Hecke Algebras and Modular Forms: Notes derived from Ribet’s 1996 Berkeley grad. course.

William Stein

September 24, 1998

ii

Preface

Disclaimer: These notes record some of what I saw in Ken Ribet’s course on Modular Forms and Hecke Operators given at U.C. Berkeley during the Spring semester 1996. They are still very rough as I wrote them during my first semester of graduate school before I knew any real mathematics.

The participants in the course were: Amod Agashe, Matt Baker, Jim Borger, Kevin Buzzard, Bruce Caskel, Robert Coleman, Janós Csirik, Annette Huber, David Jones, David Kohel, Loic Merel, David Moulton, Andrew Ogg, Arthur Ogus, Jessica Polito, Ken Ribet, Saul Schleimer, Lawren Smithline, William Stein, Takahashi, Wayne Whitney, and Hui Zui.

I wish to thank David Moulton, Joe Wetherall, and Kevin Buzzard who helped me in preparing these notes, Arthur Ogus who asked a lot of stimulating questions during the class, and of course Ken Ribet who sees clearly.

William Stein, Spring 1996, Berkeley, CA, was@math.berkeley.edu 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 (Taniyama, Shimura, Mumford-Tate) . . . . . 2

1.2 Modular Forms and Galois Representations . . . . . . . . . . . . . . . . . . . 2 1.2.1 Cusp Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.2 Hecke Operators (Mordell) . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Modular Representations and Curves 5 2.1 Arithmetic of Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Parity Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Modular Forms 9 3.1 Cusp Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Relationship With Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . 9 3.4 Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.5 Explicit Description of Sublattices . . . . . . . . . . . . . . . . . . . . . . . . 11 3.6 Action of Hecke Operators on Modular Forms . . . . . . . . . . . . . . . . . 12

4 Embedding Hecke Operators in the Dual 15 4.1 The Space of Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2 Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.3 Eigenforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Rationality and Integrality Questions 19 5.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.2 Integrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.3 Victor Miller’s Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.4 Petersson Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6 Modular Curves 23

6.1 Cusp Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6.2 Modular Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6.3 Classifying Γ(N)-structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6.4 More on Integral Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . . 24 6.5 Complex Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 6.6 Isomorphism in the Real Case . . . . . . . . . . . . . . . . . . . . . . . . . . 25 6.7 The Eichler-Shimura Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . 25 6.8 The Petterson Inner Product is Hecke Compatible . . . . . . . . . . . . . . . 27

7 Higher Weight Modular Forms 29 7.1 Definitions of T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 7.2 Double Cosets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 7.3 More General Congruence Subgroups . . . . . . . . . . . . . . . . . . . . . . 30 7.4 Explicit Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 7.5 Old and New Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

8 New Forms 33 8.1 Connection With Galois Representations . . . . . . . . . . . . . . . . . . . . 34 8.2 Semisimplicity of U_{p} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 8.3 Shimura’s Example of Nonsemisimple U_{p} . . . . . . . . . . . . . . . . . . . . 34 8.4 An Interesting Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 8.5 Observations on T_{n} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

9 Some Explicit Genus Computations 37 9.1 Computing the Dimension of S_{k}(Γ) . . . . . . . . . . . . . . . . . . . . . . . 37 9.2 Application of Riemann-Hurwitz . . . . . . . . . . . . . . . . . . . . . . . . . 37 9.3 Explicit Genus Computations . . . . . . . . . . . . . . . . . . . . . . . . . . 38 9.4 The Genus of X(N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 9.5 The Genus of X_{0}(N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 9.6 Modular Forms mod p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

10 The Field of Moduli 41 10.1 Digression on Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 10.2 When is ρ_{E} Surjective? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 10.3 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 10.4 A Descent Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 10.5 Second Look at the Descent Exercise . . . . . . . . . . . . . . . . . . . . . . 44 10.6 Action of GL_{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

11 Hecke Operators as Correspondences 47 11.1 Some Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 11.2 Hecke Operators as Correspondences . . . . . . . . . . . . . . . . . . . . . . 48 11.3 Generalities on Correspondences . . . . . . . . . . . . . . . . . . . . . . . . . 49 11.4 Jacobians of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 11.5 More on Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 11.6 Hecke Operators acting on Jacobians . . . . . . . . . . . . . . . . . . . . . . 51

11.6.1 The Albanese Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 11.6.2 The Hecke Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

11.7 The Eichler-Shimura Relation: Part I . . . . . . . . . . . . . . . . . . . . . . 53 11.8 The Eichler-Shimura Relation: Part II . . . . . . . . . . . . . . . . . . . . . 54 11.9 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 11.10More on Eichler-Shimura . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

