and a finite set T of primes of L containing the primes above those in S, such that Y (O_{K,S}) is contained in the image of π_{O_{L,T}}.

Definition 1.4. A morphism π : X ⟶ Y of curves over Spec(O_{K,S}) is said to be generically étale if it satisfies any of the following equivalent conditions:

(a) The induced map π_{C} : X(C) ⟶ Y (C) is an unramified covering of Riemann surfaces;

(b) The map π_{K} : X_{K} ⟶ Y_{K} is an étale morphism of K-varieties on the generic fibers;

(c) There exists a finite set S′ ⊃ S of primes of K such that the map π_{O_{K,S′}} : X_{O_{K,S′}} → Y_{O_{K,S′}} is a finite étale morphism of schemes over Spec(O_{K,S′}).

The following result, known as the Chevalley–Weil theorem, gives a criterion for a map π to be almost surjective.

Theorem 1.5 (Chevalley–Weil). If the morphism π is generically étale, then it is almost surjective.

Proof. Suppose that π is generically étale. By Property (c) in the definition, we may suitably enlarge S so that the map π becomes étale over Spec(O_{K,S}).

If P belongs to Y (O) = Hom(Spec(O), Y), let P^{∗}(X) = π^{−1}(P) denote the fiber of π above P. This fiber can be described as a scheme over Spec(O) by viewing P as a morphism Spec(O) ⟶ Y, and π^{−1}(P) as the schemetheoretic pull-back of π to Spec(O) via P, for which the following diagram is cartesian

P^{∗}(X) ⟶ X
↓  ↓
Spec(O) ⟶^{P} Y.

Note that π^{−1}(P) is a zero-dimensional scheme over Spec(O) of cardinality n = deg(π), which is smooth because π is étale. By the Hermite–Minkowski theorem (Theorem 1.1) there are finitely many possibilities for π^{−1}(P), as P ranges over Y (O). Hence the compositum L of their fraction fields is a finite extension of K. Let T denote the set of primes of L above those in S. Then, by construction, Y (O) is contained in π(X(O_{L,T})). It follows that π is almost surjective.

Example 1.6. The Klein and Fermat curves. The quartic curve

Y : x^{3}y + y^{3}z + z^{3}x = 0  (3)

studied by Felix Klein is a curve of genus 3 having an automorphism group G = PSL_{2}(𝔽_{7}) of order 168. By the Hurwitz bound, this is the largest number of automorphisms a curve of genus 3 may have. (A curve with this property is in fact unique up to Q̄-isomorphism.) The curve Y is also a model for the modular curve X(7). (Cf. Section 4.1 for a brief discussion of X(n).) The automorphism group PSL_{2}(7) arises from the transformations that preserve the fibers of the natural projection of Y (7) onto the j-line. In [Hur], Hurwitz proved that Y has no nontrivial rational points, as follows: let (x, y, z) be a point on the Klein quartic with integer coordinates, satisfying gcd(x, y, z) = 1. Although x, y and z have no common factor, they need not be pairwise coprime; setting

u = gcd(x, y), v = gcd(y, z), w = gcd(z, x),

one sees (after changing the signs of u, v, and/or w if necessary) that

(x, y, z) = (u^{3}w, v^{3}u, w^{3}v) =: π(u, v, w).  (4)

Substituting back into the original equation (3) and dividing by u^{3}v^{3}w^{3}, one finds that (u, v, w) is a rational point on the Fermat curve of degree 7:

X : u^{7} + v^{7} + w^{7} = 0.

Through this argument, Hurwitz shows that the map π : X ⟶ Y given by (4), a generically étale map of degree 7, is almost surjective (in fact, surjective) on rational points. This is a simple special case of Theorem 1.5. Hurwitz then applies Lamé’s result for the Fermat equation of degree 7 to conclude that the Klein quartic has no integer solutions except the trivial ones.

Note that this example gives a nontrivial diophantine relation between modular curves and Fermat curves. More sophisticated connections between these two classes of curves are discussed in Section 4.

Example 1.7. Algebraic groups. Recall that O is the ring of S-integers of a number field K. Let G be any commutative group scheme of finite type over Spec(O). Then for any integer n ≥ 1, the morphism [n] given by g ↦ g^{n} is generically étale (more precisely, étale over Spec(O[1/n])). Therefore, the Chevalley-Weil theorem implies that there is a finite extension L of K for which G(O)/nG(O) maps to the kernel of the natural map G(K)/nG(K) ⟶ G(L)/nG(L). A standard construction shows that this kernel injects into the finite group H^{1}(Gal(L/K), G[n](L)), where G[n](L) is the finite group of n-torsion-points on G(L). It follows that G(O)/nG(O) is finite. (When G = 𝔾_{m}, this statement is a weak form of Dirichlet’s S-unit theorem, while when G = A is an elliptic curve or an abelian variety, it is the weak Mordell– Weil theorem asserting that A(K)/nA(K) is finite.)

Example 1.8. It is not hard to exhibit a projective curve X of genus greater than 1 eqipped with a map π : X ⟶ ℙ_{1} which is unramified outside {0, 1, ∞}. Examples include

(a) The Fermat curve x^{n} + y^{n} = z^{n} with π(x, y, z) = x^{n}/z^{n};

(b) The modular curves X_{0}(n) and X_{1}(n) introduced in Section 3.1, with their natural maps to the j-line.

One can use the map π to show that Theorem 8 for projective curves (Faltings’ Theorem) implies the case X = ℙ_{1}−{0, 1, ∞} over Spec(O) of Theorem 8 (Siegel’s Theorem).

More generally, a celebrated theorem of Belyi asserts that any projective curve X/K can be equipped with a morphism π : X ⟶ ℙ_{1} which is unramified outside {0, 1, ∞}. (See Hugo Chapdelaine’s article in these proceedings.) This fact has been exploited by Elkies [El] to prove that the abc conjecture implies Faltings’ theorem.

Further topic: Hugo Chapdelaine’s article in these proceedings explains how the discussion of unramified coverings and the Chevalley–Weil Theorem can be adapted to treat the primitive solutions of the generalised Fermat equation x^{p} + y^{q} + z^{r} = 0. The reader who has mastered the ideas in Section 1 may skip directly to this article if so inclined.

2 Faltings’ theorem

This chapter is devoted to explaining the main ideas in Faltings’ proof of the Mordell Conjecture (Theorem 8 for projective curves over number fields).

Theorem 2.1 (Faltings). Let X be a smooth projective curve of genus ≥ 2 defined over a number field K. Then X(K) is finite.

The proof will be presented as a series of reductions.

2.1 Prelude: the Shafarevich problem

The first of these reductions, explained in Section 2.2, reduces Theorem 2.1 to a finiteness conjecture of Shafarevich. The Shafarevich problem is concerned with the collection of all arithmetic objects sharing certain common features and having “good reduction” over the ring O of S-integers of a number field K, taken, of course, up to isomorphism over K. Some key examples are:

1. the set F_{d}(O) of smooth zero-dimensional schemes over Spec(O) of cardinality d;

2. the set M_{g}(O) of smooth curves of genus g over Spec(O);

3. the set A_{g}(O) of abelian schemes of dimension g over Spec(O);

4. the set I_{g}(O) of K-isogeny classes of abelian varieties of dimension g over Spec(O).

The following question is known as the Shafarevich problem:

Question 2.2. How large are the sets above? Are they finite?

One can also ask what happens for specific values of K and S, the most interesting special case being O = Z (i.e., K = Q and S = ∅).

We now discuss these questions for the various cases listed above:

1. The set F_{d}(O) corresponds to the set of étale K-algebras (i.e., products of seperable field extensions) of rank d over K which are unramified outside S. The finiteness of F_{d}(O) is just a restatement of the Hermite– Minkowski Theorem (Theorem 1.1).

2. The set M_{0}(O) consists of the set of smooth conics over K which have good reduction outside of S. It admits a cohomological interpretation, via the exact sequence

0 ⟶ M_{0}(O) ⟶ H^{2}(K, ±1) ⟶ ⊕_{v∉S}H^{2}(K_{v}, ±1).

The fundamental results of local and global class field theory imply that M_{0}(O) is finite, and in fact, its order can be evaluated precisely:

#M_{0}(O) = 2^{#S+r−1},

where r is the number of real places of K. In particular, when K = Q and S = ∅, then M_{0}(Z) consists of one element, corresponding to the projective line ℙ_{1} over Q.

3. The set M_{1}(O) can be infinite; in fact, an infinite set of curves of genus 1 which are all isomorphic over K̄ and have good reduction outside of S can sometimes be found, even if S consists of just one prime of K. (See [Ma3], p. 241.) On the other hand, a deep conjecture of Shafarevich and Tate implies that M_{1}(O) is finite if S is empty. Also, if one replaces M_{1} by the set E of K-isomorphism classes of elliptic curves, i.e., curves of genus 1 equipped with a K-rational point, then Shafarevich [Sha] showed that E(O) is always finite.

When g > 1, the following conjecture of Shafarevich can be viewed as a one-dimensional analogue of the Hermite–Minkowski theorem (Theorem 1.1):

Conjecture 2.3. Let g ≥ 2 be an integer, and let O be the ring of S-integers of a number field K, for a finite set S of primes of K.

(a) (Shafarevich conjecture for curves). The set M_{g}(O) is finite, i.e., there are only finitely many K-isomorphism classes of curves of genus g defined over K and having good reduction outside of S.

(b) (Shafarevich conjecture for abelian varieties). The set A_{g}(O) is finite, i.e., there are only finitely many isomorphism classes of abelian varieties of dimension g defined over K and having good reduction outside of S.

(c) (Shafarevich conjecture for isogeny classes). The set I_{g}(O) is finite, i.e., there are only finitely many K-isogeny classes of abelian varieties of dimension g with good reduction outside of S.

Remark 2.4. It is a deep theorem of Fontaine [Fo] that the sets A(Z) and M_{g}(Z) are empty for g ≥ 2, i.e., there are no abelian varieties, or smooth curves of genus ≥ 2, over Spec(Z).

2.2 First reduction: the Kodaira–Parshin trick

In [Pa1], Parshin showed that part 2.3 of Conjecture 2.3 implies Theorem 2.1.

Theorem 2.5. (Kodaira–Parshin). The Shafarevich conjecture for curves implies Mordell’s conjecture.

Sketch of proof. Let X be a curve of genus g > 1 defined over a number field K. To each point P ∈ X(K) one associates a curve X_{P} and a covering map ϕ_{P} : X_{P} ⟶ X with the following properties:

1. The curve X_{P} and the map ϕ_{P} can be defined over a finite extension K′ of K which does not depend on P.

2. The genus g′ of X_{P} (and the degree of ϕ_{P}) is fixed and in particular does not depend on P.

3. The map ϕ_{P} is ramified only over the point P.

4. The curve X_{P} has good reduction outside a finite set of primes S′ of K′ which does not depend on P.

For a description of this assignment, see [Ma3], p. 243-244, [FW], p. 191-197, or [Pa1]. The reader should note that one has some leeway in constructing it, and that different versions appear in the literature.

We will describe one approach here, which consists in considering the embedding X ⟶ J of X into its jacobian that sends P to the origin of J, and letting X̃ be the pullback to X of the multiplication-by-2 map [2] : J ⟶ J. This map induces an unramified covering π : X̃ ⟶ X of degree 2^{2g}, and hence the genus of X̃ can be calculated explicitly using the Riemann–Hurwitz formula. The fiber π^{−1}(P) can be written as

π^{−1}(P) = P̃ + D,

where P̃ corresponds to the identity element of J, and hence belongs to X̃(K), and D is an effective divisor of degree 2^{2g} − 1 defined over K with support disjoint from P̃. Let J_{D} be the generalised jacobian attached to X̃ and D: the group J_{D}(K̄) is identified with the group of degree zero divisors on X̃ with support outside D, modulo the subgroup of principal divisors of the form div(f), as f ranges over the functions satisfying f(D_{0}) = 1, for all degree zero divisors D_{0} supported on D. The functor L ↦ J_{D}(K̄)^{G_{L}} (where G_{L} := Gal(L̄/L)) on finite extensions of K is representable by the algebraic group over K denoted J_{D}, which is an extension of J by a torus T over K of rank (2^{2g} − 2). In other words, there is a natural exact sequence

1 ⟶ T ⟶ J_{D} ⟶ J ⟶ 1

of commutative algebraic groups over K.

One can embed X̃ − D into J_{D} by sending a point Q to the equivalence class of the divisor (Q)−(P̃). The multiplication-by-2 map [2] on J_{D} induces a map X_{P}^{0} ⟶ X̃−D, as summarised by the following diagram with cartesian squares in which the vertical maps are induced by multiplication by 2:

\begin{aligned}J_{D} & ⟵ & X_{P}^{0} \\ ↓ & & ↓ \\ J_{D} & ⟵ & X̃ − D & ⟶ & J \\ & & ↓ & & ↓ \\ & & X & ⟶ & J. \end{aligned}  (5)

The closure X_{P} of X_{P}^{0} has the desired properties 1-4: it is defined over K, and it follows directly from the Riemann–Hurwitz formula that its genus g′ does not depend on P. Furthermore, the map X_{P}^{0} ⟶ X̃ − D is unramified, and hence X_{P} ⟶ X is ramified only over the point P. Finally, if X is smooth over Spec(O), the curve X_{P} has a smooth model over O′ := O[1/2].

The assignment P ↦ X_{P} therefore gives rise to a well-defined map

R_{1} : X(K) ⟶ M_{g′}(O′).

But this assignment is finite-to-one; for otherwise there would be a curve Y and infinitely many (by property 3) distinct maps ϕ_{P} : Y ⟶ X. This would contradict the following geometric finiteness result of De Franchis (cf. [Ma3], p. 227).

Theorem 2.6. If X and Y are curves over any field K, and Y has genus g ≥ 2, then the set Mor_{K}(X, Y) of K-morphisms from X to Y is finite.

The Shafarevich conjecture for curves, which asserts the finiteness of M_{g′}(O′), therefore implies the finiteness of X(K). This completes the proof of Theorem 2.5.

