Rational points on modular elliptic curves

Henri Darmon

Department of Mathematics, McGill University, Montreal, Quebec, Canada, H3A-2K6

Current address: Department of Mathematics, McGill University, Montreal, Quebec, Canada H3A-2K6

E-mail address: darmon@math.mcgill.ca

1991 Mathematics Subject Classification. Primary 11-02; Secondary 11F03 11F06 11F11 11F41 11F67 11F75 11F85 11G05 11G15 11G40

Key words and phrases. Elliptic curves, modular forms, Heegner points, Shimura

curves, rigid analysis, p-adic uniformisation, Hilbert modular forms, Stark-Heegner points, Kolyvagin’s theorem.

Some of the ideas discussed in this monograph have their roots in the author’s collaboration with Massimo Bertolini over the years. Exchanges with Samit Dasgupta, Peter Green, Adrian Iovita, and Adam Logan have also helped in forming and solidifying key insights. The author thanks Drs. Peter Hilton and Heath Martin of the University of Central Florida for the marvelous job they did in running the NSF-CBMS conference on which this monograph is based, as well

as the participants for their many stimulating comments and suggestions. A rough version of this text formed the basis for a graduate seminar at McGill University in the 2002-2003 academic year. The participants of this seminar – Samit Dasgupta, Hugo Chapdelaine, Antoine Gournay, and Matt Greenberg –

pointed out a number of mistakes and their suggestions have led to improvements in the manuscript. (Needless to say, the imperfections and inaccuracies which remain are to be blamed on the author alone!) Finally, it is a great pleasure to acknowledge NSERC and CICMA whose material support has greatly facilitated

the writing of this monograph.

Abstract. Based on an NSF-CBMS lecture series given by the author at the University of Central Florida in Orlando from August 8 to 12, 2001, this monograph surveys some recent developments in the arithmetic of modular elliptic curves, with special emphasis on the Birch and Swinnerton-Dyer conjecture, the construction of rational points on modular elliptic curves, and the crucial role played by modularity in shedding light on these questions.

iii

Pour Galia et Maia

Contents

Preface xi

Chapter 1. Elliptic curves 1 1.1. Elliptic curves 1 1.2. The Mordell-Weil theorem 3 1.3. The Birch and Swinnerton-Dyer conjecture 6 1.4. L-functions 7 1.5. Some known results 8 Further results and references 9 Exercises 10

Chapter 2. Modular forms 13 2.1. Modular forms 13 2.2. Hecke operators 14 2.3. Atkin-Lehner theory 16 2.4. L-series 17 2.5. Eichler-Shimura theory 18 2.6. Wiles’ theorem 20 2.7. Modular symbols 21 Further results and references 25 Exercises 25

Chapter 3. Heegner points on X_{0}(N) 29 3.1. Complex multiplication 29 3.2. Heegner points 33 3.3. Numerical examples 34 3.4. Heegner systems 35 3.5. Relation with the Birch and Swinnerton-Dyer conjecture 37 3.6. The Gross-Zagier formula 39 3.7. Kolyvagin’s theorem 40 3.8. Proof of the Gross-Zagier-Kolyvagin theorem 40 Further results 41 Exercises 41

Chapter 4. Heegner points on Shimura curves 45 4.1. Quaternion algebras 46 4.2. Modular forms on quaternion algebras 47 4.3. Shimura curves 49 4.4. The Eichler-Shimura construction, revisited 50 4.5. The Jacquet-Langlands correspondence 50 4.6. The Shimura-Taniyama-Weil conjecture, revisited 51 4.7. Complex multiplication for H/Γ_{N^{+},N^{−}} 51 4.8. Heegner systems 52 4.9. The Gross-Zagier formula 53 References 54 Exercises 54

Chapter 5. Rigid analytic modular forms 57 5.1. p-adic uniformisation 57 5.2. Rigid analytic modular forms 60 5.3. p-adic line integrals 63 Further results 65 Exercises 65

Chapter 6. Rigid analytic modular parametrisations 67 6.1. Rigid analytic modular forms on quaternion algebras 67 6.2. The Čerednik-Drinfeld theorem 68 6.3. The p-adic Shimura-Taniyama-Weil conjecture 68 6.4. Complex multiplication, revisited 69 6.5. An example 70 6.6. p-adic L-functions, d’après Schneider-Iovita-Spiess 73 6.7. A Gross-Zagier formula 74 Further results 75 Exercises 75

Chapter 7. Totally real fields 79 7.1. Elliptic curves over number fields 79 7.2. Hilbert modular forms 80 7.3. The Shimura-Taniyama-Weil conjecture 82 7.4. The Eichler-Shimura construction for totally real fields 83 7.5. The Heegner construction 83 7.6. A preview of chapter 8 84 Further results 85

Chapter 8. ATR points 87 8.1. Period integrals 87 8.2. Generalities on group cohomology 88 8.3. The cohomology of Hilbert modular groups 89 8.4. ATR points 93 References 94 Exercises 95

Chapter 9. Integration on H_{p} × H 97 9.1. Discrete arithmetic subgroups of SL_{2}(ℚ_{p}) × SL_{2}(ℝ) 98 9.2. Forms on H_{p} × H 99 9.3. Periods 101 9.4. Some p-adic cocycles 104 9.5. Stark-Heegner points 105 9.6. Computing Stark-Heegner points 106 Further results 108

Exercises 109

Chapter 10. Kolyvagin’s theorem 113 10.1. Bounding Selmer groups 113 10.2. Kolyvagin cohomology classes 117 10.3. Proof of Kolyvagin’s theorem 120 References 121 Exercises 121

Bibliography 125

Preface

This monograph is based on an NSF-CBMS lecture series given by the author at the University of Central Florida in Orlando from August 8 to 12, 2001.

The goal of this lecture series was to survey some recent developments in the arithmetic of modular elliptic curves, with special emphasis on

(1) the Birch and Swinnerton-Dyer conjecture;

(2) the construction of rational points on modular elliptic curves;

(3) the crucial role played by modularity in shedding light on these two closely related issues.

The text is divided into three parts of roughly equal length.

The first consists of chapters 1-3 and chapter 10. The first three chapters introduce the background and prerequisites for what follows: elliptic curves, modular forms and the Shimura-Taniyama-Weil conjecture, complex multiplication and the fundamental Heegner point construction whose study and generalisation is the main theme of the monograph. The notion of “Heegner system” which is spelled out in chapter 3 is used in chapter 10 to prove Kolyvagin’s theorem relating Heegner points to the arithmetic of elliptic curves, giving strong evidence for the Birch and Swinnerton-Dyer conjecture for elliptic curves of analytic rank at most one. While more advanced than chapters 1-3, chapter 10 is independent of the material in chapters 4-9 and could be read immediately after chapter 3.

Chapters 4-6 introduce variants of modular parametrisations in which modular curves are replaced by Shimura curves attached to certain indefinite quaternion algebras. A study of these parametrisations reveals an important new structure: the rigid analytic uniformisation of Shimura curves discovered by Cerednik and Drinfeld, giving rise to p-adic uniformisations of modular elliptic curves by discrete arithmetic subgroups of SL_{2}(ℚ_{p}) arising from definite quaternion algebras.

The main new contributions of this monograph are contained in chapters 7-9. These chapters give an overview of the author’s attempts to extend the theory of Heegner points and complex multiplication to certain situations where the base field is not a CM field. The notions of rigid analysis developed in chapters 5 and 6 play a key role in suggesting a p-adic variant of the theories of chapters 7 and 8. This leads, in chapter 9, to a conjectural construction of points on a modular elliptic curve over ℚ defined over ring class fields of a real quadratic field, which are expected to behave much like classical Heegner points attached to an imaginary quadratic field.

The reader is cautioned that many proofs give only the main ideas; details have often been left out or relegated to exercises, retaining (for better or for worse) the flavour of the original lecture series. Of necessity, a number of important topics had to be omitted or inadequately touched upon, for lack of time. The material covered here would be suitable for a 10 to 15-hour mini-course, or, with extra background material, for a one-semester or even a year-long seminar aimed at graduate students.

A selection of exercises is given at the end of each section. Many of these consist in working out the details of arguments sketched in the text. It is hoped that readers encountering this material for the first time will find the exercises helpful, while more sophisticated readers may elect to skip them without loss of continuity.

Henri Darmon

Montreal 2003

CHAPTER 1 Elliptic curves

1.1. Elliptic curves

Definition 1.1. An elliptic curve over a field F is a complete algebraic group over F of dimension 1.

Equivalently, an elliptic curve is a smooth projective curve of genus one over F equipped with a distinguished F -rational point, the identity element for the algebraic group law. It is a consequence of the Riemann-Roch theorem ([Si86], chap. III) that when F is of characteristic different from 2 and 3 such a curve can be described by an affine equation of the form

(1.1)  E : y^{2} = x^{3} + ax + b, with a, b ∈ F, Δ := −2^{4}(4a^{3} + 27b^{2}) ≠ 0,

in which the distinguished F -rational point is taken to be the unique point at infinity in the homogeneous equation for the corresponding projective curve. Over a field of arbitrary characteristic, an elliptic curve can still be described as a plane cubic curve, given by the somewhat more complicated equation sometimes referred to as the generalised Weierstrass normal form

(1.2)  E : y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}, with Δ ≠ 0.

In fact, elliptic curves are sometimes defined as plane cubic curves given by an equation of the form (1.1) or (1.2), the addition law on E being described geometrically via the well-known chord-and-tangent law. Such an approach has the virtue of concreteness and underscores the possibility of doing explicit calculations with elliptic curves (by computer, or even by hand) which is one of the charms of the subject. On the other hand, definition 1.1 is more conceptual and explains why elliptic curves should be singled out for special attention: they are the only projective curves that can be endowed with an algebraic group law - a structure which both facilitates and enriches their diophantine study.

For two elliptic curves over F to be isomorphic over F̄, it is necessary and sufficient that their so-called j-invariants, defined in terms of the coefficients of equation (1.1) by the formula

(1.3)  j = −\frac{2^{12}3^{3}a^{3}}{Δ}

be equal.

The structure of the group E(F) of solutions to (1.1) or (1.2) depends of course on the nature of the field F, for example:

1. When F is a finite field, the group E(F) is a finite abelian group. The study of E(F) is at the origin of many of the practical applications of elliptic curves to cryptography and coding theory.

2. If F is the real or complex numbers, then E(F) inherits from F the structure of a compact abelian group. For example, when F = ℝ, the group E(F) is abstractly isomorphic either to a circle group, or the product of a circle and ℤ/2ℤ. If F = ℂ is the field of complex numbers, then E(ℂ) is topologically isomorphic to a torus. As a complex analytic manifold it is isomorphic to the quotient of ℂ by a lattice Λ ⊂ ℂ generated by the periods of a holomorphic differential ω against the integral homology of E(ℂ). To make the isomorphism ℂ/Λ ⟶ E(ℂ) explicit, let ℘_{Λ}(z) be the Weierstrass ℘-function attached to Λ, defined by

℘_{Λ}(z) = \frac{1}{z^{2}} + ∑_{λ∈Λ−\{0\}} (\frac{1}{(z − λ)^{2}} − \frac{1}{λ^{2}} ) .

The Λ-periodic functions x = ℘_{Λ}(z) and y = ℘′_{Λ}(z) satisfy the algebraic relation

(1.4)  y^{2} = 4x^{3} − g_{2}x − g_{3},

where

(1.5)  g_{2} = 60 ∑_{λ∈Λ−\{0\}} \frac{1}{λ^{4}} , g_{3} = 140 ∑_{λ∈Λ−\{0\}} \frac{1}{λ^{6}} ,

and the map

(1.6)  Φ_{w}(z) = (℘_{Λ}(z), ℘′_{Λ}(z))

gives an isomorphism (of groups as well as complex analytic varieties) between ℂ/Λ and the elliptic curve with equation (1.4), which is isomorphic to E over ℂ. For more details, see [Si86], chap. VI.

The isomorphism between E(ℂ) and ℂ/Λ makes it transparent that the group E_{n} of points on E of order n, with coordinates in ℂ or in any algebraically closed field of characteristic zero, is isomorphic to ℤ/nℤ × ℤ/nℤ as an abstract group.

3. If F is a p-adic field (ℚ_{p}, say, or a finite extension of ℚ_{p}) then E(F) is a compact p-adic Lie group, hence an extension of a finite group by a pro-p group E_{1}(F) (cf. [Si86], chaps. IV and VII).

One may assume, after a change of variables, that E is given by an equation of the form (1.2) in which the coefficients a_{i} belong to the ring of integers O_{F} of F, and for which the associated discriminant Δ_{min} = Δ has minimal valuation in O_{F}. If Δ belongs to O_{F}^{×}, then the equation obtained by reducing (1.2) modulo a uniformiser π ∈ O_{F} defines an elliptic curve over the finite field k = O_{F} /(π). In this case one says that E/F has “good reduction”.

An important role is played in this monograph by elliptic curves having a special type of bad reduction, referred to as multiplicative reduction. This is the case where Δ_{min} ∈ O_{F} is not a unit and where the equation obtained by reducing (1.2) modulo π has an ordinary double point as its only singularity. In that case ord_{π}(j) < 0, and there is a q ∈ F^{×} which can be obtained by formally inverting the power series expressing j in terms of q

j = \frac{1}{q} + 744 + 196884q + ⋯

to express q as a power series in 1/j with integral (and hence p-adically bounded) coefficients. The curve E is isomorphic over F̄ (more precisely, over a quadratic unramified extension of F) to the curve E_{q} given by the equation

(1.7)  E_{q} : y^{2} + xy = x^{3} + a_{4}(q)x + a_{6}(q),

where

(1.8) a_{4}(q) = −s_{3}(q), a_{6}(q) = \frac{−5s_{3}(q) + 7s_{5}(q)}{12},  with s_{k}(q) = ∑_{n=1}^{∞} \frac{n^{k}q^{n}}{1 − q^{n}} .

The p-adic analytic isomorphism Φ_{Tate} : F̄^{×} ⟶ E_{q}(F̄) is obtained by setting

(1.9)  Φ_{Tate}(u) = (∑_{n∈ℤ} \frac{q^{n}u}{(1 − q^{n}u)^{2}} − 2s_{1}(q), ∑_{n∈ℤ} \frac{(q^{n}u)^{2}}{(1 − q^{n}u)^{3}} + s_{1}(q)) .

(For more details, see [Si94], ch. V.) This p-adic uniformisation theory for E yields a description of E(F̄) in the spirit of the Weierstrass theory of equation (1.6). It plays an important role in the constructions of chapters 6 and 9.

4. The study of elliptic curves over the finite, complex and p-adic fields, while of interest in its own right, is subordinate in this monograph to the case where F is a number field – the field of rational numbers or a finite extension of it. The key result and the starting point for the theory of elliptic curves over number fields is the Mordell-Weil theorem. This result was first established, in certain special cases, by Fermat himself using his method of descent, and its proof is recalled in the next section.

1.2. The Mordell-Weil theorem

Let E be an elliptic curve defined over a number field F.

Theorem 1.2 (Mordell-Weil). The Mordell-Weil group E(F) is finitely generated, i.e.,

E(F) ≃ ℤ^{r} ⊕ T,

where r ≥ 0 and T is a finite group.

Since (the modern formulation of) the proof of the Mordell-Weil theorem plays an important role in the questions studied in this monograph, particularly in chapter 10, it is worthwhile to recall here the main ideas which are behind it.

Proof of theorem 1.2. The proof is composed of two ingredients:

1. The existence of a height function h : E(F) ⟶ ℝ satisfying suitable properties:

Theorem 1.3. There exists a function

h : E(F) ⟶ ℝ

satisfying

(1) For all points Q in E(F), there is a constant C_{Q} depending only on Q, and an absolute constant C depending only on E, such that

h(P + Q) ≤ 2h(P) + C_{Q}, h(mP) ≥ m^{2}h(P) + C,

for all P ∈ E(F).

(2) For all B > 0,

{P such that h(P) < B} is finite.

2. The weak Mordell-Weil theorem:

Theorem 1.4. For any integer n ≥ 1, the group E(F)/nE(F) is finite.

These two ingredients are combined in the following descent lemma of Fermat of which the Mordell-Weil theorem is a direct consequence:

Lemma 1.5 (Fermat). Let G be an abelian group equipped with a height function satisfying the properties of theorem 1.3, and assume that the quotient G/nG is finite for some n > 1. Then G is finitely generated.

It is not our intention to focus any further on heights (which are discussed at length in [Si86] for example), or on the (elementary) descent lemma, whose proof is relegated to the exercises. The weak Mordell-Weil theorem is the most interesting ingredient from the point of view of a study of the Birch and Swinnerton-Dyer conjecture, since it is the source of the non-effectivity in the proof of the Mordell-Weil theorem.

The proof of theorem 1.4 begins with the observation that this theorem is trivially true over an algebraic closure F̄ of F, since the multiplication by n map is surjective on E(F̄). Hence the sequence

(1.10)  0 ⟶ E_{n} ⟶ E(F̄) ⟶^{n} E(F̄) ⟶ 0.

of modules equipped with their natural continuous action of G_{F} := Gal(F̄/F) is exact. Following the usual conventions of Galois cohomology, denote by

H^{i}(F, M) := H^{i}(G_{F} , M)

the group of continuous i-cocycles modulo the group of continuous i-coboundaries with values in the G_{F} -module M.(For details on these definitions, see [CF67], chapter IV, or [Si86], appendix B). Taking the Galois cohomology of the exact sequence (1.10) gives rise to the long exact cohomology sequence

0 ⟶ E_{n}(F) ⟶ E(F) ⟶^{n} E(F) ⟶^{δ} H^{1}(F, E_{n}) ⟶ H^{1}(F, E) ⟶^{n} H^{1}(F, E)

from which can be extracted the so-called descent exact sequence

(1.11)  0 ⟶ E(F)/nE(F) ⟶^{δ} H^{1}(F, E_{n}) ⟶ H^{1}(F, E)_{n} ⟶ 0.

The connecting homomorphism δ embeds the group E(F)/nE(F) into an object of Galois-theoretic nature, since H^{1}(G_{F} , E_{n}) depends only on the structure of G_{F} and of the G_{F} -module E_{n}, not on the elliptic curve E itself. For example, if all the n-division points of E(F̄) are defined over F so that G_{F} acts trivially on E_{n}, then elements in

H^{1}(F, E_{n}) = hom(G_{F} , E_{n})

are indexed by pairs (L, φ) where L is a finite Galois extension of F and φ is an identification of Gal(L/F) with a subgroup of E_{n}. If H^{1}(F, E_{n}) were a finite group, the mere existence of the exact sequence (1.11) would be enough to conclude the proof of theorem 1.4. However, this is never the case when F is a number field and n > 1. It is therefore necessary to exploit local information to pin down the image of δ in H^{1}(F, E_{n}) with greater accuracy. More precisely, for any place v of F (archimedean or not) the embedding of F into the completion F_{v} at the place v, extended to an embedding of F̄ into F̄_{v}, induces an inclusion G_{F_{v}} ⊂ G_{F}. The exact sequence (1.11) has a local counterpart with F replaced by F_{v} which fits into the commutative diagram (1.12)

0 ⟶ E(F)/nE(F) ⟶^{δ} H^{1}(F, E_{n}) ⟶ H^{1}(F, E)_{n} ⟶ 0
↓ res_{v} ↓ ↘ ∂_{v} ↓ res_{v}
0 ⟶ E(F_{v})/nE(F_{v}) ⟶^{δ} H^{1}(F_{v}, E_{n}) ⟶ H^{1}(F_{v}, E)_{n} ⟶ 0

in which the vertical arrows correspond to the natural restriction maps. Since δ(E(F)/nE(F)) is contained in the kernel of ∂_{v} for all v, it is contained in the so-called n-Selmer group of E over F defined as follows:

Definition 1.6. Let E be an elliptic curve over a number field F.

(1) The n-Selmer group of E over F, denoted Sel_{n}(E/F), is the set of classes c ∈ H^{1}(F, E_{n}) satisfying ∂_{v}(c) = 0, for all places v of F.

(2) The Shafarevich-Tate group of E/F, denoted LLI(E/F), is the set of classes c ∈ H^{1}(F, E) satisfying res_{v}(c) = 0, for all places v of F.

The exact sequence (1.11) can now be replaced by the exact sequence involving the n-Selmer group and the n-torsion in the Shafarevich-Tate group:

(1.13)  0 ⟶ E(F)/nE(F) ⟶^{δ} Sel_{n}(E/F) ⟶ LLI(E/F)_{n} ⟶ 0.

The weak Mordell-Weil theorem is then a consequence of the following general finiteness theorem for the Selmer group:

Proposition 1.7. The Selmer group Sel_{n}(E/F) is finite.

Sketch of proof of proposition 1.7. The proof can itself be divided into two stages: a local study, in which it is shown that Sel_{n}(E/F) is contained in the group H_{nΔ}^{1}(F, E_{n}) consisting of cohomology classes c ∈ H^{1}(F, E_{n}) whose restriction to the inertia group I_{v} at v is trivial, for all places v not dividing nΔ. It is only at this stage of the argument that certain facts about the geometry and arithmetic of elliptic curves (albeit, over local fields) are needed. (Cf. exercise 7.)

A global study is then needed to show that H_{nΔ}^{1}(F, E_{n}) is finite. The key ingredient in this finiteness result (cf. exercise 8) is the Hermite-Minkowski theorem asserting that there are only finitely many extensions of a given number field with bounded degree and ramification. □

To recapitulate, the proof of the Mordell-Weil theorem sketched above (and, in particular, the proof of the weak Mordell-Weil theorem) has led to the introduction of two fundamental invariants, the n-Selmer group of E/F and the Shafarevich-Tate group of E/F, fitting into the fundamental exact sequence (1.13). The weak Mordell-Weil theorem follows from the finiteness of Sel_{n}(E/F) whose proof in turn relies on the Hermite-Minkowski theorem, which is itself one of the key general finiteness results of algebraic number theory. □

Since Sel_{n}(E/F) is effectively calculable, the following question emerges naturally from the proof of theorem 1.2.

Question 1.8. How good an approximation to E(F)/nE(F) is Sel_{n}(E/F), i.e., how large can LLI(E/F)_{n} be?

The following basic conjecture implies that the group E(F)/nE(F) and the n-Selmer group Sel_{n}(E/F) only deviate by an amount which is bounded independently of n and in fact that these two groups are equal for all but finitely many primes n:

Conjecture 1.9 (Shafarevich-Tate). The group LLI(E/F) is finite.

One is now led to formulate the diophantine problem which this monograph is motivated by and largely devoted to:

Question 1.10. Given an elliptic curve E over a number field F, is there an algorithm to

(1) Test if E(F) is infinite?

(2) Compute the rank r of the Mordell-Weil group E(F)?

(3) Produce a set P_{1}, …, P_{r} of generators for E(F)/E(F)_{tor}?

These questions are ostensibly arranged in order of increasing difficulty, but some of the key mysteries are already present in (1).

1.3. The Birch and Swinnerton-Dyer conjecture

The main theoretical insights concerning question 1.10 are derived from the fundamental Birch and Swinnerton-Dyer conjecture.

Much is gained in simplicity, and comparatively little is lost in generality, by restricting to the case where F = ℚ, the field of rational numbers. In that case the elliptic curve E can be described by a so-called minimal Weierstrass equation

(1.14)  y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6},

in which the coefficients a_{i} are integers and in which Δ is minimal over all equations of this sort describing E. The invariant Δ attached to this equation is called the minimal discriminant of E. The invariant holomorphic differential form

(1.15)  ω_{E} = \frac{dx}{2y + a_{1}x + a_{3}}

is called the Néron differential attached to E.

If p does not divide Δ, then the curve over 𝔽_{p} obtained by reducing equation (1.14) is an elliptic curve whose isomorphism type does not depend on the choice of a minimal Weierstrass equation. In that case one says that E has good reduction at p and denotes by N_{p} the cardinality of the group E(𝔽_{p}) of points on E with coordinates in 𝔽_{p}. The function p ↦ N_{p} is a subtle arithmetic function – anticipating somewhat on the discussion of the next chapter, the key modularity property yields a device for gaining some control on the behaviour of this function. For now, the main (relatively) elementary information of which one disposes concerning the behaviour of N_{p} is that it can be written in the form

N_{p} = p + 1 − a_{p},

where the “error term” a_{p} satisfies Hasse’s inequality (cf. [Si86], chapter V):

|a_{p}| ≤ 2\sqrt{p}.

The basic insight behind the Birch and Swinnerton-Dyer conjecture is that the rank of E(ℚ) ought to be reflected in the finer properties of the asymptotic behaviour of the quantities N_{p} as p ⟶ ∞, and that a large rank, giving rise to an abundant supply of rational points in E(ℚ), would tend to make the numbers N_{p} rather larger than (p + 1) on average. On the basis of numerical experiments Birch and Swinnerton-Dyer were led to the following conjecture:

1.4. L-FUNCTIONS 7

Conjecture 1.11. There exists a constant C_{E} depending only on E such that

∏_{p<X} \frac{N_{p}}{p} ≃ C_{E}(log X)^{r},

where r is the rank of E(ℚ). (Here the symbol ≃ means that the ratio of the expressions apparing on the left and right tends to 1 as X ⟶ ∞.)

Conjecture 1.11 can be viewed as an example of a “local-global principle” since it asserts that the local invariants N_{p} “know about” the rank of E(ℚ), a global invariant.

One of the difficulties of conjecture 1.11 is that the product appearing on the left hand side, involving the arithmetically defined quantities N_{p}, is hard to come to terms with analytically. A better conceptual understanding of the Birch and Swinnerton-Dyer conjecture can be gained by recasting it in terms of the L-function of E/ℚ.

1.4. L-functions

To begin with it is useful to extend the definition of the coefficients a_{p}, which in the previous section were defined only for the primes p not dividing Δ, to a quantity a_{n} indexed by arbitrary positive integers n.

The definition of a_{p} is extended to the primes p of bad reduction for E according to a recipe which depends on the type of reduction of E modulo p:

Additive reduction: This occurs if E has a nodal singularity (cusp) so that its group of non-singular points is isomorphic to the additive group G_{a}. In that case set a_{p} := 0.

Split multiplicative reduction: This occurs if E has an ordinary double point as its only singularity, with tangent lines having rational slopes over 𝔽_{p}. In that case, set a_{p} := 1.

Non-split multiplicative reduction: This occurs if E has an ordinary double point as its only singularity over 𝔽_{p} with tangent lines having slopes defined over the quadratic extension of 𝔽_{p} but not 𝔽_{p} itself. In that case, set a_{p} := −1.

Closely related to the discriminant, but of even greater importance in describing the behaviour of the L-series of E, is the so-called arithmetic conductor N of E, which is (almost) characterised by the following three properties:

(1.16)  ord_{p}(N) = 0 if and only if E has good reduction at p.

In particular, N has the same set of prime divisors as the minimal discriminant Δ.

(1.17)  ord_{p}(N) = 1 if and only if E has multiplicative reduction at p.
(1.18)  ord_{p}(N) = 2 otherwise, when p > 3.

In the remaining case not covered by (1.16), (1.17) and (1.18) where p = 2 or 3 and E has additive reduction at p, a more involved recipe for calculating ord_{p}(N) is described by an algorithm of Tate [Ta72].

The L-function of E is defined as an infinite Euler product

(1.19)  L(E, s) = ∏_{p|̸N}(1 − a_{p}p^{−s} + p^{1−2s})^{−1} ∏_{p|N}(1 − a_{p}p^{−s})^{−1} =: ∑ a_{n}n^{−s},

in which the expression of L(E, s) as a Dirichlet series supplies the definition of the coefficient a_{n} when n is not prime.

Note that evaluating the Euler product formally at s = 1 gives

L(E, 1)“ = ” ∏_{p} \frac{p}{N_{p}} ,

where N_{p} is the cardinality of the group of non-singular points in E(𝔽_{p}). This equality is just formal, since by exercise 10 the Euler product defining L(E, s) only converges in the right half-plane Re(s) > 3/2.

It is believed that the behaviour of L(E, s) at s = 1 – assuming one can make sense of it, via the process of analytic continuation – should capture the asymptotics of the product ∏_{p<X} \frac{N_{p}}{p} appearing in the crude version of the Birch and Swinnerton-Dyer conjecture. This belief is made precise in the following more widely used form of the conjecture:

Conjecture 1.12 (Birch and Swinnerton-Dyer). The L-function L(E, s) extends to an entire function on ℂ and

BSD1: L(E, 1) ≠ 0 if and only if #E(ℚ) < ∞.

BSD2: The rank r of E(ℚ) is equal to the order of vanishing of L(E, s) at

s = 1. BSD3: Let R_{E} be the regulator of E, defined by

R_{E} = det(⟨P_{i}, P_{j}⟩)_{1≤i,j≤r}#E(ℚ)_{tors}^{−2},

where P_{1}, …, P_{r} is a set of independent generators for E(ℚ)/E(ℚ)_{tors}, and ⟨ , ⟩ is the canonical Néron-Tate height attached to E/ℚ, defined in [Si86], chapter VIII, §9. Let c_{p} denote the local terms defined in [Si86], Chapter VII §6, which depend only on the behaviour of E over ℚ_{p}. Then

lim_{s→1} \frac{L(E, s)}{(s − 1)^{r}} = #LLI(E/ℚ)R_{E}(∏_{p|N} c_{p})c_{∞}.

Conjecture 1.12 proposes three versions of the Birch and Swinnerton-Dyer conjecture in order of increasing strength. Statement BSD2 is the form which appears as a Clay Institute Millenium Prize problem [Wi00], but a number of the essential difficulties are already present in BSD1, particularly in the “if” statement.

1.5. Some known results

The analytic continuation of L(E, s) is often stated separately from the Birch and Swinnerton-Dyer conjecture, as an important conjecture in its own right, which has now been proved (for elliptic curves over ℚ) thanks to the work of Wiles and its subsequent refinements.

Theorem 1.13. The L-function L(E, s) extends to an entire function on ℂ and has a functional equation relating its value at s and 2 − s, of the form

Λ(E, s) = ±Λ(E, 2 − s),

where

Λ(E, s) := (2π)^{−s}Γ(s)N^{s/2}L(E, s),

and where

Γ(s) = ∫_{0}^{∞} e^{−t}t^{s−1}dt

is the Γ-function.

This theorem was proved by Wiles [Wi95] and Taylor-Wiles [TW95] in the case where E is a semistable elliptic curve; the full result was established after a series of strengthenings of Wiles’ method [Di96], [CDT99], [BCDT01]. Some of the key concepts underlying the proof of Wiles’ result – modular forms, and their associated L-series – are introduced in the next chapter.

While little is known in general about the Birch and Swinnerton-Dyer conjecture, one does have the following important partial result, which was originally proved assuming the validity of theorem 1.13, before that theorem was itself established. Reversing the chronological development, theorem 1.14 is stated here as an unconditional result.

Theorem 1.14 (Gross-Zagier, Kolyvagin). Let E be an elliptic curve over ℚ.

(1) If L(E, 1) ≠ 0, then #E(ℚ) < ∞, i.e., r = 0.

(2) If L(E, 1) = 0 and L′(E, 1) ≠ 0, then r = 1, and there is an efficient method for calculating E(ℚ).

In both cases LLI(E/ℚ) is finite.

Thus, both BSD2 of conjecture 1.12 and the Shafarevich-Tate conjecture are proved when ord_{s=1}L(E, s) ≤ 1.

One of the goals of this monograph is to discuss at least a few of the ideas that go into the proofs of theorem 1.13 and 1.14. Both rely crucially on the notion of modularity which is the focus of the next chapter.

Further results and references

Tate’s article [Ta74] presents what has by now become the standard account of the theory of elliptic curves. It forms the basis for both the classic graduate text by Silverman [Si86] and Husemoller’s book [Hu87], and the less comprehensive but more elementary undergraduate treatise [ST92]. All three references contain a discussion of the proof of the Mordell-Weil theorem. Also recommended is the short book by Cassels [Cas91] which focusses on local-global principles and on the concepts surrounding the modern formulation of the proof of the Mordell-Weil theorem.

Beyond the questions arising from the lack of effectivity in the Mordell-Weil theorem, there are a number of other tantalising open problems suggested by this result. Foremost among these is the question of whether the rank of elliptic curves over the rationals can be arbitrarily large. Mestre [Mes91] has shown that there are infinitely many elliptic curves over ℚ with rank ≥ 12, and curves with rank as large as 21 have been found by computer search. There is no number field F for which the rank of E(F) is known to get arbitrarily large as E varies over the elliptic curves defined over F, although such is expected to be the case for any number field.

There is an abundance of textbooks covering more advanced topics, such as [Si94] or Koblitz’s book [Kob93] which explores the connections between the Birch and Swinnerton-Dyer conjecture and the ancient congruent number problem of determining which integers are the areas of right-angled triangles with rational sides lengths. Also of relevance to the topics of chapter 2 is the book of Knapp [Kn92] emphasising the relation between the theory of elliptic curves and modular forms.

The conjecture of Birch and Swinnerton-Dyer was formulated in the early 60’s in a series of two articles [BSD63] and [BSD65]. The precise relation (in which a surprising factor of \sqrt{2} makes an appearance) between the asymptotics of the finite product occuring in the crude form of the Birch and Swinnerton-Dyer conjecture (conjecture 1.11) and the behaviour of L(E, s) at s = 1 is discussed in [Go] and [Con].

Exhibiting many formal analogies with the case where F is a number field is the case where F is a function field - the field k(T) of rational functions in an indeterminate T over a finite field k, or a finite extension of k(T). While largely ignored in this monograph, the study of elliptic curves over function fields exhibits many similarities with that of elliptic curves over number fields, while at the same time appearing to be more tractable. For example, the Mordell-Weil theorem holds in this context, and it is also known that the rank of elliptic curves over k(T) can become arbitrarily large, by work of Tate [TS67] and Ulmer [Ul]. (The latter work produces elliptic curves over 𝔽_{p}(T) of arbitrarily large rank whose j-invariants do not lie in the field of constants. For a variant of this construction based on the theory of Heegner points described in chapter 3, see [Da03].) More germane to this monograph, a lot more is known concerning the Birch and Swinnerton-Dyer conjecture for elliptic curves over function fields. The L-function L(E/F, s) can be defined in this context, is known to have a functional equation, and its zeroes (of which there are finitely many) satisfy an analogue of the Riemann hypothesis. Furthermore, Tate has shown that the rank of E(F) is at most the order of vanishing of L(E, s) at s = 1, and that the full Birch and Swinnerton-Dyer conjecture, in its strongest form (BSD3) holds for E/F if LLI(E/F) is finite. Together with theorem 1.14, the evidence arising from the analogy between function fields and number fields is probably the best evidence we dispose of at present for the validity of the Birch and Swinnerton-Dyer conjecture.

Exercises

(1) Prove the descent lemma (lemma 1.5) in the text.

(2) Let E be an elliptic curve over a field F. The goal of this exercise is to describe the connecting homomorphism δ : E(F)/nE(F) ⟶ H^{1}(F, E_{n}). Given a point P ∈ E(F), choose P′ ∈ E(F̄) such that nP′ = P, and set

z_{P′}(σ) = σP′ − P′, for σ ∈ G_{F} .

(a) Show that z_{P′} belongs to the group Z_{cont}^{1}(G_{F} , E_{n}) of continuous 1-cocycles on G_{F} with values in E_{n}.

(b) Show that the image of zP′ in H^{1}(F, E_{n}) depends only on P, and not on the choice of P′ that was made to define it. (Call c_{P} this image.)

(c) Show that the assignment P ↦ c_{P} yields an injective group homomorphism

E(F)/nE(F) ⟶ H^{1}(F, E_{n}),

and that δ(P) = c_{P}.

(3) Assuming that LLI(E/F) is finite, describe an effective procedure for computing E(F). (I.e., an algorithm which given (E, F) as input is guaranteed to terminate after a finite amount of time with a list of generators for E(F).)

(4) Let E be an elliptic curve over ℝ and let Λ_{E} ⊂ ℂ be a period lattice associated to a differential on E over ℝ. Show that E(ℝ) has two connected components if and only if the lattices

Λ_{E} and (Λ_{E} ∩ ℝ) + (Λ_{E} ∩ iℝ)

are equal, and that otherwise the former contains the latter with index 2.

(5) Let E be an elliptic curve defined over a number field F of odd degree. If n is odd, show that E_{n}(F) is cyclic.

(6) Let E be an elliptic curve over ℚ with minimal discriminant Δ, and let p be a prime of multiplicative reduction for E, so that E_{/ℚ_{p}} is endowed with the Tate uniformisation of equation (1.9). If ℓ is a prime which does not divide ord_{p}(Δ), show that the image of G_{ℚ} in Aut(E_{ℓ}) ≃ GL_{2}(𝔽_{ℓ}) contains a unipotent element (conjugate to a non-scalar upper triangular matrix with ones on the diagonal).

(7) Let E be an elliptic curve over a finite extension K of ℚ_{p}. Let π be a uniformiser for K and let k = O_{K}/(π) denote the residue field. Assume that E has good reduction at π.

(a) Show that the kernel of the natural reduction map

E(K) ⟶ E(k)

is a pro-p group.

(b) Conclude that if p does not divide n, then the natural reduction map E_{n}(K̄) ⟶ E_{n}(k̄) is injective.

(c) Still assuming that p |̸n, show that the action of the local Galois group G_{K} on E_{n} is unramified, i.e., that the inertia subgroup of G_{K} acts trivially on E_{n}.

(d) If E is an elliptic curve over a number field K, show that Sel_{n}(E/K) is

contained in H_{nΔ}^{1}(K, E_{n}).

(8) Let K be a number field, d a fixed integer, and S a finite set of places of K. The Hermite-Minkowski theorem asserts that there are finitely many extensions of K with degree ≤ d and unramified outside S. Assuming this basic result, show that the group H_{nΔ}^{1}(K, E_{n}) occuring in the proof of proposition 1.7 is finite.

(9) Let O be a local ring with maximal ideal 𝔪 and residue characteristic different from 2 and 3. Show that the elliptic curve with equation

y^{2} = x^{3} + ax^{2} + b, with a ∈ O^{×} and b ∈ 𝔪,

has multiplicative reduction modulo 𝔪. Show that this reduction is split (resp. non-split) if and only if a is a square (resp. a non-square) in O^{×}.

(10) Using the Hasse bound |a_{p}| < 2\sqrt{p}, show that the Euler product defining L(E, s) in equation (1.19) converges for re(s) > 3/2 and that this convergence is uniform on compact subsets in this region.

CHAPTER 2 Modular forms

The only results of any depth that can be proved about the L-function L(E, s) associated to an elliptic curve E (and hence, about the Birch and Swinnerton-Dyer conjecture formulated in the previous chapter) rely on knowing that E is modular. This chapter introduces this property and describes some of its important applications, most notably to the proof of theorem 1.13.

2.1. Modular forms

Let H be the Poincaré upper half-plane

(2.1)  H = {z ∈ ℂ such that Im(z) > 0}.

The group GL_{2}^{+}(ℝ) consisting of matrices with strictly positive determinant acts on H by Möbius transformations according to the rule

(2.2)  ( \begin{matrix}a & b \\ c & d \end{matrix} ) τ = \frac{aτ + b}{cτ + d}.

In this way GL_{2}^{+}(ℝ) acts on H by hyperbolic isometries preserving the line element ds^{2} = (dx^{2} +dy^{2})/y^{2}, where τ = x+iy. The group SL_{2}(ℤ) of matrices with integer coefficients and determinant 1 acts discretely on H. (Cf. exercises 1 and 2.)

Let Γ be any finite index subgroup of SL_{2}(ℤ).

Definition 2.1. A modular form of weight k on Γ is a holomorphic function f : H ⟶ ℂ such that

(1) f(γτ) = (cτ + d)^{k}f(τ), for all γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ.

(2) For any γ ∈ SL_{2}(ℤ), the function

f_{|γ}(τ) := (cτ + d)^{−k}f(γτ)

can be written in the form

(2.3)  ∑_{n=0}^{∞} a_{n}^{γ}q^{n/h}, where q = e^{2πiτ} ,

and h is a positive integer, the width of the cusp γ^{−1}∞ = \frac{−d}{c}. The expression ∑_{n=0}^{∞} a_{n}^{γ}q^{n/h} depends only on \frac{−d}{c} (up to multiplying q^{1/h} by an h-th roots of unity) and is called the Fourier expansion of f at \frac{−d}{c}.

A modular form f on Γ is called a cusp form if a_{0}^{γ} = 0 for all γ. Denote by S_{k}(Γ) the complex vector space of cusp forms of weight k for Γ. It is a fact (explained, for example, in [Sh71], theorem 2.24) that S_{k}(Γ) is a finite-dimensional ℂ-vector space.

Of primary importance for elliptic curves – and the only case that will be treated in any detail in this chapter – is the case where k = 2 and where

Γ = Γ_{0}(N) := \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ) such that N divides c\}

is the so-called Hecke congruence group of level N for some positive integer N. Denote by S_{2}(N) := S_{2}(Γ_{0}(N)) the space of cusp forms of weight 2 on Γ_{0}(N).

The quotient H/Γ_{0}(N) inherits from the complex structure on H the structure of a (non-compact) Riemann surface. It is useful to compactify H/Γ_{0}(N) by adjoining to it a finite set of cusps which corrrespond bijectively to the Γ_{0}(N)-orbits of points in ℙ_{1}(ℚ), with an appropriate definition of the topology and complex structure in a neighbourhood of these cusps (discussed for example in [Kn92], p. 311 or [Sh71], ch. 1). In this way the quotient H^{∗}/Γ_{0}(N) of the extended upper half-plane H^{∗} := H ∪ ℙ_{1}(ℚ) by the action of Γ_{0}(N) becomes a compact Riemann surface. Let X_{0}(N) denote the projective algebraic curve (for the time being, over ℂ) whose complex points are identified with this Riemann surface. The assignment which to f ∈ S_{2}(N) associates the expression

(2.4)  ω_{f} := 2πif(τ)dτ

identifies S_{2}(N) with the space of holomorphic differential forms on X_{0}(N)(ℂ). Thus, by the Riemann-Roch theorem, the space S_{2}(N) is a finite-dimensional complex vector space of dimension equal to the genus of X_{0}(N). This fact allows the explicit determination of the dimension of S_{2}(N) which is carried out in exercise 3.

2.2. Hecke operators

The vector space S_{2}(N) is equipped with a non-degenerate Hermitian inner product

(2.5)  ⟨f_{1}, f_{2}⟩ = ∫_{H/Γ_{0}(N)} f_{1}(τ)f_{2}(τ̄)dxdy,

known as the Petersson scalar product. It is also equipped with an action of certain Hecke operators T_{p} indexed by rational primes p and defined by the rules

(2.6)  T_{p}(f) = \{ \begin{matrix}\frac{1}{p} ∑_{i=0}^{p−1} f(\frac{τ+i}{p}) + pf(pτ), & if p |̸N \\ \frac{1}{p} ∑_{i=0}^{p−1} f(\frac{τ+i}{p}), & if p|N. \end{matrix}

These operators act linearly on S_{2}(N), and their effect on the q expansions at ∞ (cf. exercise 4) is given by the following simple formulae:

(2.7)  T_{p}(f) = \{ \begin{matrix}∑_{p|n} a_{n}q^{n/p} + p ∑ a_{n}q^{pn} & if p |̸N; \\ ∑_{p|n} a_{n}q^{n/p} & if p|N. \end{matrix}

It is convenient to extend the definition of the Hecke operators to operators T_{n} indexed by arbitrary positive integers n by equating the coefficient of n^{−s} in the identity of formal Dirichlet series

(2.8)  ∑_{n=1}^{∞} T_{n}n^{−s} := ∏_{p|̸N}(1 − T_{p}p^{−s} + p^{1−2s})^{−1} ∏_{p|N}(1 − T_{p}p^{−s})^{−1}.