12 Abelian Varieties from Modular Forms 59 12.1 Computing the Determinent of ρ_{λ} . . . . . . . . . . . . . . . . . . . . . . . . 61 12.2 Duality and Polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 12.3 The Weil Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 12.4 The Fancy Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 12.5 The Concrete Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 12.6 The Construction for X_{1}(N) . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

13 The Gorenstein Property 67 13.1 The Gorenstein Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 13.2 Proof the Gorenstein Property . . . . . . . . . . . . . . . . . . . . . . . . . . 72

13.2.1 Vague Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 13.3 Finite Flat Group Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 13.4 Reformulation of V = W problem . . . . . . . . . . . . . . . . . . . . . . . . 75 13.5 Dieudonné Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 13.6 The Proof: Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 13.7 Key Result of Boston-Lenstra-Ribet . . . . . . . . . . . . . . . . . . . . . . . 79

14 Local Properties of ρ_{λ} 81 14.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 14.2 Local Properties when p |̸N . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 14.3 Weil-Deligne Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 14.4 Local Properties when p|N . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 14.5 Definition of the Reduced Conductor . . . . . . . . . . . . . . . . . . . . . . 83 14.6 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 14.7 Adelic Representations Associated to Modular Forms . . . . . . . . . . . . . 84 14.8 More Local Properties of the ρ_{λ}. . . . . . . . . . . . . . . . . . . . . . . . . . 87

14.8.1 Possibilities for π_{p} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 14.8.2 The case ℓ = p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 14.8.3 Tate Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

15 The Weight and Serre’s Conjectures 91 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 15.2 Review of the λ-adic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 15.3 Serre’s conjecture 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

15.3.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 15.4 Serre’s conjecture 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

15.4.1 Key background points . . . . . . . . . . . . . . . . . . . . . . . . . . 93 15.5 The weight and fundamental characters . . . . . . . . . . . . . . . . . . . . . 94 15.6 The weight in Serre’s conjectures on modular representations . . . . . . . . . 98

15.6.1 θ-series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 15.6.2 Edixhoven’s paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

15.7 The extra assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

15.7.1 Companion Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 15.8 The exceptional level 1 case . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 16 Fermat’s Last Theorem 105

16.1 The application to Fermat . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 16.2 Modular Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

17 Deformations 109 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 17.2 Condition (∗) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

17.2.1 Finite flat representations . . . . . . . . . . . . . . . . . . . . . . . . 111 17.3 Classes of Liftings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

17.3.1 The case p ≠ ℓ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 17.3.2 The case p = ℓ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

17.4 Wiles’ Hecke algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

18 The Hecke Algebra T_{Σ} 115 18.1 The Hecke Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 18.2 The maximal ideal in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

18.2.1 Strip away certain Euler factors . . . . . . . . . . . . . . . . . . . . . 117 18.2.2 Make into an eigenform for U_{ℓ} . . . . . . . . . . . . . . . . . . . . . . 118

18.3 The Galois Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 18.3.1 The structure of T_{m} . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 18.3.2 The philosophy in this picture . . . . . . . . . . . . . . . . . . . . . . 120 18.3.3 Massage ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 18.3.4 Massage ρ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 18.3.5 Representations from modular forms mod ℓ . . . . . . . . . . . . . . 121 18.3.6 Representations from modular forms mod ℓ^{n} . . . . . . . . . . . . . . 122

18.4 ρ′ is of type Σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 18.5 Isomorphism between T_{m} and R_{m_{R}} . . . . . . . . . . . . . . . . . . . . . . . 123 18.6 Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 18.7 Wiles Main Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 18.8 T_{Σ} is a complete intersection . . . . . . . . . . . . . . . . . . . . . . . . . . 127 18.9 The inequality #O/η ≤ #℘_{T} /℘_{T}^{2} ≤ ℘_{R}/℘_{R}^{2} . . . . . . . . . . . . . . . . . . . 127

18.9.1 The definitions of the ideals . . . . . . . . . . . . . . . . . . . . . . . 128 18.9.2 Aside: Selmer Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 129 18.9.3 Outline of some proofs . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Chapter 1

Introduction

The main objects of study in this course are:

• Modular Forms

• Hecke Algebras

• Modular Curves

• Jacobians

• Abelian Varieties

1.1 Two Dimensional Galois Representations

The geometric objects of 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. These geometric objects in turn give rise to Galois representations.

When studying elliptic curves, the natural tool in the characteristic zero situation is to present the elliptic curve as C/L for some lattice L in C. To construct L fix a non-zero holomorphic differential ω of E over C and construct L as

{ ∫_{γ} ω | γ ∈ H_{1}(E(C), Z)}.