Remark 2.7. The reader will note that the proof of Theorem 2.5 breaks down (as it should!) when g = 1, because the set Mor_{K}(Y, X) can be (and in fact, frequently is) infinite when X has genus 1.

2.3 Second reduction: passing to the jacobian

The second step in the proof of the Mordell conjecture consists in observing that the Shafarevich conjecture for curves would follow from the corresponding statement for abelian varieties.

Proposition 2.8. The Shafarevich conjecture for curves follows from the Shafarevich conjecture for abelian varieties.

To prove Proposition 2.8, one studies the map R_{2} which associates to a curve X its jacobian J. If X is smooth over Spec(O), the same is true of J, and hence R_{2} defines a map M_{g}(O) ⟶ A_{g}(O). Key to Proposition 2.8 is the following corollary of Torelli’s theorem:

Theorem 2.9. If g ≥ 2, then the map R_{2} is finite-to-one.

Proof. Torelli’s theorem asserts that a curve X of genus ≥ 2 can be recovered by the data of its jacobian J together with the principal polarisation associated to the Riemann theta-divisor. But a given abelian variety can carry only finitely many principal polarisations. (See [CS] for a more detailed exposition of the Torelli Theorem and surrounding concepts.)

2.4 Third reduction: passing to isogeny classes

The third, crucial and more difficult reduction was carried out by Faltings himself.

Theorem 2.10. (Faltings). The Shafarevich conjecture for abelian varieties follows from the Shafarevich conjecture for isogeny classes.

As one would expect, the proof is based on showing that the natural map R_{3} : A_{g}(O) ⟶ I_{g}(O) has finite fibers. This is a consequence of the following key result:

Theorem 2.11. (Faltings) There are finitely many isomorphism classes of abelian varieties over K in a given K-isogeny class.

This result is the technical heart of Faltings’ proof, and rests on his theory of heights on moduli spaces of abelian varieties. Things become somewhat simpler if we assume that the abelian varieties in the isogeny class are semistable. This can be assumed without loss of generality because of Grothendieck’s semistable reduction theorem which asserts that every abelian variety becomes semistable after a finite extension of the ground field (for instance, one over which the points of order 3 become rational). For a finite extension K′/K, there are finitely many K-isomorphism classes of abelian varieties that are K′-isomorphic to a given abelian variety over K′, and hence the finiteness of the K-isogeny class follows from that of any K′-isogeny class.

Faltings defines a height function (now called the Faltings height) of an abelian variety. We will not dwell on the definition, but will content ourselves with stating two of its main finiteness properties:

Theorem 2.12. Let K be a number field and H be a positive constant. There are finitely many isomorphism classes of g-dimensional abelian varieties over K with height less than H.

The second finiteness property concerns the behaviour of the Faltings height on a K-isogeny class. Given a prime ℓ, the ℓ-isogeny class of an abelian variety A is the set of abelian varieties which are isogenous to A via an isogeny of ℓ-power degree. More generally, if M is any finite set of rational primes, two abelian varieties are said to be M-isogenous if they are related by a K-isogeny whose degree is a product of primes in M.

Theorem 2.13. If A is a semistable abelian variety over a number field K, then:

1. There exists a finite set M of rational primes, depending only on the isogeny class of A, such that if A ⟶ B is a K-isogeny of degree not divisible by the primes in M, then

h(A) = h(B).

2. For any finite set S of rational primes, the Faltings height is bounded on S-isogeny classes.

The proof of this theorem relies on deep results of Tate and Raynaud on group schemes and p-divisible groups; cf. Theorems 2.4 and 2.6 of [De].

For more details on the proof of theorems 2.12 and 2.13 see the expositions [CS], [FW], [Sz], [De], or [ZP]. Note that these two theorems together imply:

Proposition 2.14. Let A be a semistable abelian variety over K, and let M be as in part 1 of Theorem 2.13.

1. Up to K-isomorphism, there are finitely many abelian varieties that are K-isogenous to A via an isogeny of degree not divisible by the primes in M.

2. Given any abelian variety B over K and any finite set S of rational primes, there are finitely many abelian varieties in the S-isogeny class of B.

Proof of Theorem 2.11: Let ϕ : A ⟶ B be a K-isogeny. We can write ϕ as a composition of isogenies

A ⟶^{ϕ_{o}} B_{0} ⟶^{ϕ_{1}} B_{1},

where ϕ_{0} is of degree not divisible by the primes in M, and ϕ_{1} is an Misogeny. By part 1 of Proposition 2.14, there are finitely many possibilities for ϕ_{0} and for B_{0}. By part 2 of this proposition, for each B_{0} there are finitely many possibilities for B_{1}. Theorem 2.11 follows.

2.5 Fourth reduction: from isogeny classes to ℓ-adic representations

To an abelian variety A over K of dimension g and a prime ℓ, one can associate the ℓ-adic Tate module and ℓ-adic representations

T_{ℓ}(A) := lim_{⟵} A[ℓ^{n}], V_{ℓ}(A) := T_{ℓ}(A) ⊗ Q_{ℓ},

where the inverse limit is taken with respect to the multiplication-by-ℓ maps. The Q_{ℓ} vector space V_{ℓ}(A) is 2g-dimensional and is equipped with a Q_{ℓ}-linear action by two commuting Q_{ℓ}-algebras E and Π_{K} defined by

E = End_{K}(A) ⊗ Q_{ℓ}, Π_{K} := Z_{ℓ}[[G_{K}]] ⊗ Q_{ℓ}.

Here Z_{ℓ}[[G_{K}]] denotes the profinite group ring \underleftarrow{lim} Z_{ℓ}[Gal(L/K)], where the projective limit is taken over all finite Galois extensions L ⊂ K̄ of K.

If A and B are K-isogenous abelian varieties, they give rise to ℓ-adic representations that are isomorphic as Π_{K}-modules. In other words, the assignment A ↦ V_{ℓ}(A) yields a map

R_{4} : I_{g}(O) ⟶ \{ \begin{matrix}Isomorphism classes of \\ 2g-dimensional ℓ-adic \\ representations of Π_{K} \end{matrix} \} .

The strategy will now consist in showing that R_{4} has finite fibers, and finally in describing the image R_{4} precisely enough to show that it is finite.

We begin by introducing some further notations and recalling some background. Given a prime v of K, let I_{v} ⊂ G_{v} ⊂ G_{K} be inertia and decomposition subgroups of G_{K} attached to v. Note that the groups G_{v} and I_{v} are only well-defined up to conjugation in G_{K}, since they depend on a choice of a prime of K̄ above v. The quotient G_{v}/I_{v} is procyclic with a canonical generator Frob_{v} called the Frobenius element at v, which induces the automorphism x ↦ x^{Nv} on the residue field, where Nv denotes the norm of v (the cardinality of the associated residue field).

If V is any finite-dimensional Q_{ℓ}-vector space equipped with a continuous Π_{K}-action, we say that V is unramified at v if I_{v} acts trivially on V. When this happens, the Frobenius element Frob_{v} ∈ G_{v}/I_{v} gives an element of GL(V) which is well-defined up to conjugation in this group.

The following theorem lists some of the basic properties of V_{ℓ}(A).

Theorem 2.15. Let A be an abelian scheme over Spec(O_{K,S}). The ℓ-adic Galois representation V_{ℓ}(A) satisfies the following properties:

1. It is semisimple as a representation of E.

2. It is unramified at all v ∉ S′ := S ∪ {λ|ℓ}.

3. (Rationality) If v ∉ S′, then the characteristic polynomial of Frob_{v} has rational integer coefficients. The complex roots of this polynomial have absolute value Nv^{1/2}.

4. (Tate conjecture) The representation V_{ℓ}(A) is semisimple as a representation of Π_{K}.

Property 1 follows from the basic theory of duality for abelian varieties, and properties 2 and 3 were shown by Weil (cf. [We]). Property 4, a particular case of the Tate conjecture, is one of Faltings’ important contributions. We now explain how Faltings proved the semisimplicity of V_{ℓ}(A) over Π_{K}, adapting an idea used by Tate to prove the corresponding statement over finite fields.

Lemma 2.16. For every Π_{K}-invariant subspace W in V_{ℓ}(A), there is an element u ∈ E such that

uV_{ℓ}(A) = W.

Proof. The Z_{ℓ}-module W_{∞} = W ∩ T_{ℓ}(A) gives rise to a collection of groups W_{n} = W_{∞}/ℓ^{n}W_{∞} ⊂ A[ℓ^{n}] which are defined over K and compatible under the natural maps A[ℓ^{n+1}] ⟶ A[ℓ^{n}]. Let

α_{n} : A ⟶ A_{n} := A/W_{n},

be the natural isogeny with kernel W_{n}, and let β_{n} denote the isogeny characterised by

α_{n}β_{n} = ℓ^{n}, β_{n}α_{n} = ℓ^{n}.

Note that β_{n}(A_{n}[ℓ^{n}]) = W_{n} by construction, in light of the first identity above. By Faltings’ finiteness theorem 2.11, there exists an infinite set I = {n_{0}, n_{1}, …} ⊂ Z^{>0} for which there exist isomorphisms

ν_{i} : A_{n_{0}} ≃ A_{i}

for all i ∈ I. Now define a sequence of K-endomorphisms of A by the rule

u_{i} := β_{i}ν_{i}α_{n_{0}}.

Since End_{K}(A) ⊗ Z_{ℓ} is compact in the ℓ-adic topology, the sequence (u_{i}) has a convergent subsequence (u_{i})_{i∈J} in this topology. Let u denote the limit of such a subsequence. After eventually refining J further, we can assume that for each i ∈ J, we have natural maps

u(A[ℓ^{i}]) = u_{i}(A[ℓ^{i}]) ⟶ β_{i}(A_{i}[ℓ^{i}]) = W_{i},

with kernel and cokernel bounded independently of i, because they arise from α_{n_{0}}. It follows that

u(V_{ℓ}(A)) = W,

as was to be shown.

Corollary 2.17. The representation V_{ℓ}(A) is a semisimple Π_{K}-module.

Proof. Let W be a Π_{K}-stable subspace of V_{ℓ}(A), and let u ∈ E be an element constructed in Lemma 2.16, satisfying u(V_{ℓ}(A)) = W. Consider the right ideal uE in the algebra E. Because E is semisimple, this ideal is generated by an idempotent u_{0}. Note that u_{0}(V_{ℓ}(A)) = W. The subspace ker(u_{0}) is therefore a Π_{K}-stable complement of W in V_{ℓ}(A). Hence V_{ℓ}(A) is semisimple over Π_{K}.

In conclusion, let Rep_{S}(G_{K}, 2g) be the set of isomorphism classes of rational semisimple ℓ-adic representations of G_{K} of dimension 2g which are unramified outside of S. We have shown that R_{4} maps I_{g}(O) to Rep_{S}(G_{K}, 2g). To complete the proof of the Mordell conjecture, it remains to show:

1. The map R_{4} is finite-to-one.

2. The set Rep_{S}(G_{K}, 2g) is finite.

We will prove the first in the next section, and the second in Section 2.7.

2.6 The isogeny conjecture

The proof of the following deep conjecture of Tate is a cornerstone of Faltings’ strategy for proving the Mordell conjecture.

Theorem 2.18. (Isogeny conjecture). Let A and B be abelian varieties defined over a number field K. If V_{ℓ}(A) is isomorphic to V_{ℓ}(B) as a Π_{K}-module, then the abelian varieties A and B are isogenous.

In other words, the map R_{4} is injective.

We first note that Theorem 2.18 can be reduced to the following statement, known as the Tate conjecture for abelian varieties.

Theorem 2.19. (Tate conjecture). Let A and B be abelian varieties defined over K. Then the natural map

Hom_{K}(A, B) ⊗ Q_{ℓ} ⟶ Hom_{Π_{K}}(V_{ℓ}(A), V_{ℓ}(B))

is surjective.

To see that Theorem 2.19 implies Theorem 2.18, let j : V_{ℓ}(A) ≃ V_{ℓ}(B) be a Π_{K}-equivariant isomorphism. By Theorem 2.19, this isomorphism comes from an element u ∈ Hom_{K}(A, B)⊗Q_{ℓ}. After multiplying u by some power of ℓ, we can assume that u belongs to Hom_{K}(A, B)⊗Z_{ℓ}. Note that Hom_{K}(A, B) is dense in Hom_{K}(A, B)⊗Z_{ℓ}. Any good enough ℓ-adic approximation to u in Hom_{K}(A, B) gives the desired K-isogeny between A and B. Theorem 2.18 follows.

We next observe that Theorem 2.19 can be reduced to the following special case:

Theorem 2.20. Let A be an abelian variety over K. The natural map

End_{K}(A) ⊗ Q_{ℓ} ⟶ End_{Π_{K}}(V_{ℓ}(A))

is surjective.

The fact that Theorem 2.20 implies Theorem 2.19 can be seen by applying Theorem 2.20 to the abelian variety A × B, since

End_{K}(A × B) = End_{K}(A) ⊕ Hom_{K}(A, B) ⊕ Hom_{K}(B, A) ⊕ End_{K}(B)

and likewise for End(V_{ℓ}(A × B)) = End(V_{ℓ}(A) × V_{ℓ}(B)).

Proof of Theorem 2.20: Let ϕ be an element of End_{Π_{K}}(V_{ℓ}(A)), and let

W = {(x, ϕ(x)) ∈ V_{ℓ}(A) × V_{ℓ}(A)} ⊂ V_{ℓ}(A × A)

be the graph of ϕ. Note that W is Π_{K}-stable. Hence there is an endomorphism u ∈ End_{K}(A × A) ⊗ Q_{ℓ} = M_{2}(E) associated to W by Lemma 2.16, satisfying u(V_{ℓ}(A × A)) = W.

Let E^{0} = End_{E}(V_{ℓ}(A)) denote the commutant of E in End(V_{ℓ}(A)). For

any α ∈ E^{0}, the matrix ( \begin{matrix}α & 0 \\ 0 & α \end{matrix} ) with entries in End(V_{ℓ}(A)) commutes

with u ∈ M_{2}(E) ⊂ M_{2}(End(V_{ℓ}(A)). It follows that this matrix preserves W = image(u), and hence α commutes with ϕ. Since this argument is valid for any α ∈ E^{0}, the endomorphism ϕ belongs to the double commutant E^{00} which is equal to E by the semisimplicity of V_{ℓ}(A) as a module over E.