Let 𝕋 be the commutative subalgebra of End_{ℂ}(S_{2}(N)) generated over ℤ by the Hecke operators T_{n}, and let 𝕋^{0} denote the subalgebra generated only by those operators T_{n} with (n, N) = 1. Although presented as a ring with infinitely many generators, the algebras 𝕋 and 𝕋^{0} are in fact finitely generated in a strong sense:

Proposition 2.2. The Hecke algebras 𝕋 and 𝕋^{0} are finitely generated as ℤmodules.

Sketch of Proof. Let V = hom(S_{2}(N), ℂ) denote the vector space dual of S_{2}(N). By the theory of the Abel-Jacobi map (cf. for example [Mu], cor. 3.8) the integral homology H^{1}(X_{0}(N)(ℂ), ℤ) embeds as a sublattice Λ of V by associating to a closed cycle c on X_{0}(N)(ℂ) the functional η_{c} ∈ V defined by

η_{c}(f) = ∫_{c} ω_{f}.

The action of 𝕋 on S_{2}(N) induces an action of this algebra on V by duality, which leaves stable the lattice Λ (cf. [Kn92], props. 11.23 and 11.24). Hence 𝕋 is a subalgebra of End_{ℤ}(Λ). Since this latter ring is finitely generated as a ℤ-module, the same must be true for 𝕋 (and, a fortiori, 𝕋^{0}). □

It is worth trying to understand precisely what the rank of 𝕋 as a ℤ-module is. Let g = genus(X_{0}(N)) = dim_{ℂ}(S_{2}(N)).

Proposition 2.3. The rank of 𝕋 is at most g.

Sketch of Proof. The action of complex conjugation τ on X_{0}(N)(ℂ) induces an action of τ on Λ which commutes with that of 𝕋. Hence 𝕋 preserves the submodules Λ^{+} and Λ^{−} of Λ on which τ acts as multiplication by 1 and −1 respectively. Both Λ^{+} and Λ^{−} are of rank g and hence 𝕋 is identified with a commutative subalgebra of M_{g}(ℤ) (after choosing a basis for Λ^{+} or Λ^{−}). By exercise 5, there exists T ∈ 𝕋 such that 𝕋 contains ℤ[T] with finite index, and hence these two rings have the same rank as ℤ-modules. On the other hand, ℤ[T] is generated by 1, T, …, T^{g−1} since T satisfies its characteristic polynomial, and hence has rank at most g. □

Proposition 2.4. The rank of 𝕋 is equal to g.

Proof. Let T_{ℂ} = 𝕋 ⊗ ℂ. There is a natural ℂ-bilinear pairing

⟨ , ⟩ : T_{ℂ} × S_{2}(N) ⟶ ℂ

given by ⟨T, f⟩ = a_{1}(T f), where a_{1}(g) denotes the first Fourier coefficient of the cusp form g. It follows from equation (2.7) (cf. exercise 4) that

(2.9)  ⟨T_{n}, f⟩ = a_{n}(f).

Hence the pairing ⟨ , ⟩ is non-degenerate on the right, so that the natural map

S_{2}(N) ⟶ hom(T_{ℂ}, ℂ)

induced by ⟨ , ⟩ is injective. Hence dim_{ℂ}(T_{ℂ}) ≥ g, and therefore rank(𝕋) ≥ g. The result now follows from proposition 2.3. □

The structure of 𝕋 as a ℤ-module leads to the following important fact about modular forms in S_{2}(N) which shows the existence of many modular forms in this space having integer Fourier coefficients. The usefulness of modular forms in number theory stems from the fact that quite often these Fourier coefficients encode interesting arithmetic information.

Corollary 2.5. The space S_{2}(N) has a basis consisting of modular forms with integer Fourier coefficients.

Proof. By equation (2.9), the space of modular forms with integer Fourier coefficients is equal to the lattice dual to 𝕋 ⊂ 𝕋_{ℝ} = V^{+}. It follows that this module has rank g over ℤ. □

Denote by S_{2}(N, ℤ) and S_{2}(N, ℝ) the spaces of modular forms with integer and real Fourier coefficients respectively. Note that S_{2}(N, ℤ) is a lattice in S_{2}(N, ℝ) and that

rank_{ℤ}(S_{2}(N, ℤ)) = dim_{ℝ}(S_{2}(N, ℝ)) = dim_{ℂ}(S_{2}(N)) = g.

2.3. Atkin-Lehner theory

It is important to analyse the eigenspace decomposition of S_{2}(N) under the action of the algebras of commuting operators 𝕋 and 𝕋^{0}. To begin, we have

Lemma 2.6. If T belongs to 𝕋^{0}, then it is self-adjoint with respect to the Petersson scalar product.

Proof. See [Kn92] thm. 9.18 ad 8.22, or [Og69]. Better yet, work out exercise 6. □

It follows from lemma 2.6 combined with the spectral theorem for commuting self-adjoint operators that S_{2}(N) decomposes as an orthogonal direct sum of simultaneous eigenspaces for 𝕋^{0}:

(2.10)  S_{2}(N) = ⊕_{λ}S_{λ}^{0},

where the direct sum is taken over all ℂ-algebra homomorphisms λ : 𝕋^{0} ⟶ ℂ, and S_{λ}^{0} denotes the corresponding eigenspace in S_{2}(N). The eigenspaces S_{λ}^{0} need not be one-dimensional (cf. exercise 7). On the other hand, if λ : 𝕋 ⟶ ℂ is a ring homomorphism defined on the full Hecke algebra 𝕋, and S_{λ} is its associated eigenspace, then one does have:

Lemma 2.7 (Multiplicity one). The eigenspace S_{λ} attached to λ : 𝕋 ⟶ ℂ is one-dimensional.

Proof. This is because all the Fourier coefficients of a form f ∈ S_{λ} are completely determined by a_{1}(f), by the rule, immediate from (2.7)

a_{n}(f) = a_{1}(f)λ(T_{n}).
□

As is shown in exercise 7, the space S_{2}(N) does not decompose in general into a direct sum of the one-dimensional eigenspaces S_{λ}: the operators in 𝕋 (unlike those in 𝕋^{0}) need not act semisimply on S_{2}(N). However, there is a distinguished subspace of S_{2}(N), the so-called space of newforms, which decomposes as a direct sum of one-dimensional eigenspaces under both the actions of 𝕋 and 𝕋^{0}. More precisely, a modular form in S_{2}(N) is said to be an oldform if it is a linear combination of functions of the form f(d′z), with f ∈ S^{2}(N/d) and d′|d > 1. The new subspace of S_{2}(N), denoted S_{2}^{new}(N), is the orthogonal complement of the space S_{2}^{old}(N) of oldforms with respect to the Petersson scalar product.

Theorem 2.8 (Atkin-Lehner). The eigenspaces for the action of 𝕋^{0} on the space S_{2}^{new}(N) are all one-dimensional. The same assertion holds a fortiori for the action of 𝕋 and we have an orthogonal decomposition:

S_{2}(N) = S_{2}^{old}(N) ⊕_{λ} ℂf_{λ},

where the sum is taken over all algebra homomorphisms λ : 𝕋 ⟶ ℂ corresponding to eigenvectors in S_{2}^{new}(N), and f_{λ}(τ) = ∑_{n=1}^{∞} λ(T_{n})e^{2πinτ}.

Proof. See [AL70], or [DI95], sec. 6 for a survey. □

The simultaneous eigenvector f_{λ} is sometimes called a normalised eigenform or simply a newform of level N. Note that it satisfies a_{1}(f) = 1.

2.4. L-series

To a newform f of level N is attached the L-series

L(f, s) = ∑_{n=1}^{∞} a_{n}n^{−s},

where a_{n} := a_{n}(f) = λ(T_{n}). This L-function enjoys the following important properties:

Euler product: It admits the Euler product factorisation:

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 λ to the formal identity (2.8) relating the Hecke operators T_{n}.

Integral representation: A direct calculation (cf. exercise 8) shows that

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

where Γ(s) = ∫_{0}^{∞} e^{−t}t^{s−1}dt is the Γ-function.

Functional equation: The involution w_{N} defined on S_{2}^{new}(N) by the rule

(2.12)  w_{N}(f) = \frac{−1}{Nτ^{2}} f(\frac{−1}{Nτ^{2}} )

commutes with the Hecke operators in 𝕋^{0} and hence preserves the eigenspaces S_{λ}. It follows that for any newform f of level N,

(2.13)  w_{N}(f) = ϵf, where ϵ = ±1.

The sign ϵ is an important invariant of f. Hecke showed (cf. exercise 9) that L(f, s) satisfies the following functional equation:

(2.14)  Λ(f, s) = −Λ(w_{N}(f), 2 − s) = −ϵΛ(f, 2 − s).

2.5. Eichler-Shimura theory

The fact that the L-series L(f, s) has properties identical to those that are known (such as the Euler product factorisation of equation (1.19)) or expected (such as the functional equation) for the L-series L(E, s) attached to an elliptic curve in chapter 1 suggests that there might be a relationship between these two ostensibly different types of L-series. The following theorem of Eichler and Shimura establishes such a relationship in one direction.

Theorem 2.9. Let f be a normalised eigenform whose Fourier coefficients a_{n}(f) are integers. Then there exists an elliptic curve E_{f} over ℚ such that

L(E_{f}, s) = L(f, s).

Brief sketch of proof. The crucial construction which to such a newform f associates the elliptic curve E_{f} was discovered by Shimura. A key fact underlying Shimura’s construction is the fact that the algebraic curve X_{0}(N) has a natural model defined over ℚ. This curve, denoted X_{0}(N) as before, is the solution to the (coarse) moduli problem of classifying pairs (A, C) where A is an elliptic curve and C is a cyclic subgroup of A of order N. The element τ ∈ H/Γ_{0}(N) corresponds to the point in X_{0}(N)(ℂ) associated to the pair

(2.15)  (A_{τ} , C_{τ}) := (ℂ/⟨1, τ⟩, ⟨\frac{1}{N} ⟩) ,

and it can be checked that this correspondence is well-defined and sets up a bijection between points τ ∈ H/Γ_{0}(N) and isomorphism classes of pairs (A, C) as above. An equation for X_{0}(N) over ℚ can be written down using the modular function j defined using the j-invariant of equation (1.3) of chapter 1, by the rule:

j(τ) = j(A_{τ}).

The functions j(τ) and j(Nτ) are related by a single polynomial relation f_{N}(x, y) with coefficients in ℚ which yields a plane model for the algebraic curve X_{0}(N), albeit one which is highly singular and not easily computable in practice except for small values of N. In this model, if F is any subfield of ℂ, and τ ∈ H/Γ_{0}(N) corresponds to a point in X_{0}(N)(F), then (j(τ), j(Nτ)) belongs to F^{2}. The Hecke operators acting on S_{2}(N) arise geometrically from certain correspondences (denoted T_{p} ⊂ X_{0}(N) × X_{0}(N) by abuse of notation). More precisely, the correspondence T_{p} is given as the locus of points in X_{0}(N) × X_{0}(N) attached to pairs which are related by a cyclic p-isogeny (defined as a cyclic p-isogeny of the underlying curves which induces an isomorphism between the level N structures.) Let J_{0}(N) denote the Jacobian variety of X_{0}(N); it is an abelian variety over ℚ of dimension g = genus(X_{0}(N)) = dim(S_{2}(N)). The Hecke correspondences give rise to endomorphisms of J_{0}(N) defined over ℚ. Letting I_{f} denote the kernel of the ring homomorphism λ : 𝕋 ⟶ ℤ attached to f, the quotient J_{0}(N)/I_{f}J_{0}(N) is the desired elliptic curve E_{f}.

The key issue which arises in showing the equality of L-functions L(E_{f}, s) and L(f, s) is to relate the coefficient a_{p}(E) – obtained by counting points on E_{f} over 𝔽_{p}, or, equivalently, as the trace of the Frobenius at p acting on the p-power division points of E_{f} – to the eigenvalue of the Hecke operators T_{p}. This relationship is established by using the Eichler–Shimura congruence relation satisfied by the Hecke correspondence T_{p} in characteristic p. For example, if p does not divide N then the modular curve X_{0}(N) has an integral model with good reduction at p (cf. the discussion in section 1.5 of [DDT95]) and one has

T_{p} = F + F^{t} on X_{0}(N)_{/𝔽_{p}}

where F is the graph of the Frobenius morphism in characteristic p and F^{t} is its transpose. Deep results of Deligne and Carayol (cf. [Car91]) show that the level N of the newform f is equal to the arithmetically defined conductor of the elliptic curve E_{f}. A more complete discussion of the Eichler Shimura construction and the Eichler-Shimura congruence can be found in [Sh71], ch. 7, in [Kn92], chapter XI, or in section 1.7 of [DDT95]. □

The modular curve X_{0}(N) is embedded in its Jacobian by sending a point P to the class of the degree 0 divisor (P) − (i∞). Let

Φ_{N} : X_{0}(N) ⟶ E_{f}

be the modular parametrisation obtained by composing the embedding X_{0}(N) ⟶ J_{0}(N) with the natural projection J_{0}(N) ⟶ E_{f} arising from the Eichler-Shimura construction. The pullback Φ_{N}^{∗}(ω) of the Néron differential ω of E_{f} defined in chapter 1, (1.15), is a non-zero multiple of ω_{f}

(2.16)  Φ_{N}^{∗}(ω) = c · 2πif(τ)dτ, with c ∈ ℚ^{×}.

The rational number c is called the Manin constant attached to f. It is expected that c = 1 always, and this is known to be the case when N is square-free [Ed89].

For computational purposes, the following analytic description of the modular parametrisation

Φ_{N} : H/Γ_{0}(N) ⟶ E_{f}(ℂ)

is useful.

Proposition 2.10. Let Λ_{E} be the Néron lattice of E and let c be the Manin constant attached to E_{f}. Let Φ_{w} : ℂ/Λ_{E} ⟶ E(ℂ) be the Weierstrass uniformisation of equation (1.6) of chapter 1. Then one has

Φ_{N}(τ) = Φ_{w}(z_{τ}), where z_{τ} = c ∫_{i∞}^{τ} 2πif(z)dz = c ∑_{n=1}^{∞} \frac{a_{n}}{n} q^{n}, with q = e^{2πiτ} .

Proof. By definition of the Abel-Jacobi map and the projection J_{0}(N) ⟶ E_{f}, one sees that the image of the divisor (τ) − (i∞) is

Φ_{w} (∫_{Φ_{N} (i∞)}^{Φ_{N} (τ)} ω) = Φ_{w} (∫_{i∞}^{τ} Φ_{N}^{∗}ω)

by the change of variables formula. The result follows from (2.16). □

The Eichler-Shimura construction yields elliptic curves E for which the analytic continuation and functional equation of L(E, s) can be established. But it remains unclear, at this stage of the discussion, how special is the class of elliptic curves over ℚ that can be obtained from newforms with integer Fourier coefficients.

2.6. Wiles’ theorem

In the 1960’s, (through the work of a number of mathematicians, most prominently Shimura, Taniyama, and Weil) there gradually emerged the remarkable insight that in fact all elliptic curves (up to isogeny) should be obtainable from the Eichler-Shimura construction. The resulting conjecture, eventually known as the Shimura-Taniyama-Weil conjecture, is now a theorem thanks to a series of works building on a fundamental breakthrough of Wiles.

Theorem 2.11. Let E be an elliptic curve over ℚ of conductor N. Then there exists a newform f ∈ S_{2}(N) such that

L(E, s) = L(f, s).

Furthermore, E is isogenous to the elliptic curve E_{f} obtained from f via the Eichler-Shimura construction.

Remark 2.12. An elliptic curve over ℚ is said to be semistable if its conductor is squarefree, so that it has either good or multiplicative reduction at all primes p. The articles [Wi95] and [TW95] give a proof of the Shimura-Taniyama-Weil conjecture for this class of elliptic curves. This work was later strengthened in a series of articles ([Di96], [CDT99]) culminating in a proof of the complete conjecture in [BCDT01].

Remark 2.13. The distinguished elliptic curve E_{f} in the ℚ-isogeny class of E is commonly called the strong Weil curve attached to E (or to f).

We will say little about the ideas that go into the proof of theorem 2.11, having already devoted a survey article [DDT95] to this topic. Let us simply point out the key role played in Wiles’ approach by the continuous two-dimensional ℓ-adic representations of G_{ℚ} = Gal(ℚ̄/ℚ) attached to elliptic curves, and to modular forms. This method produces a modular form f for which one has the equality of L-series L(E, s) = L(E_{f}, s). The fact that E and E_{f} are isogenous over ℚ then follows from the isogeny conjecture proved by Faltings.

Of most relevance to the questions considered in this monograph are the following corollaries of Wiles’ result:

Corollary 2.14. The L-function L(E, s) has an analytic continuation and an integral representation of the form

(2π)^{−s}Γ(s)L(E, s) = ∫_{0}^{∞} f(it)t^{s−1}dt,

for some modular form f ∈ S_{2}(Γ_{0}(N)), and a functional equation as in theorem 1.13 of chapter 1.

Recall the sign −ϵ occuring in the functional equation of L(E, s) given by (2.13) and (2.14). When E is associated to the modular form f ∈ S_{2}(N), we will set

(2.17)  sign(E, ℚ) := −ϵ,

Note that L(E, s) vanishes to even (resp. odd) order at s = 1 when sign(E, ℚ) = 1 (resp. sign(E, ℚ) = −1).

The second corollary is the existence of an explicit complex uniformisation

Φ_{N} : H^{∗}/Γ_{0}(N) ⟶ E(ℂ),

described in proposition 2.10. (Strictly speaking, one obtains it by composing the map of that proposition with a rational isogeny E_{f} ⟶ E.) The uniformisation Φ_{N} will play a key role in the construction of certain algebraic points on E, known as Heegner points, which in turn are crucial in the proof of the theorem of Gross-Zagier-Kolyvagin. This key application of the modular parametrisation is discussed in chapter 3.

2.7. Modular symbols

Given a positive integer N, it is useful to be able to determine in practice an explicit basis of Hecke eigenforms in S_{2}(N). (One natural application, arising from theorem 2.11, would be to list the elliptic curves over ℚ of a given conductor N, taken up to isogeny.)

A useful device in carrying out the computation of S_{2}(N) is the notion of modular symbols first singled out and studied by Birch and Manin.

Definition 2.15. Let A be an abelian group. An A-valued modular symbol is a function

m : ℙ_{1}(ℚ) × ℙ_{1}(ℚ) ⟶ A, denoted m(x, y) = m{x→y},

satisfying (1) m{x→y} = −m{y→x}, for all x, y ∈ ℙ_{1}(ℚ).

(2) m{x→y} + m{y→z} = m{x→z}, for all x, y, z ∈ ℙ_{1}(ℚ).

Denote by M(A) the group of A-valued modular symbols, and simply by M the group of ℂ-valued modular symbols. The group GL_{2}(ℚ) acts naturally on M(A) by the rule

(γm){x→y} := m{γ^{−1}x→γ^{−1}y}.

Given f ∈ S_{2}(N) with associated Γ_{0}(N)-invariant differential ω_{f} := 2πif(τ)dτ, let λ_{f} be the modular symbol defined by

(2.18)  λ_{f}{x→y} := ∫_{x}^{y} ω_{f}.

Note that the cuspidality of f ensures that this integral converges. To compute it, one may choose τ ∈ H, and write

∫_{x}^{y} ω_{f} = ∫_{x}^{τ} ω_{f} + ∫_{τ}^{y} ω_{f}.

If x = ∞, then the integral from x to τ can be evaluated by using the Fourier expansion of ω_{f} at ∞, by the rule

∫_{∞}^{τ} ω_{f} = ∑_{n=1}^{∞} \frac{a_{n}}{n} e^{2πinτ} .

Otherwise, choosing a matrix γ ∈ SL_{2}(ℚ) with γx = ∞, one may reduce to the previous case by noting that

∫_{x}^{τ} ω_{f} = ∫_{∞}^{γτ} ω_{f|γ},

and using the fourier expansion of f|γ at ∞ to evaluate the latter integral.

The Γ_{0}(N)-invariance of ω_{f} implies that λ_{f} belongs to the subspace M^{Γ_{0}(N)} of Γ_{0}(N)-invariant modular symbols. This latter space is endowed with a natural action of the Hecke operators T_{p} defined, for p |̸N, by

T_{p}(m)\{x→y\} = m\{px→py\} + ∑_{j=0}^{p−1} m \{\frac{x + j}{p} →\frac{y + j}{p} \} .

In this way the map λ which to f associates λ_{f} becomes a ℂ-linear Hecke-equivariant map from S_{2}(N) to M^{Γ_{0}(N)}.

Lemma 2.16. The map λ : f ↦ λ_{f} is injective.

Proof. Suppose that λ_{f} = 0, and let F be the holomorphic function on H^{∗} defined by

F (τ) = ∫_{∞}^{τ} ω_{f}.

This function is Γ_{0}(N)-invariant, since for all γ ∈ Γ_{0}(N), one may choose any x ∈ ℙ_{1}(ℚ) and note that

F (γτ) − F (τ) = ∫_{τ}^{γτ} ω_{f} = ∫_{x}^{γx} ω_{f} = λ_{f}{x→γx} = 0.

Hence F corresponds to a holomorphic function on the compact Riemann surface X_{0}(N)(ℂ). It is therefore constant by Liouville’s theorem. (In fact, it vanishes identically, since F (i∞) = 0.) It follows that dF = ω_{f} = 0. □

Recall that g is the genus of the modular curve X_{0}(N), i.e., the complex dimension of S_{2}(N). Let s denote the number of cusps on this modular curve.

Lemma 2.17. The space M^{Γ_{0}(N)} has dimension 2g + s − 1.

Proof. Let F denote the space of ℂ-valued functions on ℙ_{1}(ℚ), and let d : F ⟶ M be the map defined by the rule

(df){x→y} = f(y) − f(x).

Clearly d is surjective and its kernel is the space of constant functions. Taking the Γ = Γ_{0}(N)-cohomology of the exact sequence

0 ⟶ ℂ ⟶^{i} F ⟶^{d} M ⟶ 0

yields the cohomology exact sequence

ℂ ⟶^{i_{0}} F^{Γ} ⟶^{d} M^{Γ} ⟶ H^{1}(Γ, ℂ) ⟶^{i_{1}} H^{1}(Γ, F).

The cokernel of i_{0} has dimension s − 1, while the kernel of i_{1} is identified with H^{1}(X_{0}(N), ℂ) ≃ ℂ^{2g}. (Cf. exercise 10.) The result follows. □

It follows from lemma 2.17 that the map λ is not surjective. The failure of surjectivity can be traced to two sources:

1. Complex conjugation acts naturally on M^{Γ_{0}(N)}, but the image of λ is not closed under this action. This is because

λ̄_{f} = λf̄,

where λf̄ denote the M-symbol attached to the anti-holomorphic differential ω̄_{f} = −2πif̄(τ)dτ̄. The homomorphism

(2.19)  λ : S_{2}(N) ⊕ \overline{S_{2}(N)} ⟶ M^{Γ_{0}(N)}

remains injective and its image has codimension s − 1.

2. A Γ_{0}(N)-invariant modular symbol m is said to be Eisenstein if there exists a Γ_{0}(N)-invariant function M : ℙ_{1}(ℚ) ⟶ ℂ such that

m{x→y} = M(y) − M(x), for all x, y ∈ ℙ_{1}(ℚ).

The space of Eisenstein modular symbols has dimension s − 1. The space of Eisenstein symbols is linearly disjoint from the image of the map λ of equation (2.19).

Lemma 2.18. The space M^{Γ_{0}(N)} is spanned by the Eisenstein modular symbols together with the symbols of the form λ_{f} and λ̄_{f}, where f ∈ S_{2}(N).

The details for this are worked out in exercise 11.

Thanks to lemma 2.18, to compute a basis of eigenforms for S_{2}(N), it suffices to find a basis for the space M^{Γ_{0}(N)}, discarding the Eisenstein modular symbols, and diagonalising the action of the Hecke operators on this space. This computation rests on the following simple observations:

First observation: We say that two elements x = \frac{a}{b} and y = \frac{c}{d} of ℙ_{1}(ℚ) are adjacent if ad − bc = ±1. (It is assumed here that the fractions representing x and y are given in lowest terms, and the convention ∞ = \frac{1}{0} is adopted.) The first observation is that an M-symbol is completely determined by its values on pairs (x, y) of adjacent elements of ℙ_{1}(ℚ). This is because any two elements x, y can be realised as the extreme terms of a finite sequence x, x_{1}, x_{2}, …, x_{n}, y, in which consecutive elements are adjacent. Clearly it is enough to show this when x = ∞, and for that, one may take x_{j} = \frac{p_{j}}{q_{j}} to be the j-th convergent in the continued fraction expansion for y.

Second observation: The group Γ_{0}(N) acts naturally on the set of ordered pairs of adjacent elements of ℙ_{1}(ℚ), and there are finitely many orbits for this action. In fact, two pairs (\frac{a}{b}, \frac{c}{d}) and (\frac{a′}{b′} , \frac{c′}{d′}) with ad − bc = a′d′ − b′c′ = 1 belong to the same orbit precisely when the ratios b/d and b′/d′ are equal in ℙ_{1}(ℤ/Nℤ). Therefore an element m ∈ M^{Γ_{0}(N)} is completely determined by the function

[ ]_{m} : ℙ_{1}(ℤ/Nℤ) ⟶ ℂ

defined by

[b : d]_{m} := m \{\frac{a}{b}→\frac{c}{d}\} , with ad − bc = 1.

This immediately gives a crude upper bound on the dimension of M^{Γ_{0}(N)}: it is at most the cardinality of ℙ_{1}(ℤ/Nℤ).

Third observation: The functions of the form [ ]_{m} on ℙ_{1}(ℤ/Nℤ), where m is a Γ_{0}(N)-invariant modular symbol, are not unrestricted but satisfy a collection of linear relations. For example, the axiom

m \{\frac{a}{b}→\frac{c}{d}\} = −m \{\frac{−c}{−d}→\frac{a}{b}\}

satisfied by modular symbols leads to the relation

(2.20)  [x]_{m} = − [\frac{−1}{x}]_{m} , for all x ∈ ℙ_{1}(ℤ/Nℤ),

while the property

m \{\frac{a}{b}→\frac{c}{d}\} = m \{\frac{a}{b}→\frac{a + c}{b + d}\} + m \{\frac{a + c}{b + d}→\frac{c}{d}\}

leads to the relation

(2.21)  [x]_{m} = [\frac{x}{x + 1}]_{m} + [x + 1]_{m}, for all x ∈ ℙ_{1}(ℤ/Nℤ).

In fact, equations (2.20) and (2.21) represent a full set of linear relations satisfied by the functions of the form x ↦ [x]_{m}. (Cf. exercise 12.)

These three observations can be combined into an explicit algorithm for finding all the Hecke eigenforms in S_{2}(N) for a given level N, the details of whose implementation are discussed for example in [Cr97]. The reader wishing to acquire some familiarity with the modular symbol method may also find it helpful to work out exercises 15 and 16.

In other applications, such as those discussed in chapter 8, one may be given an elliptic curve E whose conductor N and associated L-series coefficients a_{n}(E) can be computed (in any specified range) by examining the behaviour of E over the various completions of ℚ. One may wish to parlay this knowledge into an explicit and efficient calculation of the modular symbol λ_{E} = λ_{f} attached to f_{E}. The following theorem (a consequence of a more general result of Manin and Drinfeld) is useful for this task. Let Λ_{E} denote the so-called Néron lattice of E, generated by the periods of a Néron differential on E against the homology of E. Let t_{E} be the greatest common divisor of the integers p + 1 − a_{p}(E), where p ranges over all primes which are congruent to 1 modulo N.

Theorem 2.19. The modular symbol λ_{E} takes values in a lattice Λ, which is contained in \frac{1}{t_{E}} Λ_{E} with finite index.

Proof. If x ∈ ℙ_{1}(ℚ) and y = γx are equivalent under the action of Γ_{0}(N), then

λ_{E}{x→y} = ∫_{x}^{y} ω_{f} = ∫_{x}^{γx} ω_{f} = c^{−1} ∫_{Φ_{N} (x→γx)} ω_{E},

where Φ_{N}(x→γx) is the image under Φ_{N} of the closed path on X_{0}(N) joining x to γx. Hence this expression belongs to Λ_{E}. In the general case, one exploits the action of the Hecke operators by noting that, for all p ≡ 1 (mod N), the expression

(T_{p} − (p + 1)) λ_{E}{x→y}

can be written as a sum of expressions of the form λ_{E}{x_{j}→y_{j}}, where x_{j} and y_{j} belong to the same Γ_{0}(N)-orbit (Cf. exercise 13.) By varying the p, it follows that t_{E}λ_{E}{x→y} belongs to Λ_{E}, as was to be proved. □

The space M^{Γ_{0}(N)} is equipped with a ℂ-linear involution W_{∞} defined by

(2.22)  (W_{∞}m){x→y} = m{−x→ − y}.

A modular symbol is called a plus (resp minus) symbol if it is in the corresponding eigenspace for W_{∞}. If f belongs to S_{2}(N, ℝ), then the action of W_{∞} is related to that of complex conjugation on M by the rule:

(2.23)  (W_{∞}λ_{f}){x→y} = λ̄_{f}{x→y}.

In particular, if f is an eigenform with integer Fourier coefficients, attached to an elliptic curve E, one may write

λ_{E}{x→y} = λ_{E}^{+}{x→y}Ω^{+} + λ_{E}^{−}{x→y}iΩ^{−},

where Ω^{+} (resp Ω^{−}) is the unique positive generator of the lattice in ℝ gnerated by the real (resp imaginary) parts of elements of Λ. The functions λ_{E}^{±} are integer-valued Γ_{0}(N)-invariant modular symbols which are in the plus and minus eigenspace for W_{∞} respectively.

Further results and references

Among the many textbooks that cover the theory of modular forms, [Sh71] and [Og69] are classics. Knapp’s book [Kn92] on elliptic curves contains a fairly complete account of Eichler-Shimura Theory.

The Eichler-Shimura construction extends naturally to eigenforms f having non-rational Fourier coefficients. Since these coefficients are the eigenvalues of selfadjoint matrices with entries in ℤ, they generate an order O in a totally real field K of degree n ≤ g. The Eichler-Shimura construction, a direct generalisation of the construction in the proof of theorem 2.9, associates to f an abelian variety A_{f} of dimension n. The action of the Hecke operators on A_{f} yields a subring of End_{ℚ}(A_{f}) isomorphic to O, so that End_{ℚ}(A_{f}) ⊗ ℚ contains a field K of degree d = dim(A_{f}). Abelian varieties with this property are said to be of GL_{2}-type. It is conjectured that these are precisely (up to isogeny) the abelian varieties over ℚ that can occur as irreducible factors of J_{0}(N), but one is still far from being able to prove this natural generalisation of the Shimura-Taniyama-Weil conjecture. See [Ri94] for a discussion of this and related conjectures.

The reader wishing to understand the proof of Wiles’ theorem in the prototypical example of semi-stable elliptic curves can do no better than to consult the articles [Wi95] and [TW95] which contain the initial breakthroughs. A more pedagogically oriented exposition of these ideas is given in in [DDT95]. For a discussion of the subtle technical issues needed to extend Wiles’ breakthrough to all elliptic curves over ℚ, see [Di96], [CDT99] and [BCDT01].

Before it could be proved, the Shimura-Taniyama-Weil conjecture was supported by a certain amount of numerical evidence, and by Weil’s converse to Hecke’s theorem, asserting that if L(E, s), as well as sufficiently many of its twists by Dirichlet characters, has a functional equation of the type described in theorem 1.13, then E is isogenous to an elliptic curve E_{f} obtained from an eigenform f via the Eichler-Shimura construction. A discussion of converse theorems can be found in [Gel75] for example.

The modular symbol method is very flexible and has been implemented by Cremona [Cr97] to draw up tables of elliptic curves of conductor ≤ 5000. A different method due to Mestre (cf. [Mes91] and [Gr87]) based on the so-called Brandt matrices associated to orders in definite quaternion algebras over ℚ, is more restricted in scope (working well typically for elliptic curves of prime conductor) but appears to be more efficient than the modular symbol method in those situations where it can be readily applied.

Exercises

(1) Show that the action of a matrix γ ∈ SL_{2}(ℝ) on H by Möbius transformations preserves the hyperbolic metric on H.

(2) Let X and Y be Hausdorff topological spaces. A map π : X ⟶ Y is called proper if π^{−1}(K) is compact whenever K is compact, and open if it maps open subsets of X to open subsets of Y.

(a) Show that if π : X ⟶ Y is open and proper, and Γ ⊂ X is discrete, then π(Γ) is discrete in Y.

(b) Let G be a locally compact Hausdorff topological group and let H be a compact subgroup. If Γ is a discrete subgroup of G, show that the natural image of Γ in G/H (in the natural topology on G/H induced by the topology on G) is discrete. Show that this ceases to be true in general if H is not assumed to be compact.

(c) Show that PSL_{2}(ℤ) acts discretely on H by Möbius transformations. (I.e., the orbits for this action are discrete in the usual Euclidean topology on H.)

(3) Let Γ = SL_{2}(ℤ) and let

π : H^{∗}/Γ_{0}(N) ⟶ H^{∗}/Γ, Γ_{0}(N)τ ↦ Γτ

be the natural (branched) covering map of Riemann surfaces. For τ ∈ H^{∗}, let Γ_{τ} denote the stabiliser of τ in Γ and e_{τ} the index of Γ_{τ} ∩ Γ_{0}(N) in Γ_{τ}. Let ℙ_{1}(ℤ/Nℤ) denote the projective line over the ring ℤ/Nℤ. It is equipped with a natural action of SL_{2}(ℤ).

(a) Show that e_{τ} is equal to the ramification index of π at Γ_{0}(N)τ.

(b) Show that if τ is not equivalent under Γ to i, ρ := e^{2πi/3}, or ∞, then π is unramified at Γ_{0}(N)τ, i.e., e_{τ} = 1.

(c) Let n_{i} denote the number of fixed points of the matrix ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) acting

on ℙ_{1}(ℤ/Nℤ). Show that π is unramified at n_{i} distinct points of X_{0}(N) lying above τ = i, and has ramification index 2 at the remaining points above i.

(d) Let n_{ρ} denote the number of fixed points of the matrix ( \begin{matrix}0 & −1 \\ 1 & 1 \end{matrix} ) acting

on ℙ_{1}(ℤ/Nℤ). Show that π is unramified at n_{ρ} distinct points of X_{0}(N) lying above τ = ρ, and has ramification index 3 at the remaining points above ρ.

(e) Show that the points in π^{−1}(∞) are in bijection with the orbits for the

group Γ_{∞} = ⟨( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} )⟩ acting on ℙ_{1}(ℤ/Nℤ), and that if the point P

corresponds to the orbit O, then e_{P} = #O.

(f) Using the Riemann-Hurwitz formula, give a formula for the genus of X_{0}(N) (and hence, the dimension of S_{2}(N)) in terms of the orbit decompositions of ℙ_{1}(ℤ/Nℤ) under the actions of the three cyclic groups Γ_{i}, Γ_{ρ}, and Γ_{∞}. Make this formula explicit if N is a prime or a prime power.

(g) Show that there are no cusp forms of level N for N ≤ 10, and that S_{2}(11) is one-dimensional.

(h) Use Wiles’ theorem to conclude that there are no elliptic curves over ℚ of conductor < 11, and precisely one isogeny class of elliptic curves of conductor 11. How might one go about proving such a result, without using Wiles’ theorem?

(4) Show that the function T_{p}f described in formula (2.6) belongs to S_{2}(N), so that T_{p} is indeed a well-defined linear endomorphism of S_{2}(N). Verify formula (2.7) in the text decribing the effect of T_{p} on the Fourier expansion of f at ∞. Using this formula and the definition of the general Hecke operator T_{n}, prove equation (2.9) in the text.

(5) Let R be a commutative subring of M_{g}(ℤ). Show that there exists T ∈ R such that ℤ[T] is contained in R with finite index.

(6) Show that the Hecke operator T_{p} acting on S_{2}(N) is self-adjoint with respect to the Petersson scalar product when p |̸N.

(7) Let f be an eigenform of level N. Show that the modular forms f(z), f(pz) and f(p^{2}z) belong to S_{2}(Np^{2}) and are simultaneous eigenvectors for all the Hecke operators T_{ℓ} with ℓ not dividing Np^{2}. Show that the Hecke operator T_{p} preserves the space spanned by these three eigenforms, but that its action need not be diagonalisable.

(8) Check formula (2.11) in the text.

(9) Prove the functional equation (2.14) in the text.

(10) Complete the proof of lemma 2.17 by showing that the cokernel of i_{0} and the kernel of i_{1} are of dimension s − 1 and 2g respectively.

(11) Supply the details in the proof of lemma 2.18.

(12) Show that equations (2.20) and (2.21) represent a full set of linear relations satisfied by the functions of the form x ↦ [x]_{m}, where m is a Γ_{0}(N)-invariant modular symbol.

(13) Complete the proof of theorem 2.19 by showing that, for all p ≡ 1 (mod N),

the expression

(T_{p} − (p + 1)) λ_{E}{x→y}

can be written as a sum of expressions of the form λ_{E}{x_{j}→y_{j}}, where x_{j} and y_{j} belong to the same Γ_{0}(N)-orbit.

(14) Prove equation (2.23) relating the action of the involution W_{∞} and complex conjugation acting on modular symbols of the form λ_{f} with f ∈ S_{2}(N, ℝ).

(15) Let Γ = Γ_{0}(11) be the Hecke congruence group of level 11. Show that the space M^{Γ} of Γ-invariant modular symbols is a three-dimensional complex vector space, and write an explicit basis for these modular symbols consisting of eigenforms for the Hecke operators and for the involution W_{∞}. Use this to compute, for p ≤ 7, the number of points in E(𝔽_{p}), for E any elliptic curve over ℚ of conductor 11 (which is unique up to isogeny, by exercise 3 (h).)

(16) Carry out exercise 15 with N = 14 or 37 instead of 11.

CHAPTER 3 Heegner points on X_{0}(N)

Chapter 2 explained how thanks to Wiles’ theorem any elliptic curve E over ℚ of conductor N is attached to a normalised eigenform f ∈ S_{2}(N), leading to the proof of the functional equation satisfied by L(E, s) and the existence of the modular parametrisation

Φ_{N} : H^{∗}/Γ_{0}(N) ⟶ E(ℂ).

The most important arithmetic application of this explicitly computable parametrisation arises through the theory of complex multiplication. This theory is of great importance in its own right, allowing the analytic construction, from values of modular functions evaluated at quadratic arguments, of class fields of imaginary quadratic fields. It can also be used, in conjunction with the parametrisation Φ_{N}, to exhibit algebraic points on E defined over such class fields. These points, known as Heegner points, are an essential ingredient in the proof of theorem 1.14 of Gross-Zagier and Kolyvagin.

3.1. Complex multiplication

Let K ⊂ ℂ be a quadratic imaginary subfield of ℂ. We may write K = ℚ(ω_{D}), where D < 0 is the discriminant of K and

ω_{D} = \{ \begin{matrix}\frac{1+\sqrt{D}}{2} & if D ≡ 1 (mod 4) \\ \frac{\sqrt{D}}{2} & otherwise. \end{matrix}

It will be convenient to fix once and for all an embedding of an algebraic closure K̄ of K into ℂ.

An order in K is a subring O of K which generates K as a ℚ-vector space and is finitely generated as a ℤ-module. Every order is contained in the maximal order O_{K} = ℤ[ω_{D}], and is uniquely determined by its conductor c, a positive non-zero integer such that

O = ℤ ⊕ ℤcω_{D}.

Let A = ℂ/Λ be an elliptic curve over ℂ. Its endomorphism ring is identified with

End(A) = {α ∈ ℂ such that αΛ ⊂ Λ}.

Hence it is isomorphic to a discrete subring of ℂ. Such a ring is isomorphic either to ℤ, or to an order in a quadratic imaginary field K.

Definition 3.1. An elliptic curve A/ℂ is said to have complex multiplication if its endomorphism ring is isomorphic to an order in a quadratic imaginary field. More precisely, given such an order O, one says that A has complex multiplication by O if End(A) ≃ O.

30 3. HEEGNER POINTS ON X_{0}(N)

If A has complex multiplication by O, the corresponding period lattice of A is a projective O-module of rank one, whose isomorphism class depends only on the isomorphism type of A. Conversely, if Λ ⊂ ℂ is a projective O-module of rank one, the corresponding elliptic curve A = ℂ/Λ has complex multiplication by O. Hence there is a bijection

\{ \begin{matrix}Elliptic curves with CM by O, \\ up to isomorphism. \end{matrix} \} ⟶^{≃} \{ \begin{matrix}Rank one projective O-modules, \\ up to isomorphism. \end{matrix} \} .

The set on the right is called the Picard group, or the Class group, of O and is denoted Pic(O). The group structure on it arises from the tensor product of modules. When O = O_{K} is the maximal order, then Pic(O) is identified with the usual ideal class group of K. In any case the group Pic(O) is finite. (For more on the structure and size of the group Pic(O), see exercise 1.)

It follows from the finiteness of Pic(O) that there are finitely many isomorphism classes of elliptic curves with complex multiplication by O. Let Ell(O) be the set of all such isomorphism classes, and let A_{1}, …, A_{h} be representatives for each class in Ell(O).

Theorem 3.2. The j-invariants j(A_{i}) (with 1 ≤ i ≤ h) are algebraic numbers.

Proof. Let j = j(A_{1}). The elliptic curve A = A_{1} is isomorphic over ℂ to the curve with equation

(3.1)  y^{2} + xy = x^{3} + \frac{36}{j − 1728}x − \frac{1}{j − 1728}

Thus A can defined over ℚ(j), and in fact over the ring ℚ[j, 1/(j − 1728)]. If j is transcendental, this ring is contained in a field of transcendence degree one, and admits infinitely many distinct homomorphisms to ℂ. The corresponding specialisations of A would then yield infinitely many non-isomorphic elliptic curves with complex multiplication by O, contradicting the finiteness of Ell(O). □

We wish to gain a more precise understanding of the field generated by j = j(A_{1}), i.e., the minimal field of definition of A_{1}, as well as the action of Gal(K̄/K) on the algebraic numbers j(A_{j}).

Note that previously O = End(A) was identified with a subring of ℂ by the rule

(3.2)  α^{∗}ω_{A} = αΩ_{A}, for α ∈ O,

where ω_{A} is any regular differential on A over ℂ. More generally, if A and its endomorphisms are defined over a field L, the purely algebraic condition (3.2) makes sense over L and singles out a ring homomorphism O ⟶ L. In particular, any such L contains the fraction field K of O, and an element of O gives rise in a natural way to an element of L via its effect on the cotangent space of A over L.