1.1.1 Finite Fields (Weil, Tate)

In the 1940’s, Weil study 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. He was able to find an algebraic definition of L/nL, where n ≥ 1 and (n, char k) = 1, which is as follows. Let E[n] = \{P ∈ E(\overline{k}) : nP = 0\} = (\frac{1}{n}L)/L ≅ L/nL.

Now fix a prime ℓ, we let E[ℓ^{∞}] = \{P ∈ E(\overline{k}) : ℓ^{ν}P = 0, some ν ≥ 1\} = ∪_{ν=1}^{∞}E[ℓ^{ν}]. Tate obtained an analogous construction by defining a rank 2 free Z_{ℓ}-module Tate_{ℓ} E := \underleftarrow{lim} E[ℓ^{ν}] (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}] → E[ℓ^{ν−1}]). See [33] (III, 7). Then V_{ℓ}(E) = T_{ℓ}(E) ⊗ Q_{ℓ} is a two dimensional vector space over Q_{ℓ}. This gives the first nontrivial example of ℓ-adic étale cohomology.

1.1.2 Galois Representations (Taniyama, Shimura, Mumford-Tate)

Let E/Q be an elliptic curve and G = Gal(\overline{Q}/Q). Then E[n] = \{P ∈ E(\overline{Q}) : nP = 0\} ≅ (Z/nZ)^{2} is acted on by G and this action respects the group operation so we have a Galois representation

G ⟶^{ρ}Aut(E[n]) ≅ GL_{2}(Z/nZ)

Let K be the fixed field of ker ρ (note that K is a number field), then since Gal(K/Q) ≅ G/ ker ρ ≅ Im ρ ⊆ GL_{2}(Z/nZ) we obtain many 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 the image of ρ is often all of GL_{2}(Z/nZ) and the image is “most” of GL_{2}(Z/nZ) when E does not have complex multiplication.

1.2 Modular Forms and Galois Representations

1.2.1 Cusp Forms

Let S_{k}(N) denote the space of cusp forms of weight k and level N on the congruence subgroup Γ_{1}(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z) : a ≡ 1 (mod N), c ≡ 0 (mod N), d ≡ 1 (mod N)\}. Thus S_{k}(N) is the finite dimensional vector space consisting of all holomorphic functions f(z) on 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).

Since, in particular, f(z) = f(z + 1) we can expand f(z) as a q-series (this requires rigorous justification)

f(z) = ∑_{n=1}^{∞} c_{n}q^{n}.

A famous example is

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

τ is called the Ramanujan function. One now knows that τ is multiplicative and satisfies τ(p^{ν}) = τ(p)τ(p^{ν}) − p^{11}τ(p^{ν−1}). Δ is a normalized basis for S_{1}(1).

1.2.2 Hecke Operators (Mordell)

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 an “almost” commuting family of endomorphisms and is hence “almost” simultaneously diagonalizable. The precise meaning of “almost” and the actual structure of the Hecke algebra Q[T_{1}, …, T_{n}, …] will be studied in greater detail in the remainder of this course. Often there will exist a basis of cusp forms f = ∑_{n=1}^{∞} c_{n}q^{n} ∈ S_{k}(N) so that f_{n} is a simultaneous eigenvector for all of the Hecke operators T_{n} and, in fact, T_{n}f = c_{n}f. All of the c_{n} will be algebraic integers and the field Q(c_{1}, c_{2}, …) will be finite over Q.

A good claim can be made that the c_{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 c_{n}? How can we interpret the c_{n}? We can do this by studying the connection between Galois representations and modular forms. In 1968 work was originally begun on this by Serre, Shimura, Eichler and Deligne.

4 CHAPTER 1. INTRODUCTION

Modular Representations and Curves

2.1 Arithmetic of Modular Forms

Suppose f = ∑_{n=1}^{∞} a_{n}q^{n} is a cusp form in S_{k}(N) which is an eigenform for the Hecke operators. Then the Mellin transform associates to f the L-function L(f, z) = ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}}. Let K = Q(a_{1}, a_{2}, …), then one can show that the a_{n} are algebraic integers and K is a number field. When k = 2 Shimura associates to f an abelian variety A_{f} over Q of dimension [K : Q] on which K acts (see theorem 7.14 of [31]). Example 2.1.1 (Modular Elliptic Curves). When all of the coefficients a_{n} of the modular form f lie in Q then [K : Q] = 1 so A_{f} is a one dimensional abelian variety. A one dimensional abelian variety of nonzero genus is an elliptic curve. An elliptic curve isogenous to one arising via this construction is called modular.

