Hecke Algebras and Modular Forms: Notes from a course of Ken Ribet

William A. Stein

September 24, 1998

ii

Preface

These are the notes of a 1996 Berkeley course of Ken Ribet’s on modular forms and Hecke operators, which was given in the wake of Andrew Wiles much celebrated proof of Fermat’s last theorem.

Acknowledgement: These notes owe everything to Ribet’s beautiful lectures. I would also like to thank Kevin Buzzard, David Moulton, Helena Verrill, and Joe Wetherall who were very helpful in the preparation of these notes.

   William A. Stein (was@math.berkeley.edu)

Berkeley, California iv PREFACE

Contents

Preface iii

1 Introduction 1

1.1 Two dimensional Galois representations . . . . . . . . . . . . . . 1 1.1.1 Finite Fields (Weil, Tate) . . . . . . . . . . . . . . . . . . 1 1.1.2 Galois representations . . . . . . . . . . . . . . . . . . . . 2

1.2 Modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Cusp forms . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . 3

2 Modular Representations and Curves 5 2.1 Arithmetic of Modular Forms . . . . . . . . . . . . . . . . . . . . 5 2.2 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Parity Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Conjectures of Serre (mod ℓ version) . . . . . . . . . . . . . . . . 7 2.5 General remarks on mod p Galois representations . . . . . . . . . 7 2.6 Serre’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.7 Wiles’ Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Modular Forms 11 3.1 Cusp Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.3 Relationship With Elliptic Curves . . . . . . . . . . . . . . . . . 12 3.4 Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.5 Explicit Description of Sublattices . . . . . . . . . . . . . . . . . 14 3.6 Action of Hecke Operators on Modular Forms . . . . . . . . . . . 14

4 Embedding Hecke Operators in the Dual 17 4.1 The Space of Modular Forms . . . . . . . . . . . . . . . . . . . . 17 4.2 Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.3 Eigenforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5 Rationality and Integrality Questions 21

5.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.2 Integrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.3 Victor Miller’s Thesis . . . . . . . . . . . . . . . . . . . . . . . . 22 5.4 Petersson Inner Product . . . . . . . . . . . . . . . . . . . . . . . 23

6 Modular Curves 25 6.1 Cusp Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 6.2 Modular Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 6.3 Classifying Γ(N)-structures . . . . . . . . . . . . . . . . . . . . . 26 6.4 More on Integral Hecke Operators . . . . . . . . . . . . . . . . . 27 6.5 Complex Conjugation . . . . . . . . . . . . . . . . . . . . . . . . 27 6.6 Isomorphism in the Real Case . . . . . . . . . . . . . . . . . . . . 28 6.7 The Eichler-Shimura Isomorphism . . . . . . . . . . . . . . . . . 28 6.8 The Petterson Inner Product is Hecke Compatible . . . . . . . . 29

7 Higher Weight Modular Forms 31 7.1 Definitions of T . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 7.2 Double Cosets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 7.3 More General Congruence Subgroups . . . . . . . . . . . . . . . . 32 7.4 Explicit Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 7.5 Old and New Forms . . . . . . . . . . . . . . . . . . . . . . . . . 33

8 New Forms 35 8.1 Connection With Galois Representations . . . . . . . . . . . . . . 36 8.2 Semisimplicity of U_{p} . . . . . . . . . . . . . . . . . . . . . . . . . 36 8.3 Shimura’s Example of Nonsemisimple U_{p} . . . . . . . . . . . . . . 37 8.4 An Interesting Duality . . . . . . . . . . . . . . . . . . . . . . . . 37 8.5 Observations on T_{n} . . . . . . . . . . . . . . . . . . . . . . . . . . 38

9 Some Explicit Genus Computations 41 9.1 Computing the Dimension of S_{k}(Γ) . . . . . . . . . . . . . . . . . 41 9.2 Application of Riemann-Hurwitz . . . . . . . . . . . . . . . . . . 41 9.3 Explicit Genus Computations . . . . . . . . . . . . . . . . . . . . 42 9.4 The Genus of X(N) . . . . . . . . . . . . . . . . . . . . . . . . . 43 9.5 The Genus of X_{0}(N) . . . . . . . . . . . . . . . . . . . . . . . . . 43 9.6 Modular Forms mod p . . . . . . . . . . . . . . . . . . . . . . . . 44