The collection Ell(O) is equipped with a natural simply transitive action of Pic(O) by the rule

(3.3)  [Λ] ∗ [A] := Hom(Λ, A),

(cf. [Se67], p. 294). More concretely, if 𝔭 is any prime ideal of K whose norm is prime to c, the inclusion 𝔭 ⟶ O yields an isogeny

A = Hom(O, A) ⟶ Hom(𝔭, A),

whose kernel is identified with Hom(O, A[𝔭]) = A[𝔭], where A[𝔭] denotes the subgroup scheme of elements in A which are annihilated by all elements of 𝔭. Thus

(3.4)  [𝔭] ∗ [A] = A/A[𝔭].

It follows directly from this description (cf. exercise 6) that the action of Pic(O) on Ell(O) commutes with the natural action of G_{K} := Gal(K̄/K) on this set, so that the action of G_{K} on Ell(O) is encoded in a homomorphism

(3.5)  η : G_{K} ⟶ Pic(O), satisfying A^{σ} = η(σ) ∗ A, for all σ ∈ G_{K}.

Note that by the commutativity of Pic(O), the definition of η does not depend on the choice of base curve A made to define it.

The mere existence of η is enough to show that the j-invariants j(A_{i}) are defined over an abelian extension H := K̄^{ker η} of K.

We now describe H more precisely in terms of class field theory.

Let

𝔸_{K,f} ⊂ ∏_{ℓ≠∞}(K ⊗ ℚ_{ℓ})

denote the ring of finite adèles of K, and let

Ô := ∏_{ℓ≠∞}(O ⊗ ℤ_{ℓ})

be the closure of O in 𝔸_{K,f}. For each prime λ of K, let K_{λ} denote the completion at the corresponding non-archimedean valuation. The group K_{λ}^{×} can be viewed naturally as a subgroup of the group 𝔸_{K,f}^{×} of finite idèles attached to K. Let ι_{λ}(x) denote the idèle attached to x ∈ K_{λ}^{×}. On the global level, K (resp. K^{×}) can be viewed as a subring (resp. a subgroup) of 𝔸_{K,f} (resp 𝔸_{K,f}^{×}) via the natural diagonal embedding.

The group Pic(O) admits an adelic description, via the identification

Pic(O) = 𝔸_{K,f}^{×}/K^{×}Ô^{×},

in which the class of the idèle α corresponds to the homothety class of the lattice (α^{−1}Ô) ∩ K ⊂ ℂ.

The following is a special case of the main theorem of class field theory (cf. for example [CF67], ch, VII, thm. 5.1):

Theorem 3.3. There exists an abelian extension H_{c} of K which is unramified outside of the primes dividing c, and whose Galois group is naturally identified, via the Artin map, with Pic(O).

If 𝔭 is a prime ideal of K which is prime to c, we denote by π_{𝔭} a uniformiser of K_{𝔭}, and by [𝔭] the class in Pic(O) attached to the finite idèle ι_{𝔭}(π_{𝔭}). The Artin reciprocity law map

rec : Pic(O) ⟶ Gal(H_{c}/K)

sends the element [𝔭] to the inverse σ_{𝔭}^{−1} of the Frobenius element σ_{𝔭} at 𝔭.

The extension H_{c} whose existence is guaranteed by theorem 3.3 is called the ring class field of K attached to O, or the ring class field of K of conductor c.

Proposition 3.4. The abelian extension H is equal to the ring class field H_{c}. More precisely, for all primes 𝔭 of K which do not divide c,

η(σ_{𝔭}) = [𝔭] ∈ Pic(O).

Proof. Fix an elliptic curve A ∈ Ell(O). Let Σ be the set of primes 𝔭 of O_{K} which satisfy the following conditions:

(1) 𝔭 is unramified in H/K;

(2) The curve A has good reduction at all the primes of H above 𝔭;

(3) The prime 𝔭 does not divide the norm (from H to K) of j(A_{s}) − j(A_{t}) for all s ≠ t;

(4) The norm of 𝔭 is a rational prime p ∈ ℤ, i.e., p splits or ramifies in K/ℚ. Note that the set of primes of K satisfying these conditions has Dirichlet density one, and hence, by the Cebotarev density theorem, the corresponding Frobenius elements generate Gal(H/K). Let σ_{𝔭} be as before the Frobenius element in Gal(H/K) attached to the prime 𝔭 ∈ Σ. Choose a prime 𝔭′ of H above 𝔭 and let Ā denote the reduction of A at 𝔭′. It is defined over a finite field 𝔽 of characteristic p: the residue field of H at 𝔭′. We make two key observations (whose proofs are worked out in detail in exercise 7):

(1) The elliptic curve Ā is ordinary, i.e., there is (up to composition with an automorphism of A) a unique inseparable isogeny of degree p from Ā, given by the Frobenius morphism:

Frob : Ā ⟶ Ā^{p},

where Ā^{p} denotes the elliptic curve over 𝔽 obtained from A by applying the Frobenius map (raising to the pth power) to the coefficients of a defining equation for Ā, and Frob is the algebraic morphism of degree p sending (x, y) to (x^{p}, y^{p}).

(2) The elliptic curve A/A[𝔭] is defined over H, and the natural projection

A ⟶ A/A[𝔭]

is a purely inseparable morphism modulo 𝔭′.

It follows from these two observations that A/A[𝔭] = [𝔭]∗A is congruent, modulo 𝔭′, to σ_{𝔭}(A) ≡ Ā^{p}. Since all the A_{j} are distinct modulo 𝔭′, it follows that η(σ_{𝔭}) = [𝔭]. Since this formula holds on a set of primes of K of density one, it then holds for all the primes of K which are unramified in H; in particular H = H_{c}. □

Let M_{2}(ℤ) be the algebra of 2×2 matrices with entries in ℤ. Given any τ ∈ H, the order associated to τ is defined to be

O_{τ} := \{γ ∈ M_{2}(ℤ) such that det γ ≠ 0 and γτ = τ\} ∪ \{( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} )\} .

The elements of O_{τ} consist precisely of the matrices in M_{2}(ℤ) which have both

the column vectors ( \begin{matrix}τ \\ 1 \end{matrix} ) and ( τ̄ ; 1 ) as eigenvectors. It is transparent from this

description that O_{τ} is closed under both addition and multiplication and that it is a commutative subring of M_{2}(ℤ). The natural map which to γ ∈ O_{τ} associates the complex number z_{γ} satisying

(3.6)  γ ( \begin{matrix}τ \\ 1 \end{matrix} ) = z_{γ} ( \begin{matrix}τ \\ 1 \end{matrix} ) .

also allows O_{τ} to be viewed as a discrete subring of ℂ. Note further that he order O_{τ} is isomorphic to the endomorphism ring of the elliptic curve A_{τ} = ℂ/⟨1, τ⟩. This is because, for any γ ∈ O_{τ}, multiplication by the complex number z_{γ} of equation (3.6) preserves the lattice ⟨τ, 1⟩ and hence induces a complex endomorphism m_{γ} of A_{τ}. In fact the map γ ↦ m_{γ} identifies O_{τ} with End_{ℂ}(A_{τ}).

If O is any order in a quadratic imaginary field K ⊂ ℂ, we can write

CM(O) = {τ ∈ H/SL_{2}(ℤ) such that O_{τ} = O}.

The class group Pic(O) acts on CM(O) as follows: any class α ∈ Pic(O) can be represented by an integral ideal I ⊂ O such that O/I is cyclic. Choose such an I. The lattice (1, τ)I^{−1} is a projective O-module which contains 1 as an indivisible element. Hence we can write

(3.7)  ⟨1, τ⟩I^{−1} = ⟨1, τ′⟩,

in which the generator τ′ is well-defined modulo the action of SL_{2}(ℤ), and define

(3.8)  α ⋆ τ := τ′.

It is a routine matter (cf. exercise 5) to check that this rule does indeed endow CM(O) with an action of Pic(O) which is compatible with the action of this group on Ell(O) introduced earlier.

We may thus reformulate the main theorem of complex multiplication in a purely analytic way, as follows:

Theorem 3.5. Let K ⊂ ℂ be a quadratic imaginary field and let τ ∈ H ∩ K be an element of H which is quadratic over ℚ. Then j(τ) belongs to H, where H is the ring class field attached to the order O = O_{τ}. More precisely, for all α ∈ Pic(O), and τ ∈ CM(O),

j(α ⋆ τ) = rec(α)^{−1}j(τ).

3.2. Heegner points

Let N be a fixed positive integer and let M_{0}(N) be the ring of 2 × 2 matrices with entries in ℤ which are upper-triangular modulo N, so that Γ_{0}(N) is the group of units of determinant 1 in this ring. Adapting the terminology of the previous section, we now define the associated order of τ (relative to the level N) to be

O_{τ}^{(N)} := \{γ ∈ M_{0}(N) such that γτ = τ\} ∪ \{( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} )\} .

It is easy to see that

O_{τ}^{(N)} = O_{τ} ∩ O_{Nτ} .

The reader will note that the map from H to E(ℂ) induced by the modular parametrisation Φ_{N} is transcendental, since it is of infinite degree. Hence Φ_{N} is not normally expected to take on algebraic values when evaluated on algebraic arguments. The following theorem provides an important exception to this rule.

Theorem 3.6. Let τ be any element in H ∩ K, let O = O_{τ}^{(N)} be its associated order in M_{0}(N), and let H/K be the ring class field attached to O. Then Φ_{N}(τ) belongs to E(H).

Proof. By theorem 3.5, both j(τ) and j(Nτ) belong to a ring class field H of K associated to the order O_{τ} ∩ O_{Nτ}. Hence Φ_{N}(τ) is the image of a point in X_{0}(N)(H) (its coordinates are given by (J(τ), j(Nτ)) in the singular plane model of X_{0}(N) given by the Nth modular polynomial) by the modular parametrisation Φ_{N}. Hence Φ_{N}(τ) belongs to E(H), since the map X_{0}(N) ⟶ E induced by Φ_{N} is a map of algebraic curves defined over ℚ. (Cf. the remarks in the proof of theorem 2.9 of chapter 2.) □

Abusing notation, write O_{τ} instead of O_{τ}^{(N)} (trusting that the context will make it clear what level N is being used.) If O is any order in a quadratic imaginary field K ⊂ ℂ, we similarly rewrite

CM(O) = {τ ∈ H/Γ_{0}(N) such that O_{τ} = O}.

We then lift the action of Pic(O) to CM(O) ⊂ H/Γ_{0}(N) by setting α ⋆_{N} τ := τ′, where τ′ ∈ H is chosen so that

α ⋆ τ = τ′, α ⋆ (Nτ) = Nτ′ (mod SL_{2}(ℤ)).

Note that this property determines τ′ modulo the action of Γ_{0}(N). It is a routine matter (cf. exercise 5) to check that this rule gives a concrete description of the action of Pic(O) on CM(O) which can thus be related to the the action of Gal(H/K) on E(H) via the reciprocity law of class field theory.

Theorem 3.7 (Shimura reciprocity law). If τ belongs to CM(O) and α belongs to Pic(O), then

Φ_{N}(α ⋆ τ) = rec(α^{−1})Φ_{N}(τ).

Proof. This follows directly from the reciprocity law of theorem 3.5. □

3.3. Numerical examples

One of the charms of theorems 3.6 and 3.7 is that they lend themselves to concrete calculations and allow the construction, by analytic means, of a large supply of algebraic points on E defined over ring class fields of quadratic imaginary fields.

For example, one knows from the tables of [Cr97] (cf. also exercise 3 of chapter 2) that the elliptic curve of smallest conductor N = 11 is given by the equation

(3.9)  y^{2} + y = x^{3} − x^{2} − 10x − 20,

while the imaginary quadratic order of smallest discriminant which embeds in M_{0}(11) is O_{K} = ℤ(\frac{1+\sqrt{−7}}{2}). The field K has class number one, and the order

O = ℤ + ℤ ( \begin{matrix}−4 & −2 \\ 11 & 5 \end{matrix} )

is an order in M_{0}(11) which is isomorphic to O_{K}. (This order is unique, up to conjugation by the normaliser Γ̃_{0}(11) of Γ_{0}(11) in PGL(ℚ).) The fixed point τ for this order is

τ = \frac{−9 + \sqrt{−7}}{22}.

The Fourier coefficients a_{n}(E) of the modular form f can be calculated by counting points of E mod p, or by using the identity

f = ∑_{n=1}^{∞} a_{n}q^{n} = q ∏_{n=1}^{∞}(1 − q^{11n})^{2}(1 − q^{n})^{2} = q − 2q^{2} − q_{2}^{3}q^{4} + q^{5} + ⋯

(The calculation of these coefficients a_{n}, given an equation for E, is a built-in feature of many symbolic algebra packages such as Pari, and calculating the first 1000 coefficients takes less than a second on a small computer.) Setting q = e^{2πiτ} and computing the image of

z = ∑_{n=1}^{1000} \frac{a_{n}}{n} q^{n}

in E(ℂ) under the Weierstrass uniformisation yields a point which agrees with the point

(x, y) = (\frac{1 − \sqrt{−7}}{2}, −2 − 2\sqrt{−7})

to 35 decimal digits of accuracy. Further calculations of this kind are suggested in the exercises.

3.4. Heegner systems

Proposition 3.8. Let O be an order of discriminant prime to N. Then the set CM(O) is non-empty if and only if all the primes dividing N split in K/ℚ.

Proof. If CM(O) is non-empty, then O can be realised as a subring of M_{0}(N). Therefore, there is a ring homomorphism O ⟶ ℤ/Nℤ. Since the conductor of O is assumed to be prime to N, it follows that all ℓ dividing N are split in K. □

Because of this proposition, it is natural to require that the following Heegner hypothesis be satisfied

Hypothesis 3.9. All primes ℓ dividing N are split in K/ℚ.

Let n be any integer prime to N and let O_{n} be the order of K of conductor n. A point of the form Φ_{N}(τ), with τ ∈ CM(O_{n}), is called a Heegner point of conductor n. Let HP (n) ⊂ E(H_{n}) denote the set of all Heegner points of conductor n in E(H_{n}), where H_{n} denotes the ring class field of K of conductor n. The points in HP (n) are related by the following norm-compatibilities:

Proposition 3.10. Let n be an integer and ℓ a prime number which are prime to N. Let P_{nℓ} be any point in HP (nℓ). Then there exist points P_{n} ∈ HP (n) and (when ℓ|n) P_{n/ℓ} ∈ HP (n/ℓ) such that

Trace_{H_{nℓ}/H_{n}}(P_{nℓ}) = \{ \begin{matrix}a_{ℓ}P_{n} & if ℓ |̸n is inert in K; \\ (a_{ℓ} − σ_{λ} − σ_{λ}^{−1})P_{n} & if ℓ = λλ̄ |̸n is split in K; \\ (a_{ℓ} − σ_{λ})P_{n} & if ℓ = λ^{2} is ramified in K \\ a_{ℓ}P_{n} − P_{n/ℓ} & if ℓ|n. \end{matrix}

Proof. We content ourselves with the proof of the second norm-compatiblity relation, as the others are similar, but somewhat simpler. Let (A ⟶ A′) be the pair of N-isogenous elliptic curves corresponding to the point P_{n}. If ℓ = λλ̄ is a prime of ℚ which is split in K and prime to N, then the action of Gal(K̄/H_{n}) on A[ℓ] leaves invariant two cyclic subgroups of order ℓ: the groups C_{0} = A[λ], and C_{∞} = A[λ̄], and permutes the remaining ℓ − 1 subgroups C_{1}, …, C_{ℓ−1} transitively. In fact, this permutation action factors through a simply transitive action of Gal(H_{nℓ}/H_{n}) on {C_{1}, …, C_{ℓ−1}}. Let P_{nℓ}^{(j)} be the point in E(H_{nℓ}) corresponding to the pair (A/C_{j} →^{φ} A′/φ(C_{j})), and set P_{nℓ} = P_{nℓ}^{(1)}. The description of the Hecke operator T_{ℓ} in terms of cyclic ℓ-isogenies makes it apparent that

(3.10)  a_{ℓ}P_{n} = P_{nℓ}^{(0)} + P_{nℓ}^{(∞)} + P_{nℓ}^{(1)} + ⋯ + P_{nℓ}^{(ℓ−1)}.

On the other hand,

(3.11) P_{nℓ}^{(0)} = σ_{λ}P_{n}, P_{nℓ}^{(∞)} = σ_{λ}^{−1}P_{n}, P_{nℓ}^{(1)} + ⋯ + P_{nℓ}^{(ℓ−1)} = Trace_{H_{nℓ}/H_{n}}(P_{nℓ}).

The result follows immediately from (3.10) and (3.11). For more details and a discussion of the remaining cases see [Gr89]. □

An element τ ∈ Gal(H/ℚ) is called a reflection if its restriction to K is not the identity. Because of the dihedral nature of Gal(H/ℚ), any reflection is of order 2, and any two reflections differ by multiplication by an element of Gal(H/K). We will need the following behaviour of the points P_{n} under the action of a reflection.

Proposition 3.11. Let τ ∈ Gal(H/ℚ) be a reflection. Then there exists σ ∈ Gal(H/K) such that

τP_{n} = −sign(E, ℚ)σP_{n} (mod E(H)_{tors}),

where sign(E, ℚ) is the sign attached to E/ℚ described in equation (2.17) of chapter 2.

Proof. See [Gr84]. □

Definition 3.12. A Heegner system attached to (E, K) is a collection of points P_{n} ∈ E(H_{n}) indexed by integers n prime to N, and satisfying the norm compatibility properties of proposition 3.10 together with the behaviour under the action of reflections described in proposition 3.11.

A Heegner system is said to be non-trivial if at least one of the points P_{n} is non-torsion.

Theorem 3.13. If (E, K) satisfies the Heegner hypothesis, then there is a nontrivial Heegner system attached to (E, K).

Proof. The union of the points CM(n), as n ranges over all integers prime to N, is dense in H with respect to the complex topology, as soon as the Heegner hypothesis, which ensures that the sets CM(n) are non-empty, is satisfied. Hence the image of these points in E(ℂ) is dense with respect to the complex topology, and, in particular, infinite. Let H_{∞} denote the union of all the ring class fields of conductor prime to N. To rule out the possibility that all the points P_{n} are torsion, one uses the following lemma:

Lemma 3.14. The torsion subgroup of E(H_{∞}) is finite.

Proof. Any rational prime which is inert in K splits completely or is ramified in all ring class fields, so that the residue field of H_{∞} at such a prime q is the field 𝔽_{q^{2}} with q^{2} elements. Since the prime-to-q torsion in E(H_{∞}) injects into the finite group E(𝔽_{q^{2}}), it follows that the full torsion subgroup of E(H_{∞}) injects into E(𝔽_{q_{1}^{2}}) ⊕ E(𝔽_{q_{2}^{2}}), where q_{1} and q_{2} are two distinct rational primes which are inert in K. □

It follows from lemma 3.14 and the discussion preceeding it that at least one (in fact, infinitely many) of the points Φ_{N}(τ) is of infinite order in E(H_{∞}), proving theorem 3.13. □

The existence of a systematic supply of points defined over the ring class fields of an imaginary quadratic field represents in itself a noteworthy occurence. The next section examines the compatibility between this phenomenon and the conjecture of Birch and Swinnerton-Dyer.

3.5. Relation with the Birch and Swinnerton-Dyer conjecture

Let K be any number field, and let D_{K} be its discriminant. If v is a fractional ideal of K, denote by |v| ∈ ℚ the norm of v. If E is an elliptic curve over ℚ as before, one can consider as in the remarks closing chapter 1 the L-function L(E/K, s) of E over K. This L-series can be expressed as an Euler product taken over the primes of K:

L(E/K, s) = ∏_{v} L_{v}(E/K, s),

where L_{v}(E/K, s)^{−1} is a polynomial in |v|^{−s} of degree at most 2 given by the rule

L_{v}(E/K, s) = \{ \begin{matrix}(1 − a_{|v|}|v|^{−s} + |v|^{1−2s})^{−1} & if v |̸N; \\ (1 − a_{|v|}|v|^{−s})^{−1} & if v|N. \end{matrix}

Assume from now on that K is a quadratic field, so that one has (cf exercise 15)

(3.12)  L(E/K, s) = L(E, s)L(E′, s)

where E′ is the quadratic twist of E over K.

More generally, let

χ : Gal(H/K) ⟶ ℂ^{×}

be any character of the ring class field H of conductor c with (c, N) = 1, and set D = D_{K}c^{2}. Define the twisted L-series by the rule

L(E/K, χ, s) = ∏_{v} L_{v}(E/K, χ, s),

where L_{v}(E/K, χ, s) is given, for v |̸ND, by the formula

(3.13)  L_{v}(E/K, χ, s) = (1 − χ(σ_{v})a_{|v|}|v|^{−s} + χ(σ_{v})^{2}|v|^{1−2s})^{−1}.

(For the general formula describing the Euler factor for v|ND, see [Gr84], ch. III, sec. 19, or exercise 16.) It is useful to complete the definition of the local L-factor L_{v}(E/K, χ, s) to the infinite primes, by setting

L_{∞}(E/K, χ, s) = (2π)^{−2s}Γ(s)^{2}.

Let A = N^{2}D^{2}/ gcd(N, D).

Theorem 3.15. Let

Λ(E/K, χ, s) = A^{s/2}L_{∞}(E/K, χ, s)L(E/K, χ, s).

The L-function L(E/K, χ, s) has an analytic continuation to the entire complex plane, and satisfies a functional equation of the form

(3.14)  Λ(E/K, χ, s) = sign(E, K)Λ(E/K, χ, 2 − s),

where sign(E, K) = ±1 is a sign which depends only on E and K, not on the ring class character χ of conductor prime to N.

Remark on the proof. The proof of the analytic continuation and functional equation for L(E/K, s) and L(E/K, χ, s) relies on Rankin’s method and its extensions by Jacquet [Ja72]. When K is an imaginary quadratic field, it is explained in detail in chapter IV of [GZ84]. See also the remarks in [Gr84], ch. III, sec. 21. □

While the proof of theorem 3.15 does not matter very much for our discussion, the shape of the functional equation in this theorem – in particular, the independence on χ of the sign in equation (3.14) – has a number of striking arithmetic consequences when combined with the Birch and Swinnerton-Dyer conjecture.

For example, if H is any ring class field of K of conductor prime to N, then by exercise 17, the L-function of E/H factors as

(3.15)  L(E/H, s) = ∏_{χ} L(E/K, χ, s),

where the product is taken over all the characters of Gal(H/K). It follows from theorem 3.15 that, if sign(E, K) = −1,

(3.16)  L(E/K, χ, s) = 0 for all characters χ : Gal(H/K) ⟶ ℂ.

Hence, by (3.15)

(3.17)  ord_{s=1}L(E/H, s) ≥ [H : K].

The Birch and Swinnerton-Dyer conjecture then leads to the expectation that

(3.18)  rank(E(H)) ≥^{?} [H : K].

The following conjecture is motivated by this predicted inequality:

Conjecture 3.16. If sign(E, K) = −1, then there is a non-trivial Heegner system attached to (E, K).

We turn to a description of sign(E, K) in the special case where E is semistable over K. The analytic set attached to E over K is the set of places of K which are archimedean or at which the curve E has split multiplicative reduction. (The reason for this terminology is that S_{E,K} consists precisely of the places of K for which E(K_{v}) admits an analytic uniformisation, given by the Weierstrass and Tate theory mentioned in chapter 1 when v is archimedean and non-archimedean respectively.) The following makes theorem 3.15 more precise:

Theorem 3.17. Suppose that E is an elliptic curve over ℚ and K is a quadratic field. Then

sign(E, K) = (−1)^{#S_{E,K}} .

Idea of proof. The functional equation for L(E/K, χ, s) of [Ja72] expresses the sign in the functional equation as a product of local signs sign_{v}(E, K) indexed by the places v of K. It turns out that sign_{v}(E, K) = 1 if v does not belong to S_{E,K}, and is equal to −1 otherwise. A complete and explicit description of the “local root numbers” attached to the functional equations of L-series of elliptic curves is given in [Ro96]. □

Remark 3.18. Much of the discussion above can be transposed to the setting where ℚ is replaced by an arbitrary number field F (and where E is an elliptic curve over F, while K is a quadratic extension of F). In that setting the L-function L(E/K, χ, s) is still expected to satisfy a functional equation of a shape similar to that given by theorem 3.15, with sign(E, K) depending only on E and K, not on the ring class character χ of conductor prime to the conductor of E. On the basis of the Birch and Swinnerton-Dyer conjecture, it is natural to make the following conjecture:

Conjecture 3.19. Suppose that E is a semistable elliptic curve over a number field F and K is a quadratic extension of F. Let

sign(E, K) := (−1)^{#S_{E,K}} .

If sign(E, K) = −1, then there is a non-trivial Heegner system attached to (E, K).

Even though the statement of conjecture 3.19 is elementary and does not involve the notion of modularity, one knows at present of no method for tackling it directly without exploiting a connection between elliptic curves and automorphic forms. In fact, only in the rather limited number of cases where one can establish the analytic continuation and functional equation of L(E/K, χ, s) – by relating it to the L-series of an automorphic form on GL_{2}(F), as in the case F = ℚ covered by Wiles’ theory – does one have any means at present of relating sign(E, K) to the behaviour of this associated L-series.

To illustrate how theorem 3.17 can be applied, suppose that E is a semistable elliptic curve over ℚ and that K is a quadratic imaginary field satisfying the Heegner hypothesis with respect to E. In that case

S_{E,K} = {λ such that λ|ℓ|N and E/ℚ_{ℓ} has split multiplicative reduction} ∪ {∞}.

Since each rational prime ℓ dividing N splits in K, the primes of K for which E has split multiplicative reduction come in pairs and hence #S_{E,K} is odd, so that sign(E, K) = −1. Hence the Heegner point construction supplies a proof of conjecture 3.16 in the special case where (E, K) satisfies the Heegner hypothesis (cf. thm.3.13).

3.6. The Gross-Zagier formula

As in the previous sections, let E be an elliptic curve over ℚ and let K be an imaginary quadratic field such that (E, K) satisfies the Heegner hypothesis. Denote by {P_{n}} = {Φ_{N}(τ_{n})} the Heegner system arising from the points in HP (n). Let

P_{K} = Trace_{H_{1}/K}(P_{1}) ∈ E(K)

be the trace of a Heegner point of conductor 1 defined over the Hilbert class field of K. More generally, if χ : Gal(H_{n}/K) ⟶ ℂ^{×} is any primitive character of a ring class field extension of K of conductor n, let

P_{n}^{χ} = ∑_{σ∈Gal(H_{n}/K)} χ̄(σ)P_{n}^{σ} ∈ E(H_{n}) ⊗ ℂ.

The following result provides the essential bridge between the Heegner system {P_{n}} and the special values of the complex L-series L(E/K, s) and its twists.

Theorem 3.20 (Gross-Zagier, Zhang). Let ⟨ , ⟩_{n} denote the canonical Néron-Tate height on E(H_{n}) extended by linearity to a Hermitian pairing on E(H_{n}) ⊗ ℂ. Then

(1) ⟨P_{K}, P_{K}⟩ =^{.} L′(E/K, 1);

(2) ⟨P_{n}^{χ}, P χ̄ ; n⟩ =^{.} L′(E/K, χ, 1).

Here the symbol =^{.} denotes equality up to a non-zero fudge factor, which can in principle be made explicit. Part 1 of theorem 3.20 was proved in [GZ84] (cf. thm. 2.1 of section V.2) and part 2 is proved in [Zh01b]. The main consequence of theorem 3.20 is that the Heegner vector P_{n}^{χ} is non-zero if and only if L′(E/K, χ, 1) does not vanish.

3.7. Kolyvagin’s theorem

A non-trivial Heegner system obviously yields certain lower bounds on the size of the Mordell-Weil group of E over ring class fields of K. The following theorem reveals that a Heegner system also leads (somewhat surprisingly, at first sight) to upper bounds on the Mordell-Weil group and the Shafarevich-Tate group of E/K.

Theorem 3.21 (Kolyvagin). Let {P_{n}} be a Heegner system attached to (E, K). If P_{K} is non-torsion, then

(1) The Mordell-Weil group E(K) is of rank one, so that P_{K} generates a finite-index subgroup of E(K);

(2) The Shafarevich-Tate group of E/K is finite.

The proof of this theorem is explained in chapter 10. (The interested reader may immediately skip to this chapter which is independent of the material in chapters 4–9.)

3.8. Proof of the Gross-Zagier-Kolyvagin theorem

We now explain how the results of sections 3.6 and 3.7 can be combined to prove theorem 1.14 of chapter 1. Recall that this theorem states:

Theorem 3.22. If E is an elliptic curve over ℚ and ord_{s=1}L(E, s) ≤ 1, then

rank(E(ℚ)) = ord_{s=1}L(E, s) and #LLI(E/ℚ) < ∞.

Sketch of proof. Recall that sign(E, ℚ) denotes the sign in the functional equation for L(E, s) = L(E/ℚ, s). Suppose first that sign(E, ℚ) is equal to −1. By a result of [Wa85] (see also [MM97]), there exist infinitely many quadratic Dirichlet characters ε such that

(1) ε(ℓ) = 1 for all ℓ|N;

(2) ε(−1) = −1;
(3) L(E, ε, 1) ≠ 0.

Note that for all characters ε satisfying conditions 1 and 2 above, L(E, ε, s) vanishes to even order at s = 1. This is because the quadratic imaginary field corresponding to ε satisfies the Heegner hypothesis with respect to E, so that L(E/K, s) = L(E, s)L(E, ε, s) vanishes to odd order at s = 1. If sign(E, ℚ) is equal to 1, then for parity reasons L(E, ε, 1) = 0 for all quadratic Dirichlet characters satisfying conditions 1 and 2 above. In this case one invokes analytic results of [BFH90] and [MM91] which guarantee the existence of such a character ε for which

L′(E, ε, 1) ≠ 0.

In either case, let K be the quadratic imaginary field associated to ε. By construction,

(1) K satisfies the Heegner hypothesis relative to E;

(2) ord_{s=1}L(E/K, s) = 1, so that L′(E/K, 1) ≠ 0.

Let {P_{n}} be the Heegner system arising from the CM points on X_{0}(N) attached to K. The Gross-Zagier theorem implies that this Heegner system in non-trivial in the strong sense that P_{K} is non-torsion. Kolyvagin’s theorem then implies that E(K) has rank one, so that the quotient of E(K) by ⟨P_{K}⟩ is finite, as is the Shafarevich-Tate group of E/K. By proposition 3.11, P_{K} belongs to E(ℚ) (up to torsion) if and only if sign(E, ℚ) = −1. It follows that the rank of E(ℚ) is equal to the order of vanishing of L(E, s) at s = 1, as predicted by the Birch and Swinnerton-Dyer conjecture. Finally, the finiteness of LLI(E/K) directly implies the finiteness of LLI(E/ℚ) since the natural map LLI(E/ℚ) ⟶ LLI(E/K) induced by restriction has finite kernel. (Cf. exercise 18.) □

One cannot emphasize enough the crucial role played in this proof by the Heegner system arising from CM points on X_{0}(N).

Heegner systems are interesting objects in their own right, even beyond their striking application to the arithmetic of elliptic curves arising from the theorems of Gross-Zagier and Kolyvagin. It is therefore natural to examine the following question:

Question 3.23. Let E be an elliptic curve over ℚ and let K be a quadratic field with sign(E, K) = −1. Is it possible to construct a non-trivial Heegner system attached to (E, K)?

The next chapter presents the necessary background to give an essentially complete affirmative answer to this question (and thereby prove conjecture 3.16) in the case where K is an imaginary quadratic field. The case where K is real quadratic is more mysterious: one knows of no method for analytically constructing the class fields of real quadratic fields. In this case one must content oneself with a conjectural construction that works in certain special cases. The tools and concepts needed for this construction are introduced gradually in chapters 5, 6 and 7 and the construction is described in chapter 8.

Further results

An excellent introduction to the theory of complex multiplication is given in Serre’s article on complex multiplication (chapter XII of [CF67]). A more lengthy and leisurely introduction, containing a wealth of historical and extra material, is the book [Cox89] by Cox.

The seminal article [GZ84] and the follow-up article [GKZ87] provide a rich source of information on Heegner points, and their connections with special values of the associated Rankin L-series. Other useful background on this topic can be found in [Gr84], [Za85], [Zh01a] and [Zh01b].

A useful exposition of the proof of Kolyvagin’s theorem (which is also the focus of chapter 10) is given in [Gr89].

Exercises

(1) Let K be an imaginary quadratic field, and let O ⊂ O_{K} be the order in K of conductor c > 1. If h = #Pic(O_{K}) is the class number of K, show that

#Pic(O) = h · #\frac{(O_{K}/cO_{K})^{×}}{(ℤ/cℤ)^{×}O_{K}^{×}} .

For K = ℚ(ω_{D}) of class number one, attach to each element α ∈ (O_{K}/cO_{K})^{×} a projective O-module Λ_{α}, in such a way that Λ_{α} and Λ_{β} are isomorphic if and only if α = βu for some u ∈ (ℤ/cℤ)^{×}O_{K}^{×}. Compute Pic(O) for

(a) O = ℤ[i], ℤ[2i], ℤ[3i], and ℤ[4i];

(b) O = ℤ[ρ], ℤ[2ρ], and ℤ[3ρ], where ρ = ω_{−3}.

(c) O = ℤ[ω_{−7}] and ℤ[11ω_{−7}].

(2) Let E be an elliptic curve with complex multiplication by K. Assume that both E and the endomorphisms of E are defined over a number field H.

(a) Show that the natural image of Gal(H̄/H) in Aut(E_{n}), for any integer n, is abelian.

(b) Conclude that E does not acquire multiplicative reduction over any finite extension of H. (Hint: use exercise 6 of chapter 1.)

(c) It is know that the prime ideals dividing the denominator of the j-invariant of E are precisely the ones at which E has potentially multiplicative reduction. Conclude that if τ belongs to H ∩ K, then j(τ) is an algebraic integer.

(3) Show that j(\frac{1+i\sqrt{163}}{2}) is an integer. Compute this integer using a symbolic algebra package such as PARI. Explain the curious identity

e^{π\sqrt{163}} = 262537412640768743.9999999996...

(4) Complete the proof of theorem 3.6 in the case where the conductor of O is not assumed to be prime to N.

(5) Check that formula (3.8) does yield a well-defined action of Pic(O) on CM(O).

(6) Check that the action of Pic(O) on Ell(O) commutes with the natural action of G_{K} := Gal(K̄/K) on this set.

(7) Let E be an elliptic curve with complex multiplication by an order O in a quadratic imaginary field K, defined over an abelian extension H of K. Let 𝔭 be a prime of K of norm a rational prime p, which is unramified in H/K and modulo which E and all its Galois conjugates have good reduction. Choose a prime 𝔭′ of H above 𝔭. Let Ē be the curve obtained from E by reducing modulo 𝔭′.

(a) Show that there is a unique (up to composition with automorphisms of the image) inseparable isogeny from Ē of degree p. (Hint: show that the isogeny Ē ⟶ Ē/Ē[𝔭̄] is separable, by choosing an ideal I of norm prime to p such that 𝔭̄I = (λ) is principal. Then show that the composed map Ē ⟶ Ē/Ē[𝔭̄] ⟶ Ē/Ē[λ] = Ē is separable, by examining its effect on the tangent space of E, using equation (3.2).)

(b) By a similar argument, show that the natural projection Ē ⟶ Ē/Ē[𝔭] is

purely inseparable of degree p.

(8) For the following elliptic curves E and fundamental discriminants D, compute the class number h of D and a system (x_{1}, y_{1}), …, (x_{h}, y_{h}) of complex points approximating the Heeger points on E attached to O with an accuracy of at least 20 decimal digits. Decribe the polynomial of degree h with coordinates in ℚ satisfied by the x_{j} and the y_{j}.

(a) E : y^{2} + y = x^{3} − x^{2} − 10x − 20, of conductor 11; with D = −8, −19, −24, or −43.

(b) E : y^{2} + xy + y = x^{3} + 4x − 6, of conductor 14; with D = −31 or −68.

(c) E : y^{2} + y = x^{3} − x, of conductor 37; with D = −3, −4, −11, or −67, (9) Let E be the elliptic curve y^{2} = 4x^{3} − 28x + 25 of conductor 5077.

(a) Show (using PARI or another symbolic algebra package, say) that the sign in the functional equation of L(E, s) is −1 so that this L-series vanishes to odd order.

(b) By a direct numerical calculation, verify that the Heegner point associated to the order of discriminant −4 is torsion.

(c) Using the Gross-Zagier formula, conclude that L(E, s) has a zero of order at least 3 at s = 1.

The existence of such L-series with zeroes of high order, implied by the conjecture of Birch and Swinnerton-Dyer, plays a key role in Goldfeld’s effective solution of the Gauss class number problem. It was not known unconditionally before the work of Gross and Zagier.

(10) Show that a CM point on H/Γ_{0}(N) corresponds to an integral point on the open modular curve Y_{0}(N). (Hint: use the fact that an elliptic curve with complex multiplication has potentially good reduction at all primes.) Conclude that if the modular parametrisation Φ_{N} maps only the cusps of X_{0}(N) to the origin of E, then the curve E has a Heegner system {P_{n}} consisting entirely of integral points (relative to the minimal Weierstrass model for E).

(11) Prove a converse to the theorem of the previous exercise: the Heegner system of points {P_{n}} = {Φ_{n}(τ_{n})} consists entirely of integral points on E, if and only if the inverse image of the origin of E under Φ_{N} consists entirely of cusps on X_{0}(N).

(12) Show that the integrality property of Heegner points is satisfied by the elliptic curves of conductor < 37 and the curve labelled 37A in Cremona’s tables, but not by the curve labelled 37B in these tables.

(13) Produce an example of a Heegner point on the curve 37B which is not integral. (I am grateful to Antoine Gournay for carrying out the computer calculations necessary to formulate this exercise and the previous one.)

(14) * Give a complete list of the elliptic curves satisying the integrality property of Heegner points. (A useful reference to get started on this as yet unsolved problem is [MSw-D74].)

(15) Prove formula (3.12) in the text.

(16) The local Euler factor at p in the L-function of E/ℚ can be written as

(1 − α_{p}p^{−s})(1 − α′_{p}p^{−s}),

where, for p |̸N, we have α_{p} + α′_{p} = a_{p}(E), and α_{p}α′_{p} = p. Given a ring class character χ : Gal(H/K) ⟶ ℂ^{×}, set

(β_{p}, β′_{p}) = \{ \begin{matrix}(χ(σ_{𝔭}), χ(σ_{𝔭})^{−1}) & if p = 𝔭𝔭̄ splits in K; \\ (1, −1) & if p is inert in K; \\ (χ(σ_{𝔭}), 0) & if p = 𝔭^{2} is ramified in K but not in H; \\ (0, 0) & if p is ramified in H/K. \end{matrix}

(a) Let V be the two-dimensional Artin representation of G_{ℚ}, given by inducing χ from G_{k} to G_{ℚ}. Show that the Artin L-series L(V, s) is equal to

L(V, s) = ∏_{p}(1 − β_{p}p^{−s})^{−1}(1 − β′_{p}p^{−s})^{−1}.

(b) Show that when p does not divide N or the discriminant of H, the degree four Euler factor

(1 − α_{p}β_{p}p^{−s})(1 − α_{p}β′_{p}p^{−s})(1 − α′_{p}β_{p}p^{−s})(1 − α′_{p}β′_{p}p^{−s})

is equal to the product of the Euler factors in equation 3.13, taken over the primes v which divide p.

(c) Define the local factor at the rational prime p for L(E/K, χ, s) to be

(1 − α_{p}β_{p}p^{−s})(1 − α_{p}β′_{p}p^{−s})(1 − α′_{p}β_{p}p^{−s})(1 − α′_{p}β′_{p}p^{−s}).

Compute this Euler factor at the primes which divide N or the discriminant of H.

(17) Prove equation (3.15) in the text.

(18) Show that the natural map LLI(E/ℚ) ⟶ LLI(E/K) induced by restriction has finite kernel.

CHAPTER 4 Heegner points on Shimura curves

Any elliptic curve E over ℚ is modular, and hence is equipped with the modular parametrisation

(4.1)  Φ_{N} : H/Γ_{0}(N) ⟶ E(ℂ), where N = conductor of E

introduced in chapter 2. The theory of complex multiplication of chapter 3 allows the construction of a plentiful supply of algebraic points on E – the so called Heegner points, of the form Φ_{N}(τ), where τ ∈ H is a quadratic (imaginary) irrationality.

In particular, if K is a quadratic imaginary field satisfying the Heegner hypothesis, then for all orders O of K of conductor prime to N the set CM(O) of points in H/Γ_{0}(N) with associated order equal to O is non-empty, and it is possible to choose points τ_{n} ∈ CM(O_{n}) in such a way that the collection P_{n} = Φ_{N}(τ_{n}) forms a Heegner system in the sense of definition 3.12 of chapter 3. This Heegner system is an essential ingredient in the proof of the theorem of Gross-Zagier-Kolyvagin stated in chapter 1.

It is natural to examine what happens if the Heegner hypothesis is relaxed. For example, suppose that N = p is a prime which is inert in K. One can show (cf. exercise 1) that if τ belongs to H ∩ K, then P_{τ} := Φ_{N}(τ) belongs to E(H_{n}) for some n of the form p^{t}n′ with t ≥ 1 and (p, n′) = 1. Furthermore,

Trace_{H_{n}/H_{n′}} (P_{τ}) = 0.

Thus the Heegner point construction does not yield any points on E defined over ring class fields of conductor prime to p. This is to be expected, since S_{E,K} = {p, ∞} so that sign(E, K) is equal to 1: in this case one expects the rank of E(H_{n′}) to be small in general.

A second example which is more interesting, and which the reader may find helpful to keep in mind in a first reading of this chapter, is the one where N = pq is a product of two distinct primes p and q which are both inert in K/ℚ. In that case S_{E,K} = {p, q, ∞}, so that sign(E, K) = −1. As in the previous example, the points of the form Φ_{N}(τ) belong to E(H_{n}) where n is of the form p^{r}q^{s}n′ with r, s ≥ 1, and the trace of these points to E(H_{p^{r}n′}) or to E(H_{q^{s}n′}) are torsion. It thus appears that the modular parametrisation Φ_{N} is inadequate to produce the non-trivial Heegner system whose existence is predicted by conjecture 3.16.

To deal with this example and its obvious generalisations, it seems essential to enlarge the repertoire of modular parametrisations to include Shimura curve parametrisations as well as the more classical modular curve parametrisation of (4.1).

4.1. Quaternion algebras

A quaternion algebra over a field F is a 4-dimensional central simple algebra over F. The standard example is the ring M_{2}(F) of 2 × 2 matrices with entries in F. A quaternion algebra B over F is said to be split if it is isomorphic to M_{2}(F). More generally, if K is an extension field of F, then B is said to be split over K if B ⊗_{F} K is a split quaternion algebra over K.

Every quaternion algebra splits over some extension of F (for example, any maximal commutative subfield of B). There are, up to isomorphism, exactly two quaternion algebras over the reals: the split algebra M_{2}(ℝ) and the algebra ℍ of Hamilton’s quaternions. A similar fact is true over ℚ_{p} or any local field. All of this is elementary. (Cf. exercise 2.)

More deep is the classification of quaternion algebras over number fields, which, together with the more general classification of central simple algebras, is a cornerstone of global class field theory. (Cf. [CF67].) For any place v of F, let F_{v} denote the completion of F at v and let B_{v} := B ⊗_{F} F_{v}. One says that B is split at v if B_{v} is a split quaternion algebra. Otherwise B is said to be ramified at v.