Definition 2.1.2. Elliptic curves E_{1} and E_{2} are isogenous if there is a morphism E_{1} → E_{2} of algebraic groups, which has a finite kernel.

The following conjecture motivates much of the theory.

Conjecture 2.1.3. Every elliptic curve over Q is modular, that is, isogenous to a curve constructed in the above way.

For k ≥ 2 Serre and Deligne found a way to associate to f a family of ℓ-adic representations. Let ℓ be a prime number and K be as above, then it is well known that K ⊗_{Q} Q_{ℓ} ≅ ∏_{λ|ℓ} K_{λ}. One can associate to f a representation

ρ_{ℓ,f} : G = Gal(\overline{Q}/Q) → GL(K ⊗_{Q} Q_{ℓ})

unramified at all primes p |̸ℓN. For ρ_{ℓ,f} to be unramified we mean that for all primes P lying over p, the inertia group of the decomposition group at P is contained in the kernel of ρ_{ℓ,f}. The decomposition group D_{P} at P is the set of those g ∈ G which fix P. Let k be the residue field O/P where O is the ring of all algebraic integers. Then the inertia group I_{P} is the kernel of the map D_{P} → Gal(\overline{k}/k).

Now I_{P} ⊂ D_{P} ⊂ Gal(\overline{Q}/Q) and D_{P} /I_{P} is cyclic (being isomorphic to a subgroup of the Galois group of a finite extension of finite fields) so it is generated by a Frobenious automorphism frob_{p} lying over p. One has

Tr(ρ_{ℓ,f}(frob_{p})) = a_{p} ∈ K ⊂ K ⊗ Q_{ℓ}

and

det(ρ_{ℓ,f}) = χ_{ℓ}^{k−1}ε

where χ_{ℓ} is the ℓth cyclotomic character and ε is the Dirichlet character associated to f. There is an incredible amount of “abuse of notation” packed into this statement. First, the Frobenius frob_{P} (note P not p) is only well defined in Gal(K/Q) (so I think an unstated result is that K must be Galois), and then frob_{p} is only well defined up to conjugacy. But this works out since ρ_{ℓ,f} is well-defined on Gal(K/Q) (it kills Gal(\overline{Q}/K)) and the trace is well-defined on conjugacy classes (Tr(AB) = Tr(BA) so Tr(ABA^{−1}) = T r(B)).

2.2 Characters

Let f ∈ S_{k}(N), then for all ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z) with c ≡ 0 mod N we have

f(\frac{az + b}{cz + d}) = (cz + d)^{k}ε(d)f(z)

where ε : (Z/NZ)^{∗} → C^{∗} is a Dirichlet character mod N. If f is an eigenform for the so called “diamond-bracket operator” ⟨d⟩ so that f|⟨d⟩ = ε(d)f then ε actually takes values in K.

Led φ_{N} be the mod N cyclotomic character so that φ_{N} : G → (Z/NZ)^{∗} takes g ∈ G to the automorphism induced by g on the Nth cyclotomic extension Q(μ μ μ_{N}) of Q (where we identify Gal(Q(μ μ μ_{N})/Q) with (Z/NZ)^{∗}). Then what we called ε above in the formula det(ρ_{ℓ}) = χ_{ℓ}^{k−1}ε is really the composition

G ⟶^{φ_{N}} (Z/NZ)^{∗} ⟶^{ε}C^{∗}.

For each positive integer ν we consider the ℓ^{ν}th cyclotomic character on G,

φ_{ℓ^{ν}} : G → (Z/ℓ^{ν}Z)^{∗}.

Putting these together gives the ℓ-adic cyclotomic character

χ_{ℓ} : G → Z_{ℓ}^{∗}.

2.3 Parity Conditions

Let c ∈ Gal(\overline{Q}/Q) be complex conjugation. Then φ_{N}(c) = −1 so ε(c) = ε(−1) and χ_{ℓ}^{k−1}(c) =

(−1)^{k−1}. Now let ( \begin{matrix}a & b \\ c & d \end{matrix} ) = ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ), then for f ∈ S_{k}(N),

f(z) = (−1)^{k}ε(−1)f(z)

so (−1)^{k}ε(−1) = 1 thus

det(ρ_{ℓ,f}(c)) = ϵ(−1)(−1)^{k−1} = −1.

Thus the det character is odd so the representation ρ_{ℓ,f} is odd.