10 The Field of Moduli 47 10.1 Digression on Moduli . . . . . . . . . . . . . . . . . . . . . . . . . 48 10.2 When is ρ_{E} Surjective? . . . . . . . . . . . . . . . . . . . . . . . 48 10.3 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 10.4 A Descent Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 50 10.5 Second Look at the Descent Exercise . . . . . . . . . . . . . . . . 51 10.6 Action of GL_{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 11 Hecke Operators as Correspondences 55

11.1 Some Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 11.2 Hecke Operators as Correspondences . . . . . . . . . . . . . . . . 56 11.3 Generalities on Correspondences . . . . . . . . . . . . . . . . . . 57 11.4 Jacobians of Curves . . . . . . . . . . . . . . . . . . . . . . . . . 59 11.5 More on Hecke Operators . . . . . . . . . . . . . . . . . . . . . . 59 11.6 Hecke Operators acting on Jacobians . . . . . . . . . . . . . . . . 60

11.6.1 The Albanese Map . . . . . . . . . . . . . . . . . . . . . . 61 11.6.2 The Hecke Algebra . . . . . . . . . . . . . . . . . . . . . . 62

11.7 The Eichler-Shimura Relation: Part I . . . . . . . . . . . . . . . 62 11.8 The Eichler-Shimura Relation: Part II . . . . . . . . . . . . . . . 63 11.9 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 11.10More on Eichler-Shimura . . . . . . . . . . . . . . . . . . . . . . 67

12 Abelian Varieties from Modular Forms 69 12.1 Computing the Determinent of ρ_{λ} . . . . . . . . . . . . . . . . . 72 12.2 Duality and Polarizations . . . . . . . . . . . . . . . . . . . . . . 73 12.3 The Weil Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 12.4 The Fancy Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 12.5 The Concrete Proof . . . . . . . . . . . . . . . . . . . . . . . . . 74 12.6 The Construction for X_{1}(N) . . . . . . . . . . . . . . . . . . . . 75

13 The Gorenstein Property 77 13.1 The Gorenstein Property . . . . . . . . . . . . . . . . . . . . . . 80 13.2 Proof the Gorenstein Property . . . . . . . . . . . . . . . . . . . 83

13.2.1 Vague Comments . . . . . . . . . . . . . . . . . . . . . . . 86 13.3 Finite Flat Group Schemes . . . . . . . . . . . . . . . . . . . . . 86 13.4 Reformulation of V = W problem . . . . . . . . . . . . . . . . . . 87 13.5 Dieudonné Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 87 13.6 The Proof: Part II . . . . . . . . . . . . . . . . . . . . . . . . . . 88 13.7 Key Result of Boston-Lenstra-Ribet . . . . . . . . . . . . . . . . 91

14 Local Properties of ρ_{λ} 93 14.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 14.2 Local Properties when p |̸N . . . . . . . . . . . . . . . . . . . . . 94 14.3 Weil-Deligne Groups . . . . . . . . . . . . . . . . . . . . . . . . . 94 14.4 Local Properties when p|N . . . . . . . . . . . . . . . . . . . . . 95 14.5 Definition of the Reduced Conductor . . . . . . . . . . . . . . . . 95 14.6 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 14.7 Adelic Representations Associated to Modular Forms . . . . . . . 97 14.8 More Local Properties of the ρ_{λ}. . . . . . . . . . . . . . . . . . . 100

14.8.1 Possibilities for π_{p} . . . . . . . . . . . . . . . . . . . . . . 101 14.8.2 The case ℓ = p . . . . . . . . . . . . . . . . . . . . . . . . 101 14.8.3 Tate Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 103 15 The Weight and Serre’s Conjectures 105

15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 15.2 Review of the λ-adic case . . . . . . . . . . . . . . . . . . . . . . 105 15.3 Serre’s conjecture 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 106

15.3.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 15.4 Serre’s conjecture 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 107

15.4.1 Key background points . . . . . . . . . . . . . . . . . . . 108 15.5 The weight and fundamental characters . . . . . . . . . . . . . . 109 15.6 The weight in Serre’s conjectures on modular representations . . 113

15.6.1 θ-series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 15.6.2 Edixhoven’s paper . . . . . . . . . . . . . . . . . . . . . . 115

15.7 The extra assumption . . . . . . . . . . . . . . . . . . . . . . . . 116

15.7.1 Companion Forms . . . . . . . . . . . . . . . . . . . . . . 118 15.8 The exceptional level 1 case . . . . . . . . . . . . . . . . . . . . . 119

16 Fermat’s Last Theorem 123 16.1 The application to Fermat . . . . . . . . . . . . . . . . . . . . . . 123 16.2 Modular Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . 125

17 Deformations 127 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 17.2 Condition (∗) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