Proposition 4.1. Let S be a finite set of places of ℚ. Then there exists a quaternion algebra ramified precisely at the places in S, if and only if S has even cardinality. In this case the quaternion algebra is unique up to isomorphism.

Let Z be a finitely generated subring of F. (Of principal interest are the cases where Z = O_{F} is the ring of integers of F, or where Z is the ring of S-integers for some finite set S of places of F.)

Definition 4.2. A Z-order in B is a subring of B which is free of rank 4 as a Z-module. A maximal Z-order is a Z-order which is properly contained in no larger Z-order. An Eichler Z-order is the intersection of two maximal Z-orders.

The level of an Eichler order R = R_{1} ∩ R_{2} is the Z-module index of R in either R_{1} or R_{2}. One can show (cf. exercise 5) that this notion is independent of the description of R as an intersection of two maximal orders.

Unlike the rings of integers of number fields of which they are the non-commuta-tive counterpart, maximal Z-orders in a quaternion algebra are never unique. This is because any conjugate of a maximal order is also a maximal order. The most one can ask for in general is that a maximal Z-order be unique up to conjugation by elements of B^{×}. Such uniqueness is not true in general, but it is under the following general condition:

Definition 4.3. One says that that B and Z satisfy the Eichler condition if there is at least one archimedean prime or one prime which is invertible in Z at which B is split.

Proposition 4.4. Suppose that B and Z satisfy the Eichler condition. Then any two maximal Z-orders in B are conjugate. Likewise, any two Eichler Z-orders of the same level are conjugate.

The proof of this proposition is explained in [Vi80]. More precisely, ch. III, §5, of [Vi80] describes the set of Eichler Z-orders of a given level N in terms of an adèlic double coset space attached to B. To make this explicit, let ℤ̂ denote the usual profinite completion of ℤ and write ℚ̂ := ℤ̂ ⊗ ℚ for the ring of finite rational adèles. Fixing one Eichler Z-order R of level N in B, let

R̂ := R ⊗ ℤ̂;  B̂ := B ⊗ ℚ̂ = R̂ ⊗ ℚ

denote the “adèlisations” of R and B respectively. Then the set of Eichler Z-orders of level N in B is in natural correspondence with the coset space

B̂^{×}/ℚ̂^{×}R̂^{×},

by assigning to the coset represented by an idèle (b_{ℓ}) (indexed by rational primes ℓ) the order

(b_{ℓ})R̂(b_{ℓ}^{−1}) ∩ B.

It can be checked that this an Eichler Z-order in B of level N which depends only on the coset of (b_{ℓ}) and not on the choice of representative, and that all Eichler Zorders in B of level N are obtained in this way. It follows that the set of conjugacy classes of Eichler Z-orders of level N in B is in natural bijection with the double coset space

(4.2)  B^{×}\B̂^{×}/R̂^{×}.

Given any rational prime p, let B_{p} := B ⊗ ℚ_{p} and let R_{p} := R ⊗ ℤ_{p}. The following strong approximation theorem yields p-adic description of the double coset space appearing in (4.2):

Theorem 4.5. Let p be a prime at which the quaternion algebra B is split. Then the natural map

R[1/p]^{×}\B_{p}^{×}/R_{p}^{×} ⟶ B^{×}\B̂^{×}/R̂^{×}

which sends the class represented by b_{p} to the class of the idèle (…, 1, b_{p}, 1, …) is a bijection.

For further discussion see ch. III, §4 of [Vi80].

Any quaternion algebra B over F admits a natural four-dimensional linear representation over F by letting B act on itself by left multiplication. Given b ∈ B, the corresponding F -linear endomorphism of B has a characteristic polynomial of the form

f_{b}(x) = (x^{2} − tx + n)^{2}.

The integers t and n are called the reduced trace and the reduced norm of x respectively. (See exercise 3 and [Vi80] for more details.)

4.2. Modular forms on quaternion algebras

Let B be a quaternion algebra over ℚ which is split at ∞. (Such an algebra is called an indefinite quaternion algebra.) Fix an identification

ι : B ⊗_{ℚ} ℝ ≃ M_{2}(ℝ).

Let R be an order in B. Denote by R_{1}^{×} the group of elements of R^{×} of reduced norm 1, and let

Γ := ι(R_{1}^{×}) ⊂ SL_{2}(ℝ).

Lemma 4.6. The group Γ acts discretely on H with compact quotient.

Proof. Since R is discrete in B ⊗ ℝ, the group R_{1}^{×} is discrete in (B ⊗ ℝ)^{×}, so that Γ is a discrete subgroup of SL_{2}(ℝ). But H is identified with the coset space SL_{2}(ℝ)/SO_{2}(ℝ) where SO_{2}(ℝ) is the stabiliser of i. Since this latter group is compact, the discreteness of the action of Γ on H follows. The proof that H/Γ is compact proceeds by constructing a fundamental region for the action of Γ on H with compact closure, and uses in an essential way the assumption that Γ arises from a quaternion division algebra. It can be found for example in [Ka92], thm. 5.4.1. □

Definition 4.7. A modular form of weight k on Γ is a holomorphic function f on H such that

f(γτ) = (cτ + d)^{k}f(τ) for all γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ.

Remark 4.8. It is not necessary to assume any growth conditions, since the quotient H/Γ is already compact. In this sense the theory of modular forms attached to non-split quaternion algebras is simpler than the classical theory of forms on Γ_{0}(N). We will see shortly that the absence of cusps is also a source of extra difficulties in the theory, since the notion of Fourier expansions at the cusps is used crucially in the proof of the multiplicity one theorem of lemma 2.7, in the integral representation of L(f, s), and in proposition 2.10 giving an explicit formula for the modular parametrisation Φ_{N}.

As in the classical setting where B = M_{2}(ℚ), the main case which is relevant for elliptic curves and modular parametrisations is the one where k = 2. The space S_{2}(Γ) of forms of weight 2 on H/Γ can then be identified with the space of holomorphic differential forms on the compact Riemann surface H/Γ.

We now introduce certain subgroups of SL_{2}(ℝ) arising from quaternion algebras which will play much the same role in our discussion as the groups Γ_{0}(N) of chapter 2. Let N be a positive integer.

Definition 4.9. The factorisation N = N^{+}N^{−} is called an admissible factorisation if

(1) gcd(N^{+}, N^{−}) = 1;

(2) the integer N^{−} is squarefree, and the product of an even number of primes. A discrete subroup Γ_{N^{+},N^{−}} of SL_{2}(ℝ) can be associated to any admissible

factorisation of N as follows: let B denote the quaternion algebra ramified precisely at the primes ℓ which divide N^{−}. (Such an algebra is unique, up to isomorphism, by proposition 4.1.) Note that B is an indefinite quaternion algebra, i.e., it is split at the place ∞.

Choose a maximal order R_{0} in B - such orders are unique up to conjugation by B^{×}, by proposition 4.4. Since the algebra B is split at all the primes dividing N^{+}, and R_{0} is a maximal order, one may fix an identification

η : R_{0} ⊗ (ℤ/N^{+}ℤ) ⟶ M_{2}(ℤ/N^{+}ℤ).

Let R denote the subring of R_{0} consisting of all elements x such that η(x) is upper triangular. The subring R is an Eichler order of level N^{+} in B. Like the maximal order R_{0}, the Eichler order R is unique up to conjugation by B^{×}. After fixing as before an identification ι of B ⊗ ℝ with M_{2}(ℝ), define

Γ_{N^{+},N^{−}} = ι(R_{1}^{×}),

where R_{1}^{×} denotes as before the group of elements of reduced norm 1 in R. We now collect some basic facts about the structure of the space

S_{2}(Γ_{N^{+},N^{−}}) =: S_{2}(N^{+}, N^{−})

which are analogous to the basic properties of S_{2}(N) discussed in chapter 2.

• The space S_{2}(N^{+}, N^{−}) is naturally a Hilbert space, in which the duality is given by integration of differential one-forms (cup-product).

• It is endowed with a natural action of Hecke operators T_{p}, indexed by rational primes p, which are self-adjoint when p does not divide N. To define T_{p} in this case, let α ∈ R be an element of reduced norm p. The double coset ΓαΓ can be written as a disjoint union of left cosets

ΓαΓ = ∪_{i=0}^{p}α_{i}Γ,

and T_{p} is defined by summing the translates of f by the left coset representatives α_{i}

(4.3)  T_{p}(f(z)dz) := ∑_{i=0}^{p} f(α_{i}^{−1}z)d(α_{i}^{−1}z).

• Because the Hecke operators T_{n} commute and are self-adjoint, the space S_{2}(Γ_{N^{+},N^{−}}) is completely diagonalisable under the action of these operators.

• If f is an eigenform for the Hecke operators, its associated L-function can be defined as the product of the following local factors (at least for the primes ℓ which do not divide N:

(1 − a_{ℓ}(f)ℓ^{−s} + ℓ^{1−2s})^{−1}, where T_{ℓ}f = a_{ℓ}f.

Remark 4.10. We have not said anything about the dimensions of the various eigenspaces, and it should be remarked that here lies a complication of the theory: it is not clear that a simultaneous eigenspace for all the Hecke operators should be one-dimensional, since one lacks the notion of Fourier expansion which in the case of forms on Γ_{0}(N) allows one to recover the eigenform from a knowledge of its associated system of Hecke eigenvalues.

Nonetheless, there is a generalisation of Atkin-Lehner theory in this setting. More precisely, one can define a notion of oldforms in S_{2}(Γ_{N^{+},N^{−}}), which are forms arising from forms in S_{2}(Γ_{d^{+},N^{−}}) where d^{+} is a proper divisor of N^{+}. The space of newforms is the orthogonal complement of the space of oldforms defined in this way. It is proved in [Zh01a], §3.2.1, that the simultaneous eigenspaces in S_{2}^{new}(Γ_{N^{+},N^{−}}) for all the Hecke operators (or even merely for the good Hecke operators) are onedimensional.

We call a modular form f in such an eigenspace an eigenform on Γ_{N^{+},N^{−}}. Since f does not admit a Fourier expansion, it is also unclear by what condition one might normalise f in order to arrive at a notion of normalised eigenform. We postpone the discussion of this issue to the next chapter.

4.3. Shimura curves

The compact Riemann surface H^{∗}/Γ_{0}(N) can be interpreted as the complex points of an algebraic curve X_{0}(N) defined over ℚ; an analogous fact holds for the quotients H/Γ_{N^{+},N^{−}}. The way in which this is proved is similar: this Riemann surface admits a moduli interpretation as classifying abelian surfaces over ℚ endowed with certain extra stuctures; since this moduli problem makes sense over ℚ, it gives rise to an algebraic curve X_{N^{+},N^{−}} over ℚ whose complex points are identified with H/Γ_{N^{+},N^{−}}.

Roughly speaking, the moduli interpretation associates to τ ∈ H/Γ_{N^{+},N^{−}} an abelian surface with endomorphism ring containing the order R_{0}, and certain auxiliary level N^{+} structure. (For more details on Shimura curves and a precise definition of the moduli problem, see [BC92] or [Zh01a], ch. 1).

For example, if N^{−} = 1 so that B = M_{2}(ℚ), the maximal order R_{0} can be chosen to be M_{2}(ℤ). An abelian surface A whose endomorphism ring contains M_{2}(ℤ) decomposes as a product of an elliptic curve E with itself:

A = E × E = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) A × ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) A.

The level N structure imposed on A corresponds to the usual level N structure on E, so that in this case one recovers the usual moduli interpretation of X_{0}(N).

4.4. The Eichler-Shimura construction, revisited

Let f be an eigenform in S_{2}(Γ_{N^{+},N^{−}}) having integer Hecke eigenvalues a_{n}(f). As in the case of modular forms on Γ_{0}(N), one can associate to such an eigenform an elliptic curve over ℚ:

Theorem 4.11. There exists an elliptic curve E over ℚ such that a_{n}(E) = a_{n}(f), for all integers n such that (n, N) = 1.

Sketch of proof. The proof proceeds along lines similar to that of theorem 2.9 of chapter 2. Let 𝕋 be the algebra generated by the good Hecke operators. These operators can be realised as algebraic correspondences on the Shimura curves X_{N^{+},N^{−}} and hence give rise to endomorphisms of the Jacobian J_{N^{+},N^{−}} of X_{N^{+},N^{−}} which are defined over ℚ. On the other hand the eigenform f determines a homomorphism

φ_{f} : 𝕋 ⟶ ℤ, sending T_{n} to a_{n}(f),

Let I_{f} denote the kernel of φ_{f}. The multiplicity one result alluded to in remark 4.10 implies that the quotient

E_{f} := J_{N^{+},N^{−}}/I_{f}

is an elliptic curve. An analogue of the Eichler-Shimura congruence, this time for the correspondence T_{p} on X_{N^{+},N^{−}}^{2}, yields the equality of L-functions

L(E_{f}, s) = L(f, s).

For more details on this construction see [Zh01a], sec. 3.4. □

4.5. The Jacquet-Langlands correspondence

The Eichler-Shimura construction of the previous section, combined with Wiles’ theorem that every elliptic curve is modular, leads to the conclusion that for every admissible factorisation N^{+}N^{−} of N and for every newform g on Γ_{N^{+},N^{−}} with integer Hecke eigenvalues, there is an associated newform f on Γ_{0}(N) with the same Hecke eigenvalues as those of g at the primes ℓ not dividing N. In fact, more is true, a fact which could be established before Wiles’ proof of the Shimura-Taniyama-Weil conjecture: one does not need to assume the rationality of the Fourier coefficients of f, and the correspondence between newforms goes both ways:

4.7. COMPLEX MULTIPLICATION FOR H/Γ_{N+,N−} 51

Theorem 4.12 (Jacquet-Langlands). Let f be a newform on Γ_{0}(N), and let N = N^{+}N^{−} be an admissible factorisation of N. Then there is a newform g ∈ S_{2}(Γ_{N^{+},N^{−}}) with

L(f, s) = L(g, s) (up to finitely many Euler factors).

The proof of this theorem, which relies on techniques of non-abelian harmonic analysis, is beyond the scope of these notes, and is explained in [Gel75] (specifically, in the last chapter) and in [JL70].

4.6. The Shimura-Taniyama-Weil conjecture, revisited

The results of section 4.5 make it possible to rewrite the Shimura-Taniyama-Weil conjecture in terms of modular forms on Γ_{N^{+},N^{−}}.

Theorem 4.13. Let E/ℚ be an elliptic curve of conductor N, and let N = N^{+}N^{−} be an admissible factorisation of N. Then there exists a unique eigenform f ∈ S_{2}(Γ_{N^{+},N^{−}}) such that

T_{ℓ}(f) = a_{ℓ}(E)f, for all ℓ |̸N.

Sketch of proof. By Wiles’ theorem, there exists a newform f_{0} on Γ_{0}(N) attached to E. Theorem 4.12 produces the desired eigenform f ∈ S_{2}(Γ_{N^{+},N^{−}}). □

Theorem 4.13 supplies an essential ingredient in defining a new type of modular parametrisation

Φ_{N^{+},N^{−}} : Div^{0}(H/Γ_{N^{+},N^{−}}) ⟶ E(ℂ).

To begin, let

Φ_{N^{+},N^{−}}^{0} : Div^{0}(H) ⟶ ℂ

be the map which to a divisor D associates the line integral ∫_{D} f(z)dz. The subgroup generated by the elements of the form Φ_{N^{+},N^{−}}(D), where D is a divisor which becomes trivial in H/Γ_{N^{+},N^{−}}, is a lattice Λ_{f} in ℂ, and ℂ/Λ_{f} = E_{f}(ℂ). One thus obtains a map

Φ′_{N^{+},N^{−}} : Div^{0}(H/Γ_{N^{+},N^{−}}) ⟶ ℂ/Λ_{f} = E_{f}(ℂ).

Since E and E_{f} have the same L-function, they are isogenous over ℚ. Letting α be an isogeny E_{f} ⟶ E defined over ℚ, one then sets

Φ_{N^{+},N^{−}} = αΦ′_{N^{+},N^{−}}

.

4.7. Complex multiplication for H/Γ_{N^{+},N^{−}}

The reader will note that the one-dimensional factors of jacobians of Shimura curves do not yield any new elliptic curves over ℚ - these are already all accounted for in the jacobians of the modular curves X_{0}(N), by Wiles’ theorem. However, the larger supply of modular parametrisations

Φ_{N^{+},N^{−}} : Div^{0}(H/Γ_{N^{+},N^{−}}) ⟶ E(ℂ),

indexed by admissible factorisations of N provide new constructions of algebraic points on E, and in fact examples of Heegner systems that could not be constructed from modular curve parametrisations alone.

Following the lead of chapter 3 and defining CM points on H/Γ_{N^{+},N^{−}} as arising from τ ∈ H ∩ K, where K is a quadratic imaginary subfield of ℂ, is clearly inappropriate since the group Γ_{N^{+},N^{−}}, which depends on an identification ι of B ⊗ ℝ with M_{2}(ℝ), is only well-defined up to conjugation in SL_{2}(ℝ), a group whose action does not preserve H ∩ K. One resorts to the characterisation of CM points as those whose associated orders are orders in quadratic imaginary fields. More precisely:

Definition 4.14. Given τ ∈ H/Γ_{N^{+},N^{−}}, the associated order of τ is the set

O_{τ} := {γ ∈ R such that norm(γ) = 0 and ι(γ)(τ) = τ} ∪ {0}.

As in the case treated in chapter 3, the assignment γ ↦ z_{γ} identifies O_{τ} with a discrete subring of ℂ, so that O_{τ} is either ℤ or an order in a quadratic imaginary field K ⊂ ℂ.

Definition 4.15. A point τ ∈ H/Γ_{N^{+},N^{−}} is called a CM point if its associated order is isomorphic to an order in a quadratic imaginary field.

As in chapter 3, given an order O in a quadratic imaginary field K we write

CM(O) = {τ ∈ H/Γ_{N^{+},N^{−}} such that O_{τ} = O}.

The importance of the CM points lies in the fact that the theory of complex multiplication formulated in chapter 3 in the case of classical modular curves generalises readily to this new setting:

Theorem 4.16 (Complex multiplication for Shimura curves). Let O be an order in a quadratic imaginary field K of discriminant prime to N, and let H/K be the ring class field of K attached to O. Then

Φ_{N^{+},N^{−}}(Div^{0}(CM(O))) ⊂ E(H).

Sketch of proof. The proof uses the moduli interpretation of the points on H/Γ_{N^{+},N^{−}}. If τ belongs to CM(O), the associated abelian suface A_{τ} has endomorphism by the maximal order R_{0}, as well as by O, and these two actions commute with each other. Hence A_{τ} has endomorphism by R_{0} ⊗_{ℤ} O, and order in B ⊗ K ≃ M_{2}(K). It follows that A_{τ} is isogenous to a product A′ × A′, where A′ is an elliptic curve with complex multiplication by O. Hence A_{τ} is defined over H by the theory of complex multiplication covered in chapter 3. Further work shows that the level N^{+} structure attached to A_{τ} gives rise to a level N^{+} structure on A′ which is defined over H as well. □

4.8. Heegner systems

The following lemma reveals that the CM points arising from Shimura curve parametrisations are fundamentally new sets of points that could not be obtained by using modular curve parametrisations alone.

Lemma 4.17. Let K be a quadratic imaginary field of discriminant prime to N and let O be an order in K of conductor prime to N. Then CM(O) ≠ ∅ if and only if the following two conditions are satisfied:

(1) All the primes ℓ dividing N^{−} are inert in K.

(2) All the primes ℓ dividing N^{+} are split in K.

Proof. Since K is a quadratic subfield of the quaternion algebra B which is ramified at N^{−}, it follows that all the primes dividing N^{−} are inert in K. The fact that all primes dividing N^{+} are split in K is proved exactly as in the proof of proposition 3.8 of chapter 3. □

Lemma 4.17 leads to the proof of the following theorem:

Theorem 4.18. Let E be a semistable elliptic curve of conductor N, and let K be a quadratic imaginary field of discriminant prime to N. If sign(E, K) = −1, then there is a non-trivial Heegner system {P_{n}} attached to (E, K).

Sketch of proof. The field K determines a factorisation N = N^{+}N^{−} of N by letting N^{+} be the product of the primes which are split in K, while N^{−} is the product of the primes which are inert in K. Since sign(E, K) = −1, the set S_{E,K} has odd cardinality. On the other hand

S_{E,K} = {λ|ℓ|N^{+} such that E/ℚ_{ℓ} has split multiplicative reduction at ℓ}
∪{ℓ|N^{−}} ∪ {∞}.

The first set in the union has even cardinality, hence it follows that N^{−} is divisible by an even number of primes as well, so that N^{+}N^{−} is an admissible factorisation of N. For each integer n which is prime to N one then knows by lemma 4.17 that CM(O_{n}) is non-empty. One may choose divisors D_{n} ∈ Div^{0}(CM(O_{n})) in such a way that P_{n} := Φ_{N^{+},N^{−}}(D_{n}) forms a Heegner system. An argument analogous to the proof of theorem 3.13 of chapter 3, based on the density of the CM points in H/Γ_{N^{+},N^{−}} ensures that this Heegner system is non-trivial. □

4.9. The Gross-Zagier formula

An analogue of the Gross-Zagier formula (theorem 3.20) for Heegner points which arise from Shimura curve parametrisations was anticipated by Gross and Zagier in [Gr84] and has been recently proved by Zhang [Zh01a].

Theorem 4.19 (Zhang). If {P_{n}} is the Heegner system attached to (E, K) as above, then

⟨P_{K}, P_{K}⟩ =^{.} L′(E/K, 1).

(The symbol =^{.} is given the same meaning here as in the statement of theorem 3.20.)

We close this chapter by raising two questions which arise naturally from our discussion of Shimura curves:

(1) How does one compute numerically the parametrisation Φ_{N^{+},N^{−}} when N^{−} ≠ 1? The Fourier expansion of the modular form f attached to E in a neighbourhood of i∞ when N^{−} = 1 does not generalise in any obvious way to the setting of Shimura curves which are not equipped with cusps.

(2) The second question is the primary motivation for chapters 6,7, and 8: what construction plays the role of modular and Shimura curve parametrisations, and of the CM points on these curves, when the field K is real quadratic, and sign(E, K) = −1, in constructing the Heegner systems whose existence is predicted by conjecture 3.16?

A partial answer to question 1 can be given by exploiting a structure on Shimura curves which has no counterpart for classical modular curves: the p-adic uniformisation of these curves by certain discrete arithmetic subgroups of SL_{2}(ℚ_{p}), for p a prime dividing N^{−}. This new structure in some ways compensates for the absence of cusps and Fourier expansions, in allowing an explicit numerical description of modular forms in S_{2}(Γ_{N^{+},N^{−}}).

Question 2 lies deeper. A partial conjectural answer to it is given in chapter 8, relying on the p-adic analytic techniques introduced in the next chapter.

References

The behaviour of the Mordell-Weil groups E(H) when E is an elliptic curve of prime conductor p and H is a ring class field of a quadratic imaginary field K in which p is inert, so that sign(E, K) = 1, is studied in [BD97] where it is proved that E(H) is finite if L(E, H, 1) ≠ 0, for any ring class field H of conductor prime to p. More precisely, if G = Gal(H/K) and χ : ℤ[G] ⟶ ℂ is an algebra homomorphism, the “χ-part” E(H)^{χ} := E(H) ⊗_{χ} ℂ is trivial when L(E/K, χ, 1) ≠ 0. When sign(E, K) = 1, the non-vanishing of the factors L(E/K, χ, 1) of L(E, H, 1) is expected to occur “most of the time”. For example, work of Cornut and Vatsal [Cor02], [Va02] shows that L(E/K, χ, 1) ≠ 0 for almost all ring class characters whose conductor is a power of a fixed prime ℓ. It follows that there is no Heegner system attached to (E, K) in this situation.

The book of Vigneras [Vi80] is a good reference for the arithmetic of quaternion algebras and its connection with modular forms. The arithmetic theory of modular forms on quaternion algebras and of Shimura curves has been developed by Shimura [Sh67] building on earlier work of Eichler.

In addition to the canonical models of X_{N^{+},N^{−}} over ℚ provided by Shimura’s theory, one also has at one’s disposal integral models which have good reduction outside N. (Cf. [Car86], [Dr76]).

A useful reference for the topic of Heegner points on Shimura curves are the articles [Zh01a] and [Zh01b] which prove the Gross-Zagier formula in the context of Shimura curves in a rather general setting and contain helpful background on modular forms attached to quaternion algebras.

Exercises

(1) Let E be an elliptic curve of conductor N and let K be an imaginary quadratic field in which all the primes dividing N are inert. Let Φ_{N} : H/Γ_{0}(N) ⟶ E(ℂ) be the classical modular parametrisation attached to E.

(a) Suppose that N = p is prime. Show that that if τ belongs to H ∩ K, then P_{τ} := Φ_{N}(τ) belongs to E(H_{n}) for some n of the form p^{t}n′ with t ≥ 1 and (p, n′) = 1. Furthermore, show that

(4.4)  Trace_{H_{n}/H_{n′}} (P_{τ}) = 0.

(b) Suppose that N = pq is the product of two distinct primes. Show that the points of the form Φ_{N}(τ) are defined over ring class fields of conductor n = p^{t}q^{s}n′ with t, s ≥ 1, and that equation (4.4) continues to hold, even though sign(E, K) = −1 in this case. This justifies working with the Shimura curve parametrisation Φ_{1,pq} to produce the non-trivial Heegner system whose existence is predicted in this case.

(2) Let B be a quaternion algebra over a field F.

(a) If α ∈ B − F, show that the subalgebra K = F (α) generated by α over F is a commutative semisimple algebra of rank 2, and that it is a field if B is a division algebra. Let x ↦ x̄ denote the involution of K/F.

(b) Fix the quadratic subalgebra K ⊂ B. Show that there is an element β ∈ B^{×} satisfying βλ = λ̄β for all λ ∈ K. (Hint: Study the K-linear action of K by right multiplication on B viewed as a K-vector space under left multiplication.) Show that the element β is uniquely determined by K up to multiplication by elements of K^{×}.

(c) Show that γ = β^{2} belongs to F, and that it is uniquely determined up to multiplication by norms of non-zero elements in K.

(d) Conclude that any quaternion algebra over F is isomorphic to an algebra of the form B_{K,γ} = {a + bβ, a, b ∈ K}, where K is a quadratic semisimple algebra over F and γ ∈ F, with multiplication given by the rule

(a + bβ)(a′ + b′β) = (aa′ + bb̄′γ) + (ab′ + bā′)β.

(e) Show that the only quaternion algebras over ℝ are the split algebra M_{2}(ℝ)

and the algebra of Hamilton quaternions.

(3) Let B be a quaternion algebra over F, and let b ∈ B. Let K be a subfield of B quadratic over F and containing b. Prove that the norm and trace of b from K to F are equal to their reduced norm and trace from B.

(4) Show that a quaternion algebra becomes split over any quadratic subalgebra.

(5) Show that the level of an Eichler order R = R_{1} ∩ R_{2}, defined as the Z-module index of R in either R_{1} or R_{2}, is independent of the expression of R as the interseciton of two maximal orders R_{1} and R_{2}. (Hint: prove this first for orders in a matrix algebra over a local field.)

CHAPTER 5 Rigid analytic modular forms

The modularity of E has allowed us to construct complex uniformisations

Φ_{N} : H/Γ_{0}(N) ⟶ E(ℂ),
Φ_{N^{+},N^{−}} : Div^{0}(H/Γ_{N^{+},N^{−}}) ⟶ E(ℂ).

The CM points on these modular and Shimura curves map to a plentiful supply of algebraic points (“Heegner systems”) defined over the ring class fields of imaginary quadratic fields.

The purpose of the next two chapters is to enrich this picture further by using the uniformisation Φ_{N^{+},N^{−}} to construct explicit p-adic uniformisations of E by certain discrete arithmetic subgroups of SL_{2}(ℚ_{p}), at the primes p dividing N^{−}.

5.1. p-adic uniformisation

Let p be a prime, let | |_{p} denote the usual normalised p-adic absolute value on ℚ, and let ℚ_{p} denote the completion of ℚ with respect to this absolute value. Choose an algebraic closure ℚ̄_{p} of ℚ_{p}. Because ℚ̄_{p} has infinite degree of ℚ_{p}, it is no longer complete; the field obtained by completing ℚ̄_{p} with respect to the p-adic distance is a complete algebraically closed field which is commonly denoted ℂ_{p} and will play the role of the complex numbers in our analogy.

The p-adic upper half plane is defined (as a set) to be

(5.1)  H_{p} := ℙ_{1}(ℂ_{p}) − ℙ_{1}(ℚ_{p}).

Note that this set is really analogous to ℙ_{1}(ℂ) − ℙ_{1}(ℝ) – two copies of the Poincaré upper half-plane. In the p-adic case this set does not split naturally into two disjoint components, hence it is more appropriate to treat (5.1) as the natural generalisation of the Poincaré upper half plane.

The role of holomorphic functions on H is played by the so-called rigid analytic functions on H_{p}. These are functions that admit “nice” expressions when restricted to certain distinguished subsets of H_{p}, called affinoids.

We begin by giving prototypical examples of the regions in H_{p} (the basic affinoids, and annuli) which are the building blocks for the rigid analytic structure on H_{p}.

Let

red : ℙ_{1}(ℂ_{p}) ⟶ ℙ_{1}(𝔽̄_{p})

be the natural map given by reduction modulo the maximal ideal of the ring of integers of ℂ_{p}. Since red(ℙ_{1}(ℚ_{p})) ⊂ ℙ_{1}(𝔽_{p}), the set

A := red^{−1}(ℙ_{1}(𝔽̄_{p}) − ℙ_{1}(𝔽_{p}))
= {τ ∈ H_{p} such that |τ − t| ≥ 1, for t = 0, …, p − 1, and |τ| ≤ 1}

Figure 1. The standard affinoid and annuli, when p = 2.

is contained in H_{p}. It is an example of a standard affinoid in H_{p}. It is useful to “thicken” A by adjoining to it certain annuli (which are also subsets of H_{p}, since the p-adic absolute value on ℚ_{p}^{×} takes values in p^{ℤ}):

W_{t} = \{τ such that \frac{1}{p} < |τ − t| < 1\}, t = 0, …, p − 1,
W_{∞} = {τ such that 1 < |τ| < p}.

These regions are illustrated in figure 1, in the most easily drawn case where p = 2.

To describe more general affinoids, it is useful to introduce a basic combinatorial structure on H_{p}: the reduction map.

The target of this map is the so-called Bruhat-Tits tree of PGL_{2}(ℚ_{p}) denoted T . This is a graph whose vertices are in one-to-one correspondence with similarity classes of ℤ_{p}-lattices in ℚ_{p}^{2}. Two vertices are joined by an edge if they can be represented by lattices Λ_{1} and Λ_{2} satisfying

pΛ_{2} ⊂ Λ_{1} ⊂ Λ_{2},

in which both inclusions are proper. Since this relation is symmetrical it equips T with the structure of an unordered graph. One can show (cf. exercise 2) that T is in fact a tree all of whose vertices have valency p + 1. This tree is illustrated when p = in the figure below.

The group PGL_{2}(ℚ_{p}) acts on T in the natural way, and this yields an action on T by graph automorphisms (i.e., isometries with respect to the usual distance function on T ).

In our discussion the tree T is treated as a purely combinatorial object: a collection T_{0} of vertices indexed by homothety classes of ℤ_{p}-lattices in ℚ_{p}^{2} and a collection T_{1} of edges consisting of pairs of adjacent vertices. An ordered edge is an

5.1. p-ADIC UNIFORMISATION 59

Figure 2. The Bruhat-Tits tree of PGL_{2}(ℚ_{p}), when p = 2

ordered pair e = (v_{1}, v_{2}) of adjacent vertices. One then denotes by s(e) := v_{1} and t(v) := v_{2} the source and target of e respectively. Write E(T ) for the set of ordered edges of T .

Let v_{o} ∈ T_{0} be the distinguished vertex of T attached to the homothety class of the standard lattice ℤ_{p}^{2} ⊂ ℚ_{p}^{2}. The edges having v_{o} as endpoint correspond to index p sublattices of ℤ_{p}^{2} and thus are in canonical bijection with ℙ_{1}(𝔽_{p}). Label these edges accordingly as e_{0}, e_{1}, …, e_{p−1}, e_{∞} ∈ T_{1}.

Proposition 5.1. There is a unique map

r : H_{p} ⟶ T = T_{0} ∪ T_{1}

satisfying the following properties:

(1) r(τ) = v_{o} if and only if τ ∈ A;

(2) r(τ) = e_{t} if and only if τ ∈ W_{t};

(3) r is PGL_{2}(ℚ_{p})-equivariant, i.e.,

r(γτ) = γr(τ), for all γ ∈ PGL_{2}(ℚ_{p}).

The proof of this proposition is outlined in exercise 3.

If e = {v_{1}, v_{2}} is an edge of T , it is convenient to denote by ]e[⊂ T the singleton {e} and call it the open edge attached to e. The subset [e] := {e, v_{1}, v_{2}} of T is called the closed edge attached to e. Finally, the sets A_{[e]} := r^{−1}([e]) and W_{]e[} := r^{−1}(]e[) are called the standard affinoid and the standard annulus attached to e respectively. Note that A_{[e]} is a union of two translates of the standard affinoid A glued along the annulus W_{]e[}. The collection of affinoids A_{[e]}, as e ranges over T_{1}, gives a covering of H_{p} by standard affinoids whose pairwise intersections are either empty or of the form A_{v} := r^{−1}(v) with v ∈ V(T ). The incidence relations in this affinoid covering are thus reflected in the combinatorics of T .

Fix an affinoid A_{0} ⊂ H_{p}. A rational function having poles outside A_{0} attains its supremum on A_{0} (with respect to the p-adic metric). Hence the space of such functions can be equipped with the sup norm.

Definition 5.2. A ℂ_{p}-valued function f on H_{p} is said to be rigid-analytic if, for each edge e of T , the restriction of f to the affinoid A_{[e]} is a uniform limit, with respect to the sup norm, of rational functions on ℙ_{1}(ℂ_{p}) having poles outside A_{[e]}.

Let Γ be a discrete subgroup of SL_{2}(ℚ_{p}). Assume further that the quotient H_{p}/Γ (with its natural p-adic topology) is compact.

The sheaf of rigid analytic functions on H_{p}/Γ equips the quotient H_{p}/Γ with the structure of a rigid analytic curve over ℚ_{p}, which, by a p-adic analogue of the GAGA theorem, can be identified with an algebraic curve X over ℚ_{p} [GvdP80].

Remark 5.3. Not every curve over ℚ_{p} can be expressed as such a quotient. In fact, it can be shown that if X = H_{p}/Γ where Γ acts on T without fixed points, then it has a model over ℤ_{p} whose special fiber is a union of projective lines over 𝔽_{p} intersecting transversally at ordinary double points. The converse to this statement, due to Mumford, is a p-adic analogue of the classical complex uniformisation theorem:

Theorem 5.4 (Mumford). If X is a curve over ℚ_{p} having a model over ℤ_{p} whose special fiber consists of a union of projective lines intersecting at ordinary double points, then there is a discrete group Γ ⊂ PSL_{2}(ℚ_{p}) such that the rigid analytic curve X_{/ℂ_{p}} is isomorphic to H_{p}/Γ.

This theorem, which is discussed in [GvdP80], will not be used in the sequel, and its statement is included here only for the edification of the reader.

5.2. Rigid analytic modular forms

Let Γ ⊂ SL_{2}(ℚ_{p}) be a discrete subgroup as in the previous section.

Definition 5.5. A form of weight k on H_{p}/Γ is a rigid analytic function f on H_{p} such that

f(γτ) = (cτ + d)^{k}f(τ) for all γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ.

Denote by S_{k}(Γ) the ℂ_{p}-vector space of rigid analytic modular forms of weight k with respect to Γ. As in the discussions of chapters 3 and 4, the space S_{2}(Γ) can be identified with the space of rigid analytic differential forms on the quotient H_{p}/Γ, or, equivalently, with the space of regular differential forms on the curve X_{/ℂ_{p}}. In particular, the dimension of S_{2}(Γ) over ℂ_{p} is equal to the genus of this curve.

The classical modular forms in S_{2}(Γ_{0}(N)) studied in chapter 2 could be calculated by exploiting the connection between eigenvalues of Hecke operators and Fourier coefficients of modular forms, together with the action of Hecke operators on the homology of the modular curve made explicit in the theory of modular symbols. There is a method for constructing modular forms in S_{2}(Γ) which, while quite different (there being, for instance, no good notion of Fourier expansion for rigid analytic modular forms) is just as concrete and amenable to calculations as the modular symbol method reviewed in chapter 2. This exploits the connection between forms in S_{2}(Γ) and their associated p-adic boundary measures.

More precisely, view ℙ_{1}(ℚ_{p}) as the “boundary” of the p-adic upper half plane H_{p}. This set is endowed with its p-adic topology in which the open balls of the form

B(a, r) = {t such that |t − a| < p^{−r}}, a ∈ ℚ_{p},
B(∞, r) = {t such that |t| > p^{r}}

form a basis. These open balls are also compact, and any compact open subset of ℙ_{1}(ℚ_{p}) is a finite disjoint union of open balls of the form above.

Definition 5.6. A p-adic distribution on ℙ_{1}(ℚ_{p}) is a finitely additive function

μ : {compact open U ⊂ ℙ_{1}(ℚ_{p})} ⟶ ℂ_{p}

satisfying μ(ℙ_{1}(ℚ_{p})) = 0.

If μ is any p-adic distribution on ℙ_{1}(ℚ_{p}), and g is a locally constant function on ℙ_{1}(ℚ_{p}), then the integral ∫_{ℙ_{1}(ℚ_{p})} g(t)dμ(t) can be defined in the obvious way as a finite Riemann sum. More precisely, letting

(5.2)  ℙ_{1}(ℚ_{p}) = U_{1} ∪ ⋯ ∪ U_{m}

be a decomposition of ℙ_{1}(ℚ_{p}) as a disjoint union of open balls such that g is constant on each U_{j}, one defines

(5.3)  ∫_{ℙ_{1}(ℚ_{p})} gdμ := ∑_{j=1}^{m} g(t_{j})μ(U_{j}),

where t_{j} is any sample point in U_{j}. The distribution relation satisfied by μ ensures that this expression does not depend on the decomposition (5.2) used to define it. It is desirable to impose more stringent regularity properties on a distribution μ so that it can be integrated against a larger class of test functions.

Definition 5.7. A p-adic measure is a bounded distribution, i.e., a distribution for which there is a constant C satisfying

|μ(U)|_{p} < C, for all compact open U ⊂ ℙ_{1}(ℚ_{p}).

If λ is any continuous function on ℙ_{1}(ℚ_{p}), then the integral of λ against μ can be defined by the rule

(5.4)  ∫_{ℙ_{1}(ℚ_{p})} λ(t)dμ(t) = lim_{C={U_{α}}} ∑_{α} λ(t_{α})μ(U_{α}),

where the limit is taken over increasingly fine covers {U_{α}} of ℙ_{1}(ℚ_{p}) by disjoint compact open subsets U_{α}, and t_{α} is a sample point in U_{α}.

The connection between boundary measures and rigid analytic functions on H_{p} is given by the following lemma:

Lemma 5.8. (1) If μ is a measure on ℙ_{1}(ℚ_{p}), then

f_{μ}(z) = ∫_{ℙ_{1}(ℚ_{p})} (\frac{1}{z − t}) dμ(t)

is a rigid analytic function on H_{p}.

(2) If μ is a Γ-invariant measure on ℙ_{1}(ℚ_{p}), then f_{μ} belongs to S_{2}(Γ).

Proof. Part 1 follows directly from the definition of a rigid analytic function on H_{p}, since the integral defining f_{μ}(z) is expressed through (5.4) as a limit of rational functions having poles in ℙ_{1}(ℚ_{p}), and since the distribution and boundedness satisfied by μ ensures that this convergence is uniform on each affinoid in H_{p}. To show part 2, note that both \frac{1}{γz−t} and \frac{1}{z−γ^{−1}t} are rational functions of t having a unique simple pole at t = γz, the first with residue (−1) and the second with residue −(cz + d)^{−2}. Hence

\frac{1}{γz − t} − (cz + d)^{2} (\frac{1}{z − γ^{−1}t}) = C,

where C is a constant (possibly depending on z and γ but not on t). It follows that for any γ ∈ Γ:

f_{μ}(γz) = ∫_{ℙ_{1}(ℚ_{p})} (\frac{1}{γz − t}) dμ(t)
= (cz + d)^{2} ∫_{ℙ_{1}(ℚ_{p})} (\frac{1}{z − γ^{−1}t}) dμ(t) = (cz + d)^{2}f_{μ}(z).

The second equality uses the fact that the total measure of ℙ_{1}(ℚ_{p}) is 0, and the third exploits the invariance of μ under Γ. □

Denote by Meas(ℙ_{1}(ℚ_{p}), ℂ_{p})^{Γ} the space of all Γ-invariant measures on ℙ_{1}(ℚ_{p}). Theorem 5.9 (Schneider, Teitelbaum). The assignment μ ↦ f_{μ} is an isomor-

phism from Meas(ℙ_{1}(ℚ_{p}), ℂ_{p})^{Γ} to S_{2}(Γ).

Sketch of Proof. The construction of the inverse map can in fact be made explicit. To do this, note that an ordered edge e of T determines a subtree T_{e} of T , defined to be the largest connected subtree containing e and no other edge having s(e) as an endpoint. Let

Σ_{e} := r^{−1}(T_{e}) ⊂ H_{p},

and let Σ̄_{e} denote the closure of Σ_{e} in ℙ_{1}(ℂ_{p}). Finally let U_{e} := Σ̄_{e} ∩ ℙ_{1}(ℚ_{p}). The assignment e ↦ U_{e} sets up a correspondence between ordered edges of T and compact open balls in ℙ_{1}(ℚ_{p}). Given a rigid analytic function f on H_{p}, define a distribution μ_{f} on ℙ_{1}(ℚ_{p}) by the rule

μ_{f}(U_{e}) = lim_{j⟶∞} ∑_{x∈Σ̄_{e}} res_{x}(f_{j}(z)dz),

where f_{j} is a sequence of rational functions with poles outside A_{s(e)} which converges uniformly to f on A_{s(e)}. The residue theorem implies that μ_{f} satisfies a distribution relation, and the Γ-invariance of f(z)dz implies the corresponding Γ-invariance of μ_{f}. In effect, the boundary measure μ_{f} attached to f encodes the p-adic residues of f. The verification that f_{μ_{f}} = f is left to the reader as an exercise. (Cf. exercise 5.) □

The construction above leads to a useful description of Meas(ℙ_{1}(ℚ_{p}), ℂ_{p})^{Γ} which has the virtue of laying bare the simple combinatorial nature of this object.

Definition 5.10. A harmonic cocycle on T is a function c : E(T ) ⟶ ℂ_{p} satisfying

(1) c(e) = −c(ē), for all e ∈ E(T );

(2) ∑_{s(e)=v} c(e) = 0; ∑_{t(e)=v} c(e) = 0, for all v ∈ T_{0}.

5.3. p-ADIC LINE INTEGRALS 63

(Note that the second part of condition (2) is redundant since it follows from (1) and the first part of (2).)