2.7 The finiteness principle for rational ℓ-adic representations

Now that R_{4} has been shown to be injective, it remains to prove that the target Rep_{S}(G_{K}, 2g) is finite. The main theorem of this section is:

Theorem 2.21. (Finiteness principle for rational semisimple ℓ-adic representations). Let K be a number field and S a finite set of primes of K. Then there are finitely many isomorphism classes of rational, semisimple ℓ-adic representations of G_{K} of dimension d which are unramified outside of S.

Remark. The reader will observe that this finiteness principle is close in spirit to the Hermite–Minkowski theorem: it asserts that there are only finitely many extensions of K (albeit, of infinite degree) of a certain special kind with bounded ramification. The proof of Theorem 2.21 will in fact rely crucially on the Hermite–Minkowski theorem, as well as on the Chebotarev density theorem.

We begin by establishing the following key lemma.

Lemma 2.22. There exists a finite set T of primes of K (depending on S and d) satisfying the following two properties:

1. T is disjoint from S_{ℓ} := S ∪ {v|ℓ}.

2. Two representations ρ_{1}, ρ_{2} ∈ Rep_{S}(G_{K}, d) are isomorphic if and only

if

trace(ρ_{1}(Frob_{v})) = trace(ρ_{2}(Frob_{v})), for all v ∈ T.

Proof. Consider the set of all extensions of K of degree ≤ l^{2d^{2}} which are unramified outside S_{ℓ}. By Theorem 1.1 (Hermite–Minkowki), there are finitely many such extensions, and hence their compositum L is a finite extension of K. Let T = {v_{1}, … v_{N}} be a set of primes of K which are not in S and such that the Frobenius conjugacy classes Frob_{v_{i}} generate Gal(L/K). The existence of such a finite set follows from the Chebotarev density theorem. We claim that this set T satisfies the conclusion of Lemma 2.22. Given ρ_{1}, ρ_{2} ∈ Rep_{S}(G_{K}, d), a choice of G_{K}-stable Z_{ℓ}-lattices in the underlying representation spaces makes it possible to view each ρ_{i} as a homomorphism from Z_{ℓ}[[G_{K}]] to M_{d}(Z_{ℓ}). Let

j = ρ_{1} ⊕ ρ_{2} : Z_{ℓ}[[G_{K}]] ⟶ M_{d}(Z_{ℓ}) × M_{d}(Z_{ℓ}),

and let M denote the image of j. The induced homomorphism

j̄ : G_{K} ⟶ (M/ℓM)^{×}

factors through Gal(L/K), since the cardinality of M/ℓM is at most ℓ^{2d^{2}} and j̄ is unramified outside of S_{ℓ}. It follows that the elements

j̄(Frob_{v_{1}}), …, j̄(Frob_{v_{N}})

generate M/ℓM. By Nakayama’s lemma, the elements

j(Frob_{v_{1}}), …, j(Frob_{v_{N}})

generate M as a Z_{ℓ}-module.

In particular, if

trace(ρ_{1}(Frob_{v_{j}})) = trace(ρ_{2}(Frob_{v_{j}})), for j = 1, …, N,

then

M ⊆ Δ ⊂ M_{d}(Z_{ℓ}) × M_{d}(Z_{ℓ}),

where Δ = M_{d}(Z_{ℓ}) is embedded diagonally. Therefore one has

trace(ρ_{1}(σ)) = trace(ρ_{2}(σ)) for all σ ∈ Π_{K}.

Hence ρ_{1} and ρ_{2} have the same traces. Since they are semisimple, it follows that they are isomorphic as Π_{K}-representations.

Proof of Theorem 2.21. Let T = {v_{1}, …, v_{N}} be as in the statement of Lemma 2.22. The assignment

ρ ↦ (Tr (ρ(Frob_{v_{1}})), …, Tr (ρ(Frob_{v_{N}})))

is injective on Rep_{S}(G_{K}, d), and can only assume finitely many values, by the rationality of ρ. (More precisely, each Tr (Frob_{v_{i}}) is a rational integer of absolute value ≤ dNv_{i}^{1/2}.) Theorem 2.21 follows.

2.8 A summary of Faltings’ proof

Faltings’ proof of Mordell’s conjecture is based on a sequence of maps (here X is a curve of genus g defined over K and having good reduction outside of the finite set S of primes of K):

\{ \begin{matrix}K-rational \\ points on X \end{matrix} \} ⟶^{R_{1}} \{ \begin{matrix}Curves of genus g′ over K′ \\ with good reduction outside S′ \end{matrix} \}

⟶^{R_{2}} \{ \begin{matrix}Isomorphism classes of semistable \\ abelian varieties of dimension g′ \\ with good reduction outside S′ \end{matrix} \}

⟶^{R_{3}} \{ \begin{matrix}Isogeny classes of abelian varieties \\ of dimension g′ \\ with good reduction outside S′ \end{matrix} \}

⟶^{R_{4}} \{ \begin{matrix}Rational semisimple ℓ-adic representations \\ of dimension 2g′ unramified outside S′_{ℓ} \end{matrix} \}

1. The map R_{1} is given by Parshin’s construction, and is finite-to-one, by the geometric theorem of De Franchis.

2. The map R_{2} is defined by passing to the jacobian of a curve, and is finite-to-one by Torelli’s theorem.

3. The map R_{3} is the obvious one, and is finite-to-one, by Falting’s fundamental Theorem 2.11 on finiteness of abelian varieties in a given isogeny class.

4. The map R_{4} is defined by passing to the Tate module, and is one-to-one, thanks to the Tate conjectures proved by Faltings. The proof of the Tate conjectures is obtained by combining a strategy of Tate with the finiteness Theorem 2.11. These ideas are also used to show that the Galois representations arising in the image of R_{4} are semisimple.

5. The last set in this sequence of maps is finite by the finiteness principle for rational semisimple ℓ-adic representations, which is itself a consequence of the Chebotarev density theorem and the Hermite–Minkowski theorem.

3 Modular curves and Mazur’s theorem

The first step in the proof of the Mordell conjecture (the Kodaira–Parshin reduction) consists in transforming a question about rational points on a given curve into the Shafarevich conjecture. This new diophantine question is concerned with the moduli space of curves themselves, to which an array of techniques (notably, jacobians, ℓ-adic representations, etc.) can be applied. It is therefore apparent that the extra structures afforded by moduli spaces are of great help in studying the diophantine questions that are associated to them. So it is natural to examine more closely the simplest class of moduli spaces, which are also curves in their own right: the modular curves classifying elliptic curves with extra level structure.

3.1 Modular curves

Let p be a prime ≥ 5, and write Z for the ring Z[1/p]. The functor Y_{1}(p) which to any Z-algebra R associates the set of R-isomorphism classes of pairs (E, P) where E is an elliptic curve over Spec(R) and P is a point of order p on E_{R} is representable by a smooth affine scheme over Spec(Z) of relative dimension one, denoted Y_{1}(p).

The group (Z/pZ)^{×} acts on Y_{1}(p) by the rule t·(E, P) := (E, tP), and the quotient of Y_{1}(p) by this action is an affine scheme Y_{0}(p) over Spec(Z) which is a coarse moduli scheme classifying pairs (E, C) consisting of an elliptic curve over R and a cyclic subgroup scheme C ⊂ E of order p defined over R.

These curves admit analytic descriptions as quotients of the Poincaré upper half-plane

H = {τ ∈ C, Im(τ) > 0}

by the action of the following discrete subgroups of SL_{2}(Z):

Γ_{1}(p) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) with a − 1 ≡ c ≡ d − 1 ≡ 0 (mod p)\} ,
Γ_{0}(p) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) with c ≡ 0 (mod p)\} .

For example, the curve Y_{0}(1) is identified with Spec(Z[j]), and a birational (singular, even on the generic fiber) model for Y_{0}(p) over Spec(Z) is given by

Spec(Z[j, j′]/Φ_{p}(j, j′)),

where Φ_{p}(x, y) ∈ Z[x, y] is the canonical modular polynomial of bidegree p+1 satisfying Φ_{p}(j(τ), j(pτ)) = 0, for all τ ∈ H.

A rational point on Y_{1}(p) (resp. on Y_{0}(p)) determines an elliptic curve over Q with a Q-rational point of order p (resp. a rational subgroup of order p). The main goal of this chapter is to explain the proof of the following theorem of Mazur.

Theorem 3.1. If p > 13, then Y_{1}(p)(Q) = ∅.

Remark 3.2. Note that Theorem 3.1 can be viewed as a theorem about curves in two different ways. Firstly, it asserts that the collection of modular curves Y_{1}(p), whose genera grow with p, have no rational points once p is large enough—a type of statement that is similar in flavour to Fermat’s Last Theorem. Secondly, it leads to the uniform boundedness of the size of the torsion subgroups E(Q)_{tors} as E ranges over all elliptic curves over Q, and is therefore also a theorem about curves of genus one.

3.2 Mazur’s criterion

An important role is played in Mazur’s argument by the compactification X_{0}(p) of the affine curve Y_{0}(p). As a Riemann surface, X_{0}(p)(C) is obtained by adjoining to Y_{0}(p) a finite set of cusps which are in bijection with the orbits of Γ_{0}(p) acting on ℙ_{1}(Q) by Mobius transformations. More precisely, letting H^{∗} := H ∪ ℙ_{1}(Q), we have

X_{0}(p)(C) = Γ_{0}(p)\H^{∗} = (Γ_{0}(p)\H) ∪ {0, ∞} = Y_{0}(p)(C) ∪ {0, ∞}.

The complex structure in a neighbourhood of ∞ is defined by letting q = e^{2πiτ} be a local parameter at ∞.

The equation for the universal elliptic curve in a formal punctured neighbourhood of ∞ is given by the Tate curve

E_{q} = Z[[q]]^{×}/q^{Z} : y^{2} + xy = x^{3} + a(q)x + b(q) over Z((q)),

where

a(q) = −5 ∑_{n=1}^{∞} σ_{3}(n)q^{n}, b(q) = −\frac{1}{12} ∑_{n=1}^{∞}(7σ_{5}(n) + 5σ_{3}(n))q^{n}.

(Recall that σ_{k}(n) = ∑_{d|n} d^{k}.) The discriminant of E_{q} is equal to

Δ(E_{q}) = q ∏_{n≥1}(1 − q^{n})^{24},

and therefore E_{q} defines an elliptic curve over Z((q)).

The important q-expansion principle asserts that the parameter q is also a local parameter for the scheme X_{0}(p)_{Z} in a neighborhood of ∞. Thanks to the q-expansion principle, the completion of the local ring of X_{0}(p)_{Z} at ∞ is identified with the power series ring Z[[q]]:

Ô_{X_{0}(p),∞} = Z[[q]].

A basic technique in Mazur’s proof is to study the behaviour of certain maps on modular curves, via their behaviour in a formal neighbourhood of ∞. The following definition will be useful.

Definition 3.3. A morphism j : X ⟶ Y of schemes over Z is a formal immersion at x ∈ X(Z) if the induced map on completed local rings

j^{∗} : Ô_{Y,j(x)} ⟶ Ô_{X,x}

is surjective.

Let J_{0}(p) denote the jacobian of X_{0}(p). It is an abelian variety over Z and is equipped with an embedding

Φ : X_{0}(p) ⟶ J_{0}(p)

defined by letting Φ(x) be the class of the degree zero divisor (x) − (∞).

If J_{♯}(p) is any quotient of J_{0}(p), let j_{♯} : X_{0}(p) ⟶ J_{♯}(p) be the map obtained by composing Φ with the projection to J_{♯}(p). The following criterion of Mazur for Y_{1}(p)(Q) = ∅ is the main result of this section.

Theorem 3.4. Assume that p > 7. Suppose that there is an abelian variety quotient J_{♯}(p) of J_{0}(p) satisfying the following conditions:

(a) The map j_{♯} : X_{0}(p) ⟶ J_{♯}(p) is a formal immersion at ∞.

(b) J_{♯}(p)(Q) is finite.

Then Y_{1}(p)(Q) = ∅.

Sketch of proof. Let x̃ be a point in Y_{1}(p)(Q) corresponding to the pair (E, P), where E is an elliptic curve over Q and P ∈ E(Q) is of order p. Let E be the minimal Weierstrass model of E over Z.

The proof is divided into four steps.

Step 1. If E has potentially good reduction at the prime 3, then the special fiber E_{𝔽_{3}} is either an elliptic curve, or an extension of a finite group of connected components of cardinality 2^{a}3^{b} by the additive group 𝔾_{a/𝔽_{3}}. Such a group cannot contain a point of order p > 7, by the Hasse bound. Hence E has potentially mutiplicative reduction at 3.

Step 2. Let x ∈ X_{0}(p)(Q) be the image of x̃ under the natural map: it corresponds to the pair (E, ⟨P ⟩) consisting of the curve E and the cyclic subgroup generated by P. By Step 1, the point x reduces to one of the cusps 0 or ∞ of X_{0}(p) modulo 3. It can be asssumed without loss of generality that x reduces to ∞, by replacing (E, ⟨P ⟩) by (E/⟨P ⟩, E[p]/⟨P ⟩) otherwise.

Step 3. Consider the element j_{♯}(x) ∈ J_{♯}(p)(Q). By step 2 this element belongs to the formal group J_{♯}^{1}(p)(Q_{3}), which is torsion-free because Q_{3} is absolutely unramified. It also belongs to J_{♯}(p)(Q), which is torsion by assumption. It follows that j_{♯}(x) = 0.

Step 4. We now use the fact that j_{♯} is a formal immersion to deduce that x = ∞. To see this, let Spec(R) be an affine neighborhood of ∞ containing x. The point x gives rise to a ring homomorphism x : R ⟶ Z_{3}, which factors through the local ring Ô_{X_{0}(p),∞} = Z[[q]], so that x can be viewed as a map Z[[q]] ⟶ Z_{3}. By step 3, we have

x ∘ j_{♯}^{∗} = ∞ ∘ j_{♯}^{∗}.