17.2.1 Finite flat representations . . . . . . . . . . . . . . . . . . 129 17.3 Classes of Liftings . . . . . . . . . . . . . . . . . . . . . . . . . . 129

17.3.1 The case p ≠ ℓ . . . . . . . . . . . . . . . . . . . . . . . . 129 17.3.2 The case p = ℓ . . . . . . . . . . . . . . . . . . . . . . . . 130

17.4 Wiles’ Hecke algebra . . . . . . . . . . . . . . . . . . . . . . . . . 131

18 The Hecke Algebra T_{Σ} 133 18.1 The Hecke Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 133 18.2 The maximal ideal in R . . . . . . . . . . . . . . . . . . . . . . . 135

18.2.1 Strip away certain Euler factors . . . . . . . . . . . . . . . 136 18.2.2 Make into an eigenform for U_{ℓ} . . . . . . . . . . . . . . . 136

18.3 The Galois Representation . . . . . . . . . . . . . . . . . . . . . . 137 18.3.1 The structure of T_{m} . . . . . . . . . . . . . . . . . . . . . 138 18.3.2 The philosophy in this picture . . . . . . . . . . . . . . . 138 18.3.3 Massage ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 18.3.4 Massage ρ′ . . . . . . . . . . . . . . . . . . . . . . . . . . 139 18.3.5 Representations from modular forms mod ℓ . . . . . . . . 140 18.3.6 Representations from modular forms mod ℓ^{n} . . . . . . . 141

18.4 ρ′ is of type Σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 18.5 Isomorphism between T_{m} and R_{m_{R}} . . . . . . . . . . . . . . . . 142 18.6 Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 18.7 Wiles Main Conjecture . . . . . . . . . . . . . . . . . . . . . . . . 144 18.8 T_{Σ} is a complete intersection . . . . . . . . . . . . . . . . . . . . 147 18.9 The inequality #O/η ≤ #℘_{T} /℘_{T}^{2} ≤ ℘_{R}/℘_{R}^{2} . . . . . . . . . . . . 147 18.9.1 The definitions of the ideals . . . . . . . . . . . . . . . . . 148 18.9.2 Aside: Selmer Groups . . . . . . . . . . . . . . . . . . . . 149 18.9.3 Outline of some proofs . . . . . . . . . . . . . . . . . . . . 149 x CONTENTS

Chapter 1

Introduction

The main objects of study in this course are:

• Galois representations

• Modular forms

• Hecke algebras

• Modular curves

• Abelian varieties

1.1 Two dimensional Galois representations

The main geometric objects we will study are elliptic curves, and more generally, algebraic curves of arbitrary genus. These in turn give rise, via the Jacobian construction, to higher dimensional abelian varieties, from which we obtain Galois representations.

When studying elliptic curves over a field of characteristic zero, a natural tool is to present the curve as ℂ/L for some sublattice L of the field ℂ of complex numbers. To construct L, fix a non-zero holomorphic differential ω on E over ℂ. Then

L = { ∫_{γ} ω | γ ∈ H_{1}(E(ℂ), ℤ)}.

1.1.1 Finite Fields (Weil, Tate)

In the 1940’s, Weil studied the analogous situation for elliptic curves defined over a finite field k. He desperately wanted to find an algebraic way to describe the above correspondence between elliptic curves and lattices. He was able to find an algebraic definition of L/nL, when n is prime to the characteristic of k.

Define

E[n] := \{P ∈ E(\overline{k}) : nP = 0\}.

In the case of the complex numbers,

E[n] = (\frac{1}{n}L)/L ≅ L/nL ≈ (ℤ/nℤ) × (ℤ/nℤ).

For any prime ℓ, define

E[ℓ^{∞}] := \{P ∈ E(\overline{k}) : ℓ^{ν}P = 0, some ν ≥ 1\}
= ⋃_{ν=1}^{∞} E[ℓ^{ν}] = \underrightarrow{lim} E[ℓ^{ν}]

In an analogous way Tate constructed a rank 2 free ℤ_{ℓ}-module

T_{ℓ}(E) := \underleftarrow{lim} E[ℓ^{ν}],

where the map from E[ℓ^{ν}] → E[ℓ^{ν−1}] is multiplication by ℓ. To see that the rank is 2, check that the ℤ/ℓ^{ν}ℤ-module structure of E[ℓ^{ν}] is compatible with the maps E[ℓ^{ν}] ⟶^{ℓ} E[ℓ^{ν−1}]. For example, see [33] (III, 7). Then V_{ℓ}(E) := T_{ℓ}(E)⊗ℚ_{ℓ} is a two dimensional vector space over ℚ_{ℓ}. This is our first nontrivial example of ℓ-adic étale cohomology.