A harmonic cocycle c gives rise to a distribution μ_{c} on ℙ_{1}(ℚ_{p}) by the rule

(5.5)  μ_{c}(U_{e}) = c(e).

Conversely, c can be recovered from the associated distribution by the rule above. Under this bijection the Γ-invariant distributions correspond to Γ-invariant harmonic cocycles on T .

The finite-dimensionality of Meas(ℙ_{1}(ℚ_{p}), ℂ_{p})^{Γ} (and hence, of S_{2}(Γ)) can now be seen to follow directly from the following lemma:

Lemma 5.11. The quotient T /Γ is a finite graph.

Proof. See exercise 6. □

The space Meas(ℙ_{1}(ℚ_{p}), ℤ)^{Γ} of Γ-invariant ℤ-valued distributions on ℙ_{1}(ℚ_{p}), corresponding to ℤ-valued harmonic cocycles, yields a natural integral structure on Meas(ℙ_{1}(ℚ_{p}), ℂ_{p})^{Γ}. Its image in S_{2}(Γ), denoted by S_{2}(Γ)^{ℤ}, plays a role somewhat similar to that of modular forms with integral Fourier coefficients in the theory of modular forms on Γ_{0}(N).

5.3. p-adic line integrals

Let f be a rigid analytic function on H_{p}. The goal of this section is to define a good notion of p-adic line integral attached to such an object. This line integral should be an expression of the form ∫_{τ_{1}}^{τ_{2}} f(z)dz ∈ ℂ_{p} obeying the same formal properties of the complex line integral, namely it should be linear in f and additive in the endpoints of integration:

∫_{τ_{1}}^{τ_{2}} f(z)dz + ∫_{τ_{2}}^{τ_{3}} f(z)dz = ∫_{τ_{1}}^{τ_{3}} f(z)dz, ∀τ_{1}, τ_{2}, τ_{3} ∈ H_{p}.

If f(z)dz = dF is an exact differential on H, one would clearly like to define

(5.6)  ∫_{τ_{1}}^{τ_{2}} f(z)dz = F (τ_{2}) − F (τ_{1}).

The equation dF = f(z)dz is sufficient to define F up to a locally constant, hence constant, function in the complex setting. A difficulty arises from the circumstance that in the p-adic context, there are plenty of locally constant functions which are not constant, because H_{p} is totally disconnected. This leads to an ambiguity in the choice of F.

One may try to remedy this situation by singling out a particularly natural antiderivative of certain functions. For example, among all possible choices of function F on ℂ_{p} such that dF = \frac{dz}{z}, the p-adic logarithm defined in the open disc in ℂ_{p} of radius 1 centered at 1 by the power series

log(1 − z) = ∑_{n=1}^{∞} \frac{z^{n}}{n}

is singled out by the property of being a homomorphism from this open disc (under multiplication) to ℂ_{p}. It is customary to choose an extension of the p-adic logarithm to all of ℂ_{p} (a “branch”)

log : ℂ_{p}^{×} ⟶ ℂ_{p}

by fixing some element π ∈ ℂ_{p}^{×} satisfying |π|_{p} < 1 and decreeing that log(π) = 0 (and requiring, of course, that log be a homomorphism on ℂ_{p}^{×}). The standard choice is obtained by taking π = p, but this may not always be the most natural choice in all situations.

Having fixed a choice of p-adic logarithm, one has, for each rational differential f(z)dz on ℙ_{1}(ℂ_{p}), a formal antiderivative of the form

F (z) = R(z) + ∑_{j=1}^{t} λ_{j} log(z − z_{j}),

where R is a rational function, the λ_{j} belong to ℂ_{p}, and the z_{j} are the poles of f(z)dz. This antiderivative is unique up to an additive constant, and hence equation (5.6) can be used to write down a well-defined line integral attached to f(z)dz. Extending this definition by continuity to all rigid analytic functions leads to the following definition.

Definition 5.12. Let f be a rigid analytic function on H_{p}. Assume that its associated boundary distribution μ_{f} is a measure. Then the p-adic line integral attached to f(z)dz is defined to be

(5.7)  ∫_{τ_{1}}^{τ_{2}} f(z)dz := ∫_{ℙ_{1}(ℚ_{p})} log (\frac{t − τ_{2}}{t − τ_{1}} ) dμ_{f}(t).

Rephrasing the dicussion which precedes definition 5.12, one can seek to justify this definition a posteriori through the following formal computation:

∫_{τ_{1}}^{τ_{2}} f(z)dz = ∫_{τ_{1}}^{τ_{2}} ∫_{ℙ_{1}(ℚ_{p})} (\frac{dz}{z − t}) dμ_{f}(t) = ∫_{ℙ_{1}(ℚ_{p})} log (\frac{t − τ_{2}}{t − τ_{1}} ) dμ_{f}(t).

Note that this definition depends crucially on the choice of the branch of the p-adic logarithm that was made in fixing a primitive for dz/z.

In the special case where f is attached to a ℤ-valued distribution, i.e., where f belongs to S_{2}(Γ)^{ℤ}, it can be useful to “work multiplicatively” by formally exponentiating (5.7) above and setting:

(5.8) × ∫_{τ_{1}}^{τ_{2}} f(z)dz = × ∫_{ℙ_{1}(ℚ_{p})} (\frac{t − τ_{2}}{t − τ_{1}} ) dμ_{f}(t) := lim_{C=\{U_{α}\}} ∏_{α} (\frac{t_{α} − τ_{2}}{t_{α} − τ_{1}} )^{μ_{f} (U_{α})} ,

where the limit in the last expression is taken over increasingly fine covers C = {U_{α}} of ℙ_{1}(ℚ_{p}) by disjoint compact open subsets, and the t_{α} are sample points in U_{α}. The multiplicative integral is related to its additive counterpart by the rule

∫_{τ_{1}}^{τ_{2}} f(z)dz = log (× ∫_{τ_{1}}^{τ_{2}} f(z)dz) ,

but it carries more information, as the p-adic logarithm is not injective. Also, it is more canonical than its additive counterpart since it does not depend on a choice of a branch of the p-adic logarithm. However, its definition relies crucially on the integrality of the boundary distribution atttached to f.

The multiplicative integral can be used to define a p-adic analogue of the classical Abel-Jacobi map

Φ_{AJ} : Div^{0}(H_{p}) ⟶ hom(S_{2}(Γ)^{ℤ}, ℂ_{p}^{×}) ≃ (ℂ_{p}^{×})^{g},

following [GvdP80]. The map Φ_{AJ} is defined by sending a divisor D of degree 0 to the functional

ω ↦ × ∫_{D} ω

which belongs to hom(S_{2}(Γ)^{ℤ}, ℂ_{p}^{×}). It can be shown that Φ_{AJ} maps the group of divisors on H_{p} which become 0 on H_{p}/Γ to a lattice Λ in this group, so that it gives rise, by passing to the quotient, to an analytic map of abelian varieties over ℂ_{p}:

Φ_{AJ} : Jac(H_{p}/Γ) ⟶ hom(S_{2}(Γ)^{ℤ}, ℂ_{p}^{×})/Λ.

In the next chapter, we will identify certain special arithmetic Γ ⊂ SL_{2}(ℚ_{p}) arising from p-units in orders in definite quaternion algebras, and use them to obtain p-adic Weil uniformisations

Div^{0}(H_{p}/Γ) ⟶ E(ℂ_{p})

attached to (modular) elliptic curves E over ℚ.

Further results

The book of Gerritzen and van der Put [GvdP80] provides a good introduction p-adic Schottky groups and to Mumford’s theory of p-adic uniformisation. An account of the p-adic Poisson kernel, as well as an extension of the theory presented here to modular forms of higher weight, can be found in the articles [Te90] and [Sch84].

Exercises

(1) Show that ℚ̄_{p} is not complete in the p-adic topology by exhibiting a Cauchy sequence in this field which does not converge.

(2) Show that the graph T is a tree all of whose vertices have valency equal to p + 1.

(3) Let G = PGL_{2}(ℚ_{p}) act by Möbius transformations on H_{p}, and by left translations on T . Let G_{0} = PGL_{2}(ℤ_{p}) be the maximal compact subgroup of G and let G_{1} be the subgroup of G_{0} represented by matrices which are uppertriangular modulo p.

(a) Show that G_{0} is precisely the group of elements in G which preserve the standard afinoid A ⊂ H_{p}, and that G_{1} is the stabiliser of the annulus W_{∞} in G_{0}.

(b) Show that G_{0} (resp. G_{1}) is the stabiliser in G of the standard vertex v_{o} (resp. of the ordered edge e_{∞}).

(c) Prove proposition 5.1.

(4) Check that the expression in (5.4) converges in ℂ_{p}.

(5) Complete the proof of theorem 5.9 by showing that f_{μ_{f}} = f where μ_{f} is the boundary distribution obtained in the proof of theorem 5.9 by taking the residues of f.

(6) Endow H_{p} with the p-adic topology, and T with the topology in which a subset U is said to be open if, for any vertex v ∈ U, the edges having v as endpoint also belong to U. Show that the reduction map r : H_{p} ⟶ T is continuous for these topologies. Conclude that the quotient graph T /Γ is a finite graph if H_{p}/Γ is compact.

CHAPTER 6 Rigid analytic modular parametrisations

The parametrisations alluded to in the title of this chapter arise from the fact that, for the primes p which divide N^{−}, the Shimura curves X_{N^{+},N^{−}} introduced in chapter 4 are equipped with p-adic uniformisations discovered by Čerednik and Drinfeld which ressemble the complex analytic descriptions that were already given. These uniformisations can be used to construct p-adic modular parametrisations for elliptic curves over ℚ whose conductors satisfy suitable conditions.

6.1. Rigid analytic modular forms on quaternion algebras

Let N be a positive integer and p a prime number dividing N exactly.

Definition 6.1. A p-admissible factorisation of N is a factorisation of the form N = pN^{+}N^{−}, where

(1) The integers p, N^{+}, and N^{−} are pairwise coprime;

(2) The integer N^{−} is square-free and the product of an odd number of primes.

To each p-admissible factorisation N = pN^{+}N^{−} is attached a subgroup Γ_{N^{+},N^{−}}^{(p)} of SL_{2}(ℚ_{p}) as follows. Let B be the definite quaternion algebra ramified precisely at the primes dividing N^{−}, together with the archimedean place. Such a quaternion algebra exists by proposition 4.1. It follows from proposition 4.4 that there is a unique Eichler ℤ[1/p]-order R of level N^{+} in B, up to conjugation by B^{×}. Since B is split at the prime p, it is possible to choose an identification

ι : B ⊗ ℚ_{p} ⟶ M_{2}(ℚ_{p}).

Letting R_{1}^{×} denote the group of elements in R of reduced norm 1, the group Γ_{N^{+},N^{−}}^{(p)} is then defined to be

Γ_{N^{+},N^{−}}^{(p)} := ι(R_{1}^{×}) ⊂ SL_{2}(ℚ_{p}).

It can be proved that Γ_{N^{+},N^{−}}^{(p)} acts on H_{p} with compact quotient so that the theory of rigid analytic modular forms developed in chapter 5 applies to forms on this group. In particular the space S_{2}(Γ_{N^{+},N^{−}}^{(p)})^{ℤ} is a finitely generated ℤ-module of

rank equal to the genus of the rigid analytic curve H_{p}/Γ_{N^{+},N^{−}}^{(p)}.

The space S_{2}(Γ_{N^{+},N^{−}}^{(p)}) is in addition endowed with the action of Hecke operators T_{ℓ} which are defined as before in terms of double coset decompositions. More precisely, for each prime ℓ which does not divide N, choose an element α_{ℓ} ∈ R of reduced norm ℓ, write the double coset Γ_{N^{+},N^{−}}^{(p)}α_{ℓ}Γ_{N^{+},N^{−}}^{(p)} as a disjoint union of left cosets

Γ_{N^{+},N^{−}}^{(p)}α_{ℓ}Γ_{N^{+},N^{−}}^{(p)} = ∪_{i=1}^{ℓ+1}γ_{i}Γ_{N^{+},N^{−}}^{(p)},

and set

T_{ℓ}f(z)dz := ∑_{i=1}^{ℓ+1} f(γ_{i}^{−1}z)d(γ_{i}^{−1}z).

Definition 6.2. An eigenform f is said to be normalised if it belongs to S_{2}(Γ_{N^{+},N^{−}}^{(p)})^{ℤ} and is not divisible by any integer > 1 in this group.

Note that, if f is normalised then so is −f and there seems to be no natural way of resolving this ambiguity in signs. Note also that this normalisation applies only to eigenforms with rational residues, whose asociated Hecke eigenvalues necessarily belong to ℤ.

6.2. The Čerednik-Drinfeld theorem

The rigid analytic curve H_{p}/Γ_{N^{+},N^{−}}^{(p)} shares some common features with the quotient H/Γ_{N^{+},N^{−}} that arose in decribing the complex uniformisation of the Shimura curve X_{N^{+},N^{−}p}, most notably the presence of a large ring of correspondences given by Hecke operators indexed by the primes ℓ that do not divide N. The following theorem of Čerednik and Drinfeld reveals that this analogy runs deeper:

Theorem 6.3. The rigid analytic quotient H_{p}/Γ_{N^{+},N^{−}}^{(p)} is isomorphic (as an algebraic curve over ℂ_{p}) to X_{N^{+},N^{−}p}.

Remark 6.4. More precisely, the quotient H_{p}/Γ_{N^{+},N^{−}}^{(p)} is identified with the ℂ_{p}-points of an algebraic curve X defined over ℚ_{p}. The curve X becomes isomorphic to X_{N^{+},N^{−}p} over the unramified quadratic extension of ℚ_{p}.

Theorem 6.3 asserts that the quotients H_{p}/Γ_{N^{+},N^{−}}^{(p)} and H/Γ_{N^{+},N^{−}p} decribe – over ℂ_{p} and ℂ respectively – the same algebraic curve, even though the groups Γ_{N^{+},N^{−}}^{(p)} and Γ_{N^{+},N^{−}p} are defined in terms of different quaternion algebras. It was first proved by Čerednik [Ce76] building on ideas of Ihara [Ih68], (the same which partly inspired the point of view adopted in chapter 9; cf also [Ih79]). A more conceptual proof relying on an interpretation of H_{p} as classifying certain formal groups was later given by Drinfeld [Dr76]. The standard reference for the Čerednik-Drinfeld theorem, and particularly for Drinfeld’s approach, is the exposition given in [BC92].

6.3. The p-adic Shimura-Taniyama-Weil conjecture

Let E be an elliptic curve of conductor N and let pN^{+}N^{−} be a p-admissible factorisation of N. Let Γ_{N^{+},N^{−}}^{(p)} ⊂ SL_{2}(ℚ_{p}) be the discrete subgroup arising from this factorisation.

Theorem 6.5 (“Rigid Shimura-Taniyama-Weil ”). There exists a unique (up to sign) normalised eigenform f ∈ S_{2}(Γ_{N^{+},N^{−}}^{(p)})^{ℤ} such that

T_{ℓ}(f) = a_{ℓ}(E)f,

for all ℓ |̸N.

Sketch of proof. By the Shimura-Taniyama-Weil theorem in the case of Shimura curves (theorem 4.13 of chapter 4) there is an eigenform g on H/Γ_{N^{+},N^{−}p} attached to E, which is unique up to multiplication by ℂ^{×}; but theorem 6.3 implies that the Hecke modules S_{2}(Γ_{N^{+},N^{−}}^{(p)}) and S_{2}(Γ_{N^{+},N^{−}p}) give rise to the same systems of eigenvalues for the Hecke operators, and with the same multiplicities, since they are both identified with the space of regular differentials on the curve X_{N^{+},N^{−}p} (over ℂ_{p} and ℂ respectively). Therefore there is associated to g a Hecke eigenform f ∈ S_{2}(Γ_{N^{+},N^{−}}^{(p)}) which is unique up to multiplication by an element of ℂ_{p}^{×}. Since the Hecke eigenvalues are integers, f can be rescaled to be a normalised eigenform in S_{2}(Γ_{N^{+},N^{−}}^{(p)})^{ℤ}, and it is then uniquely determined up to sign. □

Remark 6.6. It is not strictly necessary to invoke the Čerednik-Drinfeld theorem in this proof. In fact, the Jacquet-Langlands correspondence – which in the case at hand follows from an earlier result of Eichler – makes it possible to establish directly the existence of a Γ_{N^{+},N^{−}}^{(p)}-invariant measure μ_{f} (or, equivalently, of the associated harmonic cocycle c_{f}) attached to f, given the knowledge that E corresponds to a classical modular form on H/Γ_{0}(N).

Theorem 6.5 implies the existence of a rigid analytic Weil uniformisation

Φ_{N^{+},N^{−}}^{(p)} : Div^{0}(H_{p}/Γ_{N^{+},N^{−}}^{(p)}) ⟶ E(ℂ_{p})

which is defined as follows. Firstly, the function

Div^{0}(H_{p}) ⟶ ℂ_{p}^{×}, D ↦ × ∫_{D} f(z)dz

maps the group of divisors which become trivial in H_{p}/Γ_{N^{+},N^{−}}^{(p)} to a lattice in ℂ_{p}^{×} generated by an element q ∈ ℚ_{p}^{×}. The Tate curve E_{q} := ℂ_{p}^{×}/q^{ℤ} is related to E by an isogeny β which is defined over ℂ_{p}, and letting

Φ_{Tate} : ℂ_{p}^{×} ⟶ E_{q}(ℂ_{p})

be the Tate uniformisation one sets

(6.1)  Φ_{N^{+},N^{−}}^{(p)}(D) := β (Φ_{Tate} (× ∫_{D} f(z)dz)) .

To actually calculate this map only requires a knowledge of the residues attached to f(z)dz (and of the isogeny β). Section 6.5 illustrates through an example how this can sometimes be achieved in practice.

6.4. Complex multiplication, revisited

The notion of CM points on H_{p}/Γ_{N^{+},N^{−}}^{(p)}, and a p-adic analytic Heegner point construction, can be formulated for the rigid analytic modular parametrisation Φ_{N^{+},N^{−}}^{(p)} in a way which is pleasingly similar to the complex setting described in chapter 4.

Definition 6.7. The associated order of τ ∈ H_{p} is the set

O_{τ} := {γ ∈ R such that ι(γ)(τ) = τ} ∪ {0}.

As before, one can show that O_{τ} is isomorphic either to ℤ[1/p] or to a ℤ[1/p]- order in a quadratic subfield K of B. Also, O_{τ} is equipped with a canonical inclusion into ℂ_{p} which to γ associates the eigenvalue of ι(γ) acting on the column vector (τ, 1).

Remark 6.8. Since B is a definite quaternion algebra, note that the field K is necessarily a quadratic imaginary field.

Definition 6.9. A point τ ∈ H_{p}/Γ_{N^{+},N^{−}}^{(p)} is called a CM point if its associated order is a ℤ[1/p]-order in a quadratic imaginary field.

Fixing such an order O, we write

CM(O) := {τ ∈ H_{p}/Γ_{N^{+},N^{−}}^{(p)} such that O_{τ} = O.}

Lemma 6.10. Let O be an order of discriminant prime to N, with fraction field K. Then CM(O) is non empty if and only if

(1) K is a quadratic imaginary extension of ℚ;

(2) all the primes dividing N^{−}p are inert in K;

(3) all the primes dividing N^{+} are split in K.

Proof. The proof is almost identical to that of lemma 4.17 of chapter 4, except that the roles of the places p and ∞ are interchanged. The reader may find it instructive to fill in the details of the proof. (See exercise 5.) □

Since p is inert in K, the Picard group of rank one projective modules over O is equal to the Picard group of the order O ∩ O_{K} of K. Let H be the ring class field of K attached to O. Note that its conductor is prime to p. Hence p (viewed as a prime of K) splits completely in H/K.

The following can be viewed as a p-adic variant of the main theorem of complex multiplication for Shimura curves as formulated in theorem 4.16 of chapter 4.

Theorem 6.11. Let O be a ℤ[1/p]-order in a quadratic imaginary field K, of discriminant prime to N, and let H/K be the associated ring class field, viewed as a subfield of ℂ_{p}. Then

Φ_{N^{+},N^{−}}^{(p)}(Div^{0}(CM(O))) ⊂ E(H).

Idea of proof. Under the moduli interpretation of X_{N^{+},N^{−}p} given by Drinfeld’s theory, points τ ∈ CM(O) correspond to points in X_{N^{+},N^{−}p} which are moduli of abelian surfaces with endomorphism by M_{2}(O_{0}). These surfaces are isomorphic to a product A × A of an elliptic curve A with CM by O_{0}, with itself. Given this fact, the result follows from the usual theory of complex multiplication. Further details are explained in [BD98]. □

6.5. An example

We now describe an example to illustrate how the rigid analytic modular parametrisations can be computed and used to find algebraic points on elliptic curves, in practice. Let E be the elliptic curve of conductor N = 14 given by the minimal Weierstrass equation

(6.2)  y^{2} + xy + y = x^{3} + 4x − 6.

The triple (p, N^{+}, N^{−}) = (7, 1, 2) is a 7-admissible factorisation of 14.

Let B denote the algebra of rational Hamilton quaternions

B = ℚ + ℚi + ℚj + ℚk,

a quaternion algebra over ℚ which is ramified precisely at 2 and ∞. Let R_{0} be Hurwitz’s ring of integral quaternions:

R_{0} = ℤ + ℤi + ℤj + ℤω,

where ω = \frac{1+i+j+k}{2} . The main facts about R_{0} that will be needed are

Theorem 6.12 (Hurwitz). The ring R_{0} is the unique maximal order in B up to conjugation in B^{×}. Every left R_{0}-ideal is principal.

Proof. See for example [Gr87], §1 and 2. □

In particular, the ring R := R_{0}[1/7] is the unique maximal ℤ[1/7]-order in B up to conjugation in B^{×}. Fix an isomorphism ι of B_{7} := B ⊗ ℚ_{7} with M_{2}(ℚ_{7}) which has the property that ι^{−1}(M_{2}(ℤ_{7})) = R_{0} ⊗ ℤ_{7}. To fix ideas, we may take

ι(i) = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , ι(j) = ( \begin{matrix}ρ & ρ + 1 \\ ρ + 1 & −ρ \end{matrix} ) , ι(k) = ( \begin{matrix}ρ + 1 & −ρ \\ −ρ & −ρ − 1 \end{matrix} ) ,

where ρ = lim_{n⟶∞} 2^{7^{n}} is a primitive cube root of unity in ℤ_{7}.

Let T_{7} = T = T_{0} ∪ T_{1} denote the Bruhat-Tits tree of PGL_{2}(ℚ_{7}) introduced in chapter 5, and let Γ = ι(R_{1}^{×}) ⊂ PSL_{2}(ℚ_{7}). This group acts naturally on T ; the following lemma yields a precise description of the quotient graph T /Γ.

Lemma 6.13. The group Γ has precisely two orbits acting on T_{0}. Likewise the set Γ\T_{1} also has cardinality two.

Proof. Recall the distinguished vertex v_{o} attached to the standard lattice ℤ_{7}^{2}. The map γ ↦ γv_{0} identifies the coset space PGL_{2}(ℚ_{7})/PGL_{2}(ℤ_{7}) with T_{0}. Hence Γ\T_{0} can be identified with

(6.3)  ι(R_{1}^{×})\PGL_{2}(ℚ_{7})/PGL_{2}(ℤ_{7}) = R_{1}^{×}\B_{7}^{×}/R_{0,7}^{×}ℚ_{7}^{×},

where as before R_{0,7} := R_{0} ⊗ ℤ_{7} and R_{1}^{×} denotes the group of elements of reduced norm 1 in R. Since R_{1}^{×}\R^{×}/R_{0,7}^{×}ℚ_{7}^{×} has cardinality two – with cosets consisting of elements of R^{×} whose determinant is a unit of ℤ_{7} multiplied by an even (odd) power of 7 – the set on the right of (6.3) admits a natural two-to-one map to the coset space

R^{×}\B_{7}^{×}/R_{0,7}^{×}ℚ_{7}^{×}.

Noting that R = R_{0}[1/7] and that the prime 7 is split in the quaternion algebra B, theorem 4.5 implies that this coset space is equal to

(6.4)  B^{×}\B̂^{×}/R̂_{0}^{×}.

But by the discussion preceeding the statement of theorem 4.5 of chapter 4, the double coset space of (6.4) is in bijection with the set of conjugacy classes of maximal orders in B. This set has cardinality one by theorem 6.12 above. In fact, the quotient Γ\T_{0} admits a simple description: one orbit consists of vertices which are at an even distance from the vertex v_{o}, and the second orbit consists of vertices which are at an odd distance from v_{0}. To study Γ\T_{1}, note that any edge of T is equivalent under Γ to an edge having v_{o} as endpoint, i.e., to one of the edges e_{0}, … e_{6}, e_{∞} naturally indexed by ℙ_{1}(𝔽_{7}) that were introduced in chapter 5. Since the stabiliser of v_{0} in Γ is equal to ι(R_{0}^{×}), it follows that Γ ⊂ T_{1} is naturally identified with the orbit space

ι(R_{0}^{×})\ℙ_{1}(𝔽_{7}).

The group R_{0}^{×} is a group of order 24, generated by i, j, and ω, and ι(R_{0}^{×}) acts on ℙ_{1}(𝔽_{7}) in the obvious way. This action breaks ℙ_{1}(𝔽_{7}) into two orbits of the same cardinality, {∞, 0, 2, 3} and {1, 4, 5, 6}. Hence Γ\T_{1} has two orbits, E_{1} and E_{2}, defined by

e ∈ \{ \begin{matrix}E_{1} if e is Γ-equivalent to e_{0}, e_{2}, e_{3}, or e_{∞}. \\ E_{2} if e is Γ-equivalent to e_{1}, e_{4}, e_{5}, or e_{6}. \end{matrix}
□

Lemma 6.13 allows the determination of the full set of Γ-invariant harmonic cocycles on T . Firstly, such a cocycle c is completely determined by its values on the ordered edges e′_{0}, …, e′_{∞} having the same endpoints as e_{0}, …, e_{∞} respectively, and ordered by setting s(e′_{j}) = v_{o}. Secondly, the harmonicity and Γ-invariance conditions force the relations

c(e′_{0}) = c(e′_{2}) = c(e′_{3}) = c(e′_{∞}) = −c(e′_{1}) = −c(e′_{4}) = −c(e′_{5}) = −c(e′_{6}).

Conversely, there is a non-zero Γ-invariant cocycle satisfying the relations above. Normalise it so that c(e′_{0}) = 1. The ℤ-module S_{2}(Γ)^{ℤ} has rank one, and is generated by the rigid analytic modular form f whose residues are given by the harmonic cocycle c.

The first few Hecke operators acting on c can be evaluated explicitly yielding the following list of Hecke eigenvalues:

\begin{matrix}ℓ & 3 & 5 & 11 & 13 & 17 & 19 & 23 \\ a_{ℓ}(f) & −2 & 0 & 0 & −4 & 6 & 2 & 0 \end{matrix}

It can be checked (using Pari, or consulting the tables in [Cr97]) that a_{ℓ}(f) is equal to the coefficient a_{ℓ}(E) attached to the (unique, up to isogeny) elliptic curve E of conductor 14 in (6.2), giving a partial numerical verification of theorem 6.5 in this case.

The explicit determination of the harmonic cocycle c encoding the residues of f makes it possible to evaluate numerically the p-adic line integrals attached to f. To illustrate how this can be combined with the theory of complex multiplication to construct algebraic points on E, let K = ℚ(\sqrt{−11}) be the imaginary quadratic field of smallest discriminant in which both 2 and 7 are inert, and let ω_{11} = (1+\sqrt{−11})/2 denote a generator for its ring of integers. After fixing an embedding Ψ : O_{K} ⟶ R and letting τ and τ′ denote the two fixed points in H_{7} of ιΨ(K^{×}), a computer calculation (carried out to 5 digits of 7-adic accuracy) shows that

J = × ∫_{τ′}^{τ} f(z)dz = 13149 + 2287ω_{11} (mod 7^{5}).

The image of this integral under the Tate uniformisation attached to E over ℚ_{7} is

(6.5) Φ_{Tate}(J) = (10696, 6528 + 9861ω_{11}) ≡ (\frac{7}{11}, \frac{−(41 + 116ω_{11})}{121}) (mod 7^{5}).

The latter expression is a global point on E(K).

Although this calculation does not actually prove that equality holds in (6.5), it does illustrate how the ideas presented in the last few chapters can be used in practice to find algebraic points on elliptic curves by p-adic analytic methods. (A second example is detailed in exercise 3.) Calculations of this sort form the basis form the basis for the conjecture of chapter 9 yielding Heegner systems attached to real quadratic fields.

6.6. p-ADIC L-FUNCTIONS, D’APRÈS SCHNEIDER-IOVITA-SPIESS 73

6.6. p-adic L-functions, d’après Schneider-Iovita-Spiess

Recall from chapter 2 that if f ∈ S_{2}(Γ_{0}(N)) is a normalised eigenform, then one has the corresponding integral representation for its L-series:

(6.6)  Λ(f, s) := (2π)^{−s}Γ(s)L(f, s) = ∫_{0}^{∞} f(iy)y^{s−1}dy

which yields the analytic continuation and functional equation for L(f, s). If f is a modular form on a quaternion algebra it is natural to ask for a similar expression describing L(f, s) as a Mellin transform attached to f. On a superficial level, Hecke’s construction does not generalise, since it relies on the notion of Fourier expansions which are meaningful only for modular forms attached to the split quaternion algebra M_{2}(ℚ).

To arrive at the desired generalisation, it is helpful to view the Mellin transform of f as an integral along the real points of a torus arising from the split quadratic algebra ℚ × ℚ ⊂ M_{2}(ℚ). More precisely, let K ≃ ℚ × ℚ be the quadratic algebra of diagonal matrices in M_{2}(ℚ). This algebra is optimally embedded with respect to the subalgebra M_{0}(N), in the sense that K ∩ M_{0}(N) ≃ ℤ × ℤ is the maximal order of K. The fixed points of K^{×} acting by Möbius transformations on H^{∗} are 0 and ∞. Let z be a rational function with divisor (0) − (∞). Then Λ(f, s) is simply the integral of this function, raised to the power s − 1, betwen the two fixed points 0 and ∞, against a measure naturally associated to f. By analogy, since a non-split quaternion algebra does not contain the algebra ℚ × ℚ as a quadratic subalgebra, one might attempt to define L(f, s) as a Mellin transform of f along the (real, or p-adic) points of a global torus in B^{×} coming from the units of a quadratic subalgebra K ⊂ B. Over the reals, this does not seem to lead to a useful analytic object, since the group (K ⊗ ℝ)^{×}/O_{K}^{×}ℝ^{×} is a compact group, isomorphic to a circle, which has discrete Pontryagin dual. In the p-adic setting, however, it is natural to consider the space of continuous ℂ_{p}-valued characters of the compact p-adic group (K ⊗ ℚ_{p})^{×}/O_{K}^{×}ℚ_{p}^{×}. This “p-adic dual” is not discrete and is in fact endowed with a non-trivial topological and p-adic analytic structure.

To simplify our discussion, we will only treat the case where the prime p is inert in K/ℚ. (The case where p is split in K is treated in exercise 4.) In that case the group ι(K_{p}^{×}) acting on H_{p} has two fixed points α and ᾱ which belong to H_{p} ∩ K

and are interchanged under the action of Gal(K_{p}/ℚ_{p}). Let (\frac{z−α}{z−ᾱ}) be a rational function with divisor (α) − (ᾱ). It is tempting to define, by analogy with (6.6)

(6.7)  L_{p}(f, s)“ =^{?} ” ∫ α ; ᾱ f(z) (\frac{z − α}{z − ᾱ})^{s−1} dz.

If s belongs to ℤ, then ω_{g} = f(z) (\frac{z−α}{z−ᾱ})^{s−1} dz is a rigid analytic differential form on H_{p} − {α, ᾱ} with associated boundary distribution on ℙ_{1}(ℚ_{p}) given by

μ_{g}(t) = μ_{f}(t) (\frac{t − α}{t − ᾱ})^{s−1} .

Although the integral appearing in (6.7) does not converge, if we simply ignore the terms arising from the end points α and ᾱ we obtain the following candidate for a generalisation of (6.6), in which the argument of K has been inserted to emphasize the essential dependence of this definition on the chosen quadratic subalgebra:

(6.8)  L_{p}^{naive}(f, K, s) := ∫_{ℙ_{1}(ℚ_{p})} log (\frac{t − α}{t − ᾱ}) (\frac{t − α}{t − ᾱ})^{s−1} dμ_{f}(t).

This expression does converge, and even interpolates to s ∈ ℤ_{p}. Moreover, it is the derivative with respect to s of an even simpler expression:

Definition 6.14. The Schneider-Iovita-Spiess L-function attached to f and K is the expression

L_{p}(f, K, s) := ∫_{ℙ_{1}(ℚ_{p})} (\frac{t − α}{t − ᾱ})^{s−1} dμ_{f}(t).

Remark 6.15. This type of definition was proposed by Schneider in [Sch84], with the role of K being played by the local split algebra ℚ_{p} × ℚ_{p} embedded in B_{p} ≃ M_{2}(ℚ_{p}). Since this embedding has no global origin it is unclear what relation (if any) Schneider’s construction bears with classical special values and with other types of p-adic L-functions. The idea of requiring that K arise from a global quadratic subalgebra of B (thus giving more rigidity to Schneider’s construction) was arrived at independently by Iovita and Spiess. (Cf. for example [BDIS02] for a more thorough discussion.)

The p-adic Mellin transform L_{p}(f, K, s) is a more tractable object than its classical counterpart of (6.6), as the following theorem illlustrates. Let E^{K} be the elliptic curve over ℚ obtained by twisting E by the quadratic character attached to K.

Theorem 6.16. The order of vanishing of L_{p}(f, K, s) is greater than both the rank of E(ℚ) and the rank of E^{K}(ℚ).

This theorem lies beyond the scope of our discussion. A proof, as well as a conjecture describing the precise order of vanishing of L_{p}(f, K, s) and the philosophy underlying it, is given in [BD03].

Remark 6.17. In contrast, the mechanism whereby large rank of E(ℚ) forces extra vanishing in the complex L-function L(f, s) is not understood at all. To take stock of the ignorance surrounding this question, note that the following remains open: Does there exist a curve E for which ord_{s=1}L(E, s) > 3? On the other hand theorem 6.16 yields examples of elliptic curves for which one can show that ord_{s=1}L_{p}(E, K, s) > 23, thanks to the elliptic curves curves of large rank that have been produced by Mestre and his school.

6.7. A Gross-Zagier formula

Suppose for simplicity that K has class number one and that

Ψ(K) ∩ R = Ψ(O), where O = O_{K}[1/p].

Let f be a modular eigenform on H_{p}/Γ_{N^{+},N^{−}}^{(p)} with integer residues and let E be the strong Weil curve associated to it. The following gives an arithmetic interpretation of the special value L′_{p}(f, K, 1) in the spirit of the Gross-Zagier formula (theorem 3.20 of chapter 3).

Theorem 6.18. There exists a global point P_{K} ∈ E(K) such that

βΦ_{Tate}(exp(L′_{p}(f, K, 1))) = P_{K} (mod E(K_{p})_{tors}).

Proof. Let α and ᾱ be the fixed points of ιΨ(K^{×}) acting on H_{p}, and let

P_{K} = Φ_{N^{+},N^{−}}^{(p)}((α) − (ᾱ)) ∈ E(ℂ_{p}).

Since (α) − (ᾱ) belongs to Div^{0}(CM(O)), and K is its own Hilbert class field, it follows from theorem 6.11 that P_{K} belongs to E(K). On the other hand,

L′_{p}(f, K, 1) = ∫_{ℙ_{1}(ℚ_{p})} log (\frac{t − α}{t − ᾱ}) dμ_{f}(t) = ∫ α ; ᾱ f(z)dz.

Hence

exp(L′_{p}(f, K, 1)) = × ∫ α ; ᾱ f(z)dz (mod (K_{p}^{×})_{tors}).

Applying the Tate uniformisation and the isogeny β to both sides yields the desired result, in light of (6.1). □

Further results

The first proof of the Čerednik-Drinfeld theorem was obtained by Čerednik [Ce76] building on work of Ihara [Ih68]. Drinfeld’s proof [Dr76] which also gives a moduli interpretation to the p-adic upper half plane is explained in [BC92].

The numerical example studied in section 6.5 is taken from [BD96]. More details and the general context for the Schneider-Iovita-Spiess approach to p-adic L-functions are explained in [BD01] and [BDIS02]. The proof of theorem 6.16 (which follows from a more general Iwasawa-theoretic “main conjecture” for certain p-adic L-functions which include the rigid analytic L-function discussed above as a special case) is given in [BD03].

Exercises

(1) Let A be an abelian surface with quaternionic multiplication by R, so that R acts on A by endomorphisms which are defined over ℚ. We say that A has complex multiplication by an order O in a quadratic field K if there is an inclusion O ⊂ End_{R}(A), where End_{R}(A) denoted the algebra of endomorphisms of A which commute with R. Show that if A has complex multiplication by O, it is isomorphic over ℚ to a product E × E of elliptic curves with complex multiplication by O.

(2) Let B be the algebra of Hamilton’s quaternions over ℚ, and let

R_{0} = ℤ[i, j, k, \frac{1 + i + j + k}{2} ]

be Hurwitz’s maximal order. For any odd prime p, let R = R_{0}[1/p], choose an identification ι of B ⊗ ℚ_{p} with M_{2}(ℚ_{p}), and set Γ = ι(R_{1}^{×}). Show that Γ\T_{p} is a graph with two vertices. Give an example of a group Γ ⊂ B^{×} for which the quotient Γ\T_{p} has more than two vertices.

(3) There are two isogeny classes of elliptic curves of conductor 26, which are labelled 26A and 26B in the Cremona tables; their minimal Weierstrass equations are

26A : y^{2} + xy + y = x^{3} − 5x − 8, 26B : y^{2} + xy + y = x^{3} − x^{2} − 3x + 3.

Let B and R_{0} be as in section 6.5, and let R = R_{0}[1/13]. Choose an embedding ι of B_{13} into M_{2}(ℚ_{13}) by setting

ι(i) = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , ι(j) = ( \begin{matrix}ρ & ρ + 1 \\ ρ + 1 & −ρ \end{matrix} ) , ι(k) = ( \begin{matrix}ρ + 1 & −ρ \\ −ρ & −ρ − 1 \end{matrix} ) ,

where ρ = lim_{n⟶∞} 3^{13^{n}} is a primitive cube root of unity in ℤ_{13}. Let Γ = ι(R_{1}^{×}) and let T_{13} = T_{0} ∪ T_{1} be the Bruhat-Tits tree of PGL_{2}(ℚ_{13}).

(a) Show that there are exactly two orbits for the action of Γ on T_{0}, and that there are three orbits for the action of Γ on T_{1}. More precisely show that these three orbits E_{1}, E_{2} and E_{3} are characterised by the property

e ∈ \{ \begin{matrix}E_{1} if e is Γ-equivalent to e_{∞}, e_{0}, e_{3}, or e_{4}, \\ E_{2} if e is Γ-equivalent to e_{1}, e_{7}, e_{11}, or e_{12}, \\ E_{3} if e is Γ-equivalent to e_{2}, e_{5}, e_{6}, e_{8}, e_{9}, or e_{10}. \end{matrix}

Draw the quotient graph T_{13}/Γ.

(b) Show that the space of Γ-invariant harmonic cocycles on the tree T_{13} is two-dimensional, by showing that such a harmonic cocycle is completely determined by its values on the ordered edges e′_{∞}, e′_{1} and e′_{2} associated to e_{∞}, e_{1} and e_{2} and having v_{o} as source.

(c) Compute the action of the Hecke operator T_{3} on a basis of the space of Γ-invariant harmonic cocycles on T_{13}. Show that a basis of eigenvectors for this action is given by the harmonic cocycles c_{1} and c_{2}, where

c_{1}(e′_{∞}) = 1,  c_{1}(e′_{1}) = −1,  c_{1}(e′_{2}) = 0.
c_{2}(e′_{∞}) = 3,  c_{2}(e′_{1}) = 3,  c_{2}(e′_{2}) = −4.

(d) Let f_{j} be the rigid analytic modular form in S_{2}(Γ) attached to the cocycle c_{j}. Show that f_{1} is associated to the elliptic curve 26A and that f_{2} to the elliptic curve 26B in the correspondence of theorem 6.5.

(e) Use the theory of complex multiplication to find points on these elliptic curves defined over ℚ(\sqrt{−11}) by evaluating the appropriate 13-adic integral.

(4) This exercise studies the Schneider-Iovita-Spiess p-adic L-function L_{p}(f, K, s) in the case where p is split in the quadratic imaginary field K ⊂ B.

(a) Show that the two fixed points α and ᾱ of ι(K^{×}) acting on ℙ_{1}(ℂ_{p}) by Möbius transformations belong to the boundary ℙ_{1}(ℚ_{p}) of H_{p}.

(b) If O is a ℤ[1/p]-order of K, show that O_{1}^{×} is of rank one. Let ϵ be a generator for this group.

(c) Show that the subgroup of Γ = ι(R_{1}^{×}) which fixes α and ᾱ is of rank one. Let γ ∈ SL_{2}(ℚ_{p}) be a generator of this group.

(d) Fix an embedding of K into ℚ_{p} and let ϵ_{p} be the image of ϵ in ℚ_{p}. Let log : ℚ_{p}^{×} ⟶ ℚ_{p} be a branch of the p-adic logarithm, chosen so that log(ϵ_{p}) = 0. Define x^{s} = exp(s log(x)). Show that the measure

μ(t) = (\frac{t − α}{t − ᾱ})^{s−1} dμ_{f}(t)

is well-defined on ℙ_{1}(ℚ_{p}) − {α, ᾱ} and is invariant under the action of γ.

(e) Let D be a fundamental region for the action of γ on ℙ_{1}(ℚ_{p}) − {α, ᾱ}. Show that the expression

L_{p}(f, K, s) := ∫_{D} (\frac{t − α}{t − ᾱ})^{s−1} dμ_{f}(t)

converges for s ∈ ℤ_{p} and is independent of the choice of D.

(f) Show that

L′_{p}(f, K, 1) = ∫_{τ}^{γτ} f(z)dz,

where τ is any element of H_{p}. Show that this expression is a rational multiple of log(q), where q is the Tate period attached to E_{/ℂ_{p}}.

(5) Prove lemma 6.10.

CHAPTER 7 Totally real fields

To a modular elliptic curve E over ℚ of conductor N three different types of modular parametrisation have been attached so far:

• The classical modular curve parametrisation

Φ_{N} : H/Γ_{0}(N) ⟶ E(ℂ)

of chapter 2;

• The Shimura curve parametrisation

Φ_{N^{+},N^{−}} : Div^{0}(H/Γ_{N^{+},N^{−}}) ⟶ E(ℂ)

of chapter 5;

• The rigid analytic parametrisation

Φ_{N^{+},N^{−}}^{(p)} : Div^{0}(H_{p}/Γ_{N^{+},N^{−}}^{(p)}) ⟶ E(ℂ_{p})

arising from the theory of Čerednik and Drinfeld.

In all cases it has been possible to develop a notion of CM points on the appropriate (complex or p-adic) upper half-plane, whose image under the corresponding modular parametrisation yields points on E defined over abelian extensions of certain quadratic imaginary fields.