It follows that x = ∞, since j_{♯}^{∗} was assumed to be surjective, contradicting the initial assumption that x belongs to Y_{0}(p).

Mazur’s criterion reduces Theorem 3.1 to the problem of exhibiting a quotient J_{♯}(p) of J_{0}(p) satisfying the conditions of Theorem 3.4.

3.3 The jacobian J_{0}(p)

The fact that makes it possible to analyse the jacobian J_{0}(p) precisely, and exhibit a nontrivial quotient of it with finite Mordell–Weil group, arises from two related ingredients.

(a) Hecke operators. If n is an integer that is not divisible by p, the modular curve X_{0}(np) is equipped with two maps π_{1}, π_{2} to X_{0}(p), defined by

π_{1}(E, C) = (E, C[p]), π_{2}(E, C) = (E/C[n], C/C[n]).

The pair (π_{1}, π_{2}) gives rise to an embedding of X_{0}(pn) in the product X_{0}(p) × X_{0}(p). The image in this product, denoted T_{n}, is an algebraic correspondence on X_{0}(p) defined over Q, which gives rise to an endomorphism of J_{0}(p) defined over Q. On the level of divisors, T_{n} is described by

T_{n}(E, C) = ∑_{E⟶E′}(E′, C′),  (6)

where the sum is taken over the cyclic isogenies φ : E ⟶ E′ of degree n, and C′ = φ(C). Let T denote the subring of End_{Q}(J_{0}(p)) generated by the Hecke operators T_{n}. It is finitely generated (as a ring, and even as a module) over Z. Our basic approach to constructing J_{♯}(p) is to use the endomorphisms in T to decompose the abelian variety J_{0}(p) (up to Q-isogeny) into smaller pieces which can then be analysed individually. If R is any ring, let T_{R} denote the R-algebra T ⊗ R.

(b) Modular forms. If R is any Z-algebra, let S_{2}(p, R) denote the space of regular differentials on X_{0}(p)_{R}. Restriction to the formal neighborhood Spec(R[[q]]) of ∞ ∈ X_{0}(p) gives rise to a map (called the q-expansion map)

q-exp : S_{2}(p, R) ⟶ R[[q]]dq.

When R = C, the space S_{2}(p, C) is identified with the vector space of homomorphic functions f : H ⟶ C for which

(i) the differential 2πif(τ)dτ is invariant under Γ_{0}(p), i.e.,

f (\frac{aτ + b}{cτ + d}) = (cτ + d)^{2}f(τ), for all ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(p).

(ii) 2πif(τ)dτ extends to a holomorphic differential on the compactified modular curve X_{0}(p). In particular, it admits a Fourier expansion of the form

f(τ) = ∑_{n=1}^{∞} a_{n}e^{2πinτ},

so that q-exp(2πif(τ)dτ) = ∑_{n=1}^{∞} a_{n}q^{n} \frac{dq}{q}.

The action of the Hecke operators T_{n} on J_{0}(p)_{R} induces an action on the cotangent space S_{2}(p, R), which can be described explicitly on the level of the q-expansions. For example, if ℓ ≠ p is prime,

T_{ℓ} (∑_{n=1}^{∞} a_{n}q^{n} \frac{dq}{q} ) = (∑_{ℓ|n} a_{n}q^{n/ℓ} + ℓ ∑_{n=1}^{∞} a_{n}q^{nℓ}) \frac{dq}{q} .  (7)

There is an extra Hecke operator T_{p} defined via an algebraic correspondence X_{0}(p^{2}) ⊂ X_{0}(p) × X_{0}(p) which admits the following simpler formula for its action on q-expansions:

T_{p} (∑_{n=1}^{∞} a_{n}q^{n} \frac{dq}{q} ) = (∑_{p|n} a_{n}q^{n/p}) \frac{dq}{q} .  (8)

The definition of T_{ℓ} for ℓ prime can then be extended to all integers n by the multiplicativity relations implicit in the following identity of formal Dirichlet series:

∑_{n≥1} T_{n}n^{−s} = (1 − T_{p}p^{−s})^{−1} ∏_{ℓ≠p}(1 − T_{ℓ}ℓ^{−s} + ℓ^{1−2s})^{−1}.  (9)

In other words,

T_{mn} = T_{m}T_{n} if gcd(m, n) = 1, T_{ℓ^{n+1}} = a_{ℓ}T_{ℓ^{n}} − ℓT_{ℓ^{n−1}}.

Proposition 3.5. The algebra T_{Q} is a commutative semisimple algebra of dimension g := dim_{Q} S_{2}(p, Q) = genus(X_{0}(p)).

Sketch. The fact that T_{Q} is commutative follows from the explicit description of the operators T_{n} as correspondences given in (6) (or, if one prefers, from equation (7) describing its effect on q-expansions). The semisimplicity arises from the fact that the operators T_{n} are self-adjoint with respect to the Hermitian pairing on S_{2}(p, C) (Petersson scalar product) defined by

⟨ω_{1}, ω_{2}⟩ = \frac{1}{2i} ∫_{Γ_{0}(p)\H} ω_{1} ∧ ω̄_{2}.

(We mention in passing that in general, the operator T_{ℓ} acting on S_{2}(N, C) need not be self-adjoint when ℓ|N, but it is self-adjoint when restricted to the space of so-called newforms. We are using implicitly the fact that S_{2}(p, C) is equal to its subspace of newforms.) One computes the dimension of T_{Q} by showing that the bilinear pairing

T_{Q} × S_{2}(p, Q) ⟶ Q, (T, f) := a_{1}(T f)

is left and right nondegenerate, and in fact positive definite. The details are left to the reader (who may also consult Section 2.2 of [Da2] for more details).

As a consequence of Proposition 3.5 and its proof, one has the decomposition

T_{Q} = K_{1} × ⋯ × K_{t}

of T_{Q} into a product of totally real fields, with ∑_{j=1}^{t}[K_{j} : Q] = n. The factors K_{j} are indexed by:

(a) The points ϕ_{1}, …, ϕ_{t} of Spec(T_{Q}), viewed as algebra homomorphisms ϕ_{j} : T_{Q} ⟶ Q̄ (taken modulo the natural action of G_{Q} = Gal(Q̄/Q)).

(b) The distinct G_{Q}-equivalences classes f_{1}, …, f_{t} of eigenforms for T, normalised so that a_{1}(f_{j}) = 1. The q-expansions of these eigenforms are described by

f_{j} = ∑_{n=1}^{∞} ϕ_{j}(T_{n})q^{n}.

The quotient A_{f} attached to f is defined by letting

A_{f} := J_{0}(p)/I_{f}, where I_{f} := ker(T ⟶ K_{f}).

With these notations, the main result of this section is the following Eichler-Shimura decomposition, which asserts that J_{0}(p) is isogenous to a product of Q-simple factors indexed by the (G_{Q}-orbits of) normalised eigenforms f_{j} (j = 1, …, t).

Theorem 3.6. The abelian variety J_{0}(p) is Q-isogenous to the product

∏_{i=1}^{t} A_{f_{i}},

of Q-simple abelian varieties A_{f_{i}}. The varieties A_{f} that occur in this decomposition have the following properties:

(a) dim(A_{f}) = [K_{f} : Q];

(b) The natural image of T_{Q} in End_{Q}(A_{f}) ⊗ Q is isomorphic to K_{f}.

For more details on this decomposition see Chapter 2 of [Da2].

Thanks to Theorem 3.6, we are reduced to the following question:

Question 3.7. Find a criterion involving the normalised eigenform f for the quotient A_{f} to have finite Mordell–Weil group.

3.4 The Birch and Swinnerton-Dyer conjecture

The key to bounding the rank of A_{f}(Q) (and showing that this rank is zero, for a sufficiently large collection of normalised eigenforms f) lies in studying the so-called Hasse–Weil L-series attached to A_{f}.

Let A be an abelian variety (of dimension d, say) defined over Q. The Hasse–Weil L-series of A is most conveniently defined in terms of the ℓ-adic representation V_{ℓ}(A) that was introduced in Section 2.5. If p ≠ ℓ is a prime, the Frobenius element acts naturally on on the space V_{ℓ}(A)^{I_{p}} of vectors in V_{ℓ}(A) that are fixed under the action of the inertia group at p. (Recall that V_{ℓ}(A)^{I_{p}} = V_{ℓ}(A) if A has good reduction at p ≠ ℓ.) By the rationality of the representation V_{ℓ}(A), the characteristic polynomial

F_{p}(T) := det(1 − Frob_{p}|_{V_{ℓ}(A)^{Ip}}T)

has integer coefficients. Furthermore, it does not depend on the choice of ℓ, and can therefore be defined for all p. This makes it possible to define the Hasse–Weil L-series as a function of the complex variable s, by the infinite product

L(A, s) = ∏_{p} F_{p}(p^{−s})^{−1}.

Using the rationality of the Galois representation V_{ℓ}(f) in the sense of Theorem 2.15, one can show that the infinite product defining L(A, s) converges uniformly on compact subsets of {s ∈ C|ℜ(s) > 3/2}, and hence defines an analytic function in this region.

Concerning the behaviour of L(A, s) and its connection to the arithmetic of A over Q, there are the following two fundamental conjectures:

Conjecture 3.8. The L-series L(A, s) has an analytic continuation to the entire complex plane and a functional equation of the form

Λ(A, s) := (2π)^{−ds}Γ(s)^{d}N^{s/2}L(A, s) = ±Λ(A, 2 − s),

where N is the conductor of A.

In particular, if Conjecture 3.8 is true, the process of analytic continuation gives meaning to the behaviour of L(A, s) in a neighborhood of the central critical point s = 1 for the functional equation, and in particular, the order of vanishing of L(A, s) at s = 1 is defined. The Birch and Swinnerton-Dyer conjecture relates this order of vanishing to the arithmetic of A over Q:

Conjecture 3.9. If A is an abelian variety over Q, then

rank(A(Q)) = ord_{s=1}(L(A, s)).

In particular, A(Q) is finite if L(A, 1) ≠ 0.

Both Conjectures 3.8 and 3.9 are far from being proved in general. But much more is known when A = A_{f} occurs in the Eichler–Shimura decomposition of the modular jacobian J_{0}(N), as will be explained in the next section.

3.5 Hecke theory

A newform of level N is a normalised eigenform f = ∑_{n≥1} a_{n}q^{n} \frac{dq}{q} on Γ_{0}(N) whose associated sequence (a_{n})_{(n,N)=1} of Fourier coefficients is different from that of any eigenform g on Γ_{0}(d) with d|N and d ≠ N.

To each newform f = ∑_{n≥1} a_{n}q^{n} \frac{dq}{q} ∈ S_{2}(N, C), one can associate an L-series

L(f, s) := ∑_{n=1}^{∞} a_{n}n^{−s}.

This L-series enjoys the following properties, which were established by Hecke:

(a) Euler product: It admits the Euler product factorisation given by

L(f, s) = ∏_{p∤N}(1 − a_{p}p^{−s} + p^{1−2s})^{−1} ∏_{p|N}(1 − a_{p}p^{−s})^{−1},

as can be seen by applying φ_{f} to the formal identity (9) expressing the Hecke operators T_{n} in terms of the operators T_{ℓ} for ℓ prime.

(b) Integral representation: The L-series L(f, s) can be represented as an integral transform of the modular form f, by the formula:

Λ(f, s) := (2π)^{−s}Γ(s)N^{s/2}L(f, s) = N^{s/2} ∫_{0}^{∞} f(it)t^{s−1}dt,  (10)

where Γ(s) = ∫_{0}^{∞} e^{−t}t^{s−1}dt is the Γ-function. In particular, because f is of rapid decay at the cusps, this integral converges absolutely to an analytic function of s ∈ C.

(c) Functional equation: The involution w defined on S_{2}(N, C) by the

rule

w(f)(τ) = \frac{1}{Nτ^{2}} f (\frac{−1}{Nτ} )  (11)

commutes with the Hecke operators and hence preserves its associated eigenspaces. It follows that for the eigenform f,

w(f) = εf, where ε = ±1.  (12)

The L-series L(f, s) satisfies the functional equation

Λ(f, s) = −Λ(w(f), 2 − s) = −εΛ(f, 2 − s).  (13)

It is a direct calculation to derive this functional equation from the integral representation of Λ(f, s).

For the next result, we view f as an element of S_{2}(N, K_{f}). (Recall that K_{f} is the totally real field generated by the Fourier coefficients of f.) Any complex embedding σ : K_{f} ↪ C yields an eigenform f^{σ} with complex coefficients, to which the Hecke L-function L(f^{σ}, s) may be attached. The following result relates the L-series of Hasse–Weil and of Hecke.

Theorem 3.10. Let A_{f} be the abelian variety associated to the newform f ∈ S_{2}(N, C) by the Eichler–Shimura construction. Then

L(A_{f}, s) = ∏_{σ:K_{f} ⟶C} L(f^{σ}, s).

In particular, Conjecture 3.8 holds for A_{f}.

The main ingredient in the proof of Theorem 3.10 is the Eichler–Shimura congruence which relates the Hecke correspondence T_{p} ⊂ X_{0}(N)^{2} in characteristic p to the graph of the Frobenius morphism and its transpose. For more details and references see Chapter 2 of [Da2].

Theorem 3.10 reveals that one has better control of the arithmetic of the abelian varieties A_{f}—Conjecture 3.8 remains open for the general abelian variety A over Q. In fact, one has the following strong evidence for Conjecture 3.9 for the abelian varieties A_{f}.

Theorem 3.11. If L(A_{f}, 1) ≠ 0, then A_{f}(Q) is finite.

The main ingredients that go into the proof of Theorem 3.11 are

1. The theory of Heegner points on modular curves;

2. The theorem of Gross–Zagier expressing the canonical heights of the images of these points in A_{f} in terms of special values of L-series closely related to L(f, s);

3. A theorem of Kolyvagin which relates the system of Heegner points and the arithmetic of A_{f} over Q.

These ingredients will be discussed in somewhat more detail in Section 5 devoted to elliptic curves and the Birch and Swinnerton-Dyer conjecture.