1.1.2 Galois representations

Some of the pioneers in the study of Galois representations were Taniyama, Shimura, Artin, and Tate.

Let E/ℚ be an elliptic curve over the rational numbers ℚ. The absolute Galois group G = Gal(\overline{ℚ}/ℚ) acts on the set E[n]. Since E is defined over ℚ, this action respects the group operations and so we obtain a Galois representation:

ρ : G → Aut(E[n]) ≅ GL_{2}(ℤ/nℤ).

Let K be the field cut out by the ker(ρ). Then K is a finite Galois extension of ℚ since E[n] is a finite set and ker(ρ) is a normal subgroup. Since

Gal(K/ℚ) ≅ G/ ker ρ ≅ Im ρ ⊆ GL_{2}(ℤ/nℤ)

we obtain, in this way, subgroups of GL_{2}(ℤ/nℤ) as Galois groups.

Shimura showed that, if we start with the elliptic curve

E : y^{2} + y = x^{3} − x^{2},

then for “most” n the image of ρ is all of GL_{2}(ℤ/nℤ). More generally, the image is “most” of GL_{2}(ℤ/nℤ) when E does not have complex multiplication. (E is said to have complex multiplication if its endomorphism ring strictly larger than ℤ.)

1.2 Modular forms

Modular forms are analytic objects, having an inordinate amount of symmetry. Many spectacular theorems and deep conjectures link Galois representations with modular forms.

1.2.1 Cusp forms

Define

Γ_{1}(N) := \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ) : a ≡ d ≡ 1, c ≡ 0 (mod N)\}.

Let S_{k}(N) denote the space of cusp forms of weight k and level N for Γ_{1}(N). Thus S_{k}(N) is the complex vector space consisting of all holomorphic functions f(z) on the upper half-plane 𝔥 = {z ∈ ℂ : Im(z) > 0} vanishing at ∞ and satisfying

f(\frac{az + b}{cz + d}) = (cz + d)^{k}f(z) for all ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{1}(N).

In particular, since ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) ∈ Γ_{1}(N), so that f(z) = f(z + 1), we can expand f(z) as a series in q(z) = e^{2πiz}

f(z) = ∑_{n=1}^{∞} c_{n}q^{n}.

It can be shown that S_{k}(N) has finite dimension.

A famous example of a cusp form is

Δ = q ∏_{n=1}^{∞}(1 − q^{n})^{24} = ∑_{n=1}^{∞} τ(n)q^{n}

which is a basis for S_{12}(1). The coefficients τ(n) define the Ramanujan τfunction. One now knows that τ is multiplicative and satisfies

τ(p^{ν+1}) = τ(p)τ(p^{ν}) − p^{11}τ(p^{ν−1}).

This can be proved using the theory of Hecke operators.

1.2.2 Hecke Operators

Mordell defined, for n ≥ 1, operators T_{n} on S_{k}(N) called Hecke operators. These proved very fruitful. The set of such operators forms a commuting family of endomorphisms and is hence “almost” simultaneously diagonalizable. The precise meaning of “almost” and the actual structure of the Hecke algebra 𝕋 = ℚ[T_{1}, T_{2}, …] will be studied in greater detail in the remainder of this course. Often, there will exist a basis {f_{1}, …, f_{r}} ⊂ S_{k}(N) of cusp forms so that each f = f_{i} = ∑_{n=1}^{∞} a_{n}q^{n} is a simultaneous eigenvector for all the Hecke operators T_{n} and, in fact, T_{n}f = a_{n}f. The a_{n} are then necessarily algebraic integers being roots of the characteristic polynomial of T_{n}, and much less trivially, the field ℚ(a_{1}, a_{2}, …) is finite over ℚ.

A good claim can be made that the a_{n} are often interesting integers because they exhibit remarkable properties. For example, τ(n) ≡ ∑_{d|n} d^{11} (mod 691). How can we study the a_{n}? How can we interpret the a_{n}? One way is to study the connections between Galois representations and modular forms, many of which were first discovered by Serre, Shimura, Eichler and Deligne.

152 CHAPTER 1. INTRODUCTION

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156 BIBLIOGRAPHY

Index

congruence subgroup, 2 cusp forms, 2 Deligne, 3 Eichler, 3 Galois representations, 1 Hecke operator, 3 Jacobian, 1 modular curves, 1 Serre, 3 Shimura, 2, 3