In this chapter we turn to the question of what happens if the ground field ℚ is replaced by a more general number field F. This question is not motivated merely by the pursuit of generalisation. Rather, an examination of the number field case suggests a broader perspective on modular parametrisations, a perspective that will be germane to the conjectural theory of Heegner points attached to real quadratic fields presented in chapter 9.

7.1. Elliptic curves over number fields

Let E be an elliptic curve over over a number field F. The conductor of E is now an integral ideal N of O_{F}. Let |𝔫| denote the norm of the (fractional or integral) ideal 𝔫. For each prime 𝔭 of F, define an integer a(𝔭) by the rule:

(7.1)  a(𝔭) = 1 + |𝔭| − #E(O_{F} /𝔭), if 𝔭 |̸N,

and a(𝔭) = 0 (resp. 1, −1) if E has additive (resp split, non-split multiplicative) reduction at 𝔭. To f and N is associated a space of “automorphic forms”

S_{2}(N) ⊂ L^{2}(GL_{2}(F)\GL_{2}(𝔸_{F})),

equipped as in the case where F = ℚ with an action of Hecke operators indexed by the primes of F.

We will not go here into the details of the precise definition of this space of automorphic forms, as this would take us too far afield.

A generalisation of the Shimura-Taniyama-Weil conjecture predicts that there exists an automorphic form f ∈ S_{2}(N) such that

T_{ℓ}f = a_{ℓ}(E)f for all primes ℓ of F not dividing N.

Our hope (formulated vaguely here, and made more precise in the next three chapters) is that the existence of such an f provides a handle on the arithmetic of E_{/F}, useful not just in proving analyticity properties for the L-function L(E/F, s) and algebraicity results for its special values, but also in constructing algebraic points on E, as in the prototypical case where F = ℚ. Essential difficulties arise because there is no systematic generalisation of the notion of modular curve in the number field setting. For instance, here is what happens in the simplest (and perhaps most interesting) case where F is a quadratic field:

1. In the case (discussed more thoroughly in the next sections) where F is real, the form f is a Hilbert modular form defined on a Hilbert modular surface. An analogue of the Eichler-Shimura construction can often be given by applying the Jacquet-Langlands correspondence and passing to a form on an appropriate Shimura curve, but this is not always possible: for instance it fails when f is attached to an elliptic curve with everywhere good reduction over F. The theory presented in this and the next chapter proposes a construction of algebraic points on E directly in terms of the periods of f which would apply to all cases where F is real quadratic.

2. If F is imaginary quadratic, the form f corresponds to a differential form on the upper-half space ℂ × ℝ^{>0} invariant under the action of a discrete arithmetic subgroup Γ ⊂ SL_{2}(ℂ) (a so-called Bianchi group, cf. [EGM98]). By obvious dimension considerations, the hyperbolic three-manifold on which f is defined cannot correspond to the complex points of an algebraic variety and one is at a loss to propose, even conjecturally, an appropriate generalisation of the Eichler-Shimura construction in this setting (in spite of [HST93], [T94] which succeeds in attaching to f a compatible system of ℓ-adic Galois representations having the same properties as those attached to E). The case where F is imaginary quadratic is not touched upon at all in these notes, although a natural extension of the theory proposed in chapter 9 coud be expected to yield conjectural p-adic analytic constructions of non-trivial Heegner systems attached to elliptic curves defined over such fields.

7.2. Hilbert modular forms

We specialise the discussion – and are correspondingly more precise – in the case where F is a totally real field of degree n + 1 over ℚ. Let ι_{0}, …, ι_{n} be the distinct real embeddings of F. The (n + 1)-tuple of embeddings ι = (ι_{0}, …, ι_{n}) induces embeddings

F ⟶ ℝ^{n+1}, M_{2}(F) ⟶ M_{2}(ℝ)^{n+1}, PSL_{2}(F) ⟶ PSL_{2}(ℝ)^{n+1},

which will all be denoted by ι by a slight abuse of notation. If γ is an element of F, or a matrix with entries in F, it will occasionally be convenient to write γ_{j} or ι_{j}(γ) for the image of γ under the j-th real embedding of F.

Note that the group SL_{2}(ℝ)^{n+1} acts naturally on the product H_{0}×H_{1}×⋯×H_{n} of (n + 1) copies of the complex upper half-plane indexed in the same way as the places of F. Given \underline{τ} = (τ_{0}, …, τ_{n}) ∈ H^{n+1} and \underline{γ} = (γ_{0}, …, γ_{n}) ∈ SL_{2}(ℝ)^{n+1}, we will write \underline{γ}\underline{τ} = (γ_{0}τ_{0}, …, γ_{n}τ_{n}).

Lemma 7.1. The group Γ = ι(PSL_{2}(O_{F})) acts discretely on H^{n+1}.

Proof. Note that the image of O_{F} under ι is discrete in ℝ^{n+1}. Hence so is the image of M_{2}(O_{F}) in M_{2}(ℝ^{n+1}), and therefore the group Γ := ι(PSL_{2}(O_{F})) is a discrete subgroup of PSL_{2}(ℝ)^{n+1}. Since H^{n+1} is the quotient of PSL_{2}(ℝ)^{n+1} by the compact subgroup O_{2}(ℝ)^{n+1} the result follows by the same argument that was used in the proof of lemma 4.6 of chapter 4. □

In order to work as much as possible using classical rather than adèlic notations, it is useful to make the following simplifying assumption:

Hypothesis 7.2. The field F has narrow class number one.

Thus it is assumed that every ideal of O_{F} is principal and has a totally positive generator, so that there exist units ϵ_{0}, ϵ_{1}, …, ϵ_{n} with the property that

ι_{k}(ϵ_{j}) > 0, if k ≠ j, ι_{j}(ϵ_{j}) < 0.

Definition 7.3. A Hilbert modular form of weight (k_{0}, …, k_{n}) on Γ is a holomorphic function

f : H_{0} × ⋯ × H_{n} ⟶ ℂ

such that

f(\underline{γ}\underline{τ}) = (c_{0}τ_{0} + d_{0})^{k_{0}} ⋯ (c_{n}τ_{n} + d_{n})^{k_{n}}f(\underline{τ}) for all \underline{γ} = (( \begin{matrix}a_{j} & b_{j} \\ c_{j} & d_{j} \end{matrix} ))_{j} ∈ Γ.

Of special relevance to elliptic curves is the case where

Γ = ι(Γ_{0}(N)), where Γ_{0}(N) := \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(O_{F}) such that N|c\} ,

and (k_{0}, …, k_{n}) = (2, …, 2). The Hilbert modular form f is then said to be of parallel weight 2 with respect to Γ, and the space of all such forms is denoted S_{2}(N), as in the case where F = ℚ.

The theory of Hilbert modular forms exhibits many features analogous to the classical situation where F = ℚ. Most notable perhaps is the existence of Fourier

expansions. More precisely, since every matrix of the form ( \begin{matrix}1 & λ \\ 0 & 1 \end{matrix} ) with λ ∈ O_{F}

belongs to Γ_{0}(N), we have:

f(\underline{τ} + \underline{λ}) = f(\underline{τ}), for all λ ∈ O_{F} .

Therefore f is periodic under translation by elements of ι(O_{F}) and admits a Fourier expansion

f(\underline{τ}) = ∑_{ν∈O_{F}} a_{ν}(f)e^{2πi(\underline{τ}·(\underline{ν}/\underline{d}))},

where d is a totally positive generator of the different of F, and

\underline{τ} · (\underline{ν}/\underline{d}) := τ_{0}ν_{0}/d_{0} + τ_{1}ν_{1}/d_{1} + ⋯ + τ_{n}ν_{n}/d_{n}.

The Fourier coefficients a_{ν}(f), indexed by elements ν of O_{F}, satisfy the following basic properties:

• The Koecher principle implies that, if ι_{j}(ν) < 0 for some j, then a_{ν}(f) = 0: in other words, a_{ν}(f) is non-zero only on totally positive elements. (Cf. for example [Gar90] §1.4.)

• The invariance of f under the matrices of the form ( \begin{matrix}ϵ & 0 \\ 0 & ϵ^{−1} \end{matrix} ), where

ϵ is a unit in O_{F}, implies that a_{ν}(f) = a_{ϵ^{2}ν}(f) for all such units. Under the condition that F has narrow class number one, every totally positive unit of O_{F} is a square of a unit and hence a_{ν}(f) (for ν totally positive) depends only on the ideal generated by ν, not on ν itself: thus ν ↦ a_{ν}(f) can be viewed as a function on the ideals of O_{F}.

As in the definitions of chapter 2, it is possible to define the Fourier expansion of f at an arbitrary cusp γ^{−1}∞ (with γ ∈ SL_{2}(O_{F})) by expanding f|_{γ} as a Fourier series. Let a_{0}^{γ} denote the constant coefficient in this expansion.

Definition 7.4. If a_{0}^{γ}(f) = 0, for all γ ∈ SL_{2}(O_{F}), then f is said to be a cusp form.

We shall denote by S_{2}(N) the space of cusp forms of parallel weight (2, …, 2) on (H_{0} × ⋯ × H_{n})/Γ_{0}(N). The space S_{2}(N) is equipped with many of the familiar structures present in the case where F = ℚ, such as the existence of an inner product, and a collection of mutually commuting self-adjoint Hecke operators T_{ℓ} indexed by the primes ℓ |̸N of O_{F}, leading to a definition of a simultaneous eigenform for these Hecke operators. If f is such an eigenform, then a_{ℓ}(f) is equal to the eigenvalue of the Hecke operator T_{ℓ} acting on f, as in the classical case. For more details the reader is invited to consult the book of Bump [Bu97].

7.3. The Shimura-Taniyama-Weil conjecture

Recall the running assumption that F is a totally real field of narrow class number one. If E is an elliptic curve over F, the definition of the coefficients a(𝔭) attached to E given in equation (7.1) can be extended to all integral ideals in the usual way through the equality of Dirichlet series

∏_{λ|̸N}(1 − a(λ)|λ|^{−s} + |λ|^{1−2s})^{−1} ∏_{λ|N}(1 − a(λ)|λ|^{−s})^{−1} = ∑_{ν} a(ν)|ν|^{−s},

where the products are taken over prime ideals and the sum is taken over integral ideals of F. We may also view a(ν) as a function on the totally positive elements of O_{F} in the obvious way.

The natural extension of the Shimura-Taniyama conjecture for elliptic curves over totally real fields can be formulated concretely as follows:

Conjecture 7.5. Let E be an elliptic curve of conductor N over F. Then the series defined by

f(\underline{τ}) = ∑_{ν>>0} a(ν)e^{2πi(\underline{τ}·(\underline{ν}/\underline{d}))}

is an eigenform in S_{2}(N).

The ideas of Wiles, as generalised by a number of other mathematicians, notably Diamond and Skinner, lead to the proof of many special cases of the Shimura-Taniyama-Weil conjecture over totally real fields, as the following result indicates:

Theorem 7.6 (Skinner-Wiles). Let E be a semistable elliptic curve of conductor N over a totally real field F. Assume further that

• Either [F : ℚ] is odd, or N ≠ 1.

• The extension of F generated by the coordinates of the 3-division points of E has Galois group GL_{2}(𝔽_{3}) over F.

Then conjecture 7.5 holds for E.

The semi-stability assumption and the second condition are present for technical reasons having to do with difficulties that are encountered in certain lifting theorems of Wiles when the mod 3 Galois representation attached to E is reducible. The first condition arises from a more serious (and, from the point of view of this monograph, more interesting) difficulty that is not present in the case of curves over ℚ: the Eichler-Shimura construction does not always admit a suitable generalisation to the context of totally real fields. This question is discussed further in the next section.

7.4. The Eichler-Shimura construction for totally real fields

Theorem 7.7. Let f be a Hilbert modular eigenform in S_{2}(N) with rational Hecke eigenvalues. Assume that [F : ℚ] is odd, or that there is a prime 𝔭 of F dividing N exactly. Then there is an elliptic curve E/F of conductor N such that

a_{λ}(E) = a_{λ}(f), for all λ |̸N.

Sketch of the construction. Let B be the quaternion algebra over F ramified precisely at the archimedean places v_{1}, …, v_{n}, if n + 1 = [F : ℚ] is odd, or at v_{1}, …, v_{n}, 𝔭, if n + 1 is even. Let R be an Eichler order of level N (resp. N/𝔭) in B if [F : ℚ] is odd (resp. even). Since B is split at the place v_{0}, one may choose an identification

ι_{0} : B ⊗_{v_{0}} ℝ ⟶ M_{2}(ℝ).

Let Γ := ι_{0}(R_{1}^{×}). This group acts discretely on H with compact quotient and H/Γ can be interpreted as the complex points of a Shimura curve X which is defined over F. An appropriate generalisation of the theory of Jacquet-Langlands to the context of modular forms on B produces a modular form g ∈ S_{2}(H/Γ) attached to f. The elliptic curve E_{f} is constructed analytically as a quotient of Jac(X) following a generalisation to totally real fields of the theory alluded to in section 4.4. For a detailed description of the theory of Shimura curves attached to quaternion algebras over totally real fields, and their use in defining a variant of the Eichler-Shimura construction in this context, see chapters 1 and 3 of [Zh01a] and the references contained therein. □

The cases where the Eichler-Shimura construction can be carried out yield a Shimura curve parametrisation

Div^{0}(H/Γ) ⟶ E(ℂ),

where Γ is the group introduced in the discussion of theorem 7.7. It should be noted, however, that not all elliptic curves over a totally real field F can be uniformised by a Shimura curve in this way. For instance, the theory excludes elliptic curves with everywhere good reduction defined over a totally real field of even degree.

7.5. The Heegner construction

The Heegner point construction generalises precisely to the situations where E is the quotient of the Jacobian of a Shimura curve, i.e., can be obtained from the Eichler-Shimura construction.

For in that case, letting Γ be the subgroup of SL_{2}(ℝ) arising as in the previous section from a quaternion algebra B split at precisely one archimedean place v_{0} of F, define for each τ ∈ H/Γ the associated order of τ to be

O_{τ} := {γ ∈ R such that det(γ) ≠ 0 and γτ = τ} ∪ {0}.

Lemma 7.8. The order O_{τ} is either equal to O_{F}, or to an order in a quadratic CM extension of F.

Proof. By the same reasoning as in section 3.2, the order O_{τ} is a commutative subring of B containing O_{F}. Its fraction field K is therefore a commutative subfield of B, hence is either equal to F or to a quadratic extension of F. Suppose the latter occurs. For all the archimedean places v_{j} of F, one has K ⊗_{v_{j}} ℝ = ℝ × ℝ or ℂ. The former cannot happen for j = 1, …, n since B ⊗_{v_{j}} ℝ = ℍ has no zero divisors. It is also the case that K ⊗_{v_{0}} ℝ = ℂ because the action of K^{×} on H_{0} has a fixed point. □

Given any order O in a CM extension K of F, denote by CM(O) the set of τ ∈ H/Γ whose associated order O_{τ} is equal to O, and denote by H the ring class field of K attached to O.

Theorem 7.9. If τ belongs to CM(O), then Φ_{N^{+},N^{−}}(Div^{0}(CM(O))) is contained in E(H).

Remarks on the proof. The idea here is to give a moduli interpretation to the Shimura curve H/Γ (or at least to a finite covering of it) and thereby interpret the CM points on this quotient as coming from the moduli of certain abelian varieties with CM by O which are then defined over the appropriate ring class field. See ch. 2 of [Zh01a] for more details. □

The occasional absence of a Shimura curve parametrisation raises the problem of constructing algebraic points on E in such situations - for example, when E is an elliptic curve over a totally real field F of even degree. Note that for such an E we have:

Proposition 7.10. Assume conjecture 3.19. If K is any CM extension of F, then sign(E, K) = 1.

Proof. For under these hypotheses the analytic set S_{E,K} = {v_{0}, …, v_{n}} consists only of the archimedean places of F (viewed as complex places of K). Since n + 1 is even, the result follows from conjecture 3.19. □

It follows that if E is an elliptic curve with everywhere good reduction over a totally real field of even degree, then one should expect no Heegner systems attached to any CM extension of F, so that the theory of complex multiplication is presumably unable to produce any Heegner system attached to E – a difficulty which runs parallel to the fact that E does not appear in the Jacobian of any Shimura curve over F.

7.6. A preview of chapter 8

Let us return to the setting where f is a Hilbert modular form of parallel weight 2 on (H_{0} × ⋯ × H_{n})/Γ_{0}(N) attached to the elliptic curve E over F. Given any τ ∈ H_{0}, write Γ_{τ} for the stabiliser of τ in Γ = Γ_{0}(N).

Lemma 7.11. The group Γ_{τ} is an abelian group of rank at most n.

Proof. Let O_{τ} ⊂ M_{0}(N) be the associated order of τ in M_{0}(N), i.e., the algebra of matrices which preserve the line spanned by the column vector (τ, 1). Then O_{τ} is isomorphic either to O_{F} or to an O_{F} -order in a quadratic extension K of F (i.e,. a subring of K which contains O_{F}, is finitely generated as an O_{F} -module, and generates K as a ℚ-algebra). In the former case Γ_{τ} is trivial, and in the latter case it is identified with the group of elements of O_{τ}^{×} whose norm from K to F is equal to one. Since the torus K^{×} ⊂ GL_{2}(F) has a fixed point on H_{0}, it follows that K ⊗_{v_{0}} ℝ ≃ ℂ, so that the place v_{0} lies below a complex place of K. Hence K has at most 2n + 1 archimedean places; by the Dirichlet unit theorem

rank_{ℤ}(O_{τ}^{×}) ≤ 2n; rank_{ℤ}(O_{F}^{×}) = n.

Since the norm map O_{τ}^{×} ⟶ O_{F}^{×} has finite cokernel, the result follows. □

Remark 7.12. Equality is attained in lemma 7.11 precisely when K is an extension of F which is complex at v_{0} and real at all the other archimedean places of F. Such an extension will be called an almost totally real (ATR) extension of F. This notion depends on the chosen ordering v_{0}, …, v_{n} of the real embeddings of F, or at least on singling out the distinguished place v_{0}. If K is an ATR extension, we will customarily fix as part of the data an extension of v_{0} to a complex embedding of K, making it possible to view K as a subfield of the complex numbers.

Definition 7.13. An ATR point is a point τ ∈ H_{0} such that rank(Γ_{τ}) = n. Equivalently, τ is ATR if it belongs to H_{0} ∩ K for some ATR extension K of F.

Let H′_{0} denote the set of ATR points on H_{0}.

The basic insight to be developed in the next chapter is that there ought to be a natural substitute

Φ′_{N} : H′_{0}/Γ ⟶ E(ℂ)

for the Weil uniformisation of chapter 2. This new type of uniformisation, as in the case n = 0 which it generalises, is constructed from periods of the differential (n + 1)-form

ω_{f} := (2πi)^{n+1}f(\underline{τ})d\underline{τ}

attached to E. The precise definition of Φ′_{N}, based on cohomological properties of the groups Γ and Γ_{τ}, is given in the next chapter. The main conjecture that will emerge, a natural generalisation of the theory of Heegner points, states roughly that if τ ∈ H_{0} ∩ K is an ATR point, then Φ′_{N}(τ) is a global point defined over an appropriate ring class field of K.

Further results

The book of Bump [Bu97], particularly the first chapter, provides an excellent introduction to the theory of Hilbert modular forms. Other good references are the books by Garrett [Gar90] and Freitag [Frei90].

A useful account of the theory of Shimura curves attached to totally real fields, and of Heegner points arising from such Shimura curve parametrisations is the article of Zhang [Zh01a] which also proves the generalisation of the formula of Gross and Zagier in this context.

CHAPTER 8 ATR points

We retain the setting of the previous chapter. Thus F is a totally real field of narrow class number 1 and of degree n + 1 over ℚ, and E is an elliptic curve over F of conductor N, attached to a Hilbert modular form f ∈ S_{2}(N) of level N and parallel weight two. Let

ω_{f} = (2πi)^{n+1}f(τ_{0}, …, τ_{n})dτ_{0} ⋯ dτ_{n}

be the holomorphic dfferential (n + 1)-form on (H_{0} × ⋯ × H_{n})/Γ attached to f.

8.1. Period integrals

For j = 0, …, n, let x_{j} and y_{j} ∈ H_{j} be points on the j-th upper half-plane indexed by the real embeddings ι_{j} of F. Write

(8.1)  ∫_{x_{0}}^{y_{0}}∫_{x_{1}}^{y_{1}}⋯∫_{x_{n}}^{y_{n}}ω_{f} ∈ ℂ

for the usual multiple integral attached to the diferential form ω_{f}. It can be calculated numerically by exploiting the Fourier expansion of f, and is given by the expression

|d| ∑_{ν>>0} a(ν)/|ν| ∏_{j=0}^{n} (e^{2πi \frac{ν_{j}}{d_{j}} y_{j}} − e^{2πi \frac{ν_{j}}{d_{j}} x_{j}}) .

In formulating our conjectures it is important to replace ω_{f} by a non-homolorphic differential ω_{f}^{+} with the same associated Hecke eigenvalues. To do this let Σ = {±1}^{n} and for each σ = (σ_{1}, …, σ_{n}) ∈ Σ, let γ_{σ} ∈ M_{0}(N)^{×} be an element whose determinant is a unit of O_{F}^{×} which is positive at ι_{0} and the places ι_{j} for which σ_{j} = 1, and negative at the places ι_{j} for which σ_{j} = −1. (Such units exist, thanks to the narrow class number one assumption that was made for F.) For τ_{j} ∈ H_{j} write

τ_{j}^{σ} = \{ \begin{matrix}γ_{σ}τ_{j} & if σ_{j} = 1 or j = 0, \\ γ_{σ}τ̄_{j} & if σ_{j} = −1, \end{matrix}

and set

ω_{f}^{σ} = (2πi)^{n+1}f(τ_{0}^{σ}, τ_{1}^{σ}, …, τ_{n}^{σ})dτ_{0}^{σ}dτ_{1}^{σ} ⋯ dτ_{n}^{σ}.

Finally write:

(8.2)  ω_{f}^{+} := \sqrt{|d|}^{−1} ∑_{σ∈Σ} ω_{f}^{σ}.

Properties of the multiple integral: The multiple integral of equation (8.1) enjoys the obvious additivity and Γ-invariance properties, such as

(8.3) (∫_{x_{0}}^{y_{0}}⋯∫_{x_{j}}^{t_{j}}⋯∫_{x_{n}}^{y_{n}}ω_{f}) + (∫_{x_{0}}^{y_{0}}⋯∫_{t_{j}}^{y_{j}}⋯∫_{x_{n}}^{y_{n}}ω_{f}) = ∫_{x_{0}}^{y_{0}}⋯∫_{x_{j}}^{y_{j}}⋯∫_{x_{n}}^{y_{n}}ω_{f},

for all t_{j} ∈ H_{j} and for all j = 0, …, n. Also,

∫_{γx_{0}}^{γy_{0}}⋯∫_{γx_{n}}^{γy_{n}}ω_{f} = ∫_{x_{0}}^{y_{0}}⋯∫_{x_{n}}^{y_{n}}ω_{f}, for all γ ∈ Γ.

The same identities continue to hold with ω_{f} replaced by ω_{f}^{+}. The integral attached to ω_{f}^{+} also enjoys an invariance property under the larger group M_{0}(N)^{×}, which contains Γ with index 2^{n+1}. (Cf. exercise 1.) More precisely, extend the action of Γ on H_{j} to an action of M_{0}(N)^{×} by setting, for all τ ∈ H_{j}:

(8.4)  γτ = \{ \begin{matrix}γ_{j}τ & if det(γ_{j}) > 0; \\ γ_{j}τ̄ & if det(γ_{j}) < 0. \end{matrix}

Let c : ℂ ⟶ ℂ denote complex conjugation. For γ ∈ M_{0}(N^{×}), set s_{γ} = 0 (resp. s_{γ} = 1) if det(γ_{0}) > 0 (resp. if det(γ_{0}) < 0).

Lemma 8.1. For all x_{j}, y_{j} ∈ H_{j}, with 0 ≤ j ≤ n, and all γ ∈ M_{0}(N)^{×},

∫_{γ_{0}x̄_{0}}^{γ_{0}ȳ_{0}} ∫_{γ_{1}x_{1}}^{γ_{1}y_{1}}⋯∫_{γ_{n}x_{n}}^{γ_{n}y_{n}}ω_{f}^{+} = c^{s_{γ}} (∫_{x_{0}}^{y_{0}}⋯∫_{x_{n}}^{y_{n}}ω_{f}^{+}) .

Proof. This is left to the reader as an exercise. (Cf. exercise 1.) □

Remark 8.2. Let Γ̃ be the group of elements γ ∈ M_{0}(N)^{×} for which det(γ_{0}) is positive. It follows from lemma 8.1 that the differential ω_{f}^{+} is invariant under the action of Γ̃.

8.2. Generalities on group cohomology

We begin by briefly recalling some standard notations and terminology from group cohomology.

Let M be a ℤ-module with trivial Γ-action. The space of M-valued r-cochains on Γ, denoted C^{r}(Γ, M), is the set of functions f : Γ^{r} ⟶ M. Let

d : C^{r}(Γ, M) ⟶ C^{r+1}(Γ, M)

be the differential of degree r, defined as in [CF67], ch. 4:

(df)(g_{1}, …, g_{r+1}) = g_{1}f(g_{2}, …, g_{r+1}) + ∑_{j=1}^{r}(−1)^{j}f(g_{1}, …, g_{j}g_{j+1}, …, g_{r+1})
+(−1)^{r+1}f(g_{1}, …, g_{r}).

The space of M-valued r-cocycles on Γ, denoted Z^{r}(Γ, M), is the submodule of C^{r}(Γ, M) of functions f satisfying

df = 0.

The space of M-valued r-coboundaries, denoted B^{r}(Γ, M), is the submodule of C^{r}(Γ, M) consisting of functions of the form df, for some f ∈ C^{r−1}(Γ, M). Since dd = 0, the module B^{r}(Γ, M) is contained in Z^{r}(Γ, M). The quotient module

H^{r}(Γ, M) = Z^{r}(Γ, M)/B^{r}(Γ, M)

is called the r-th cohomology group of Γ with coefficients in M.

An alternate definition of H^{r}(Γ, M) in terms of homogeneous cochains will be useful for the calculations in this chapter. A homogeneous r-cochain is a function f : Γ^{r+1} ⟶ M satisfying

f(sg_{0}, …, sg_{r}) = sf(g_{0}, …, g_{r}) for all s ∈ Γ.

The space of homogeneous r-cochains is denoted C_{hom}^{r}(Γ, M): these groups form a complex in which the differential map d : C_{hom}^{r}(Γ, M) ⟶ C_{hom}^{r+1}(Γ, M) is given by the simpler formula

(df)(g_{0}, …, g_{r+1}) = ∑_{j=0}^{r+1}(−1)^{j}f(g_{0}, …, g_{j−1}, g_{j+1}, …, g_{r+1}).

One defines the group of homogeneous r-coboundaries B_{hom}^{r}(Γ, M) (resp. the group of homogeneous r-cocycles Z_{hom}^{r}(Γ, M)) as the image of d (resp. the kernel of d).

Since a homogeneous r-cochain is determined by its values on r-tuples of the form (1, g_{1}, g_{1}g_{2}, g_{1}g_{2}g_{3}, …, g_{1} … g_{r}), one may associate to a homogeneous rcochain f a non-homogeneous r-cochain θf by the rule

(θf)(g_{1}, …, g_{r}) = f(1, g_{1}, g_{1}g_{2}, …, g_{1} ⋯ g_{r}).

The commutativity of the diagram

C_{hom}^{r}(Γ, M) ⟶^{d} C_{hom}^{r+1}(Γ, M)
↓ θ  ↓ θ
C^{r}(Γ, M) ⟶^{d} C^{r+1}(Γ, M)

shows that the r-th cohomology of Γ with coefficients in M can also be computed as

H^{r}(Γ, M) = Z_{hom}^{r}(Γ, M)/B_{hom}^{r}(Γ, M).

8.3. The cohomology of Hilbert modular groups

Let τ be any element of H_{0}. To the Γ-invariant differential form ω_{f}^{+} and the point τ we associate a non-homogeneous (n + 1)-cochain

κ_{τ} ∈ C^{n+1}(Γ, ℂ)

by choosing a base point \underline{x} = (x_{1}, …, x_{n}) in H_{1} × ⋯ × H_{n}, and setting

(8.5)  κ_{τ} (α_{0}, α_{1}, …, α_{n}) = ∫_{τ}^{α_{0}τ}∫_{α_{0}x_{1}}^{α_{0}α_{1}x_{1}}∫_{α_{0}α_{1}x_{2}}^{α_{0}α_{1}α_{2}x_{2}}⋯∫_{α_{0}···α_{n−1}x_{n}}^{α_{0}···α_{n}x_{n}}ω_{f}^{+}.

Lemma 8.3. The (n + 1)-cochain κ_{τ} is an (n + 1)-cocycle, i.e., it belongs to Z^{n+1}(Γ, ℂ).

Proof. We show this by induction on n, the case n = 1 being checked by a direct computation. For the general case, let κ′_{τ} (g_{0}, …, g_{n+1}) be the homogeneous (n + 1)-cochain attached to κ_{τ}, given by the formula:

κ′_{τ} (g_{0}, …, g_{n+1}) = ∫_{g_{0}τ}^{g_{1}τ} ∫_{g_{1}x_{1}}^{g_{2}x_{1}} ⋯ ∫_{g_{n}x_{n}}^{g_{n+1}x_{n}} ω_{f}^{+}.

Fixing arbitray arguments g_{n+1} and g_{n+2} ∈ Γ, let κ′′_{τ} : Γ^{n+1} ⟶ ℂ be the function given by

κ′′_{τ}(h_{0}, …, h_{n}) = ∫_{h_{0}τ}^{h_{1}τ} ∫_{h_{1}x_{1}}^{h_{2}x_{1}} ∫_{h_{2}x_{2}}^{h_{3}x_{2}} ⋯ ∫_{h_{n−1}x_{n−1}}^{h_{n}x_{n−1}} ∫_{g_{n+1}x_{n}}^{g_{n+2}x_{n}} ω_{f}^{+}.

The additivity property of the multiple integral combined with the induction hypothesis implies that

(8.6)  dκ′′_{τ}(h_{0}, …, h_{n+1}) = 0, for all h_{0}, …, h_{n+1} ∈ Γ.

On the other hand, a direct calculation shows that

dκ′_{τ} (g_{0}, …, g_{n+2}) = ∑_{j=0}^{n}(−1)^{j}κ′_{τ} (g_{0}, …, g_{j−1}, g_{j+1}, …, g_{n+2})
+ (−1)^{n+1}κ′_{τ} (g_{0}, …, g_{n}, g_{n+2}) + (−1)^{n+2}κ′_{τ} (g_{0}, …, g_{n+1})

= dκ′′_{τ}(g_{0}, …, g_{n+1}) + (−1)^{n} {∫_{g_{0}τ}^{g_{1}τ} ∫_{g_{1}x_{1}}^{g_{2}x_{1}} … ∫_{g_{n−1}x_{n−1}}^{g_{n}x_{n−1}} ∫_{g_{n+1}x_{n}}^{g_{n+2}x_{n}} ω_{f}^{+}

− ∫_{g_{0}τ}^{g_{1}τ} ∫_{g_{1}x_{1}}^{g_{2}x_{1}}… ∫_{g_{n−1}x_{n−1}}^{g_{n}x_{n−1}} ∫_{g_{n}x_{n}}^{g_{n+2}x_{n}}ω_{f}^{+} + ∫_{g_{0}τ}^{g_{1}τ} ∫_{g_{1}x_{1}}^{g_{2}x_{1}}… ∫_{g_{n−1}x_{n−1}}^{g_{n}x_{n−1}} ∫_{g_{n}x_{n}}^{g_{n+1}x_{n}}ω_{f}^{+}}

But this last expression is equal to zero, by (8.6) and (8.3). □

In analysing the dependence of κ_{τ} on the choice of the base point \underline{x}, it is convenient to temporarily denote by κ_{τ,\underline{x}} the (n + 1)-cocycle defined in equation (8.5). Let Γ_{τ} denote the stabiliser of τ in Γ.

Lemma 8.4. Let \underline{x} and \underline{y} be any two base points in H_{1} × ⋯ × H_{n}. There exists an n-cochain ρ_{\underline{x},\underline{y}} ∈ C^{n}(Γ, ℂ) such that

(1) κ_{τ,\underline{x}} − κ_{τ,\underline{y}} = dρ_{\underline{x},\underline{y}},

(2) ρ_{\underline{x},\underline{y}}(g_{1}, …, g_{n}) = 0, for all g_{1}, …, g_{n} ∈ Γ_{τ} .

Proof. As in the previous proof it is easiest to see this by working with the homogeneous n + 1-cocycles κ′_{τ,\underline{x}} and κ′_{τ,\underline{y}} attached to κ_{τ,\underline{x}} and κ_{τ,\underline{y}} respectively. Assume without loss of generality that \underline{x} and \underline{y} differ in a single coordinate, x_{j} say. Then the expression for (κ′_{τ,\underline{x}} − κ′_{τ,\underline{y}})(g_{0}, …, g_{n+1}) is given by

(κ′_{τ,\underline{x}} − κ′_{τ,\underline{y}}) = ∫_{g_{0}τ}^{g_{1}τ} ⋯ ∫_{g_{j−1}x_{j−1}}^{g_{j}x_{j−1}} ∫_{g_{j}x_{j}}^{g_{j}y_{j}} ∫_{g_{j+1}x_{j+1}}^{g_{j+2}x_{j+1}} ⋯ ∫_{g_{n}x_{n}}^{g_{n+1}x_{n}} ω_{f}^{+}
− ∫_{g_{0}τ}^{g_{1}τ}⋯ ∫_{g_{j−1}x_{j−1}}^{g_{j}x_{j−1}}∫_{g_{j+1}x_{j}}^{g_{j+1}y_{j}}∫_{g_{j+1}x_{j+1}}^{g_{j+2}x_{j+1}}⋯ ∫_{g_{n}x_{n}}^{g_{n+1}x_{n}} ω_{f}^{+}.

A direct calculation (cf. exercise 2) reveals that this expression is equal to dρ_{\underline{x},\underline{y}}, where ρ_{\underline{x},\underline{y}} : Γ^{n+1} ⟶ ℂ is defined by

(8.7) ρ_{\underline{x},\underline{y}}(g_{0}, …, g_{n}) = ∫_{g_{0}τ}^{g_{1}τ} ∫_{g_{1}x_{1}}^{g_{2}x_{1}} ⋯ ∫_{g_{j−1}x_{j−1}}^{g_{j}x_{j−1}} ∫_{g_{j}x_{j}}^{g_{j}y_{j}} ∫_{g_{j}x_{j+1}}^{g_{j+1}x_{j+1}} ⋯ ∫_{g_{n−1}x_{n}}^{g_{n}x_{n}} ω_{f}^{+}.

The result follows immediately from this formula. □

Remark 8.5. Lemma 8.4 and its proof imply, in particular, that the natural image of κ_{τ} in H^{n+1}(Γ, ℂ) is an invariant of ω_{f}^{+} which does not depend on the choice of base point \underline{x}, or, for that matter, of τ, that was made to define it.

A lattice Λ ⊂ ℂ is called a trivialising lattice for the cocycle κ_{τ} if the natural image of κ_{τ} in H^{n+1}(Γ, ℂ/Λ) is trivial. The key conjecture 8.6 below postulates the existence of a trivialising lattice for κ_{τ} which can be defined explicitly in terms of the periods attached to E. More precisely, choose a Néron differential ω_{E} on E/F and let Ω_{j}^{+}, (j = 1, …, n) denote the real period attached to the elliptic curve E_{j} and the differential ι_{j}(ω_{E}). Let Λ_{0} denote the period lattice attached to the differential ι_{0}(ω_{E}) on E_{0}. Note that ω_{E} is only well-defined up to multiplication by a unit in F; hence the periods Ω_{j}^{+} are defined up to multiplication by a non-zero element of ι_{j}(O_{F}^{×}). Since the norms of these elements are ±1, the lattice

Λ := Ω_{1}^{+} ⋯ Ω_{n}^{+}Λ_{0}

is independent of the choice of a Néron differential ω_{E}.

Notice also that Λ is homothetic to Λ_{0}. Let

(8.8)  Φ_{0} : ℂ/Λ ⟶ E_{0}(ℂ)

be the Weiserstrass uniformisation which is inverse to the map given by integration of the differential form Ω_{1}^{+} ⋯ Ω_{n}^{+}ι_{0}(ω_{E}).

Conjecture 8.6. There exists an integer t_{E} (depending only on E, and not on τ) such that the cocycle t_{E}κ_{τ} has Λ as a trivialising lattice.

Note that by remark 8.5, such an integer t_{E}, if it exists, can be chosen independently of τ.

Remark 8.7. In the case n = 0, conjecture 8.6 merely asserts that an integer multiple of the cocycle κ_{τ} given by

κ_{τ} (γ) = ∫_{τ}^{γτ} ω_{f}

becomes trivial modulo Λ, the Néron latice of E. But this is well-known: for instance if E is the strong Weil curve attached to the classical modular curve parametrisation, one may take t_{E} to be the Manin constant attached to E, an integer which is conjecturally always equal to one.

Remark 8.8. Conjecture 8.6 is related, in the case where n = 1, to work of Oda [Od82] on periods for Hilbert modular surfaces.

Let κ̄_{τ} denote the natural image of t_{E}κ_{τ} in Z^{n+1}(Γ, ℂ/Λ). Assuming conjecture 8.6, the cocycle κ̄_{τ} is an (n+1)-coboundary, i.e., it belongs to B^{n+1}(Γ, ℂ/Λ). Thus it is possible to write

(8.9)  κ̄_{τ} = dξ̃_{τ} , with ξ̃_{τ} ∈ C^{n}(Γ, ℂ/Λ).

Note that equation (8.9) makes ξ̃_{τ} well-defined only up to elements of Z^{n}(Γ, ℂ/Λ). The following theorem shows that the element ξ̃_{τ} – or at least, some multiple of it – is in fact well-defined up to n-coboundaries.

Theorem 8.9. Suppose that n is odd. Then H^{n}(Γ, ℝ/ℤ) has finite exponent.

Proof. This follows from an extension due to Harder [Ha75] of a result of Matsushima and Shimura [MS78] valid in the case where Γ acts on H^{n+1} with compact quotient. More precisely, theorem 6.3 of chapter III, §6 of [Frei90] states that

H^{n}(Γ, ℝ) = 0.

Hence H^{n}(Γ, ℝ/ℤ) injects (via the connecting homomorphism arising from the exact sequence 0 ⟶ ℤ ⟶ ℝ ⟶ ℝ/ℤ ⟶ 0) into the kernel of the map H^{n+1}(Γ, ℤ) ⟶ H^{n+1}(Γ, ℝ). This kernel consists of the torsion in H^{n+1}(Γ, ℤ) which is finite (since it is finitely generated) and the result follows. □

Assume now that n is odd, and let e_{Γ} denote the exponent of H^{n}(Γ, ℂ/Λ) which is finite by theorem 8.9. (This boundedness of the exponent is the single – but crucial – stage of the construction where it becomes necessary to assume n ≠ 0.) The n-cochain

ξ_{τ} := e_{Γ}ξ̃_{τ}

is then well-defined, modulo

e_{Γ}Z^{n}(Γ, ℂ/Λ) ⊂ B^{n}(Γ, ℂ/Λ).

Then let θ_{τ} ∈ C^{n}(Γ_{τ} , ℂ/Λ) be the restriction of ξ_{τ} to Γ_{τ}^{n}.

Lemma 8.10. The n-cochain θ_{τ} is an n-cocycle.

Proof. This follows from the fact, immediate from (8.5), that the (n + 1)- cochain κ_{τ} vanishes identically on Γ_{τ}^{n+1}. □

Lemma 8.11. The natural image of θ_{τ} in H^{n}(Γ_{τ} , ℂ/Λ) does not depend on the choice of ξ̃_{τ} made to define it.

Proof. This is because ξ_{τ} is well-defined up to n-coboundaries on Γ. □

Lemma 8.12. The natural image of θ_{τ} in H^{n}(Γ_{τ} , ℂ/Λ) does not depend on the choice of base points \underline{x} that were made to define κ_{τ} and κ̄_{τ}.

Proof. This follows from lemma 8.4. (Cf. exercise 3.) □

Conclusion: To any point τ ∈ H_{0} has been associated (at least when n is odd) a cohomology class

θ_{τ} ∈ H^{n}(Γ_{τ} , ℂ/Λ).

This object depends only on the orbit of τ under the action of the group Γ̃ of remark 8.2. More precisely, suppose that τ ∈ H_{0} and τ′ = ατ belong to the same orbit for this group. Then Γ_{τ′} = αΓ_{τ} α^{−1}, and we have

Lemma 8.13. The classes of θ_{τ} and θ_{τ′} admit representative cocycles θ̃_{τ} and θ̃_{τ′} (in Z^{n}(Γ_{τ} , ℂ/Λ) and Z^{n}(Γ_{τ′}, ℂ/Λ), respectively) such that

θ̃_{τ′}(αg_{1}α^{−1}, …, αg_{n}α^{−1}) = θ̃_{τ} (g_{1}, …, g_{n}),

for all g_{1}, …, g_{n} ∈ Γ_{τ}.

Proof. See exercise 5. □

8.4. ATR points

Let K be a quadratic extension of F which is ATR (relative to the chosen ordering v_{0}, …, v_{n} of the archimedean places of F). Let τ ∈ H_{0} ∩ K be an ATR point, so that the group Γ_{τ} is a free abelian group of rank n. In that case we have

Proposition 8.14. The cohomology group H^{n}(Γ_{τ} , ℂ/Λ) is canonically isomorphic to ℂ/Λ.

Proof. This follows directly from the fact that Γ_{τ} ≃ ℤ^{n} can be made to act freely on ℝ^{n} with quotient T^{n} = R^{n}/ℤ^{n}, so that the cohomology of ℤ^{n} is identified with the cohomology of a compact connected oriented n-manifold. □

Let J_{τ} ∈ ℂ/Λ be the invariant attached to θ_{τ} by proposition 8.14. It follows from lemma 8.13 that

J_{ατ} = J_{τ} , for all α ∈ Γ.

Conclusion: To any AT R point τ ∈ H_{0} has been associated a canonical invariant J_{τ} ∈ ℂ/Λ, which depends only on the Γ̃-orbit of τ ∈ H_{0}. Recall the Weierstrass uniformisation Φ_{0} of equation (8.8) and set

P_{τ} := Φ_{0}(J_{τ}) ∈ E_{0}(ℂ).

We can now define the map Φ′_{N} alluded to at the end of chapter 7 by the rule

Φ′_{N}(τ) = P_{τ} .

We now formulate a conjecture, analogous to the classical Shimura reciprocity law of chapter 3, which implies that the point Φ′_{N}(τ) (with τ ∈ H_{0} ∩ K an ATR point) is an algebraic point on E defined over an abelian extension of K which can be described precisely in terms of class field theory.

The following assumption, while not essential, will be made to simplify the discussion, and is analogous to the Heegner hypothesis of chapter 3:

Hypothesis 8.15. All the primes dividing the conductor N of E are split in K/F.