3.6 The winding quotient

The criterion for the finiteness of A_{f}(Q) supplied by Theorem 3.11 allows us to construct a quotient J_{♯}(p) which is in some sense the “largest possible” quotient with finite Mordell–Weil group.

We construct J_{♯}(p), following Merel, by letting e_{0} be the vertical path from 0 to i∞ on H; its image in X_{0}(p) gives an element in the relative homology H_{1}(X_{0}(p)(C), Z; {cusps}). By a result of Manin–Drinfeld, the element e_{0} gives rise to an element e in the rational homology ℍ := H_{1}(X_{0}(p)(C), Q). This element is referred to as the winding element.

The Hecke algebra T = T_{Q} acts on ℍ by functoriality of correspondences. Let e_{f} denote the image of e in ℍ/I_{f}ℍ. The integral formula (10) for L(f, s) shows that

e_{f} ≠ 0 if and only if L(f, 1) ≠ 0.

Hence it is natural to define

J_{e}(p) := J_{0}(p)/I_{e}, where I_{e} := Ann_{T}(e).

Theorem 3.12. The Mordell–Weil group J_{e}(p)(Q) is finite.

Proof. Up to isogeny, J_{e}(p) decomposes as

J_{e}(p) ∼ ∏_{e_{f} ≠0} A_{f} = ∏_{L(f,1)≠0} A_{f}.

Theorem 3.11 implies that A_{f}(Q) is finite for all the f that appear in this decomposition. The theorem follows.

In order to exploit Mazur’s criterion with J_{♯}(p) = J_{e}(p), and thereby prove Theorem 3.1, it remains to show that the natural map j_{e} : X_{0}(p) ⟶ J_{e}(p) is a formal immersion at ∞. (So that in particular J_{e}(p) is nontrivial, which is not clear a priori from its definition!) This is done in the article of Marusia Rebolledo in these proceedings (cf. Theorem 4 of Section 2.3 of [Reb]). Rebolledo’s article goes significantly further by showing that the natural map from the d-th symmetric power X_{0}(p)^{(d)} of X_{0}(p) sending (P_{1}, …, P_{d}) to the image in J_{♯}(p) of the divisor class (P_{1}) + ⋯ + (P_{d}) − d(∞) is a formal immersion at (∞, …, ∞), as soon as p is sufficiently large relative to d.

Remark 3.13. Our presentation of Mazur’s argument incorporates an important simplification due to Merel, which consists in working with the winding quotient J_{e}(p) whose finiteness is known thanks to Theorem 3.11. At the time of Mazur’s original proof described in [Ma1], Theorems 3.11 and 3.12 were not available, and Mazur’s approach worked with the so-called Eisenstein quotient J_{eis}(p). This quotient contains a rational torsion subgroup of order n = numerator(\frac{p−1}{12}), and one of the key results in [Ma1] is to establish the finiteness of J_{eis}(Q) by an n-descent argument. In Merel’s approach, Mazur’s somewhat delicate “Eisenstein descent” is in effect replaced by Kolyvagin’s descent based on Heegner points and the theorem of Gross–Zagier.

3.7 More results and questions

By various refinements of the techniques discussed above, Mazur was able to classify all possible rational torsion subgroups of elliptic curves over Q and obtained the following results:

Theorem 3.14. Let T be the torsion subgroup of the Mordell–Weil group of an elliptic curve E over Q. Then T is isomorphic to one of the following 15 groups:

Z/mZ for 1 ≤ m ≤ 10 or m = 12,
Z/2mZ × Z/2Z for  1 ≤ m ≤ 4.

For the proof, see [Ma1], p. 156. We note in passing that all possibilities for T that are not ruled out by Mazur’s theorem do in fact occur infinitely often: the associated modular curves are of genus 0 and have a rational point.

Mazur’s theorem implies that rational points of order p on elliptic curves cannot occur for p > 7. One can ask similar questions for rational subgroups. In this direction, Mazur proved the following result in [Ma2].

Theorem 3.15. Suppose that there is an elliptic curve E over Q with a rational subgroup of prime order p. Then p ≤ 19 or p = 37, 43, 67, or 163.

The four exceptional values of p in Theorem 3.15 correspond to discriminants of imaginary quadratic fields of class number one. The corresponding elliptic curves with complex multiplication can be defined over Q and have a rational subgroup of order p.

Theorem 3.15 implies that for large enough p, the Galois representation

ρ_{E,p} : G_{Q} ⟶ Aut(E[p])

is always irreducible. One can also ask whether, for large enough p, this Galois representation is in fact necessarily surjective. The existence of elliptic curves with complex multiplication, for which ρ_{E,p} is never surjective when p ≥ 3, precludes an affirmative answer to this question. Discarding elliptic curves with complex multiplication, the following conjecture (which appears in [Se1], p. 299, §4.3, phrased more prudently as an open question) can be proposed:

Conjecture 3.16. (Surjectivity conjecture) If E is an elliptic curve over Q without complex multiplication, and p ≥ 19 is prime, then the Galois representation associated to E[p] is surjective.

The surjectivity conjecture remains open, more than 30 years after [Ma1]. The hypothetical cases that are the most difficult to dispose of are those where the image of ρ_{E,p} is contained in the normalizer of a Cartan subgroup, particularly a nonsplit Cartan subgroup.

It is also natural to search for analogues of Theorem 3.14 over number fields other than Q; a remarkable breakthrough was achieved on this problem by S. Kamienny and Merel around 1992 ([Ka], [Me]).

Theorem 3.17. Let K be a number field. Then the size of E(K)_{tors} is bounded by a constant B(K) which depends only on K. In fact, this constant can be made to depend only on the degree of K over Q.

The proof of this theorem is explained in the article by Marusia Rebolledo [Reb] in these proceedings.

We finish with a conjecture that can be viewed as a “mod p analogue” of Theorem 2.18 (Tate’s isogeny conjecture).

Conjecture 3.18. There exists an integer M such that, for all p ≥ M, any two elliptic curves E_{1} and E_{2} over Q are isogenous if and only if E_{1}[p] ≃ E_{2}[p] as G_{Q}-modules.

This conjecture appears to be difficult. It is not even clear what the best value M might be, assuming it exists. (Calculations of Cremona [Cr] based on his complete tables of elliptic curves over Q of conductor ≤ 30, 000 show that necessarily M > 13.) We mention Conjecture 3.18 here because it implies strong results about ternary diophantine equations analogous to Fermat’s Last Theorem, thanks to the methods explained in Chapter 4.

4 Fermat curves

The purpose of this section is to discuss the Fermat curves

F_{n} : x^{n} + y^{n} = z^{n},

and the proof of Fermat’s Last Theorem, that these curves have no nontrivial rational points when n ≥ 3. Fermat’s Last Theorem has the same flavour as Mazur’s Theorem 3.1, since it determines all of the rational points in a naturally arising infinite collection of algebraic curves. Although Fermat curves are simpler to write down as explicit equations, they do not admit a direct moduli interpretation, and therefore turn out to be harder to analyse than modular curves. In fact, the eventual solution of Fermat’s Last Theorem is based on an elaborate reduction of the study of Fermat curves to diophantine questions about modular curves. In particular, Theorem 3.1—its statement, as well as some of the techniques used in its proof—play an essential role in the proof of Fermat’s Last Theorem.

4.1 Motivation for the strategy

Hugo Chapdelaine’s article in these proceedings discusses the more general problem of classifying the primitive integer solutions of the generalised Fermat equation

x^{p} + y^{q} + z^{r} = 0,  (14)

and sets up a “dictionary” relating

\{ \begin{matrix}Strategies for studying \\ primitive solutions of \\ x^{p} + y^{q} + z^{r} = 0 \end{matrix} \} and \{ \begin{matrix}Unramified coverings \\ of ℙ_{1} − \{0, 1, ∞\} \\ of signature (p, q, r) \end{matrix} \} .

The idea explained in [Chap] is that, given an unramified covering

π : X ⟶ ℙ_{1} − {0, 1, ∞},

one can study (14) by

1. Attempting to classify the possible fibers of π over the points in

Σ_{p,q,r} = \{\frac{a^{p}}{c^{r}} , with a^{p} + b^{q} = c^{r} and (a, b, c) primitive\} ⊂ ℙ_{1}(Q).

Since the ramification in these fibers is bounded, there can only be finitely many, by the Hermite–Minkowski theorem. In particular, the compositum of these extensions is a finite extension of Q, denoted L.

2. Understanding the L-rational points on the curve X.

To apply these principles to the classical Fermat equation, one is led to consider unramified coverings of ℙ_{1} − {0, 1, ∞} of signature (p, p, p). Among such coverings, one finds:

1. the Fermat curve F_{p} : x^{p} + y^{p} = z^{p} itself, equipped with the natural projection π : (x, y, z) ↦ t = \frac{x^{p}}{z^{p}} of degree p^{2}. For this π, it is clear that π(F_{p}(Q)) ⊃ Σ_{p,p,p}; but this merely leads to a tautological reformulation of the original question.

2. There are many coverings of signature (p, p, p) with solvable Galois groups, and studying these leads to classical attempts to prove Fermat’s Last Theorem by factoring x^{p} +y^{p} over the p-th cyclotomic fields. This circle of ideas led to many interesting questions on cyclotomic fields and their class groups, but has proved unsucessful (so far) in settling Fermat’s Last Theorem.

A third type of covering is obtained from modular curves. These coverings, which are nonsolvable, arises naturally in light of the strong results obtained in Section 3.

More precisely, let Y (n) be the open modular curve that classifies elliptic curves with full level n structure, i.e., pairs

(E, ι : Z/nZ × μ_{n} ⟶ E[n])

where ι is an identification which induces an isomorphism

∧^{2}ι : ∧^{2}(Z/nZ × μ_{n}) = μ_{n} ≃ ∧^{2}(E[n]) = μ_{n}.

Over the base Z = Z[1/2], the curve Y (2)_{Z} is identified with

Spec(Z[λ, 1/λ, 1/(λ − 1)]) = (ℙ_{1} − {0, 1, ∞})_{Z},

where λ is the parameter that occurs in the Legendre family

E_{λ} : y^{2} = x(x − 1)(x − λ).

The natural covering map π : Y (2p) ⟶ Y (2) is an unramified covering of signature (p, p, p), with Galois group SL_{2}(Z/pZ)/⟨±1⟩. Given λ = \frac{a^{p}}{c^{p}} ∈ Σ_{p,p,p}, the fiber π^{−1}(λ) is contained in the field of definition of the field of p-division points of the elliptic curve

y^{2} = x(x − 1)(x − a^{p}/c^{p}).  (15)

In practice, it is more convenient to work with the closely related Frey curve,

E_{a,b,c} : y^{2} = x(x − a^{p})(x − c^{p}),

which differs from (15) by a quadratic twist, and replace the study of the fiber of π at λ with considerations involving the mod p Galois representation

ρ_{a,b,c} : G_{Q} ⟶ Aut(E_{a,b,c}[p]) ≃ GL_{2}(Z/pZ).

We normalise (a, b, c) so that a ≡ 3 (mod 4) and c is even. (This can always be done, by permuting a, b and c and changing their signs if necessary.) With this normalisation, the minimal discriminant, conductor, and j-invariant associated to E_{a,b,c} are

Δ = 2^{−8}(abc)^{2p},  N = ∏_{ℓ|abc} ℓ,  j = \frac{2^{8}(b^{2p} + a^{p}c^{p})^{3}}{(abc)^{2p}} .  (16)

In particular, the elliptic curve E_{a,b,c} is semistable: it has either good or (split or nonsplit) multiplicative reduction at all primes. (The reader may wish to consult Section 2 of the article by Pierre Charollois in this proceedings volume, which discusses the local invariants of Frey curves in greater detail.)

4.2 Galois representations associated to Frey curves

The following theorem states the main local properties of the Galois representation ρ_{a,b,c}.

Theorem 4.1. The representation ρ = ρ_{a,b,c} has the following properties.

(a) It is unramified outside 2 and p;

(b) The restriction of ρ to a decomposition group D_{2} at 2 is of the form

ρ_{a,b,c}|_{D_{2}} = ( \begin{matrix}χ_{cyc}ψ & κ \\ 0 & ψ^{−1} \end{matrix} ) ,

where χ_{cyc} : G_{Q_{2}} ⟶ (Z/pZ)^{×} is the mod p cyclotomic character, and ψ is an unramified character of order 1 or 2.

(c) The restriction of ρ to D_{p} comes from the Galois action on the points of a finite flat group scheme over Z_{p}.

Proof. (a) Let ℓ ≠ 2, p be a prime. The analysis of the restriction of ρ = ρ_{a,b,c} to D_{ℓ} can be divided into three cases:

Case 1: The prime ℓ does not divide abc. In that case, it is a prime of good reduction for E_{a,b,c}, and the action of D_{ℓ} on E_{a,b,c}[p] is therefore unramified, by the criterion of Néron–Ogg–Shafarevich.

Case 2: The prime ℓ divides abc. It is therefore a prime of multiplicative reduction for E_{a,b,c}. Hence the curve E_{a,b,c}, or a twist of it over the unramified quadratic extension of Q_{ℓ}, is isomorphic to the Tate curve 𝔾_{m}/q_{ℓ}^{Z} over Q_{ℓ}. More precisely, replacing E_{a,b,c} by its twist if necesary, we have an identification which respects the action of G_{Q_{ℓ}} on both sides:

E(Q̄_{ℓ}) ≃ Q̄_{ℓ}^{×}/⟨q_{ℓ}⟩,  (17)

where q_{ℓ} ∈ Q_{ℓ}^{×} is the ℓ-adic Tate period, which is obtained by reverting the power series with integer coefficients

j = \frac{1}{q} + 744 + 196884q + ⋯

that expresses j in terms of q, to obtain a power series

q = Tate(1/j) = 1/j + ⋯ ∈ (1/j)Z[[1/j]]^{×}.