Remark 2.3.1 (Vague Question). How can one recognize representations like ρ_{ℓ,f} “in nature”? Mazur and Fontaine have made relevant conjectures. The Shimura-Taniyama conjecture can be reformulated by saying that for any representation ρ_{ℓ,E} comming from an elliptic curve E there is f so that ρ_{ℓ,E} ≅ ρ_{ℓ,f}.

2.4. CONJECTURES OF SERRE (MOD ℓ VERSION) 7

2.4 Conjectures of Serre (mod ℓ version)

Suppose f is a modular form, ℓ ∈ Z prime, λ a prime lying over ℓ, and the representation

ρ_{λ,f} : G → GL_{2}(K_{λ})

(constructed by Serre-Deligne) is irreducible. Then ρ_{λ,f} is conjugate to a representation with image in GL_{2}(O_{λ}), where O_{λ} is the ring of integers of K_{λ}. Reducing mod λ gives a representation

\overline{ρ}_{λ,f} : G → GL_{2}(F_{λ})

which has a well-defined trace and det, i.e., the det and trace don’t depend on the choice of conjugate representation used to obtain the reduced representation. One knows from representation theory that if such a representation is semisimple then it is completely determined by its trace and det (more precisely, the characteristic polynomials of all of its elements – see chapter ??). Thus if \overline{ρ}_{λ,f} is irreducible (and hence semisimple) then it is unique in the sense that it does not depend on the choice of conjugate.

2.5 General remarks on mod p Galois representations

[[This section was written by Joseph Loebach Wetherell.]]

First, what are semi-simple and irreducible representations? Remember that a representation ρ is a map from a group G to the endomorphisms of some vector space W (or a free module M if we are working over a ring instead of a field, but let’s not worry about that for now). A subspace W′ of W is said to be invariant under ρ if ρ takes W′ back into itself. (The point is that if W′ is invariant, then ρ induces representations on both W′ and W/W′.) An irreducible representation is one where the only invariant subspaces are 0 and W. A semi-simple representation is one where for every invariant subspace W′ there is a complementary invariant subspace W′′ – that is, you can write ρ as the direct sum of ρ|_{W′} and ρ|_{W′′}.

Another way to say this is that if W′ is an invariant subspace then we get a short exact sequence

0 → ρ|_{W/W′} → ρ → ρ|_{W′} → 0.

Furthermore ρ is semi-simple if and only if every such sequence splits.

Note that irreducible representations are semi-simple.

One other fact is that semi-simple Galois representations are uniquely determined (up to isomorphism class) by their trace and determinant.

Now, since in the case we are doing, G = Gal(\overline{Q}/Q) is compact, it follows that the image of any Galois representation ρ into GL_{2}(K_{λ}) is compact. Thus we can conjugate it into GL_{2}(O_{λ}). Irreducibility is not needed for this.

Now that we have a representation into GL_{2}(O_{λ}), we can reduce to get a representation \overline{ρ} to GL_{2}(F_{λ}). This reduced representation is not uniquely determined by ρ, since we had a choice of conjugators. However, the trace and determinant are invariant under conjugation, so the trace and determinant of the reduced representation are uniquely determined by ρ.

So we know the trace and determinant of the reduced representation. If we also knew that it was semi-simple, then we would know its isomorphism class, and we would be done. So we would be happy if the reduced representation is irreducible. And in fact, it is easy to see that if the reduced representation is irreducible, then ρ must also be irreducible. Now, it turns out that all ρ of interest to us will be irreducible; unfortunately, we can’t go the other way and claim that ρ irreducible implies the reduction is irreducible.

2.6 Serre’s Conjecture

Serre has made the following conjecture which is still open at the time of this writing.

Conjecture 2.6.1 (Serre). All irreducible representation of G over a finite field which are odd, i.e., det(σ(c)) = −1, c complex conjugation, are of the form \overline{ρ}_{λ,f} for some representation ρ_{λ,f} constructed as above.

Example 2.6.2. Let E/Q be an elliptic curve and let σ_{ℓ} : G → GL_{2}(F_{ℓ}) be the representation induced by the action of G on the ℓ-torsion of E. Then det σ_{ℓ} = φ_{ℓ} is odd and σ_{ℓ} is usually irreducible, so Serre’s conjecture would imply that σ_{ℓ} is modular. From this one can, assuming Serre’s conjecture, prove that E is modular.

Definition 2.6.3. Let σ : G → GL_{2}(F) (F is a finite field) be a represenation of the Galois group G. The we say that the representions σ is modular if there is a modular form f, a prime λ, and an embedding F ↪ \overline{F}_{λ} such that σ ≅ \overline{ρ}_{λ,f} over \overline{F}_{λ}.