Fix an ideal N of O_{K} whose norm to O_{F} is equal to N. Recall that M_{0}(N) is the O_{F} -algebra of 2 × 2 matrices with entries in O_{F} which are upper-triangular modulo N. Denote by

η : M_{0}(N) ⟶ O_{F} /NO_{F}

the homomorphism sending a matrix to its upper left-hand entry taken modulo N. Let

Ψ : K ⟶ M_{2}(F)

be an embedding of F -algebras. Such an embedding is said to be optimal if

Ψ(K) ∩ M_{0}(N) = Ψ(O_{K}).

Let τ ∈ H_{0} denote the unique fixed point for Ψ(K^{×}) acting on H_{0}. The embedding Ψ is said to be normalised if Ψ(α) acts on the column vector (τ, 1) as multiplication by α, and if the kernel of ηΨ (restricted to O_{K}) is equal to N.

The group Γ̃ acts naturally by conjugation on the set of normalised optimal embeddings.

Let H denote the Hilbert class field of K, and let G denote its Galois group over K. This group G is identified with the Picard group of O_{K}.

The group G acts naturally on the collection of Γ̃-conjugacy classes of normalised optimal embeddings of K into M_{2}(F) (cf. exercise 6). Given σ ∈ G, and a normalised embedding Ψ, let σ ⋆ Ψ denote a representative for the Γ̃-orbit of the image of the embedding Ψ by the action of σ.

Lemma 8.16. The group G acts simply transitively on the set of Γ̃-conjugacy classes of normalised optimal embeddings of K into M_{2}(F).

Proof. See exercise 6. □

Given a normalised embedding Ψ with fixed point τ ∈ H_{0}, one may set

P_{Ψ} = Φ′_{N}(τ).

Note that this point depends only on the Γ̃-conjugacy class of Ψ.

We view the Hilbert class field H of K as a subfield of ℂ via a complex embedding which extends the fixed embedding v_{0} of K into ℂ.

Conjecture 8.17. The local point P_{Ψ} is the image of a global point in E(H). More precisely

P_{σ⋆Ψ} = σ^{−1}(P_{Ψ}), for all σ ∈ H.

Remark 8.18. The simplest case of the construction (and the one which runs the most parallel to the p-adic construction of chapter 9) is the case where n = 1. The field F is then a real quadratic field and K is a quadratic extension of F having one complex and two real places. The space X is a Hilbert modular surface attached to the Hecke congruence subgroup of level N in SL_{2}(O_{F}).

Remark 8.19. Specialising even further, suppose that F is a real quadratic field of narrow class number one, and that E is an elliptic curve with everywhere good reduction over F, so that N = 1. Let K be a quadratic extension of F with one complex and two real places. In this situation conjecture 8.17 yields an analytic construction of points on E(H), where H is the Hilbert class field of K. This setting is of interest because (1) if E is not isogenous to its Galois conjugate then E does not appear as a quotient of the Jacobian of any modular or Shimura curve, so that the theory of complex multiplication does not suggest any modular construction of Heegner systems on E; and (2) this situation is simple enough to be amenable to machine calculations. For some numerical verifications that have been performed in this case, see [DL03].

Remark 8.20. The only ATR extensions of F = ℚ are the imaginary quadratic fields, where one is placed in the usual setting of the theory of complex multiplication. There are “not enough” archimedean places of ℚ for any new phenomena to manifest themselves through conjecture 8.17. In particular, since a real quadratic field is not an ATR extension, the generalisation of the Heegner point construction to the context of real quadratic fields falls outside the scope of the theory developed in the last two chapters. To encompass this case it is necessary to combine this theory with some of the notions arising from rigid analysis that were surveyed in chapters 5 and 6. This will be the goal of the next chapter.

References

A reference for periods of Hilbert modular forms (at least in the case of Hilbert modular surfaces) is the book of Oda [Od82].

For an introduction to the cohomology of groups, and in particular to the cohomology of discrete subgroups of Lie groups, see [Br94]. General finiteness results on the cohomology of discrete groups (particularly arithmetic groups) can be found in the articles [Se70] and [Se71] by Serre. The cohomology of Hilbert modular groups, building on the techniques of Matsushima-Shimura [MS78] and Harder [Ha75] is studied in [Frei90].

For further discussion of conjecture 8.17, see [BDG03] and [DL03]. The article [DL03] presents numerical evidence for conjecture 8.17 when E has everywhere good reduction over a real quadratic field, and discusses the algorithms that were used to calculate Φ′_{N} in this case.

Exercises

(1) Assuming that F has narrow class number one, show that Γ is a normal subgroup of M_{0}(N)^{×} with quotient isomorphic to (ℤ/2ℤ)^{n+1}. Write down representatives for the cosets of Γ in M_{0}(N)^{×}. (Hint: they can be chosen to be diagonal matrices.) Use these representatives to prove lemma 8.1.

(2) Complete the proof of lemma 8.4 by showing that the homogeneous n-cochain ρ_{\underline{x},\underline{y}} of equation 8.7 satisfies dρ = κ_{τ,\underline{x}} − κ_{τ,\underline{y}}.

(3) Provide the details of the proof of lemma 8.12.

(4) Let G = ℤ^{n} and let M be a G-module with trivial G-action. Show that the isomorphism between H^{n}(G, M) and M can be written in terms of nonhomogeneous n-cochains by choosing a system of generators γ_{1}, …, γ_{n} of G and sending the class of the cocycle θ to the quantity

J_{θ} := ∑_{σ∈S_{n}} sgn(σ)θ_{τ} (γ_{σ1}, γ_{σ2}, …, γ_{σn}),

where sgn(σ) = ±1 is the sign of the permutation σ.

(5) Prove lemma 8.13, and conclude that the invariant J_{τ} attached to an ATR point τ ∈ H_{0} depends only on the Γ̃-orbit of τ, so that Φ′_{N} descends to a map

Φ′_{N} : H′_{0}/Γ̃ ⟶ E(ℂ).

(6) Let K be a fixed ATR extension of F in which all the prime divisors of N are split. Adapt the ideas of sections 3.1 and 3.2 to define a natural action of G = Pic(O_{K}) on the collection of Γ̃- conjugacy classes of normalised optimal embeddings of K into M_{2}(F) (relative to M_{0}(N), and a choice of ideal N ⊂ O_{K}). Show that this action is simply transitive.

CHAPTER 9 Integration on H_{p} × H

A number of different modular parametrisations have been introduced so far, and their attendant theories of special (CM, or ATR) points used to construct Heegner systems attached to modular elliptic curves.

In chapters 2 and 3, the classical modular parametrisation

Φ_{N} : H/Γ_{0}(N) ⟶ E(ℂ)

attached to a modular elliptic curve over ℚ of conductor N, combined with the theory of complex multiplication, was used to define global points on E over ring class fields of quadratic imaginary fields satisfying a suitable Heegner hypothesis.

In chapters 4 and 7, the Shimura curve parametrisations

Φ_{N^{+},N^{−}} : Div^{0}(H/Γ_{N^{+},N^{−}}) ⟶ E(ℂ)

were used to construct more general Heegner systems attached to elliptic curves over ℚ and quadratic imaginary extensions of ℚ. When suitably generalised to groups coming from quaternion algebras over a totally real field F, Shimura curve parametrisations may be used to construct Heegner systems attached to elliptic curves over F and certain CM extensions of F.

In chapter 6, the rigid analytic uniformisation

Φ_{N^{+},N^{−}}^{(p)} : Div^{0}(H_{p}/Γ_{N^{+},N^{−}}^{(p)}) ⟶ E(ℂ_{p})

based on the Čerednik-Drinfeld theory of p-adic uniformisation of Shimura curves was then exploited to construct the Heegner systems of chapter 4 by p-adic analytic means.

Finally, when E is a modular elliptic curve of conductor 𝔫 defined over a totally real field F of degree n + 1 > 1, and f is the associated Hilbert modular form on (H_{0} × ⋯ × H_{n})/Γ_{0}(𝔫), modular or Shimura curve parametrisations are not always available. Chapters 7 and 8 introduced a natural substitute

Φ′_{N} : H′_{0}/Γ_{0}(𝔫) ⟶ E(ℂ)

defined on the set H′_{0} of ATR points in H_{0} in terms of certain integrals attached to f. This substitute can be used – conjecturally – to construct points on elliptic curves defined over abelian extensions of ATR extensions of F.

This chapter proposes a synthesis of the ideas of chapters 6 and 8 to define (conjecturally, as in chapter 8) Heegner systems attached to pairs (E, K) where E is a modular elliptic curve over ℚ and K is a real quadratic field for which sign(E, K) = −1.

98 9. INTEGRATION ON H_{p} × H

Note that, if sign(E, K) = −1, then there is at least one prime divisor p of N which is either inert or ramified in K/ℚ, by theorem 3.17 of chapter 3. The main idea of the construction is to fix such a prime divisor and make it play the same role as the distinguished archimedean place v_{0} in the theory of the two previous chapters.

9.1. Discrete arithmetic subgroups of SL_{2}(ℚ_{p}) × SL_{2}(ℝ)

Suppose henceforth that the conductor N of E is of the form pM, with p a prime not dividing M. In this case E has multiplicative reduction at p by (1.17) of chapter 1. Let q be the Tate period attached to E/ℚ_{p} and let

Φ_{Tate} : ℂ_{p}^{×} ⟶ E(ℂ_{p})

be the Tate uniformisation attached to E as in equation (1.9) of chapter 1. Set w = a_{p}, so that

(9.1)  w = \{ \begin{matrix}1 & if E has split multiplicative reduction at p. \\ −1 & if E has non-split multiplicative reduction at p. \end{matrix}

Let Γ ⊂ SL_{2}(ℤ[1/p]) be the group of matrices which are upper triangular modulo M. This group acts naturally both on H and on the p-adic upper half plane H_{p} of chapter 5. It acts on each of these upper half planes with dense orbits, but its action on the product H_{p} × H is discrete. (Cf. exercise 1.) Note that Γ is the group of elements of determinant one in the ring R = M_{0}(M)[1/p] of matrices with entries in ℤ[1/p] which are upper-triangular modulo M. Let Γ̃ = R^{×} and let α_{∞} and α_{p} be elements of Γ̃ satisfying

(9.2)  det(α_{∞}) = −1, det(α_{p}) = p.

A point τ belonging to H or H_{p} is called a special point if its stabiliser in Γ is infinite. Let H′ and H′_{p} denote the set of special points in H and H_{p} respectively. It can be seen that H′ consists precisely of the points in H ∩ K, where K ranges over all the imaginary quadratic extensions of ℚ in which p is split. In [Ih68], Ihara describes a natural correspondence between H′ and certain points on X_{0}(M) in characteristic p, this correspondence being derived via the classical theory of complex multiplication developed in chapter 3. (Cf. exercise 2.)

On the other hand, the set H′_{p} is an object that appears to have been less studied. One can write

H′_{p} = ∪_{K}(H_{p} ∩ K),

where K ranges over all real quadratic subfields of ℂ_{p} which are not contained in ℚ_{p} (so that p is inert or ramified in K).

The goal of this chapter is to explain in detail how the following program can be carried out:

(1) Attach to the elliptic curve E a “mock Hilbert modular form of weight (2, 2)” on (H_{p} × H)/Γ, denoted ω.

(2) Define a ℂ_{p}-valued integration theory attached to ω, behaving formally like the integrals attached to a Hilbert modular form in chapter 8. The ideas of p-adic analysis and p-adic integration explained in chapter 5, together with the Manin-Drinfeld theorem described in section 2.7, are the main ingredients in this definition.

9.2. FORMS ON H_{p} × H 99

(3) Use these periods to define a parametrisation

Φ_{N}^{′(p)} : H′_{p}/Γ ⟶ E(ℂ_{p})

by mimicking the formal aspects of the definition of the parametrisation Φ′_{N} given in chapter 8;

(4) Make a precise conjecture predicting that the points Φ_{N}^{′(p)}(τ) ∈ E(ℂ_{p}) are defined over certain ring class fields of real quadratic fields.

9.2. Forms on H_{p} × H

Motivated by the analogy with Hilbert modular forms, we seek an appropriate notion of “form of weight (2, 2) on (H_{p} × H)/Γ”. In an informal sense, such an object would be a Γ-invariant expression of the form

“ω = f(z_{p}, z)dz_{p}dz”,

where z_{p} is a p-adic and z is a complex variable. If one requires that the function f be rigid analytic in the first variable and holomorphic in the second, one runs into difficulties in making such a notion mathematically precise. While a sensible definition of ω seems elusive, the p-adic boundary distribution that was attached to a rigid analytic modular form in chapter 5 is a notion that can be transposed in a natural way to the current setting.

Recall the Bruhat-Tits tree T of PGL_{2}(ℚ_{p}), whose set T_{0} of vertices is in bijection with the set of ℚ_{p}^{×}-homothety classes of rank two ℤ_{p}-modules in ℚ_{p}^{2}. Recall that T_{1} and E(T ) are the set of unordered and ordered edges of T respectively, and that s(e) and t(e) denote the source and target vertex of e ∈ E(T ). Let ē denote the oriented edge obtained from e by interchanging its source and target.

The following definition, motivated by the considerations of chapter 5 and 6, is tailored to capture the notion of the “p-adic boundary distribution attached to ω”.

Definition 9.1. A cusp form of weight 2 on (T × H)/Γ is a function

f : E(T ) × H ⟶ ℂ

satisfying

(1) f(γe, γz) = (cz + d)^{2}f(e, z), for all γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ.

(2) The function f is harmonic, i.e., for each vertex v of T ,

∑_{s(e)=v} f(e, z) = 0,

and for each oriented edge e of T , f(ē, z) = −f(e, z).

(3) For each oriented edge e of T , the function f_{e}(z) := f(e, z) is a cusp form of weight 2 (in the usual sense) on the group Γ_{e} := Stab_{Γ}(e).

Property 1 is suggested by the desired Γ-invariance of ω, and property 2 by the ideas in the proof of theorem 5.9 of chapter 5. Note that an element of the space S_{2}(T , Γ) of cusp forms of weight 2 on (T ×H)/Γ is completely described by a collection {f_{e}} of cusp forms on Γ_{e}, indexed by the ordered edges e of T , satisfying the compatibility relation

(9.3)  f_{γe}(γz)d(γz) = f_{e}(z)dz, for all γ ∈ Γ,

together with the harmonicity condition 2.

Let e_{o} be the distinguished oriented edge of T whose stabiliser in Γ is equal to Γ_{0}(N), and whose source is fixed by Γ_{0}(M). The group Γ (resp. Γ̃) acts transitively on the unoriented (resp. oriented) edges of T (cf. exercise 1), and for each e the stabiliser subgroups Γ_{e} are conjugate in Γ to either Γ_{e_{o}} = Γ_{0}(N) or Γē_{0}.

The matrices α_{∞} and α_{p} of equation (9.2) generate the quotient Γ̃/Γ, and can be chosen in such a way that they preserve the unordered edge attached to e_{o}, so that

(9.4)  α_{∞}e_{o} = e_{o}, α_{p}e_{o} = ē_{o}.

The involution W_{p} on the space S_{2}(T , Γ) is defined by the rule

(W_{p}f)(e, z)dz = f(α_{p}e, α_{p}z)d(α_{p}z).

This definition does not depend on the choice of α_{p} that was made. If α_{p} is in addition normalised to satisfy (9.4) (a convention that will be adopted from now on) then it can also be used to define the usual Atkin-Lehner involution on the space S_{2}(N) = S_{2}(Γ_{0}(N)) of classical cusp forms of level N (denoted W_{p} as well by an abuse of notation) by the rule

(9.5)  (W_{p}f_{0})(z)dz = f_{0}(α_{p}z)d(α_{p}z) for f_{0} ∈ S_{2}(N).

Lemma 9.2 below reveals that the space S_{2}(T , Γ) is intimately connected with S_{2}(N). In order to state it precisely, let

φ_{s} : X_{0}(N) ⟶ X_{0}(M)

be the projection arising from the inclusion Γ_{0}(N) ⊂ Γ_{0}(M), and let φ_{t} = φ_{s}W_{p} Making an abuse of notation, denote by the same symbols φ_{s} and φ_{t} the two degeneracy maps from S_{2}(N) to S_{2}(M) induced from φ_{s} and φ_{t} by pushforward of differential forms. More precisely, choosing a system of coset representatives for Γ_{0}(N) in Γ_{0}(M):

(9.6)  Γ_{0}(M) = γ_{1}Γ_{0}(N) ∪ ⋯ ∪ γ_{p+1}Γ_{0}(N),

one has

(9.7) φ_{s}(f)(z)dz = ∑_{j=1}^{p+1} f(γ_{j}^{−1}z)d(γ_{j}^{−1}z), φ_{t}(f)(z)dz = ∑_{j=1}^{p+1} f(αγ_{j}^{−1}z)d(αγ_{j}^{−1}z).

The kernel of

(9.8)  φ_{s} ⊕ φ_{t} : S_{2}(N) ⟶ S_{2}(M) ⊕ S_{2}(M)

is called the subspace of p-new forms, denoted S_{2}^{new−p}(N).

Lemma 9.2. The function which to f(e, z) associates f_{o}(z) := f_{e_{o}}(z) induces an isomorphism from S_{2}(T , Γ) to S_{2}^{new−p}(N).

Proof. See exercise 3 or the proof of lemma 1.3 of [Da01]. □

The Hecke operators T_{ℓ} (ℓ |̸N) act on the space S_{2}(T , Γ) via the identifications of lemma 9.2. One can also give a direct description in the spirit of equations (2.6) of chapter 2 and (4.3) of chapter 4. For each prime ℓ, write the double coset

Γ ( \begin{matrix}ℓ & 0 \\ 0 & 1 \end{matrix} ) Γ as a disjoint union of left cosets:

(9.9)  Γ ( \begin{matrix}ℓ & 0 \\ 0 & 1 \end{matrix} ) Γ = ∪_{j}γ_{j}Γ.

Then T_{ℓ}f is given by

(9.10)  (T_{ℓ}f)(e, z)dz = ∑_{j} f(γ_{j}^{−1}e, γ_{j}^{−1}z)d(γ_{j}^{−1}z).

Let f_{0} be the normalized newform on Γ_{0}(N) associated to E by theorem 2.11 of chapter 2. The form f_{0} is an eigenvector for the involution W_{p} acting on on S_{2}(N). It is known that

(9.11)  W_{p}f_{0} = −wf_{0},

where w is the sign that was attached to E in (9.1). Let f be the form in S_{2}(T , Γ) which is related to f_{0} by lemma 9.2, so that f_{e_{o}} = f_{0}. Given γ ∈ Γ̃, let |γ|_{p} = ord_{p}(det(γ)) ∈ ℤ/2ℤ. Write R_{+}^{×} for the group of elements in Γ of positive determinant.

Lemma 9.3. The form f satisfies the following transformation rule under the group R_{+}^{×} ⊃ Γ:

f(γe, γz)d(γz) = w^{|γ|_{p}}f(e, z)dz, for all γ ∈ R_{+}^{×}.

Proof. The lemma is clearly true for γ ∈ Γ. Since α_{p} generates the quotient R_{+}^{×}/Γ, and f is determined by its behaviour on {e_{o}} × H, it suffices to check the lemma for γ = α_{p} and e = e_{o}. In that case one has:

f(α_{p}e_{0}, α_{p}z)d(α_{p}z) = f(ē_{0}, α_{p}z)d(α_{p}z) = −(W_{p}f_{0})(z)dz
= wf_{0}(z)dz = wf(e_{o}, z)dz,

where the penultimate equality follows from (9.11). □

9.3. Periods

It will be convenient (but not essential) to make the following simplifying assumption:

Assumption 9.4. The elliptic curve E is unique in its ℚ-isogeny class.

By this assumption, E is isomorphic to the strong Weil curve in its isogeny class. Let

Φ_{N} : H^{∗}/Γ_{0}(N) ⟶ E(ℂ)

be the strong Weil parametrisation attached to E. Letting ω_{E} denote the Néron differential of E, one has

(9.12)  φ^{∗}(ω_{E}) = c · 2πif_{0}(z)dz,

where c is the Manin constant, introduced in (2.16) of section 2.5.

Choose elements x, y in the extended upper half-plane H^{∗} := H ∪ ℙ_{1}(ℚ). The function κ̃_{f}{x→y} : E(T ) ⟶ ℂ defined by

(9.13)  κ̃_{f}{x→y}(e) := c · 2πi ∫_{x}^{y} f_{e}(z)dz

is a complex-valued harmonic cocycle on T , as follows immediately from the harmonicity properties of f itself. As in equation (5.5) of section 5.2, the harmonic cocycle κ̃_{f}{x→y} gives rise to a complex-valued distribution μ̃_{f}{x→y} on the boundary ℙ_{1}(ℚ_{p}) of H_{p} by the rule

(9.14)  μ̃_{f}{x→y}(U_{e}) = κ̃_{f}{x→y}(e).

(Here U_{e} ⊂ ℙ_{1}(ℚ_{p}) is the compact open subset attached to e ∈ E(T ) as in chapter 5.) Using equation (5.3) of chapter 5, the distribution μ̃_{f}{x→y} can be integrated against locally constant complex-valued functions on ℙ_{1}(ℚ_{p}). For the purposes of p-adic integration, it is desirable that κ̃_{f}{x→y} take on integral or at least p-adic integral values and thereby correspond to a measure against which locally analytic functions can be integrated.

Such an integrality can be achieved by invoking the theory of modular symbols presented in section 2.7, provided that x and y belong to ℙ_{1}(ℚ). For in this case, the value of κ̃_{f}{x→y}(e) can be expressed in terms of the modular symbols

(9.15)  λ̃_{f_{0}}{x→y} := c · 2πi ∫_{x}^{y} f_{0}(z)dz.

(Here λ̃_{f_{0}} is a slight modification of the M-symbol denoted λ_{f_{0}} in equation (2.18) of section 2.7.) More precisely, choosing γ ∈ R_{+}^{×} such that γe = e_{o},

(9.16)  κ̃_{f}{x→y}(e) = w^{|γ|_{p}} λ̃_{f_{0}}{γx→γy}.

Let Λ_{E} denote the Néron lattice attached to the elliptic curve E. Theorem 2.19 of section 2.7 combined with assumption 9.4 implies the following corollary:

Corollary 9.5. For all x, y ∈ ℙ_{1}(ℚ), the harmonic cocycle κ̃_{f}{x→y} takes its values in Λ_{E}, and hence gives rise to a Λ_{E}-valued measure μ_{f}{x→y} on ℙ_{1}(ℚ_{p}).

If E(ℝ) has two components then Λ_{E} is generated by a positive real period Ω_{+} and a purely imaginary period Ω_{−}. If E(ℝ) has one connected component, then Ω is contained with index two in the lattice spanned by Ω_{+} and Ω_{−}, where Ω_{+} (resp. Ω_{−}) denotes the real (resp. imaginary) half-period attached to E. In either case, thanks to corollary 9.5, one can write

(9.17)  κ̃_{f}{x→y}(e) = κ_{f}^{+}{x→y}(e) · Ω_{+} + κ_{f}^{−}{x→y}(e) · Ω_{−},

with κ_{f}^{±}{x→y}(e) ∈ ℤ. Choose a sign w_{∞} = ±1 and let κ_{f}{x→y} denote κ_{f}^{+}{x→y} (resp. κ_{f}^{−}{x→y}) if w_{∞} = 1 (resp. w_{∞} = −1). Write λ_{f_{0}} for the corresponding ℤ-valued modular symbol attached to f_{0}. Let μ_{f}{x→y} denote the ℤ-valued distribution on ℙ_{1}(ℚ_{p})) attached to κ_{f}{x→y}, so that, with the notations of (9.16),

(9.18)  μ_{f}{x→y}(U_{e}) := κ_{f}{x→y}(e) = w^{|γ|_{p}}λ_{f_{0}}{γx→γy}.

Given γ ∈ Γ̃, set

|γ|_{∞} = \{ \begin{matrix}0 & if det(γ) > 0; \\ 1 & if det(γ) < 0. \end{matrix}

It is worth recording the following lemma which describes the behaviour of κ_{f} under the full group Γ̃.

Lemma 9.6. For all γ ∈ Γ̃, x, y ∈ ℙ_{1}(ℚ), and e ∈ E(T ),

κ_{f}{γx→γy}(γe) = w_{∞}^{|γ|_{∞}}w^{|γ|_{p}}κ_{f}{x→y}(e).

Proof. If γ belongs to R_{+}^{×}, this follows directly from the transformation properties of f given in lemma 9.3. It therefore suffices to prove the lemma for a single

γ which generates Γ̃/R_{+}^{×}, say the matrix γ ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ), and for e = e_{o}. Then we have

(9.19)  κ̃_{f}\{γx→γy\}(γe_{o}) = ∫_{−x}^{−y} f_{0}(z)dz = λ̃_{f_{0}}\{−x→ − y\} = \overline{λ̃_{f_{0}}\{x→y\}},

where the last equality follows from (2.22) and (2.23) of section 2.7. The lemma follows at once, given the definition of λ_{f} in terms of λ̃_{f} and w_{∞}. □

Because the values μ_{f}{x→y}(U_{e}) are integral and hence p-adically bounded as e varies in E(T ), the distribution μ_{f}{x→y} defines a p-adic measure on ℙ_{1}(ℚ_{p}). In particular, if h is any bounded locally analytic ℂ_{p}-valued function on ℙ_{1}(ℚ_{p}), the integral

(9.20)  ∫_{ℙ_{1}(ℚ_{p})} h(t)dμ_{f}{x→y}(t) ∈ ℂ_{p}

can be defined as in equation (5.4) of chapter 5.

Inspired by definition 5.12 of chapter 5, the following definition, depending similarly on a choice of log, imposes itself naturally.

Definition 9.7. Let z_{1} and z_{2} be elements of H_{p}, and let x, y ∈ ℙ_{1}(ℚ).

(9.21)  ∫_{z_{1}}^{z_{2}}∫_{x}^{y}ω := ∫_{ℙ_{1}(ℚ_{p})} log (\frac{t − z_{2}}{t − z_{1}} ) dμ_{f}\{x→y\}(t) ∈ ℂ_{p}.

The following lemma shows that this definition is well-behaved:

Lemma 9.8. The double integrals of definition 9.7 satisfy the following properties:

(9.22)  ∫_{z_{1}}^{z_{3}}∫_{x}^{y}ω = ∫_{z_{1}}^{z_{2}}∫_{x}^{y}ω + ∫_{z_{2}}^{z_{3}}∫_{x}^{y}ω, for all z_{1}, z_{2}, z_{3} ∈ H_{p};
(9.23) ∫_{z_{1}}^{z_{2}}∫_{x_{1}}^{x_{3}}ω = ∫_{z_{1}}^{z_{2}}∫_{x_{1}}^{x_{2}}ω + ∫_{z_{1}}^{z_{2}}∫_{x_{2}}^{x_{3}}ω, for all x_{1}, x_{2}, x_{3} ∈ ℙ_{1}(ℚ);
(9.24) ∫_{γz_{1}}^{γz_{2}}∫_{γx}^{γy}ω = w^{|γ|_{p}}w_{∞}^{|γ|_{∞}} ∫_{z_{1}}^{z_{2}}∫_{x}^{y}ω, for all γ ∈ R^{×}.

Proof. The first and second identity are a direct consequence of definition 9.7, while the third follows from lemma 9.6. See exercise 4 for details. □

We emphasize once again that the symbol ω is used as a placeholder and does not refer to an independently defined mathematical object. Only the form f in S_{2}(T , Γ) is defined, but this is enough to make sense of definition 9.7. The notation suggests that the left hand side of definition 9.7 be viewed as a period for a form of weight (2, 2) on (H_{p} × H)/Γ, with the complex period Ω_{+} or Ω_{−} “factored out”.

To obtain stronger formulae, it is preferable to avoid choosing a p-adic logarithm, exploiting the fact that κ_{f}{x→y} is ℤ-valued to make the same multiplicative refinement as in chapter 5 to define

(9.25)  × ∫_{z_{1}}^{z_{2}}∫_{x}^{y}ω := × ∫_{ℙ_{1}(ℚ_{p})} (\frac{t − z_{2}}{t − z_{1}} ) dμ_{f}\{x→y\}(t) ∈ ℂ_{p}^{×}.

Here × ∫ denotes the multiplicative integral, in which limits of products replace the usual limits of Riemann sums, as in equation (5.8) of chapter 5.

Properties analogous to those of lemma 9.8, with addition replaced by multiplication, hold for the multiplicative integral:

Lemma 9.9. The double multiplicative integral of definition 9.7 satisfies the following properties:

(9.26)  × ∫_{z_{1}}^{z_{3}}∫_{x}^{y}ω = × ∫_{z_{1}}^{z_{2}}∫_{x}^{y}ω × × ∫_{z_{2}}^{z_{3}}∫_{x}^{y}ω, for all z_{1}, z_{2}, z_{3} ∈ H_{p};
(9.27) × ∫_{z_{1}}^{z_{2}}∫_{x_{1}}^{x_{3}}ω = × ∫_{z_{1}}^{z_{2}}∫_{x_{1}}^{x_{2}}ω × × ∫_{z_{1}}^{z_{2}}∫_{x_{2}}^{x_{3}}ω, for all x_{1}, x_{2}, x_{3} ∈ ℙ_{1}(ℚ);

(9.28) × ∫_{γz_{1}}^{γz_{2}}∫_{γx}^{γy}ω = (× ∫_{z_{1}}^{z_{2}}∫_{x}^{y}ω)^{w^{|γ|p} w_{∞}^{|γ|∞}} , for all γ ∈ R^{×}.

Proof. The proof is identical to that of lemma 9.8; see exercise 4. □

9.4. Some p-adic cocycles

Given any τ ∈ H_{p}, guided by equation (8.5) of the previous chapter in the case n = 1, we define a 2-cochain κ_{τ} ∈ Z^{2}(Γ, ℂ_{p}^{×}) by choosing a base point x ∈ ℙ_{1}(ℚ) and setting

κ_{τ} (γ_{1}, γ_{2}) = × ∫_{τ}^{γ_{1}τ}∫_{γ_{1}x}^{γ_{1}γ_{2}x}ω.

One verifies (by the same calculations that were performed in the previous chapter) that κ_{τ} is a two-cocycle and that its image in H^{2}(Γ, ℂ_{p}^{×}) depends only on the form f, not on the base points x and τ that were used to define it. The following is a p-adic analogue of conjecture 8.6 of chapter 8.

Conjecture 9.10. Let q ∈ ℚ_{p}^{×} be Tate’s p-adic period attached to E. The natural image of κ_{τ} in H^{2}(Γ, ℂ_{p}^{×}/q^{ℤ}) is 0.

A part of conjecture 9.10 is established in [Da01], where it is shown to be related to conjectures of Mazur, Tate and Teitelbaum [MTT] on values of derivatives of p-adic L-functions, conjectures that were later established by Greenberg and Stevens [GS93]. Assume from now on that conjecture 9.10 holds for E.

Corollary 9.11. There exists a one-cochain

ξ̃_{τ} ∈ C^{1}(Γ, ℂ_{p}^{×}/q^{ℤ})

such that κ_{τ} = dξ̃_{τ} (mod q^{ℤ}).

We note that the the one-cochain ξ̃_{τ} is well-defined modulo

Z^{1}(Γ, ℂ_{p}^{×}/q^{ℤ}) = hom(Γ, ℂ_{p}^{×}/q^{ℤ}).

We now invoke the following counterpart of theorem 8.9 of chapter 8:

Theorem 9.12. The abelianisation of Γ is finite. In particular the cohomology group H^{1}(Γ, ℂ_{p}^{×}/q^{ℤ}) is finite.

Proof. See [Ih68], chapter 3, §1-7, or exercise 9. □

Let e_{Γ} denote the exponent of H^{1}(Γ, ℂ_{p}^{×}/q^{ℤ}). Then the cochain

ξ_{τ} := e_{Γ}ξ̃_{τ}

is a well-defined element of C^{1}(Γ, ℂ_{p}^{×}/q^{ℤ}).

The cochain ξ_{τ} depends, of course, on the choice of base points x and τ. To analyse its dependence on x, observe that if y is another base point in ℙ_{1}(ℚ), and κ_{τ}^{x} and κ_{τ}^{y} are the respective associated 2-cocyles, then as in the proof of lemma 8.4,

κ_{τ}^{x} − κ_{τ}^{y} = dρ_{τ}^{x,y},

where ρ_{τ}^{x,y} ∈ C^{1}(Γ, ℂ_{p}^{×}/q^{ℤ}) is defined by

ρ_{τ}^{x,y}(γ) = × ∫_{τ}^{γτ}∫_{γx}^{γy}ω.

In particular, this cochain vanishes on the stabiliser subgroup Γ_{τ} ⊂ Γ of τ.

It follows that the restriction θ_{τ} of ξ_{τ} to Γ_{τ} is independent of the choices of x and ξ̃_{τ} that were made in defining it.

The definition of θ_{τ} makes it clear that this one-cochain is in fact a well-defined homomorphism from Γ_{τ} to ℂ_{p}^{×}/q^{ℤ}. Hence we have associated (as in the case n = 1 of chapter 8) a well-defined element

θ_{τ} ∈ hom(Γ_{τ} , ℂ_{p}^{×}/q^{ℤ})

to any τ ∈ H_{p}.

9.5. Stark-Heegner points

Of course, the invariant θ_{τ} can only be interesting if the stabiliser Γ_{τ} is nontrivial, i.e., if τ belongs to H′_{p}. This occurs precisely when τ belongs to K ∩ H_{p}, where K ⊂ ℂ_{p} is a real quadratic field in which p is inert or ramified.

More specifically, we will assume that K is a real quadratic subfield of ℂ_{p} satisfying the following “modified Heegner hypotheses” relative to N, analogous to those imposed in chapter 3, except that the roles of the places p and ∞ have been interchanged:

(1) p is inert in K;

(2) all the primes dividing M are split in K. Note that one then has

S_{E,K} = {p, ∞_{1}, ∞_{2}} ∪ {λ|M},

which has odd cardinality, so that sign(E, K) = −1 in this case.

For any τ ∈ H_{p} ∩ K we define as in the previous section the associated order O_{τ} to be the ring of matrices in M_{0}(M)[1/p] which preserve the line spanned by the column vector (τ, 1). This order is isomorphic to a ℤ[1/p]-order in K and hence the group O_{τ,1}^{×} of elements of O_{τ} of determinant one has rank one. In fact, the stabiliser Γ_{τ} of τ in Γ is identified with O_{τ,1}^{×}/⟨±1⟩ and is therefore a cyclic group of infinite order. Let γ_{τ} denote a generator of Γ_{τ}.

The element

J_{τ} = θ_{τ} (γ_{τ}) = ξ_{τ} (γ_{τ}) ∈ ℂ_{p}^{×}/q^{ℤ}

is a canonical invariant in ℂ_{p}^{×}/q^{ℤ} attached to τ ∈ H′_{p}, a p-adic analogue of the invariant that was also denoted by J_{τ} in chapter 8. (Of course, the most direct analogy is with the case n = 1 of the construction of chapter 8.)

Let

P_{τ} = Φ_{Tate}(J_{τ}) ∈ E(ℂ_{p}),

and define:

Φ_{N}^{′(p)}(τ) := P_{τ} .

Fixing τ ∈ H′_{p}, let H^{+} denote the narrow ring class field of K attached to the order O_{τ}, and let H denote the usual ring class field attached to that order. The field H^{+} is an extension of H of degree at most 2 which is trivial precisely when O_{τ}^{×} contains an element of norm −1. The Galois group of H^{+}/H is generated by complex conjugation c, defined using any complex embedding of H^{+}. View H and H^{+} as subfields of ℂ_{p} by fixing an embedding H^{+} ⊂ ℂ_{p}.

Conjecture 9.13. Let τ ∈ H′_{p} be a special point, Then

P_{τ} = Φ^{′(p)}(τ) belongs to E(H^{+}),

and

cP_{τ} = w_{∞}P_{τ} .

This conjecture, like the main conjecture of chapter 8, can be made more precise by formulating a Shimura reciprocity law for the points P_{τ}. The reader is referred to section 5.2 of [Da01] where this is spelled out in detail, or better yet, invited to work through exercise 6.

9.6. Computing Stark-Heegner points

As in the constructions that were described in chapter 8, the main difficulty in making the definition of J_{τ} explicit and computable is the need to produce a 1-cochain ξ̃_{τ} satisfying the relation

dξ̃_{τ} = κ_{τ} .

We now outline a method for computing ξ̃_{τ}, at least in an important special case.

Recalling the notations of section 2.7, for any abelian group A with trivial Γaction let M(A) denote the group of A-valued modular symbols as in definition 2.15 of chapter 2. The group Γ acts on M(A) by the rule

γ · m{x→y} = m{γ^{−1}x→γ^{−1}y}.

Note that in general Γ does not act transitively on ℙ_{1}(ℚ). To remedy this one chooses a base point x ∈ ℙ_{1}(ℚ) and writes M_{0}(A) for the group of functions on (Γx) × (Γx) arising from the restriction of a modular symbol on ℙ_{1}(ℚ) × ℙ_{1}(ℚ).

A point τ ∈ H_{p} gives rise to a one-cochain c_{τ} ∈ C^{1}(Γ, M_{0}(ℂ_{p}^{×})) by the rule

c_{τ} (γ){x→y} = × ∫_{τ}^{γτ}∫_{x}^{y}ω.

It can be checked by a direct computation that c_{τ} is a one-cocycle.

Let c̄_{τ} be the natural image of c_{τ} in H^{1}(Γ, M_{0}(ℂ_{p}^{×})). Let F(ℂ_{p}^{×}) denote the space of ℂ_{p}^{×}-valued functions on Γx, equipped with the natural Γ-action. Consider the exact sequence of Γ-modules

(9.29)  0 ⟶ ℂ_{p}^{×} ⟶^{i} F(ℂ_{p}^{×}) ⟶^{Δ} M_{0}(ℂ_{p}^{×}) ⟶ 0,

where i is the obvious inclusion into the space of constant functions, and Δ is defined by

Δg{x→y} := g(y) − g(x).

Taking the Γ-cohomology of (9.29) yields a connecting homomorphism

δ : H^{1}(Γ, M_{0}(ℂ_{p}^{×})) ⟶ H^{2}(Γ, ℂ_{p}^{×}),

and it can be checked that

δc̄_{τ} = κ̄_{τ}^{#},

where κ̄_{τ}^{#} is the cohomology class represented by the 2-cocycle

κ_{τ}^{#}(g_{0}, g_{1}) := κ_{τ} (g_{1}^{−1}, g_{0}^{−1}).

It follows that δc̄_{τ} is a coboundary precisely when κ̄_{τ} is, and the same holds after replacing ℂ_{p}^{×} by ℂ_{p}^{×}/q^{ℤ} for any q ∈ ℂ_{p}^{×}. Therefore the following conjecture (made under hypothesis 9.4) is a natural strengthening of conjecture 9.10.

Conjecture 9.14. The natural image of c_{τ} in H^{1}(Γ, M_{0}(ℂ_{p}^{×}/q^{ℤ})) is trivial.

This conjecture implies the existence of a modular symbol

η̃_{τ} ∈ M_{0}(ℂ_{p}^{×}/q^{ℤ})

with the property that

(9.30)  × ∫_{τ}^{γτ}∫_{x}^{y}ω = η̃_{τ} {γ^{−1}x→γ^{−1}y} ÷ η̃_{τ} {x→y} (mod q^{ℤ}).

Of course the modular symbol η̃_{τ} is only well-defined modulo H^{0}(Γ, M_{0}(ℂ_{p}^{×}/q^{ℤ})). It can be shown (cf. exercise 7) that this group injects into hom(Γ, ℂ_{p}^{×}/q^{ℤ}), hence is annihilated by the integer e_{Γ} introduced in the previous section. Setting

η_{τ} = e_{Γ} · η̃_{τ} ,

we see that this modular symbol is independent of the choice of η̃_{τ} satisfying (9.30).

We define the “semi-indefinite integral”

× ∫ ^{τ}∫_{x}^{y}e_{Γ}ω := η_{τ} {x→y} ∈ ℂ_{p}^{×}/q^{ℤ},

for any τ ∈ H_{p} and x, y ∈ Γx_{0}. This expression satisfies all the properties suggested by the notation (cf. exercise 8)

(9.31)  × ∫ ^{τ}∫_{x}^{y}e_{Γ}ω × × ∫ ^{τ}∫_{y}^{z}e_{Γ}ω = × ∫ ^{τ}∫_{x}^{z}e_{Γ}ω, for all x, y, z ∈ ℙ_{1}(ℚ);
(9.32) × ∫ ^{τ_{2}}∫_{x}^{y}e_{Γ}ω ÷ × ∫ ^{τ_{1}}∫_{x}^{y}e_{Γ}ω = (× ∫_{τ_{1}}^{τ_{2}}∫_{x}^{y}ω)^{e_{Γ}} , for all τ_{1}, τ_{2} ∈ H_{p};
(9.33)  × ∫ ^{γτ}∫_{γx}^{γy}e_{Γ}ω = × ∫ ^{τ}∫_{x}^{y}e_{Γ}ω for all γ ∈ Γ.

More importantly, the cochain ξ_{τ} of the previous section can be expressed in terms of the semi-indefinite integral attached to η_{τ}:

Proposition 9.15. For all γ ∈ Γ,

ξ_{τ} (γ) = × ∫ ^{τ}∫_{x}^{γx}e_{Γ}ω.

Proof. Let ξ_{τ}^{o} denote the expression on the right hand side. In light of the property characterising ξ_{τ} it suffices to show that

dξ_{τ}^{o}(g_{0}, g_{1}) = κ_{τ} (g_{0}, g_{1})^{e_{Γ}}.

But

dξ_{τ}^{o}(g_{0}, g_{1}) = × ∫ ^{τ}∫_{x}^{g_{1}x}e_{Γ}ω ÷ × ∫ ^{τ}∫_{x}^{g_{0}g_{1}x}e_{Γ}ω × × ∫ ^{τ}∫_{x}^{g_{0}x}e_{Γ}ω
= × ∫ ^{τ}∫_{x}^{g_{1}x}e_{Γ}ω × × ∫ ^{τ}∫_{g_{0}g_{1}x}^{g_{0}x}e_{Γ}ω

= × ∫ ^{τ}∫_{x}^{g_{1}x}e_{Γ}ω ÷ × ∫ ^{g_{0}^{−1}τ}∫_{x}^{g_{1}x}e_{Γ}ω

= (× ∫_{g_{0}^{−1}τ}^{τ}∫_{x}^{g_{1}x}ω)^{e_{Γ}} = (× ∫_{τ}^{g_{0}τ}∫_{g_{0}x}^{g_{0}g_{1}x}ω)^{e_{Γ}} .

The proposition follows. □

The calculation of J_{τ} = ξ_{τ} (γ_{τ}) is thus reduced to computing the indefinite integral

(9.34)  × ∫ ^{τ}∫_{x}^{y}e_{Γ}ω.

We present a method for doing this, at least in the special case where M = 1 so that Γ = PSL_{2}(ℤ[1/p]). (This corresponds to the case where E has prime conductor p.) Note that in this case one may choose e_{Γ} = 1 (cf. exercise 10).

First observation. By the remark in the paragraph titled “first observation” in section 2.7, it is enough to compute (9.34) when x and y are adjacent cusps in the sense of that paragraph.

Second observation: Any pair of adjacent cusps is Γ-equivalent to (0, ∞). By the Γ-equivariance property of (9.33), it is enough to be able to compute the expression

× ∫ ^{τ}∫_{0}^{∞}e_{Γ}ω (as a function of τ ∈ H_{p}).