In particular, note that, by (16),

ord_{ℓ}(q_{ℓ}) = ord_{ℓ}(1/j) = ord_{ℓ}(Δ) ≡ 0 (mod p).  (18)

The explicit description of the G_{Q_{ℓ}}-module E(Q̄_{ℓ}) given by (17) implies that

E(Q̄_{ℓ})[p] ≃ {ζ_{p}^{a}q_{ℓ}^{b/p}, 0 ≤ a, b ≤ p − 1},

where ζ_{p} is a primitive pth root of unity in Q̄_{ℓ}^{×}. In the basis (ζ_{p}, q_{ℓ}^{1/p}) for E[p], the restriction of ρ = ρ_{a,b,c} to D_{ℓ} can be written as

ρ(σ) = ( \begin{matrix}χ_{cyc}(σ)ψ(σ) & κ(σ) \\ 0 & ψ^{−1}(σ) \end{matrix} ) ,  (19)

where χ_{cyc} is the p-th cyclotomic character giving the action of D_{ℓ} on the p-th roots of unity, and ψ is an unramified character of order at most 2 (which is trivial precisely when E has split multiplicative reduction at ℓ.) Furthermore, the cocycle κ is unramified, by (18): this is because the extension Q_{ℓ}(ζ_{p}, q_{ℓ}^{1/p}) through which ρ_{a,b,c}|_{G_{Q_{ℓ}}} factors is unramified. Part (a) of Theorem 4.1 follows.

(b) When ℓ = 2, the elliptic curve E_{a,b,c} has multiplicative reduction at 2, and hence is identified with a Tate curve over Q_{2}. The result then follows from (19) with ℓ = 2.

(c) When ℓ = p does not divide abc, the Galois representation ρ_{a,b,c} arises from the p-torsion of an elliptic curve with good reduction at p, and is hence from a finite flat group scheme over Z_{p}. In the case where p|abc (which corresponds to what was known classically as the second case of Fermat’s Last Theorem) one has a similar conclusion: essentially, the condition ord_{p}(q_{p}) ≡ 0 (mod p) limits the ramification of ρ_{a,b,c} at p and implies that E_{a,b,c}[p] extends to a finite flat group scheme over Z_{p}, in spite of the fact that E_{a,b,c} itself does not have a smooth model over Z_{p}.

The following theorem gives a global property of the representation ρ_{a,b,c}.

Theorem 4.2 (Mazur). The Galois representation ρ_{a,b,c} is irreducible.

Proof. This follows (at least when p is large enough) from Theorem 3.15. We will now give a self-contained proof which rests on the ideas developed in the proof of Theorem 3.4.

Suppose that ρ_{a,b,c} is reducible. Then E = E_{a,b,c} has a rational subgroup C of order p, and the pair (E, C) gives rise to a rational point x on the modular curve X_{0}(p). Let ℓ ≠ p be an odd prime that divides abc. Then E has multiplicative reduction at ℓ. Therefore, the point x reduces to one of the cusps 0 or ∞ of X_{0}(p) modulo ℓ. It can be asssumed without loss of generality that x reduces to ∞, as in Step 2 of the proof of Theorem 3.4. Now recall the natural projection Φ_{e} : J_{0}(p) ⟶ J_{e}(p) of J_{0}(p) to its winding quotient J_{e}(p), and the resulting map j_{e} : X_{0}(p) ⟶ J_{e}(p). The element j_{e}(x) belongs to the formal group J_{e}^{1}(p)(Q_{ℓ}), which is torsion-free, and to J_{e}(p)(Q), which is torsion by Theorem 3.12. Hence j_{e}(x) = 0. We now use the fact that j_{e} is a formal immersion to deduce that x = ∞, as in Step 4 of the proof of Theorem 3.4 (with 3 replaced by ℓ).

Remark 4.3. The importance of the diophantine study of modular curves described in Chapter 3 in the proof of Fermat’s Last Theorem, via Theorem 4.2, cannot be overemphasised. It is sometimes underplayed in expositions of Fermat’s Last Theorem which tend to focus on the ingredients that were supplied later.

Thanks to Theorems 4.1 and 4.2, Fermat’s Last Theorem is now reduced to the problem of “classifying” the irreducible two-dimensional mod p representations satisfying the strong restrictions on ramification imposed by Theorem 4.1—or in some sense, to make Theorem 1.1 precise for the class of extensions of Q arising from such representations. The control we have over questions of this type (which in general seem very hard) arises from the deep and largely conjectural connection that is predicted to exist between Galois representations and modular forms.

4.3 Modular forms and Galois representations

Let f = ∑_{n} a_{n}q^{n} be a newform in S_{2}(N, C). Let K_{f} denote as before the finite extension of Q generated by the Fourier coefficients of f, so that f belongs to S_{2}(N, K_{f}). The Fourier coefficients of f belong to the ring O_{f} of integers of K_{f}. Let 𝔭 be a prime ideal of O_{f} and let K_{f,𝔭} denote the completion of K_{f} at 𝔭.

Theorem 4.4. There exists a Galois representation

ρ_{f,𝔭} : G_{Q} ⟶ GL_{2}(K_{f,𝔭})

such that 1. The representation ρ_{f,𝔭} is unramified ouside Np.

2. The characteristic polynomial of ρ_{f,𝔭}(Frob_{ℓ}) is equal to x^{2} − a_{ℓ}x + ℓ, for all primes ℓ not dividing p.

3. The representation ρ_{f,𝔭} is odd, i.e., the image of complex conjugation has eigenvalues 1 and −1.

Sketch of proof. Let A_{f} be the abelian variety quotient of J_{0}(N) associated to f by the Eichler–Shimura construction (Theorem 3.6). Its endomorphism ring End_{Q}(A_{f}) contains T/I_{f}, which is an order in K_{f}. In this way, the Galois representation V_{p}(A_{f}) is equipped with an action of K_{f} ⊗ Q_{p} which commutes with the action of G_{Q}. Let

V_{f,𝔭} = V_{p}(A_{f}) ⊗_{K_{f}} K_{f,𝔭}.

It is a two-dimensional K_{f,𝔭}-vector-space, equipped with a continuous linear action of G_{Q}. The fact that it has the desired properties, particularly property 2, is a consquence of the Eichler–Shimura congruence that was used to prove the equality of L-series given in Theorem 3.10. See Chapter 2 of [Da2] for further details and references.

4.4 Serre’s conjecture

Modular forms can also be used to construct two-dimensional representations of G_{Q} over finite fields. More precisely, let O_{f,𝔭} be the ring of integers of K_{f,𝔭}. Since G_{Q} is compact and acts continuously on V_{f,𝔭}, it preserves an O_{f,𝔭}-stable sublattice V_{f,𝔭}^{0} ⊂ V_{f,𝔭} of rank two over O_{𝔭,p}. Let 𝔽_{𝔭} := O_{f,𝔭}/𝔭 be the residue field of O_{f} at 𝔭. The action of G_{Q} on the two-dimensional 𝔽_{𝔭}-vector space W_{f,𝔭} := V_{f,𝔭}^{0}/𝔭V_{f,𝔭}^{0} gives rise to a two-dimensional mod 𝔭 representation

ρ̄_{f,𝔭} : G_{Q} ⟶ GL_{2}(𝔽_{𝔭}).

Like its 𝔭-adic counterpart, this representation is unramified outside of pN and also satisfies parts 2 and 3 of Theorem 4.4.

In [Se2], Serre associates to any two-dimensional Galois representation

ρ : G_{Q} ⟶ GL_{2}(𝔽)  (20)

with coefficients in a finite field 𝔽 two invariants N(ρ) and k(ρ), called the Serre conductor and Serre weight of ρ, respectively. The Serre conductor N(ρ) is only divisible by primes distinct from the characteristic of 𝔽 at which ρ is ramified. When ρ = ρ̄_{f,𝔭} arises from a modular form, the Serre conductor N(ρ) always divides (but is not necessarily equal to) the level N of f. In particular, using parts 1 and 2 of Theorem 4.1, one can show that

N(ρ_{a,b,c}) = 2.  (21)

The recipe for defining k(ρ) is somewhat more involved, but depends only on the restriction of ρ to the decomposition group (in fact, the inertia group) at p. It will suffice, for the purposes of this survey, to note that when ρ arises from the p-division points of a finite flat group scheme over Z_{p}, then Serre’s recipe gives k(ρ) = 2. Hence, by part 3 of Theorem 4.1,

k(ρ_{a,b,c}) = 2.  (22)

In [Se2], Serre conjectures that any odd irreducible two-dimensional mod p Galois representation ρ as in (20) necessarily arises from an appropriate modular form mod p of weight k(ρ) and level N(ρ). This conjecture has recently been proved by Khare and Wintenberger (cf. Theorem 1.2 of [KW]) in the case where N(ρ_{a,b,c}) is odd, and follows in the general case from a similar method, using a result of Kisin [Ki].

Theorem 4.5. Let ρ be an odd, irreducible two-dimensional mod p representation of G_{Q}. Then there exists an eigenform f of weight k(ρ) on Γ_{1}(N(ρ)), and a prime 𝔭|p of the field K_{f} such that ρ is isomorphic to ρ̄_{f,𝔭} as a representation of G_{Q}.

Proof of Fermat’s Last Theorem. Let (a, b, c) be a primitive nontrivial solution of Fermat’s equation x^{p}+y^{p} = z^{p}, and consider the Galois representation ρ = ρ_{a,b,c} associated to the p-division points of the associated Frey curve. It follows from Theorem 4.2 that ρ is an odd, irreducible mod p representation of G_{Q}. Its Serre conductor and weight are N(ρ) = 2 and k(ρ) = 2 by (21) and (22). Therefore Theorem 4.5 implies the existence of a nontrivial cusp form in S_{2}(2, C). This leads to a contradiction, because there are no such cusp forms: the modular curve X_{0}(2) has genus zero and hence has no regular differentials. This contradiction implies Fermat’s Last Theorem.

4.5 The Shimura–Taniyama conjecture

Historically, the proof of Theorem 4.5 by Khare and Wintenberger came almost 10 years after Wiles proved Fermat’s Last theorem. In essence, Wiles proved enough of Theorem 4.5 to cover the Galois representations ρ_{a,b,c} arising from hypothetical solutions of Fermat’s equation.

More precisely, the articles [Wi] and [TW] proved the following result, known as the Shimura-Taniyama conjecture for semistable elliptic curves:

Theorem 4.6. Let E be a semistable elliptic curve over Q of conductor N. Then there is a normalised eigenform f in S_{2}(N, Z) such that V_{p}(E) is isomorphic to V_{p}(A_{f}).

The proof of this theorem—or even, an outline of its main ideas—is beyond the scope of this survey. For details the reader is invited to consult [DDT] for example.

Theorem 4.6 implies that ρ_{a,b,c} arises from a modular form in S_{2}(N, C), where N = ∏_{ℓ|abc} ℓ. The Serre conjecture (Theorem 4.5) for ρ_{a,b,c} then follows from an earlier theorem of Ribet (which also played an important role in Wiles’ original approach to proving Theorem Theorem 4.6.)

Theorem 4.7. Suppose that p is odd. Let ρ be an irreducible mod p Galois representation which arises from a modular form (of some weight and level). Then it also arises from an eigenform of weight k(ρ) and level N(ρ).

Aside from the fact that it proves Fermat’s Last Theorem, the importance of Theorem 4.6 can be justified on several other levels.

Firstly, the methods used to prove Theorem 4.6 were subsequently refined in [BCDT] to prove the full Shimura–Taniyama conjecture: all elliptic curves over Q are modular. This result is of great importance in understanding the arithmetic of elliptic curves over Q, as will be explained in more detail in the next chapter.

Secondly—and this is a theme that we will not begin to do justice to, because it falls outside the scope of this survey—Wiles’ method for proving Theorem 4.6 has led to a general, flexible method for establishing relationships between Galois representations and modular forms. It was by building on these techniques that Khare and Wintenberger proved Serre’s conjecture (Theorem 4.5). Over the years, many other conjectures of this type have been proved building on the proof of Theorem 4.6: for instance, special cases of Artin’s conjecture relating representations with finite image to modular forms of weight one (cf. for example [Tay2] and the references contained therein), and a proof of the Sato–Tate conjecture for elliptic curves over Q in [Tay1], [HST], and [CHT].

Closer to the themes that have been developed in this chapter, we mention a natural generalisation of Theorem 4.6 concerning abelian varieties of GL_{2}- type. An abelian variety A over Q is said to be of GL_{2}-type if End_{Q}(A) ⊗ Q contains a field K with [K : Q] = dim(A). The reason for this terminology is that such an A gives rise, for each prime ideal 𝔭 of K, to a two-dimensional Galois representation

ρ_{A,𝔭} : G_{Q} ⟶ GL_{2}(K_{𝔭})

arising from the action of G_{Q} on V_{p}(A)⊗_{K} K_{𝔭}. The abelian varieties A_{f} arising from the Eichler-Shimura construction are examples of abelian varieties of GL_{2}-type. A conjecture of Fontaine and Mazur predicts that all abelian varieties of GL_{2}-type arise as quotients of jacobians of modular curves. It can be shown that this generalisation of the Shimura-Taniyama conjecture follows from Theorem 4.5. (Cf. for example [Se2] or the introduction of [Ki].)

4.6 A summary of Wiles’ proof

There are some enlightening parallels to be drawn between the proof of Fermat’s Last Theorem and Faltings’ proof of the Mordell conjecture as summarised in Section 2.8. Like Faltings’ proof, the proof of Fermat’s Last theorem is based on a sequence of maps, resulting in a sequence of transformations leading from the original problem to questions about other types of structures, such as Galois representations, and ultimately modular forms. These reductions are summarised in the diagram below.

\{ \begin{matrix}Integer solutions \\ (a,b,c) of \\ x^{p} + y^{p} = z^{p} \end{matrix} \} ⟶^{R_{1}} \{ \begin{matrix}Semistable elliptic curves \\ of conductor N = abc \\ and discriminant 2^{−8}(abc)^{2p} \end{matrix} \}

⟶^{R_{4}} \{ \begin{matrix}Irreducible galois representations \\ ρ : G_{Q} ⟶ GL_{2}(𝔽_{p}) \\ with N(ρ) = 2 and k(ρ) = 2. \end{matrix} \}

⟶^{R_{5}} \{ \begin{matrix}Cusp forms in \\ S_{2}(2, Z/pZ). \end{matrix} \} .