2.7 Wiles’ Perspective

Suppose E/Q is an elliptic curve and ρ_{ℓ,E} : G → GL_{2}(Z_{ℓ}) the associated ℓ-adic representation on the Tate module T_{ℓ}. Then by reducing we obtain a mod ℓ representation

\overline{ρ}_{ℓ,E} = σ_{ℓ,E} : G → GL_{2}(F_{ℓ}).

If we can show this representation is modular for infinitely many ℓ then we will know that E is modular.

Theorem 2.7.1 (Langland’s and Tunnel). If σ_{2,E} and σ_{3,E} are irreducible, then they are modular.

This is proved by using the fact that GL_{2}(F_{2}) and GL_{2}(F_{3}) are solvable so we may apply “base-change”.

Theorem 2.7.2 (Wiles). If ρ is an ℓ-adic representation which is irreducible and modular mod ℓ with ℓ > 2 and certain other reasonable hypothesis are satisfied, then ρ itself is modular.

Modular Forms

Our goal is to explain modular forms as functions of lattices or of elliptic curves. Good references are Serre [24] and Katz [1].

3.1 Cusp Forms

First suppose N = 1, then we must define S_{k} = S_{k}(1). Let Γ_{1}(1) = SL_{2}(Z), then S_{k} consists of all functions f holomorphic on the upper half plane H and such that for all ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z) one has

f(\frac{aτ + b}{cτ + d}) = (cτ + d)^{k}f(τ),

and f vanishes at infinity. Thus, in particular, f(τ + 1) = f(τ) and so f passes to a well defined function of q = e^{2πiτ}. So f(q) is a function on {z : 0 < |z| < 1} and the condition that f(τ) vanishes at infinity is that f(q) extends to a holomorphic function on {z : |z| < 1} and f(0) = 0. In this case, we may write f(q) = ∑_{n=1}^{∞} a_{n}q^{n}.

3.2 Lattices

A lattice L ⊂ C is a subring L = Zω_{1} + Zω_{2} for which ω_{1}, ω_{2} ∈ C are lineary independent over R. Without loss, we may assume that ω_{1}/ω_{2} ∈ H.

Let

R = {lattices in C} = {(E, ω) : E is an elliptic curve, ω ∈ Ω_{E}^{1}}

and

M = {(ω_{1}, ω_{2}) : ω_{1}, ω_{2} ∈ C, Im(ω_{1}/ω_{2}) > 0}.

There is a left action of SL_{2}(Z) on M

( \begin{matrix}a & b \\ c & d \end{matrix} ) : (ω_{1}, ω_{2}) ↦ (aω_{1} + bω_{2}, cω_{1} + dω_{2})

and SL_{2}(Z) \M ≅ R.

3.3 Relationship With Elliptic Curves

There is a map L ↦ C/L from lattices to complex tori which, by Weierstrass theory, correspond to elliptic curves defined over C along with a distinguished differential ω = dz.

Conversely, if E/C is an elliptic curve, we can obtain the corresponding lattice by fixing a differential ω and taking the lattice to be the image of the map

H_{1}(E(C), Z) ⟶^{integration} C

which takes γ ∈ H_{1} to ∫_{γ} ω ∈ C.

There is a map M/C → H defined by (ω_{1}, ω_{2}) ↦ ω_{1}/ω_{2}. This gives an isomorphism

R/C^{∗} = (SL_{2}(Z) \M)/C^{∗} ⟶SL_{2}(Z) \H

and R/C^{∗} = { elliptic curves /C (without differentials)}.

If f : H → C we define F : M → C by F (ω_{1}, ω_{2}) = f(ω_{1}/ω_{2}). Suppose now that F is a lattice function and sattisfies the homogeneity condition F (λL) = λ^{−k}F (L). Then

f(\frac{aτ + b}{cτ + d}) = F (Z\frac{aτ + b}{cτ + d} + Z)
= F ((cτ + d)^{−1}(Z(aτ + b) + Z(cτ + d)))
= (cτ + d)^{k}F (Z(aτ + b) + Z(cτ + d))
= (cτ + d)^{k}F (Z + τZ)
= (cτ + d)^{k}f(τ)

so functions of lattices with the homogeneity condition come from functions f ∈ M_{k}. Thus, if f ∈ M_{k} and F is the corresponding lattice function then

F (Zω_{1} + Zω_{2}) = F (ω_{2}(Z + Z\frac{ω_{1}}{ω_{2}} )) = ω_{2}^{−k}F (Z + Z\frac{ω_{1}}{ω_{2}} ) = ω_{2}^{−k}f(\frac{ω_{1}}{ω_{2}} ),

so we can recover F from f.