Third observation: An elementary computation (cf. [DG01], or exercise 11) shows that

(9.35)  × ∫ ^{τ}∫_{0}^{∞}e_{Γ}ω = (× ∫_{1− \frac{1}{τ}}^{τ−1}∫_{0}^{∞}ω)^{e_{Γ}} .

In this way the calculation of ξ_{τ} is reduced to that of the double multiplicative integrals defined in section 9.3.

In more general situations (both in the S-arithmetic case of this chapter, and in the Hilbert modular setting of the previous chapter), the calculation of the cochain ξ_{τ} presents interesting difficulties, and it would be useful to develop feasible algorithms for performing this computation.

Further results

The theory of integration on H_{p} × H and its application to defining “Stark-Heegner points” attached to real quadratic fields is explained in [Da01]. Numerical evidence for conjecture 9.13, resting on the explicit definition of the map Φ_{N}^{′(p)} described in section 9.6, is given in [DG01]. Some of the ideas explained in this chapter are also covered (with a different emphasis focussing more strongly on padic L-functions) in [BD01] and [BDG03].

For various perspectives on the action of subgroups of PSL_{2}(ℤ[1/p]) on H_{p}×H, see [Ih68], [Ih79], [Se80], and [Sta87].

Exercises

(1) Let Γ be the finite index subgroup of SL_{2}(ℤ[1/p]) consisting of the matrices which are upper triangular modulo M.

(a) Show that this group acts on H and on the p-adic upper half plane H_{p} with dense orbits, but that the action of Γ on H_{p} × H is discrete.

(b) Let T = T_{0} ∪ T_{1} denote the Bruhat-Tits tree of H_{p}, equipped with its natural Γ-action. Show that there are precisely two (resp one) orbits for the action of Γ on T_{0} (resp. T_{1}).

(c) Show that the stabiliser of a vertex of T in Γ is conjugate to the group

Γ_{0}(M), while the stabiliser of an edge is conjugate to the group Γ_{0}(N).

(2) Let Γ = SL_{2}(ℤ[1/p]) be the group acting on H×H_{p} by Möbius transformations. A point τ ∈ H is called a special point if its stabiliser in Γ is infinite.

(a) Show that if τ is a special point, then τ belongs to H ∩ K, where K is a quadratic imaginary subfield of ℂ in which p is split. Write p = 𝔭𝔭̄.

(b) Let j be the usual modular j-function. Show that if τ is special, then j(τ) belongs to the ring Oℚ̄ of algebraic integers. (Hint: use the theory developed in chapter 3, particularly exercise (2) of that chapter.)

(c) Let 𝔭̄ be an ideal of Oℚ̄ above p and let

red : Oℚ̄ ⟶ 𝔽̄_{p}

denote the corresponding reduction map. Given a special point τ in H, define the weight of τ by letting α_{1} and α_{2} ∈ K denote the eigenvalues of a generator for the stabiliser subgroup Γ_{τ}, and setting

wt(τ) = max(|ord_{𝔭}(α_{1})|, |ord_{𝔭}(α_{2})|).

Letting n = wt(τ), show that

η(τ) := red(j(τ))

belongs to 𝔽_{p^{n}} and to no smaller extension of 𝔽_{p}, where n is the weight of τ.

(d) * Let SS ⊂ ℙ_{1}(𝔽_{p^{2}}) denote the set of supersingular j-invariants in characteristic p, and let H′ denote the set of special points of H. Show that the map

η : H′/Γ ⟶ 𝔽̄_{p} − SS

is a bijection. (See also [Ih68], chapter 5.)

(3) Prove lemma 9.2 in the text.

(4) Prove that the double multiplicative integral satisfies the properties given in lemma 9.9.

(5) Let H_{p}^{unram} denote the set of points in H_{p} which map to T_{0} under the reduction map defined in chapter 5. Define a ℂ-valued double integral attached to a form in S_{2}(T , Γ) by the rule

× ∫_{τ_{1}}^{τ_{2}}∫_{τ_{3}}^{τ_{4}}ω := ∑_{e:r(τ_{3})→r(τ_{4})} ∫_{τ_{1}}^{τ_{2}} f_{e}(z)dz, τ_{1}, τ_{2} ∈ H, τ_{3}, τ_{4} ∈ H_{p}^{unram},

the the sum being taken over the ordered edges in the path joining r(τ_{3}) to r(τ_{4}).

(a) Show that this integral satisfies all the formal properties of the ℂ_{p}^{×}-valued integral stated in lemma 9.8.

(b) Mimic the constructions of chapters 8 and 9 to define a map

Φ′_{N} : H′/Γ ⟶ E(ℂ).

(Hint: After defining κ_{τ} in the obvious way, the analogue of conjectures 8.6 and 9.10 can be proved using theorem 2.19 of section 2.7. This should yield an explicit formula for a one-cochain ξ_{τ} ∈ ℂ^{1}(Γ, ℂ/Λ_{E}) satisfying dξ_{τ} = e_{Γ}κ_{τ} (mod Λ_{E})).

(c) State the analogue of conjecture 9.13 in this setting, and prove it using the theory developed in chapter 3. (For more details on this exercise see section 4 of [BDG03].)

(6) Formulate precisely a Shimura reciprocity law in the style of the conjectures of chapter 8 for the points P_{τ} defined in this chapter.

(7) Construct an injective homomorphism from H^{0}(Γ, M_{0}(A)) to hom(Γ, A) (for any abelian group A).

(8) Show that the indefinite multiplicative integral satisfies the properties listed in equations (9.31), (9.32), and (9.33). (Conclude that the same is true for the properties listed in lemma 9.8.)

(9) Let Γ ⊂ PSL_{2}(ℤ[1/p]) be the group of matrices which are upper triangular modulo M. Let A be any Γ-module, and let F(T_{0}, A) and F(T_{1}, A) denote the module of A-valued functions on the set of vertices and unordered edges of T respectively.

(a) Construct an exact sequence of Γ-modules

0 ⟶ A ⟶ F(T_{0}, A) ⟶ F(T_{1}, A) ⟶ 0.

(b) Let Γ′_{0}(M) = α_{p}Γ_{0}(M)α_{p}^{−1}, where α_{p} is the element of Γ of determinant p satisfying α_{p}e_{o} = ē_{o}. Construct an exact sequence

⋯ ⟶ H^{i}(Γ_{0}(M), A) ⊕ H^{i}(Γ′_{0}(M), A) ⟶ H^{i}(Γ_{0}(N), A) ⟶^{δ}

⟶ H^{i+1}(Γ, A) ⟶ H^{i+1}(Γ′_{0}(M), A) ⊕ H^{i+1}(Γ′_{0}(M), A) ⟶ ⋯

(Hint: Take the Γ-cohomology of the exact sequence displayed in (a) and apply Shapiro’s lemma.)

(c) Use this exact sequence to analyse the abelianisation of Γ.

(10) Let Γ = PSL_{2}(ℤ[1/p]). Show that the space M(A)^{Γ} is trivial, for any abelian group A. Compute the abelianisation of Γ in this case.

(11) Prove equation (9.35) in the text.

(12) Let O = ℤ[τ] be the real quadratic order defined by

τ = \{ \begin{matrix}\sqrt{n^{2} + 1} & if n ≥ 2 is even; \\ \frac{n+\sqrt{n^{2}+4}}{2} & if n is odd. \end{matrix}

EXERCISES 111

This order has fundamental unit η of norm −1 given by

η = \{ \begin{matrix}n + \sqrt{n^{2} + 1} & if n ≥ 2 is even; \\ \frac{n+\sqrt{n^{2}+4}}{2} & if n is odd. \end{matrix}

Show that

J_{τ} = × ∫_{−η}^{η′}∫_{0}^{∞}ω,

where η′ is the Galois conjugate of η.

CHAPTER 10 Kolyvagin’s theorem

Let E be an elliptic curve over ℚ and let K be a quadratic extension of ℚ satisfying sign(E, K) = −1. A theme of the last few chapters has been the construction of so-called “Heegner Systems” attached to the pair (E, K). Recall that if {P_{n}}_{(n,N)=1} is such a Heegner system, one sets P_{K} := Trace_{H/K}(P_{1}).

The goal of this chapter is to explain the main ideas behind the proof of Kolyvagin’s Theorem whose statement, given in chapter 3, we now recall:

Theorem 10.1 (Kolyvagin). If P_{K} is a point of infinite order, then

(1) The rank of E(K) is equal to 1.

(2) The Shafarevich-Tate group of E over K is finite.

Like the proof of the weak Mordell-Weil theorem sketched in chapter 1, the proof of theorem 10.1 proceeds by studying E(K)/pE(K) – or rather, the p-Selmer group Sel_{p}(E/K) – for a suitably chosen “descent prime” p. It will be convenient to assume that p satisfies the following conditions:

(1) p does not divide 6N;

(2) The natural homomorphism from G_{ℚ} to Aut(E_{p}) ≃ GL_{2}(𝔽_{p}) is surjective. We will then prove the following “mod p” version of Kolyvagin’s theorem:

Theorem 10.2. If the image of P_{K} in E(K)/pE(K) is non-zero, then the p-Selmer group Sel_{p}(E/K) is generated by δ(P_{K}). In particular E(K) has rank one and LLI(E/K)_{p} is trivial.

Remark 10.3. Theorem 10.2 does not imply the full strength of theorem 10.1, because of the restrictions that are made on p. However, if E has no complex multiplications, these restrictions exclude only finitely many primes p, by a theorem of Serre [Se72]:

Theorem 10.4 (Serre). Let E be an elliptic curve defined over a number field F. Assume that E has no complex multiplications, i.e., that EndF̄(E) = ℤ. Then the natural homomorphism from G_{F} to Aut(E_{p}) is surjective, for all but finitely many primes p.

Remark 10.5. Note that theorem 10.2 applied to a single prime p which does not divide P_{K} and satisfies conditions (1) and (2) above yields the most interesting consequence of Kolyvagin’s theorem: for instance it is enough to prove part (1) of theorem 10.1. It therefore makes sense to focus on its proof, which conveys the main ideas while avoiding some technical complications that appear in the full proof of theorem 10.1. This is what will be done in the remainder of chapter 10.

10.1. Bounding Selmer groups

We begin by presenting a general approach for bounding the size of Selmer groups which plays an important role in Kolyvagin’s argument (and also in Wiles’ proof of the Shimura-Taniyama-Weil conjecture; for this latter point, see [DDT95], section 2.3).

Let K be any number field and let G_{K} = Gal(K̄/K) denote its absolute Galois group endowed with the Krull topology. Let M be any finite module equipped with a continuous action of G_{K}. Recall the Galois cohomology group

H^{1}(K, M) = H^{1}(G_{K}, M),

defined as the group of continuous one-cocycles on G_{K} with values in M, modulo the group of 1-coboundaries on G_{K}.

A prime v of K is said to be good for M if

(1) M is unramified at v;

(2) v does not divide the cardinality of M. Let I_{v} be the inertia subgroup of G_{v} = G_{K_{v}}, viewed as a subgroup of G_{K} by fixing an embedding of K̄_{v} into K̄. Let K_{v}^{nr} denote the maximal unramified extension of K_{v}; its Galois group is identified with G_{v}/I_{v} and is isomorphic to ℤ̂, with a canonical topological generator given by the frobenius element Frob_{v} at v.

If v is good, one disposes of the short exact inflation-restriction sequence

0 ⟶ H^{1}(K_{v}^{nr}/K_{v}, M) ⟶^{inf} H^{1}(K_{v}, M) ⟶^{∂_{v}} H^{1}(I_{v}, M)^{G_{Kv}} ⟶ 0.

The image of the group H^{1}(K_{v}^{nr}/K_{v}, M) under inflation is called the finite or unramified part of H^{1}(K_{v}, M), and is denoted H_{f}^{1}(K_{v}, M). The quotient

H_{s}^{1}(K_{v}, M) := H^{1}(K_{v}, M)/H_{f}^{1}(K_{v}, M) = H^{1}(I_{v}, M)^{G_{Kv}}

is called the singular part or singular quotient of H^{1}(K_{v}, M). Following a terminology suggested by Mazur, the natural projection

∂_{v} : H^{1}(K_{v}, M) ⟶ H_{s}^{1}(K_{v}, M)

will be referred to as the residue map at v. If c ∈ H^{1}(K, M) is a global cohomology class, we will also denote by c_{v} its natural image in H^{1}(K_{v}, M) under the restriction map to G_{v}, and set, by abuse of notation, ∂_{v}(c) := ∂_{v}(c_{v}).

Definition 10.6. A set of Selmer conditions attached to M and K is a collection of subgroups L_{v} ⊂ H^{1}(K_{v}, M) for each place v of K, such that

L_{v} = H_{f}^{1}(K_{v}, M) for all but finitely many v.

Note that this definition makes sense, since for a given M all but finitely many places v are good and therefore the groups H_{f}^{1}(K_{v}, M) are defined for them.

Definition 10.7. Let L = {L_{v}}_{v} be a set of Selmer conditions attached to M and K. The Selmer group attached to the triple (K, M, {L_{v}}), denoted H_{L}^{1}(K, M), is the set of c ∈ H^{1}(K, M) such that c_{v} belongs to L_{v}, for all v.

The principal example of interest to us is the one where M = E_{p} is the module of p-division points of an elliptic curve and where

L_{v} = δ_{v}(E(K_{v})/pE(K_{v})).

The fact that L_{v} is a set of Selmer conditions follows from the proof of proposition 1.7 (more precisely, the part of that proof that is developed in exercise 7 of chapter 1). By definition,

H_{L}^{1}(K, E_{p}) = Sel_{p}(E/K),

where Sel_{p}(E/K) is the p-Selmer group of chapter 1 attached to E/K.

We have the following general finiteness property for Selmer groups: Proposition 10.8. If L is any set of Selmer conditions for M, then

#H_{L}^{1}(K, M) < ∞.

Proof. This is a direct generalisation of exercise 8 and is left to the reader. □

We now present a technique for bounding the orders of Selmer groups, whereby this question is turned into the problem of manfacturing classes in H^{1}(K, M) with prescribed residues.

For this it is necessary to introduce some further notions arising from Tate local duality in Galois cohomology. Let M^{∗} := hom(M, G_{m}) denote the Kummer dual of M, equipped with its natural G_{K}-action

(σf)(m) := σf(σ^{−1}m), for σ ∈ G_{K}, f ∈ M^{∗}, m ∈ M

arising from the action of G_{K} on M and on the roots of unity. The cup product combined with the calculation of the local Brauer group in local class field theory yields a pairing

⟨ , ⟩_{v} : H^{1}(K_{v}, M) × H^{1}(K_{v}, M^{∗}) ⟶ H^{2}(K_{v}, G_{m}) = ℚ/ℤ.

Theorem 10.9 (Tate). The pairing ⟨ , ⟩_{v} is a non-degenerate bilinear pairing. If v is a good prime for M, it is also good for M^{∗} and the groups H_{f}^{1}(K_{v}, M) and H_{f}^{1}(K_{v}, M^{∗}) are orthogonal complements of each other under this pairing.

Proof. For a discussion and proof of this theorem, see [Ta62], §2 and [Mi86], ch, I. (See also exercise 3 for a discussion in the case where v is a good prime for M.) □

Given L_{v} ⊂ H^{1}(K_{v}, M), let L_{v}^{∗} ⊂ H^{1}(K_{v}, M^{∗}) denote the exact annihilator of L_{v} under the local Tate pairing. It follows immediately from Tate’s theorem that if the groups L_{v} form a set of Selmer conditions for M, then the groups L_{v}^{∗} are a collection of Selmer conditions for H^{1}(K, M^{∗}).

Definition 10.10. The Selmer group attached to M^{∗} and L^{∗} := {L_{v}^{∗}}_{v} is called the dual Selmer group of H_{L}^{1}(K, M).

While the orders of H_{L}^{1}(K, M) and its dual Selmer group H_{L^{∗}}^{1}(K, M^{∗}) are subtle global invariants which are typically difficult to compute, the ratio of these orders is a product of simple local terms which can be calculated in practice without difficulty. This is the content of the following key result.

Theorem 10.11 (Duality theorem for Selmer groups).

\frac{#H_{L}^{1}(K, M)}{#H_{L^{∗}}^{1}(K, M^{∗})} = \frac{#H^{0}(K, M)}{#H^{0}(K, M^{∗})} ∏_{v} \frac{#L_{v}}{#H^{0}(K_{v}, M)}.

See [DDT95], §2.3 for an outline of the proof, which relies on the full force of global class field theory, and specifically on the Poitou-Tate nine-term exact sequence given in thm. 4.10 of ch. I of [Mi86].

Note the analogy between theorem 10.11 and the Riemann-Roch theorem. For this reason the term appearing on the right-hand side of theorem 10.11 will be denoted χ_{L}(K, M) and called the Euler characteristic attached to H_{L}^{1}(K, M).

Fix a collection {L_{v}} of Selmer conditions for M. Let S be any finite set of good primes for M, chosen so that

L_{v} = H_{f}^{1}(K_{v}, M) for all v ∈ S.

Definition 10.12. The relaxed Selmer group at S, denoted H_{(S)}^{1}(K, M), is the set of classes c ∈ H^{1}(K, M) such that

c_{v} belongs to L_{v}, for all v ∉ S.

Definition 10.13. The restricted Selmer group at s, denoted H_{[S]}^{1}(K, M), is the set of classes in c ∈ H_{L}^{1}(K, M) such that

c_{v} = 0 for all v ∈ S.

There are obvious inclusions

H_{[S]}^{1}(K, M) ⊂ H_{L}^{1}(K, M) ⊂ H_{(S)}^{1}(K, M).

It is also clear that H_{(S)}^{1}(K, M) and H_{[S]}^{1}(K, M^{∗}) are dual Selmer groups in the sense of definition 10.10. Hence theorem 10.11, applied to both H_{L}^{1}(K, M) and H_{(S)}^{1}(K, M) yields the following useful identity:

(10.1)  \frac{#H_{(S)}^{1}(K, M)}{#H_{[S]}^{1}(K, M^{∗})} = χ_{L}(K, M) · # (⊕_{v∈S}H_{s}^{1}(K_{v}, M)) .

A finite set S of good primes for M^{∗} is said to control the Selmer group H_{L^{∗}}^{1}(K, M^{∗}) if H_{[S]}^{1}(K, M^{∗}) = 0, i.e., if the natural map

H_{L^{∗}}^{1}(K, M^{∗}) ⟶ ⊕_{v∈S}H_{f}^{1}(K_{v}, M^{∗})

is injective.

The following proposition is a key ingredient in the proof of Kolyvagin’s theorem, as well as in many of the arguments which bound orders of Selmer groups:

Theorem 10.14. Suppose that χ_{L}(K, M) = 1, and let S be a set of good primes which controls H_{L^{∗}}^{1}(K, M^{∗}). Then the cardinality of H_{L}^{1}(K, M) is equal to the cardinality of the cokernel of the residue map ∂_{S} := ⊕_{v∈S}∂_{v}:

∂_{S} : H_{(S)}^{1}(K, M) ⟶ ⊕_{v∈S}H_{s}^{1}(K_{v}, M).

Proof. Consider the tautological exact sequence

(10.2)  0 ⟶ H_{L}^{1}(K, M) ⟶ H_{(S)}^{1}(K, M) ⟶^{∂_{S}} ⊕_{v∈S}H_{s}^{1}(K_{v}, M).

The assumptions that χ_{L}(K, M) = 1 and that S controls H_{L^{∗}}^{1}(K, M^{∗}) implies, in light of (10.1), that the two groups appearing on the right of (10.2) have the same cardinality; the result follows. □

Thanks to theorem 10.14, the problem of bounding H_{L}^{1}(K, M) is reduced to that of producing, for a well-chosen set S of primes which controls H_{L^{∗}}^{1}(K, M^{∗}), a supply of cohomology classes in H_{(S)}^{1}(K, M) whose residues at the primes of S can be controlled so that the order of the cokernel of ∂_{S} can be estimated. In Euler system arguments the goal is often to estimate the cokernel of ∂_{S} in terms of a quantity connected with the behaviour of a suitable L-function attached to M and K. (See for instance [Da02] for a discussion of this point of view.)

10.2. Kolyvagin cohomology classes

Returning to the case which is germane to the proof of Kolyvagin’s theorem, let E be an elliptic curve over any number field K, and let p be a prime of good reduction for E. One sets

M = E_{p}, L_{v} = δ_{v}(E(K_{v})/pE(K_{v})),

so that by definition H_{L}^{1}(K, M) = Sel_{p}(E/K), the p-Selmer group attached to E that was defined in chapter 1. Write Sel_{p}^{∗}(E/K) := H_{L^{∗}}^{1}(K, E_{p}) for the dual Selmer group attached to Sel_{p}(E/K). In this case, the non-degenerate alternating Weil pairing E_{p} × E_{p} ⟶ μ_{p} yields an identification of E_{p} with its Kummer dual E_{p}^{∗}, so that both Sel_{p}(E/K) and Sel_{p}^{∗}(E/K) can be viewed as contained in H^{1}(K, E_{p}).

Proposition 10.15. The Euler characteristic χ_{L}(K, E_{p}) is equal to 1. (Hence the groups Sel_{p}(E/K) and Sel_{p}^{∗}(E/K) have the same cardinality.)

Proof. The global term #H^{0}(K, E_{p})/#H^{0}(K, E_{p}^{∗}) appearing in the formula for χ_{L}(K, E_{p}) of theorem 10.11 is equal to 1 since E_{p} and E_{p}^{∗} are isomorphic as G_{K}- modules. (In fact, by the assumptions that were made on p, both the numerator and the denominator in this expression are equal to 1.) Note that the local term χ_{v} := #L_{v}/#H^{0}(K_{v}, E_{p}) is of the form

(10.3)  #(G ⊗ 𝔽_{p})/#G[p], where G = E(K_{v}).

The function which to an abelian group G associates the expression in (10.3) is additive in exact sequences and trivial on finite groups. If v does not divide p∞, then E(K_{v}) is an extension of a finite group by a pro-ℓ group, with ℓ ≠ p. Hence

(10.4)  χ_{v} = 1, for all v |̸p∞.

Let d be the degree of K over ℚ. By a direct calculation,

(10.5)  ∏_{v|∞} χ_{v} = p^{−d}.

Finally, if v divides p, then ∏_{v|p} E(K_{v}) is isomorphic (by the theory of the formal group logarithm; cf. [Si86], ch. IV and VII) to an extension of a finite group by a group which is abstractly isomorphic to ℤ_{p}^{d}. Hence

(10.6)  ∏_{v|p} χ_{v} = p^{d}.

Equations (10.4), (10.5) and (10.6) imply that χ_{L}(K, E_{p}) = 1, as was to be shown. □

Remark 10.16. The Selmer group Sel_{p}(E/K) and its dual Sel_{p}^{∗}(E/K), viewed as subgroups of H^{1}(K, E_{p}), are in fact equal. The most natural way to prove this is by local methods, by showing that L_{v} is its own orthogonal complement under the local Tate pairing on H^{1}(K_{v}, E_{p}). (When v divides neither p nor the conductor of E, so that v is a good prime for E_{p}, this fact has already been explained in a more general setting.) We will not insist on the equality of Sel_{p}(E/K) and its dual, since we will not make use of it in the proof of theorem 10.2. In fact, maintaining the notational distinction between Sel_{p}(E/K) and Sel_{p}^{∗}(E/K) is more illustrative of how a general Euler system argument for bounding H_{L}^{1}(K, M) might proceed, in cases where M is not identified with its Kummer dual. (Such as the study of the adjoint or symmetric square representations attached to E_{p} that arises in Wiles’ work on the Shimura-Taniyama-Weil conjecture.)

For the remainder of this chapter, let us specialise further to the case where E is an elliptic curve over ℚ and K is a quadratic field satisfying sign(E, K) = −1. Let p be a prime satisfying the conditions preceeding the statement of theorem 10.2.

Definition 10.17. A rational prime ℓ is called a Kolyvagin prime relative to (E, K, p) if

(1) ℓ does not divide 2NDisc(K)p, and is inert in K;

(2) p divides the fourier coefficient a_{ℓ} exactly;

(3) p^{2} divides ℓ + 1;

(4) if K is a real quadratic field, then the fundamental unit u_{K} of O_{K}^{×} is a p^{2}-th power in (O_{K}/ℓ)^{×}.

The following proposition asserts that there are plenty of Kolyvagin primes. In fact, sufficiently many can be produced to control Sel_{p}^{∗}(E/K).

Proposition 10.18. There are infinitely many Kolyvagin primes. In fact, there exists a finite set S of Kolyvagin primes which controls Sel_{p}^{∗}(E/K).

Sketch of proof. Let M = K(E_{p^{2}}, u_{k}^{1/p^{2}}) be the finite extension of ℚ obtained by adjoining to K the coordinates of the points of order p^{2} in E and the p^{2}-th roots of u_{K} if K is real quadratic. The condition that ℓ be a Kolyvagin prime can be rephrased as a condition on the Frobenius element at ℓ in the Galois extension M/ℚ (cf. exercise 4) and so it follows that there are infinitely many Kolyvagin primes: in fact, these form a subset of the primes of ℚ having positive Dirichlet density. The construction of a finite set S of Kolyvagin primes which controls Sel_{p}^{∗}(E/K) also follows from a careful application of the Chebotarev density theorem, this time to the finite extension M_{S} of M which is “cut out” by all the elements in Sel_{p}^{∗}(E/K). The details of the proof are involved and the reader may wish to consult [BD03], theorem 3.4, or better yet, work through exercise 4 where they are spelled out. □

We now describe the construction of certain cohomology classes κ(ℓ) indexed by the Kolyvagin primes. Let H_{ℓ} denote the ring class field of K of conductor ℓ, and set

G_{ℓ} = Gal(H_{ℓ}/H), G̃_{ℓ} = Gal(H_{ℓ}/K).

Note that G_{ℓ} is a cyclic group which is canonically identified, via class field theory, with

G_{ℓ} = (O_{K}/ℓ)^{×}/(ℤ/ℓℤ)^{×}⟨u_{K}⟩.

It follows from the definition of a Kolyvagin prime that n_{ℓ} = #G_{ℓ} is divisible by p^{2}. Let Ñ_{1} be an arbitrary lift to ℤ[G̃_{ℓ}] of the norm element ∑_{σ∈Gal(H/K)} σ under the natural projection ℤ[G̃_{ℓ}] ⟶ ℤ[Gal(H/K)]. Denote by σ_{ℓ} a generator of G_{ℓ}, and define the following elements in ℤ[G_{ℓ}] and ℤ[G̃_{ℓ}]:

(10.7)  N_{ℓ} = ∑_{σ∈G_{ℓ}} σ, D_{ℓ} = ∑_{i=0}^{n_{ℓ}−1} iσ_{ℓ}^{i}, Ñ_{ℓ} = N_{ℓ}Ñ_{1}, D̃_{ℓ} = D_{ℓ}Ñ_{1}.

Note that these elements are related by the following basic identities

(σ_{ℓ} − 1)D_{ℓ} = n_{ℓ} − N_{ℓ}, (σ_{ℓ} − 1)D̃_{ℓ} = n_{ℓ}Ñ_{1} − Ñ_{ℓ}.

All four of the group ring elements defined in (10.7) act as ℤ-linear operators on E(H_{ℓ}). Let P_{ℓ} ∈ E(H_{ℓ}) be the point of level ℓ in the Heegner system attached to (E, K), and set

Q_{ℓ} = D̃_{ℓ}P_{ℓ}.

Proposition 10.19. The natural image of Q_{ℓ} in (E(H_{ℓ})/pE(H_{ℓ})) is invariant under the Galois group G̃_{ℓ}.

Proof. The image of the point D_{ℓ}P_{ℓ} in E(H_{ℓ})/pE(H_{ℓ}) is invariant under G_{ℓ}. To see this, note that p divides both n_{ℓ} and a_{ℓ} by the definition of a Kolyvagin prime, and that

(10.8)  (σ_{ℓ} − 1)D_{ℓ}P_{ℓ} = (n_{ℓ} − N_{ℓ})P_{ℓ} = n_{ℓ}P_{ℓ} − a_{ℓ}P_{1}.

The image of Q_{ℓ} in E(H_{ℓ})/pE(H_{ℓ}) is the sum of the Gal(H/K)-translates of the image of D_{ℓ}P_{ℓ} in the group (E(H_{ℓ})/pE(H_{ℓ}))^{G_{ℓ}}. Hence it is invariant under G̃_{ℓ}. □

Remark 10.20. Applying the operator Ñ_{1} to (10.8) gives:

(10.9)  (σ_{ℓ} − 1)Q_{ℓ} = n_{ℓ}Ñ_{1}P_{ℓ} − a_{ℓ}P_{K}.

Because E_{p}(H_{ℓ}) is trivial (cf. exercise 1), multiplication by p is injective on E(H_{ℓ}). Thus one can define a cohomology class κ̄(ℓ) ∈ H^{1}(G̃_{ℓ}, E(H_{ℓ}))_{p} by the rule

(10.10)  κ̄(ℓ)(σ) = \frac{(σ − 1)Q_{ℓ}}{p} .

Let κ̄(ℓ) also denote, by abuse of notation, the natural image of this class in H^{1}(K, E)_{p} under inflation. Finally, denote by κ(ℓ) any lift of κ̄(ℓ) to H^{1}(K, E_{p}) by the map arising in the Kummer descent exact sequence. Note that the residues of κ(ℓ) depend only on κ̄(ℓ) – more precisely, ∂_{v}(κ(ℓ)) ≠ 0 if and only if κ̄(ℓ)_{v} ≠ 0.

Proposition 10.21. The class κ(ℓ) belongs to H_{(ℓ)}^{1}(K, E_{p}), the relaxed Selmer group at ℓ. Furthermore

(1) ∂_{ℓ}(κ(ℓ)) ≠ 0 if and only if the image of P_{K} in E(K_{ℓ})/pE(K_{ℓ}) is non-zero.

(2) The class κ(ℓ) belongs to H^{1}(K, E_{p})^{ϵ}, where ϵ = sign(E, ℚ).

Proof. The fact that ∂_{v}(κ(ℓ)) = κ̄(ℓ)_{v} = 0 when v |̸ℓ follows from the fact that H_{ℓ}/K is unramified outside ℓ, so that the restriction of the cocycle κ̄(ℓ) to any inertia group outside ℓ is identically 0.

To prove (1), note that σ_{ℓ} generates the decomposition group at ℓ in Gal(H_{ℓ}/K). Hence ∂_{ℓ}(κ(ℓ)) is non-zero if and only if κ̄(ℓ)(σ_{ℓ}) is non-zero in E(K_{ℓ})/pE(K_{ℓ}). But by equation (10.9),

κ̄(ℓ)(σ_{ℓ}) = \frac{(σ_{ℓ} − 1)Q_{ℓ}}{p} = −\frac{a_{ℓ}}{p} P_{K} in E(K_{ℓ})/pE(K_{ℓ}),

the second equality holding because p^{2} divides n_{ℓ}. Since p divides a_{ℓ} exactly, the result follows. To prove part (2), one uses the fact that if τ is any lift of the generator of Gal(K/ℚ) to Gal(H_{ℓ}/ℚ), then since this latter group is a generalised dihedral group,

τσ = σ^{−1}τ for all σ ∈ G̃_{ℓ}.

Hence, by exercise 5,

τD_{ℓ} = −D_{ℓ}τ (mod n_{ℓ}).

By definition of a Heegner system (cf. definition 3.12 of chapter 3, particularly the property that is asserted in proposition 3.11)

τD_{ℓ}P_{ℓ} = −D_{ℓ}τP_{ℓ} = ϵσD_{ℓ}P_{ℓ} (mod n_{ℓ}E(H_{ℓ}))

for some σ ∈ G̃_{ℓ}. Part (2) follows after applying Ñ_{1} to this identity. □

In addition to the classes κ(ℓ) depending on a single Kolyvagin prime ℓ, it is important to have at one’s disposal certain classes κ(ℓ_{1}ℓ_{2}) depending (in a symmetrical way) on two distinct Kolyvagin primes ℓ_{1} and ℓ_{2}. These are defined exactly in the same way as the classes κ(ℓ) above, but using the point

Q_{ℓ_{1}ℓ_{2}} = D_{ℓ_{1}}D_{ℓ_{2}} Ñ_{1}P_{ℓ_{1}ℓ_{2}}

arising from the point P_{ℓ_{1}ℓ_{2}} in the Heegner system. The properties of this class are summarised below:

Proposition 10.22. The class κ(ℓ_{1}ℓ_{2}) belongs to H_{(ℓ_{1}ℓ_{2})}^{1}(K, E_{p}), the relaxed Selmer group at {ℓ_{1}, ℓ_{2}}. Furthermore,

(1) ∂_{ℓ_{2}}(κ(ℓ_{1}ℓ_{2})) ≠ 0 if and only if v_{ℓ_{2}}(κ(ℓ_{1})) ≠ 0.

(2) κ(ℓ_{1}ℓ_{2}) belongs to H^{1}(K, E_{p})^{−ϵ}, where ϵ =sign(E, ℚ).

The proof of proposition 10.22 is identical to that of proposition 10.21 and is left as an exercise. (Cf. exercise 6.)

10.3. Proof of Kolyvagin’s theorem

The following strengthening of proposition 10.18 is needed in the proof of Kolyvagin’s theorem:

Proposition 10.23. There exists a finite set S = {ℓ_{1}, …, ℓ_{t}) of Kolyvagin primes with the property that

(1) S controls Sel_{p}^{∗}(E/K).

(2) The image of P_{K} in E(K_{ℓ_{i}})/pE(K_{ℓ_{i}}) is non-zero, for j = 1, …, t.

(3) The value v_{ℓ_{j}}(κ(ℓ_{1})) is non-zero, for j = 2, …, t.

Proof. This follows from the ideas detailed in exercise 4. □

End of proof of Kolyvagin’s theorem. Let S be a set of primes satisfying the conclusion of proposition 10.23. By property (1) stated in this proposition, combined with theorem 10.14, it is enough to bound from above the cokernel of the map

(10.11)  ∂_{S} : H_{(S)}^{1}(K, E_{p}) ⟶ ⊕_{ℓ∈S}H_{s}^{1}(K_{ℓ}, E_{p}).

Note that if ℓ is a Kolyvagin prime, the Frobenius element σ_{ℓ} attached to ℓ in Gal(K̄/K) acts trivially on E_{p}, so that E_{p}(K_{ℓ}) = E_{p}. A direct calculation (cf. part (c) of exercise 3) shows that

H_{f}^{1}(K_{ℓ}, E_{p}) = E_{p}/(σ_{ℓ} − 1)E_{p} = E_{p}, H_{s}^{1}(K_{ℓ}, E_{p}) = hom_{G_{K_{ℓ}}} (μ_{p}, E_{p}) = E_{p}.

Hence the group on the right of (10.11) is an 𝔽_{p}-vector space of dimension 2t, and each eigenspace under the action of τ has dimension t over 𝔽_{p}. We bound the cokernel of ∂_{S} one eigenspace at a time.

Step 1. Restricting the map ∂_{S} to the ϵ-eigencomponents,

∂_{S}^{ϵ} : H_{(S)}^{1}(K, E_{p})^{ϵ} ⟶ ⊕_{j=1}^{t}H_{s}^{1}(K_{ℓ_{j}}, E_{p})^{ϵ},

one sees that the vectors ∂_{S}^{ϵ}(κ_{ℓ_{1}}), …, ∂_{S}^{ϵ}(κ_{ℓ_{t}}) are linearly independent, by proposition 10.21 and property (2) of proposition 10.23. Hence ∂_{S}^{ϵ} is surjective; it follows that Sel_{p}(E/K)^{ϵ} is trivial. In particular E(ℚ) (resp. E(K)/E(ℚ)) is finite if sign(E, ℚ) = 1 (resp. if sign(E, ℚ) = −1), which is consistent with the Birch and Swinnerton-Dyer conjecture over ℚ.

Step 2. By proposition 10.22 and property (3) of proposition 10.23, it likewise follows that the t−1 vectors ∂_{S}(κ(ℓ_{1}ℓ_{2})), …, ∂_{S}(κ(ℓ_{1}ℓ_{t})) are linearly independent and belong to the −ϵ-eigenspace for the action of Gal(K/ℚ) on ⊕_{ℓ∈S}H_{s}^{1}(K_{ℓ}, E_{p}). Hence the map ∂_{S}^{−ϵ} has a cokernel of dimension at most 1. It follows that Sel_{p}(E/K)^{−ϵ} is generated by the non-zero vector δ(P_{K}) so that

dim_{𝔽_{p}}(Sel_{p}(E/K)^{−ϵ}) = 1.

The results obtained in steps 1 and 2 imply that Sel_{p}(E/K) is a one-dimensional 𝔽_{p}-vector space generated by δ(P_{K}). This completes the proof of theorem 10.2. □

References

Kolyvagin’s theorem is proved in [Kol88] and [Kol89]. Our exposition of Kolyvagin’s method was influenced by the presentation in in [Gr89] and by the point of view taken in [DDT95] and [Da02]. Other useful references to the general machinery of Euler systems and Kolyvagin’s argument can be found in [Kol90] and [Ru00] for example.

Exercises

(1) Suppose that p is a descent prime satisfying the hypotheses of chapter 10. Show that if L is any solvable extension of K then E_{p}(L) is trivial.

(2) Suppose that p is a descent prime satisfying the hypotheses of chapter 10. Show that if L = K(E_{p}), the restriction map

H^{1}(K, E_{p}) ⟶ H^{1}(L, E_{p}) = hom(G_{L}, E_{p})

is injective.

(3) Let K_{v} denote the completion of a number field K at a place v and let K_{n}^{nr} denote its maximal unramified extension. Suppose that v is a good place for M.

(a) Show that H^{2}(K_{v}^{nr}/K_{v}, (K_{v}^{nr})^{×}) is trivial. Conclude that if c_{1} and c_{2} belong to H_{f}^{1}(K_{v}, M) and H_{f}^{1}(K_{v}, M^{∗}) respectively, then

⟨c_{1}, c_{2}⟩_{v} = 0.

(b) Show that v is a good place for M^{∗}.

(c) Let σ_{v} ∈ Gal(K_{v}^{nr}/K_{v}) be the Frobenius element at v, and let m = #M. Explicitly produce (without using the Tate local pairing) canonical identifications

H_{f}^{1}(K_{v}, M) = M/(σ_{v} − 1)M, H_{s}^{1}(K_{v}, M^{∗}) = hom(μ_{m}, M)[σ_{v} − 1].

Conclude that H_{f}^{1}(K_{v}, M) and H_{s}^{1}(K_{v}, M^{∗}) are in natural duality, so that in particular they have the same cardinality.

(d) Assuming the non-degeneracy of the local Tate pairing between H^{1}(K_{v}, M) and H^{1}(K_{v}, M), complete the proof of theorem 10.9 when v is a good prime for M.

(4) Let E be an elliptic curve over ℚ and let K be a quadratic extension of ℚ. Let p be a descent prime satisfying the condtions that were imposed in the introduction of chapter 10.

(a) Show that the exension L = K(E_{p^{2}}) is Galois over ℚ with Galois group

Gal(L/ℚ) = Gal(K/ℚ) × Gal(ℚ(E_{p^{2}})/ℚ) = Gal(K/ℚ) × Aut(E_{p^{2}}),

so that elements in Gal(L/ℚ) are identified with pairs (τ^{j}, T) with j ∈ {0, 1} and T ∈ GL_{2}(ℤ/p^{2}ℤ).

(b) Show that there are infinitely many primes ℓ satisfying conditions (1) to (3) in definition 10.17 of a Kolyvagin prime.

(c) Suppose that K is a real quadratic field, and let u_{K} be a fundamental unit for K. Show that the extension M = L(u_{k}^{1/p^{2}}) obtained by adjoining to M a p^{2}-th root of u_{K} is Galois over ℚ. Show that its Galois group can be described as the semi-direct product

Gal(M/ℚ) = μ_{p^{2}}×|Gal(L/ℚ),

where the action of Aut(E_{p^{2}}) on μ_{p}^{2} is the natural one, and the generator τ of Gal(K/ℚ) is made to act on μ_{p^{2}} as −1. Thus elements of Gal(M/ℚ) can be indexed by triples (ζ, τ^{j}, T) with ζ a p^{2}-th root of unity, j ∈ {0, 1}, and T ∈ Aut(E_{p^{2}}) ≃ GL_{2}(ℤ/p^{2}ℤ).

(d) Let T ∈ GL_{2}(ℤ/p^{2}ℤ) be a matrix with eigenvalues a and −1/a, where a ≡ 1 (mod p) but not modulo p^{2}. Show that any rational prime ℓ which is unramified in M/ℚ and whose frobenius element Frob_{ℓ}(M/ℚ) satisfies

Frob_{ℓ}(M/ℚ) = (1, −1, T)

is a Kolyvagin prime. Conclude that there are infinitely many Kolyvagin primes.

(e) Let s be any non-zero element of H^{1}(K, E_{p}). Assume that s belongs to a specific eigenspace for the action of τ ∈ Gal(K/ℚ), so that

τs = δs, for some δ ∈ {1, −1}.

Let M_{s} be the extension of M cut out by the image s̄ of s under restriction to H^{1}(M, E_{p}) = Hom(Gal(M̄/M), E_{p}). Show that the extension M_{s} is Galois over ℚ, and that Gal(M_{s}/ℚ) is identified with the semi-direct product

Gal(M_{s}/ℚ) = E_{p}×|Gal(M/ℚ),

where the quotient Gal(M/ℚ) acts on the abelian normal subgroup E_{p} of Gal(M_{s}/ℚ) by the rule

(ζ, τ^{j}, T)(v) = δ^{j}T̄v.

(Here T̄ denotes the natural image of T in Aut(E_{p}).)

(f) Show that the group Gal(M_{s}/ℚ) contains an element of the form (v, 1, τ, T), where the automorphism T is as in (d), and the vector v ∈ E_{p} is non-zero and belongs to the δ-eigenspace for T̄.

(g) Let ℓ |̸N be a rational prime which is unramified in M_{s}/ℚ and satisfies

Frob_{ℓ}(M_{s}/ℚ) = (v, 1, τ, T).

Show that ℓ is a Kolyvagin prime, and that, if λ is the (unique) prime of K above ℓ, we have s_{λ} ≠ 0. Conclude that there exist infinitely many Kolyvagin primes ℓ such that ∂_{λ}(s) = 0 and s_{λ} ≠ 0.

(h) Let H be any finite-dimensional subspace of H^{1}(K, E_{p}). Using (g), show that there is a finite set S of Kolyvagin primes with the property that the natural map induced by restriction

H ⟶ ⊕_{v}H^{1}(K_{v}, E_{p})

is injective. Conclude proposition 10.18.

(i) Prove proposition 10.23.

(5) Following the notations in the proof of proposition 10.21, show that:

τD_{ℓ} = −D_{ℓ}τ (mod n_{ℓ}), τD_{ℓ_{1}}D_{ℓ_{2}} = D_{ℓ_{1}}D_{ℓ_{2}}τ (mod gcd(n_{ℓ_{1}}, n_{ℓ_{2}})).

(In particular, these identities hold modulo p^{2}.)

(6) Give the details of the construction of the cohomology classes κ(ℓ_{1}ℓ_{2}) depending on two Kolyvagin primes, and prove the properties asserted in proposition 10.22.

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