1. The map R_{1} is defined via the Frey curve, and is reminiscent of the Kodaira–Parshin construction of Section 2.2. An important difference is that the set of primes of bad reduction of the Frey curve associated to (a, b, c) is not bounded independently of (a, b, c). In fact, the set of primes of bad reduction for E consists exactly of the primes that divide abc.

2. The map R_{4} plays a role analogous to the passage to the ℓ-adic representations in Faltings’ proof. An important difference here is that we consider mod p representations (with coefficients in a finite field) rather than p-adic representations. The justification for doing this is given by Theorem 4.1, which shows that the mod p representation ρ attached to E_{a,b,c} has bounded ramification. Note that the corresponding p-adic representation would be ramified precisely at the primes dividing pabc. It is an exercise to show that the map R_{4} is finite-to-one when p ≥ 7. (Hint: use Faltings’ theorem 2.1, and the fact that X(p) has genus > 1 when p ≥ 7.) It is even believed that R_{4} is injective once p is large enough (cf. Conjecture 3.18), but this assertion is still unproved.

3. The map R_{5} is a new ingredient that has no counterpart in Faltings’ proof of Mordell’s conjecture, and exploits the deep “dictionary” that is expected to exist between Galois representations and modular forms— in this case, the Serre conjecture proved by Khare and Wintenberger.

4. The final step in the argument exploits the fact that there are no modular forms of weight two and level two. This last point may seem like a “lucky accident” in the proof of Fermat’s Last Theorem. Indeed the presence of modular forms of higher level presents an obstruction for the method based on Frey curves to yield results on more general ternary diophantine equations of Fermat type. However, see the article by Charollois in this volume [Char], where a refinement of the techniques described in this chapter leads to a strikingly general result on the generalised Fermat equation ax^{p} + by^{p} + cz^{p} = 0.

Remark 4.8. One of the consequences of Conjecture 3.18 is that the generalised Fermat equation ax^{n} + by^{n} + cz^{n} = 0 (with a, b, c fixed) has no primitive integer solutions (x, y, z) with xyz ≠ 0, ±1, once n is large enough. (The reader who masters the ideas in the article by Pierre Charollois in this proceedings volume will be able to prove this assertion.)

5 Elliptic curves

After surveying curves of genus > 1, we turn our attention to curves of genus 1. A projective curve of genus 1 over a field K, equipped with a distinguished K-rational point over that field, is endowed with a natural structure of a commutative algebraic group over K for which the distinguished element becomes the identity. Such a curve is called an elliptic curve.

If E is an elliptic curve defined over a number field K, then the Mordell– Weil Theorem (cf. Theorem 7 of the introduction) asserts that the group E(K) of K-rational points on E is finitely generated. Let r(E, K) denote the rank of this finitely generated abelian group. Many of the important questions in the theory of elliptic curves revolves around calculating this invariant, and understanding its behaviour as E or K vary.

Question 5.1. Is there an effective algorithm to calculate r(E, K), given E and K?

Showing that Fermat’s method of descent yields such an effective algorithm is intimately connected to the Shafarevich–Tate conjecture asserting the finiteness of the Shafarevich–Tate group LLI(E/K) of E/K.

One can also fix a base field (the most natural, and interesting, case being the case where K = Q) and ask

Question 5.2. Is the rank r(E, K) unbounded, as E ranges over all elliptic curves defined over K?

One can also fix an elliptic curve E and enquire about the variation of r(E, K) as K ranges over different number fields.

The main tool available at present to study r(E, K) is the relationship between the rank and the Hasse–Weil L-series predicted by the Birch and Swinnerton-Dyer conjecture (Conjecture 3.9).

Assume that E is an elliptic curve over Q. Thanks to Theorem 4.6 (and its extension to all elliptic curves over Q given in [BCDT]), the Hasse–Weil L-series L(E, s) is equal to L(f, s) for some newform f of weight two. In particular, L(E, s) has an analytic continuation to the entire complex plane, and a functional equation.

The main result we will discuss in this chapter is the following:

Theorem 5.3. Let E be an elliptic curve over Q, and let L(E, s) be its Hasse–Weil L-series. If r := ord_{s=1}L(E, s) ≤ 1, then r(E, Q) = r and LLI(E/Q) is finite.

5.1 Modular parametrisations

Let E be an elliptic curve over Q of conductor N. Recall the modular curve X_{0}(N) that was introduced in Section 3.1. The following theorem, which produces a dominant rational map from such a curve to E, plays a crucial role in the proof of Theorem 5.3. Theorem 5.4. There exists a nonconstant map of curves over Q

φ : X_{0}(N) ⟶ E.

Proof. By Theorem 4.6, there is a normalised eigenform f in S_{2}(N, Z) satisfying L(E, s) = L(f, s). Let A_{f} be the quotient of J_{0}(N) associated to f via the Eichler–Shimura construction. By assumption, the Galois representations V_{p}(E) and V_{p}(A_{f}) are isomorphic. Hence the isogeny conjecture (Theorem 2.18) implies the existence of an isogeny α : A_{f} ⟶ E defined over Q. Composing such an isogeny with the natural surjective morphism J_{0}(N) ⟶ A_{f} gives a nonconstant map Φ : J_{0}(N) ⟶ E. The modular parametrisation φ is defined by setting φ(x) := Φ((x) − (∞)).

It is useful to describe briefly how the modular parametrisation φ can be computed analytically. The pull-back φ^{∗}(ω_{E}) is a nonzero rational multiple of the diferential form

ω_{f} := 2πif(τ)dτ = ∑_{n=1}^{∞} a_{n}q^{n} \frac{dq}{q} .

Denote by Λ_{f} the collection of periods of ω_{f} (integrals of ω_{f} against smooth closed one-chains C in X_{0}(N)(C)):

Λ_{f} := {∫_{C} ω_{f}, where ∂C = 0} .

It is a lattice in C, and A_{f}(C) ≃ C/Λ_{f}. Let us replace E by A_{f}, so that α = 1. It is suggestive (for later generalisations) to view φ as a map

φ : Div^{0}(X_{0}(N)) ⟶ E.

This map is defined on Div^{0}(X_{0}(N)(C)) by the rule

φ(Δ) := ∫_{∂(C)=Δ^{C s.t.}} ω_{f} (mod Λ_{f}),  (23)

where the integral is taken over any smooth one-chain C whose boundary is Δ. The invariant φ(Δ) ∈ C/Λ_{f} is viewed as a point on E(C) via the Weierstrass uniformisation.

5.2 Heegner points

Perhaps the most important arithmetic application of the modular parametrisation arises from the fact that X_{0}(N) is endowed with a systematic supply of algebraic points defined over abelian extensions of imaginary quadratic fields—the so-called CM-points. These points correspond, in the moduli interpretation of X_{0}(N), to pairs (A, C) where A is an elliptic curve whose endomorphism ring O = End(A) is an order in a quadratic imaginary field K. Such an elliptic curve is said to have complex multiplication by K, and the corresponding points on X_{0}(N) are called CM-points attached to K. Let CM(K) denote the set of all CM points in X_{0}(N) attached to K. It satisfies the following properties.

1. The set CM(K) is dense in X_{0}(N)(C) (relative to the Zariski topology, and also the complex topology).

2. Let K^{ab} denote the maximal abelian extension of K. Then CM(K) is contained in X_{0}(N)(K^{ab}).

3. Analytically, CM(K) = Γ_{0}(N)\(H ∩ K).

Definition 5.5. The collection of points

HP (K) := {φ(Δ)}_{Δ∈Div^{0}(CM(K))} ⊂ E(K^{ab})

is called the system of Heegner points on E attached to K.

The usefulness of Heegner points arises from two facts:

1. They can be related to L-series, thanks to the theorem of Gross–Zagier and its generalisations.

2. They can be used to bound Mordell–Weil groups and Shafarevich–Tate groups of elliptic curves, following a descent method that was discovered by Kolyvagin.

Heegner points and L-series.

For simplicity, suppose that the imaginary quadratic field K satisfies the following so-called Heegner hypothesis:

Hypothesis 5.6. There exists a ideal N of norm N in O_{K} with cyclic quotient.

This hypothesis is used to construct a distinguished element in HP(K). More precisely, let h denote the class number of K, and let H be its Hilbert class field. By the theory of complex multiplication, there are precisely h distinct (up to isomorphism over C) elliptic curves A_{1}, …, A_{h} having endomorphism ring equal to O_{K}. The j-invariants of these curves belong to H, and are permuted simply transitively by the action of Gal(H/K). It is therefore possible to choose A_{1}, …, A_{h} in such a way that they are defined over H, and permuted by the action of Gal(H/K).

The pairs (A_{i}, A_{i}[N]) (with 1 ≤ i ≤ h) correspond to points P_{i} in X_{0}(N)(H). Let

P_{K} := φ((P_{1}) + ⋯ + (P_{h}) − h(∞)) ∈ E(K).  (24)

The fact that the point P_{K} has an explicit moduli description makes it possible to establish some of its key properties. For example, let P̄_{K} denote the image of P_{K} under complex conjugation. Then it can be shown that

P̄_{K} = wP_{K} (mod E(K)_{tors}),  (25)

where w ∈ {±1} is the negative of the sign in the functional equation for L(E, s) = L(f, s). (Cf. Chapter 3 of [Da2].) This provides a simple connection between the behaviour of P_{K} and the L-series L(E, s).

We note that, in many cases where Hypothesis 5.6 is satisfied (for example, when all the primes dividing N are split in the quadratic imaginary field K), the sign in the functional equation for the Hasse–Weil L-series L(E/K, s) is −1, so that L(E/K, 1) = 0. It then becomes natural to consider the first derivative L′(E/K, 1) at the central critical point. The following theorem of Gross and Zagier establishes an explicit link between P_{K} and this quantity.

Theorem 5.7. Let ⟨f, f⟩ denote the Petersson scalar product of f with itself, and let h(P_{K}) denote the Néron–Tate canonical height of P_{K} on E(K). There is an explicit nonzero rational number t such that

L′(E/K, 1) = t · ⟨f, f⟩ · h(P_{K}).  (26)

In particular, the point P_{K} is of infinite order if and only if L′(E/K, 1) ≠ 0.

The proof of Theorem 5.7 given in [GZ] proceeds by a direct calculation in which both sides of (26) are computed explicitly, compared, and found to be equal.

Remark 5.8. Let P_{n} be a point in CM(K) corresponding to an elliptic curve with endomorphism ring equal to the order O_{n} of conductor n in K. Such a point can be defined over the ring class field H_{n} of K of conductor n, whose Galois group G_{n} := Gal(H_{n}/K) is canonically identified with the class group Pic(O_{n}) by class field theory. If χ : G_{n} ⟶ C^{×} is a complex character, one can generalise (24) to define

P_{χ} := φ ( ∑_{σ∈G_{n}} χ(σ)P_{n}^{σ}) ∈ E(H_{n}) ⊗ C.  (27)

A generalisation of Theorem 5.7 due to S. Zhang (cf. for example [Zh1], [Zh2], [Ho1] and [Ho2]) relates the height of P_{χ} to the derivative of the twisted L-series L(E/K, χ, s) at s = 1.

When L′(E/K, 1) ≠ 0, the method of Heegner points gives an efficient method for producing a point of infinite order in E(K). The following proposition asserts the existence of many K for which the L-series does not vanish.

Proposition 5.9. Suppose that r := ord_{s=1}L(E, s) ≤ 1. Then there exist infinitely many quadratic imaginary fields K satisfying Hypothesis 5.6 for which

ord_{s=1}(L(E/K, s)) = 1.

The proof of this proposition is explained in [MM].

Heegner points and arithmetic: Kolyvagin’s descent

Theorem 5.7 implies that if L′(E/K, 1) ≠ 0, then P_{K} is of infinite order and hence r(E, K) ≥ 1. The following theorem of Kolyvagin gives a bound in the other direction as well.

Theorem 5.10 (Kolyvagin). Suppose that P_{K} is of infinite order in E(K). Then r(E, K) = 1, and LLI(E/K) is finite.

For a proof of this theorem, see [Gr] or Chapter 10 of [Da2]. Let us just mention here that Kolyvagin’s proof makes essential use of the fact that point P_{K} does not come alone, but rather is part of a norm-compatible system of points in E(K^{ab}) arising from the (infinite) collection of points in HP(K). These points are used to construct global cohomology classes in H^{1}(K, E[p]) whose local behaviour can be controlled precisely and related to P_{K}. Under the assumption that P_{K} is of infinite order, this system of ramified cohomology classes is enough to bound the p-Selmer group of E/K and show that r(E, K) = 1.

5.3 Proof of Theorem 5.3

We will now explain how the properties of P_{K} and HP(K) described in the previous section can be combined to prove Theorem 5.3:

Proof of Theorem 5.3. Assume that r ≤ 1. By Proposition 5.9, there is a quadratic imaginary field K satisfying Hypothesis 5.6, for which

ord_{s=1}(L(E/K, s)) = 1.

Fix such a K, and consider the point P_{K}. Since L′(E/K, 1) ≠ 0, Theorem 5.7 implies that P_{K} is of infinite order. Theorem 5.10 then shows that

r(E, K) = 1, and LLI(E/K) is finite.

Let E′ denote the quadratic twist of E over K. We then have

1 = r(E, K) = r(E, Q) + r(E′, Q).

To be able to ignore finer phenomena associated to torsion in E(K), it is convenient to replace P_{K} by its image in E(K) ⊗ Q. Since E(K) ⊗ Q is generated by P_{K}, it follows that

r(E, Q) = \{ \begin{matrix}0 & if P̄_{K} = −P_{K}, \\ 1 & if P̄_{K} = P_{K}. \end{matrix}

Theorem 5.3 now follows from (25).

Remark 5.11. The proof of Theorem 5.3 carries over with only minor changes when E is replaced by the abelian variety quotient A_{f} attached to an arbitrary eigenform f of weight 2 on Γ_{0}(N). This is how Theorem 3.11 is proved:

L(A_{f}, 1) ≠ 0 ⟹ A_{f}(Q) is finite.