3.4 Hecke Operators

Define a map T_{n} from the free abelian group generated by all C-lattices into itself by

T_{n}(L) = ∑_{(L:L′)=n} L′.

Then if F is a function on lattices define T_{n}F by

(T_{n}F)(L) = n^{k−1} ∑_{(L:L′)=n} F (L′).

Since (n, m) = 1 implies T_{n}T_{m} = T_{nm} and T_{p^{k}} is a polynomial in Z[T_{p}] the essential case to consider is n prime.

Suppose L′ ⊂ L with (L : L′) = n, then L/L′ is killed by n so nL ⊂ L′ ⊂ L and

L′/nL ⊂ L/nL ≅ (Z/nZ)^{2}.

Thus the subgroups of (Z/nZ)^{2} of order n correspond to the sublattices L′ of L of index n. When n = ℓ is prime there are ℓ + 1 such subgroups. (The subgroups correspond to nonzero vectors in F_{ℓ} modulo scalar equivalence and there are \frac{ℓ^{2}−1}{ℓ−1} of them.)

Suppose L′ ⊂ L is a sublattice of index ℓ and let L′′ = ℓ^{−1}L′. Note that ℓL ⊂ L′ so L ⊂ ℓ^{−1}L′ = L′′ and L is a sublattice of L′′ of index ℓ. Thus, assuming F satisfies the homogeneity condition F (λL) = λ^{−k}F (L),

ℓ^{k−1} ∑_{L′} F (L′) = \frac{1}{ℓ} ∑_{L′′} F (L′′)

which helps explain the extra factor of n^{k−1} in our definition of T_{n}F – we are “averaging” over the sublattices (note that there are ℓ + 1 terms yet we divide by ℓ so we aren’t exactly averaging).

We now give a geometric description of the ℓth Hecke operator. Let L ⊂ L′′ be lattices with (L′′ : L) = ℓ and let E = C/L, E′′ = C/L′′ be the elliptic curves corresponding to L, L′′, respectively. Then E[ℓ] = \frac{1}{ℓ}L/L contains H = L′′/L which may be thought of as a line [Ed: I don’t know why!]. Then the Hecke operator is

E ↦ \frac{1}{ℓ} ∑_{lines H⊂E[ℓ]} E/H.

Let π̂ be the isogeny dual to π : E → E/H. Then in terms of pairs (E, ω) we have

(E, ω) ↦ \frac{1}{ℓ} ∑_{H⊂E[ℓ],#H=ℓ}(E/H, π_{∗}ω) = ℓ^{k−1} ∑_{H⊂E[ℓ]}(E/H, π̂^{∗}(ω)).

We consider modular forms f on Γ_{1}(1) = SL_{2}(Z), that is, holomorphic functions on H ∪ {∞} which satisfy

f(τ) = f(\frac{aτ + b}{cτ + d})(cτ + d)^{−k}

for all ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z). Using a Fourier expansion we write

f(τ) = ∑_{n=0}^{∞} a_{n}e^{2πiτn},

and say f is a cusp form if a_{0} = 0. There is a correspondence between modular forms f and lattice functions F satisfying F (λL) = λ^{−k}F (L) given by F (Zτ + Z) = f(τ).

3.5 Explicit Description of Sublattices

The nth Hecke operator T_{n} of weight k is defined by

T_{n}(L) = n^{k−1} ∑_{\begin{matrix}L′⊂L \\ (L:L′)=n \end{matrix}} L′.

What are the L′ explicitly? Note that L/L′ is a group of order n and

L′/nL ⊂ L/nL = (Z/nZ)^{2}.

Write L = Zω_{1}+Zω_{2}, let Y_{2} be the cyclic subgroup of L/L′ generated by ω_{2} and let d = #Y_{2}. Let Y_{1} = (L/L′)/Y_{2}, then Y_{1} is generated by the image of ω_{1} so it is a cyclic group of order a = n/d. We want to exhibit a basis of L′. Let ω′_{2} = dω_{2} ∈ L′ and use the fact that Y_{1} is generated by ω_{1} to write aω_{1} = ω′_{1} + bω_{2} for some integer b and some ω′_{1} ∈ L′. Since b is only well-defined modulo d we may assume 0 ≤ b ≤ d − 1. Thus

(\begin{matrix}ω′_{1} \\ ω′_{2} \end{matrix}) = (\begin{matrix}a & b \\ 0 & d \end{matrix})(\begin{matrix}ω_{1} \\ ω_{2} \end{matrix})

and the change of basis matrix has determinent ad = n. Since

Zω′_{1} + Zω′_{2} ⊂ L′ ⊂ L = Zω_{1} + Zω_{2}

and (L : Zω′_{1} + Zω′_{2}) = n (since the change of basis matrix has determinent n) and (L : L′) = n we see that L′ = Zω′_{1} + Zω′_{2}.