The reader will recall the key role played by this theorem in the proof of Theorem 3.1 and (even more importantly) in Merel’s proof of the uniform boundedness conjecture for elliptic curves explained in Marusia Rebolledo’s article in these proceedings.

5.4 Modularity of elliptic curves over totally real fields

Because of the crucial role played by the system HP(K) in the proof of Theorem 5.3, it is natural to ask whether such structures are present in more general settings. For example, we would like to prove analogues of Theorem 5.3 for elliptic curves defined over number fields other than Q. The class of number fields for which this program is best understood is the class of totally real fields.

More precisely, let F be a totally real field of degree n, and let E be an elliptic curve over F, of conductor N. Assume, for simplicity, that F has narrow class number one, so that in particular the conductor can now be viewed as a totally positive element of O_{F} rather than just an ideal.

The group Γ_{0}(N; O_{F}) ⊂ SL_{2}(O_{F}) is defined as the group of matrices that are upper triangular modulo N. The n distinct real embeddings v_{1}, …, v_{n} : F ⟶ R of F allow us to view Γ_{0}(N; O_{F}) as a subgroup of SL_{2}(R)^{n}. This subgroup acts discretely on the product H^{n}, and the analytic quotient Γ_{0}(N; O_{F})\H^{n} represents the natural generalisation of modular curves to this setting:

1. This quotient is identified with the complex points of an n-dimensional algebraic variety Y_{0}(N; O_{F}) defined over F. This variety can be compactified by adjoining a finite set of cusps, much as in the setting n = 1 of classical modular curves. A suitable desingularisation of the resulting projective variety is denoted X_{0}(N; O_{F}), and is called a Hilbert modular variety. Hilbert modular varieties are basic examples of higher dimensional Shimura varieties.

2. The variety X_{0}(N; O_{F}) is equipped with natural Hecke correspondences T_{λ} indexed by the prime ideals of O_{F}.

3. These correspondences induce linear actions on the n-th deRham cohomology H_{dR}^{n}(X_{0}(N; O_{F})), and the eigenvalues of the Hecke operators are expected to encode the same type of arithmetic information as in the case where F = Q.

To amplify this last point and make it more precise, we state the following generalisation of the Shimura–Taniyama conjecture to elliptic curves over F:

Conjecture 5.12. Let E be an elliptic curve over F of conductor N. There exists a closed (in fact, holomorphic) differential form ω ∈ H_{dR}^{n}(X_{0}(N; O_{F})) satisfying

T_{λ}(ω) = a_{λ}(E)ω,

for all primes λ |̸N of O_{F}.

Remark 5.13. In some cases, the methods of Wiles for proving the modularity of elliptic curves over Q have been extended to the setting of elliptic curves over totally real fields, and many cases of Conjecture 5.12 can be made uncondtional.

5.5 Shimura curves

When n > 1, the holomorphic differential form ω whose existence is predicted by Conjecture 5.12 cannot be used to directly produce an analogue of the modular parametrisation. In this sense, there is no immediate generalisation of Theorem 5.4, which plays such a crucial role in the construction of HP(K) when n = 1.

To extend the notion of Heegner points, it is necessary to introduce another generalisation of modular curves: the so-called Shimura curves associated to certain quaternion algebras over F.

A quaternion algebra B over F is said to be almost totally definite if

B ⊗_{v_{1}} R ≃ M_{2}(R), B ⊗_{v_{j}} R ≃ ℍ, for 2 ≤ j ≤ n.

We can associate to any order R in B a discrete subgroup

Γ := v_{1}(R^{×}) ⊂ SL_{2}(R),

which acts discretely on H by Mobius transformations. When F = Q and B = M_{2}(Q) is the split quaternion algebra, one recovers the analytic description of the modular curves X_{0}(N). Otherwise, the analytic quotient Γ ⊂ H is a compact Riemann surface which can be identified with the complex points of an algebraic curve X possessing a canonical model over F. The curve X can be related (following a construction of Shimura) to the solution of a moduli problem and is also equipped with a supply CM(K) ⊂ X(K^{ab}) of CM points, associated this time to any quadratic totally imaginary extension K of F.

An elliptic curve E over F is said to be arithmetically uniformisable if there is a nonconstant map defined over F, generalising Theorem 5.4,

φ : Div^{0}(X) ⟶ E.

The theory of Jacquet–Langlands gives a precise (partly conjectural) understanding of the class of elliptic curves that should be arithmetically uniformisable:

Theorem 5.14. Let E be an elliptic curve over F which is not isogenous to any of its Galois conjugates. Then E is arithmetically uniformisable if and only if

1. E is modular in the sense of Conjecture 5.12;

2. E has potentially semistable reduction at a prime of F, or can be defined over a field F of odd degree.

The collection HP(K) := φ(Div^{0}(CM(K))) ⊂ E(K^{ab}), for suitable totally complex quadratic extensions K/F, can be used to obtain results analogous to Theorem 5.3 for elliptic curves over totally real fields. See [Zh1] where general results in this direction are obtained.

The articles [Vo] and [Gre1] in this volume describe Shimura curves and the associated parametrisations in more detail, from a computational angle. The article [Vo] discusses explicit equations for Shimura curves of low degree, and [Gre1] explains how to approach the numerical calculation of the systems HP(K) of Heegner points via p-adic integration of the associated modular forms, exploiting the theory of p-adic uniformisation of these curves due to Cerednik and Drinfeld.

5.6 Stark–Heegner points

Heegner points arising from Shimura curve parametrisations do not completely dispell the mystery surrounding the Birch and Swinnerton-Dyer conjecture for (modular) elliptic curves over totally real fields, since (even assuming the modularity Conjecture 5.12) there remain elliptic curves over F that are not arithmetically uniformisable.

The simplest example of such an elliptic curve is one that has everywhere good reduction over a totally real field F of even degree, and is not isogenous to any of its Galois conjugates. (More generally, one can also consider any quadratic twist of such a curve.) For these elliptic curves, there is at present very little evidence for the Birch and Swinnerton-Dyer conjecture, and in particular the analogue of Theorem 5.3 is still unproved when ord_{s=1}L(E/F, s) = 1. (In the case where L(E/F, 1) ≠ 0, see the work of Matteo Longo [Lo].)

The notion of Stark–Heegner points represents an attempt to remedy this situation (albeit conjecturally) by exploiting the holomorphic differential nform ω whose existence is predicted by Conjecture 5.12 rather than resorting to a Shimura curve parametrisation. We note that the holomorphic form ω can be written

ω = f(τ_{1}, …, τ_{n})dτ_{1} ⋯ dτ_{n},

where f is a (holomorphic) Hilbert modular form of parallel weight 2 on

Γ_{0}(N), satisfying, for all matrices ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N),

f (\frac{a_{1}τ_{1} + b_{1}}{c_{1}τ_{1} + d_{1}} , …, \frac{a_{n}τ_{n} + b_{n}}{c_{n}τ_{n} + d_{n}} ) = (c_{1}τ_{1} + d_{1})^{2} ⋯ (c_{n}τ_{n} + d_{n})^{2}f(τ_{1}, …, τ_{n}).

We let any unit ϵ ∈ O_{F}^{×} act on H^{n} by the rule:

ϵ ⋆ τ_{j} = \{ \begin{matrix}ϵ_{j}τ_{j} & if ϵ_{j} > 0; \\ ϵ_{j}τ̄_{j} & if ϵ_{j} < 0. \end{matrix}

For any subset S ⊂ {2, …, n} of cardinality m, we can then define a closed differential n-form of type (n − m, m) by choosing a unit ϵ of O_{F}^{×} which is negative at the places of S, and positive at the other embeddings, and setting

ω_{S} = f(ϵ ⋆ τ_{1}, …, ϵ ⋆ τ_{n})d(ϵ ⋆ τ_{1}) … d(ϵ ⋆ τ_{n}).

Finally we set

ω_{E} := ∑_{S⊂{2,...,n}} ω_{S}.

The following conjecture is due to Oda [Oda].

Conjecture 5.15. The set of periods

Λ_{f} := {∫_{C} ω_{E} for C ∈ H_{n}(X_{0}(N, F)(C), Z)} ⊂ C

is a lattice which is commensurable with the period lattice of E_{1} := v_{1}(E).

Conjecture 5.15 can be used to define a generalisation of the modular parametrisation of equation (23). This map is defined on homologically trivial (n − 1)-cycles on X_{0}(N; O_{F})(C) by the rule

φ(Δ) := ∫_{∂(C)=Δ^{C s.t.}} ω_{f} (mod Λ_{f}).  (28)

The interest of this generalisation of (23) is that it is possible to define a collection of distinguished topological (n−1)-cycles on which φ is conjectured to take algebraic values.

These cycles, which play the same role that Heegner divisors of degree zero played in the case where n = 1, are defined in terms of certain quadratic extensions K of F. Such a quadratic extension is said to be almost totally real if

K ⊗_{v_{1}} R ≃ C, K ⊗_{v_{j}} R ≃ R ⊕ R for 2 ≤ j ≤ n.

Let ι : K ⟶ M_{2}(F) be an F -algebra embedding, and let K_{1}^{×} be the group of elements whose norm to F is equal to 1. The torus v_{1}(ι(K_{1}^{×})) acts on H with a unique fixed point τ_{1}, and ι(K_{1}^{×}) acts on the region {τ_{1}} × H^{n−1} without fixed points. The orbit of any point in this region under the action of ι((K ⊗_{F} R)_{1}^{×}) is a real (n − 1)-dimensional manifold Z_{ι} ⊂ {τ_{1}} × H^{n−1} which is homeomorphic to R^{n−1}. The group G_{ι} := ι(K^{×}) ∩ Γ_{0}(N, O_{F}) is an abelian group of rank n − 1, corresponding to a finite index subgroup of the group of relative units in K/F. Consider a fundamental region for the action of G_{ι} on Z_{ι}. The image Δ_{ι} of such a region in the Hilbert modular variety X_{0}(N; O_{F}) is a closed (n − 1)-cycle, which is topologically isomorphic to a real (n − 1)-dimensional torus.

Conjecture 5.16. Assume that Δ_{ι} is homologically trivial. Then the point φ(Δ_{ι}) ∈ E_{1}(C) is an algebraic point, and is in fact the image of a point in E(K^{ab}) under any embedding K^{ab} ⟶ C extending v_{1} : F ⟶ R.

Remark 5.17. The original formulation of Conjecture 5.16 given in [DL] was phrased in terms of group cohomology. The definition of φ(Δ_{ι}) used in Conjecture 5.16, which suggests an analogy between φ and higher Abel-Jacobi maps, was formulated only later, in [CD] (in a context where cusp forms are replaced by Eisenstein series; the elements φ(Δ_{ι}) can then be related to Stark units). The equivalence between Conjecture 5.16 and the main conjecture of [DL] is explained in [CD].

Conjecture 5.16 can be formulated more precisely, in a way that makes a prediction about the fields of definition of the points φ(Δ_{ι}). It is expected that the system of points

HP(K) := {φ(Δ_{ι})}_{ι:K⟶M_{2}(F)},

as ι ranges over all possible embeddings, gives rise to an infinite collection of algebraic points in E(K^{ab}) with properties similar to those of the system of Heegner points defined in Section 5.2. Such a system of points (if its existence, and basic properties, could be established, a tall order at present!) would lead to a proof of Theorem 5.3 for all (modular) elliptic curves defined over totally real fields, not just those that are arithmetically uniformisable.

For more details on Conjecture 5.16, a more precise formulation, and numerical evidence, see Chapter 8 of [Da2], or [DL]. For an explanation of the relation between Conjecture 5.16 and the conjectures of [DL], see [CD].

The Stark-Heegner points attached to Hilbert modular forms that were defined and studied in [DL] and [CD] can be viewed as the basic prototype for the general notion of Stark-Heegner points. Here are some further variants that have been explored so far in the literature:

1. If E is an elliptic curve over Q of conductor N = pM with p ∤ M, a p-adic analogue of the map φ of equation (28)—described in terms of group cohomology rather than singular cohomology, following the same approach and in [DL]—is defined in [Da1], by viewing E as uniformised by the “mock Hilbert surface”

Γ_{0}(M; Z[1/p])\(H_{p} × H),

where H_{p} := C_{p} −Q_{p} is the p-adic upper half plane, and Γ_{0}(M; Z[1/p]) is the group of matrices in SL_{2}(Z[1/p]) which are upper-triangular modulo M. The resulting map φ associates a point in P_{ι} ∈ E(Q̄_{p}) to any embedding ι : K ⟶ M_{2}(Q) when K is a real quadratic field in which p is inert. The system {P_{ι}} ⊂ E(Q̄_{p}), as ι ranges over all embeddings of K into M_{2}(Q), is expected to yield a system of points in E(K^{ab}) with the same properties as the Heegner points attached to an imaginary quadratic base field. This construction is not expected to yield new cases of the Birch and Swinnerton-Dyer over the base field Q—this conjecture is completely known when ord_{s=1}L(E, s) ≤ 1, thanks to Theorem 5.3. However, it would give new cases of this conjecture over certain abelian extensions of real quadratic fields, and, more importantly perhaps, it suggests an explicit analytic construction of class fields of real quadratic fields. For more details on Stark-Heegner points attached to real quadratic fields, see [Da1] or Chapter 9 of [Da2]. The article [DP] describes efficient algorithms for calculating the points φ(Δ_{ι}), and uses them to gather numerical evidence for the conjectures of [Da1], while [BD] provides some theoretical evidence.

2. The article [Tr] formulates and tests numerically a Stark–Heegner construction that leads to conjectural systems of algebraic points on elliptic curves defined over a quadratic imaginary base field. The details of the construction of [Tr] are explained in the article [Gre2] by Matt Greenberg in this proceedings volume. We remark that there is not a single example of an elliptic curve E genuinely defined over such a field (i.e., which is not isogenous to its Galois conjugate) for which Theorem 5.3 (or even just the Shafarevich–Tate conjecture) has been proved.

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