Thus there is a one-to-one correspondence between sublattices L′ ⊂ L of index n and matrices ( \begin{matrix}a & b \\ 0 & d \end{matrix} ) with ad = n and 0 ≤ b ≤ d − 1. In particular, when n = p is prime there p + 1 of these. In general, the number of such sublattices equals the sum of the positive divisors of n.

3.6 Action of Hecke Operators on Modular Forms

Now assume f(τ) = ∑_{m=0}^{∞} c_{m}q^{m} is a modular form with corresponding lattice function F. How can we describe the action of the Hecke operator T_{n} on f(τ)? We have

T_{n}F (Zτ + Z) = n^{k−1} ∑_{\begin{matrix}a,b,d \\ ab=n \\ 0≤b<d \end{matrix}} F ((aτ + b)Z + dZ)

= n^{k−1} ∑ d^{−k}F (\frac{aτ + b}{d}Z + Z)

= n^{k−1} ∑ d^{−k}f(\frac{aτ + b}{d})

= n^{k−1} ∑_{a,d,b,m} d^{−k}c_{m}e^{2πi( \frac{aτ+b}{d})m}

= n^{k−1} ∑_{a,d,m} d^{1−k}c_{m}e^{\frac{2πiamτ}{d}} \frac{1}{d} ∑_{b=0}^{d−1}(e^{\frac{2πim}{d}})^{b}

= n^{k−1} ∑_{\begin{matrix}ad=n \\ m′≥0 \end{matrix}} d^{1−k}c_{dm′}e^{2πiam′τ}

= ∑_{\begin{matrix}ad=n \\ m′≥0 \end{matrix}} a^{k−1}c_{dm′}q^{am′}.

In the second to the last expression we let m = dm′, m′ ≥ 0, then used the fact that the sum \frac{1}{d} ∑_{b=0}^{d−1}(e^{\frac{2πim}{d}})^{b} is only nonzero if d|m.

Thus

T_{n}f(q) = ∑_{\begin{matrix}ad=n \\ m≥0 \end{matrix}} a^{k−1}c_{dm}q^{am}

and if μ ≥ 0 then the coefficient of q^{μ} is

∑_{\begin{matrix}a|n \\ a|μ \end{matrix}} a^{k−1}c_{\frac{nμ}{a^{2}}} .

Remark 3.6.1. When k ≥ 1 the coefficients of q^{μ} for all μ belong to the Z-module generated by the c_{m}.

Remark 3.6.2. Setting μ = 0 gives the constant coefficient of T_{n}f which is

∑_{a|n} a^{k−1}c_{0} = σ_{k−1}(n)c_{0}.

Thus if f is a cusp form so is T_{n}f. (T_{n}f is holomorphic since its original definition is as a finite sum of holomorphic functions.)

Remark 3.6.3. Setting μ = 1 shows that the coefficient of q in T_{n}f is ∑_{a|1} 1^{k−1}c_{n} = c_{n}. As an immediate corollary we have the following important result.

Corollary 3.6.4. Suppose f is a cusp form for which T_{n}f has 0 as coefficient of q for all n ≥ 1, then f = 0.

Remark 3.6.5. When n = p is prime we get an interesting formula for the action of T_{p} on the q-expansion of f. One has

T_{p}f = ∑_{μ≥0} ∑_{\begin{matrix}a|n \\ a|μ \end{matrix}} a^{k−1}c_{\frac{nμ}{a^{2}}} q^{μ}.

Since n = p is prime either a = 1 or a = p. When a = 1, c_{pμ} occurs in the coefficient of q^{μ} and when a = p, we can write μ = pλ and we get terms p^{k−1}c_{λ} in q^{λp}. Thus

T_{n}f = ∑_{μ≥0} c_{pμ}q^{μ} + p^{k−1} ∑_{λ≥0} c_{λ}q^{pλ}.

14 CHAPTER 3. MODULAR FORMS

Embedding Hecke Operators in the Dual

4.1 The Space of Modular Forms

Let Γ = Γ_{1}(1) = SL_{2}(Z) and for k ≥ 0 let

M_{k} = {f = ∑_{n=0}^{∞} a_{n}q^{n} : f is a modular form for Γ}

⊂ S_{k} = {f = ∑_{n=1}^{∞} a_{n}q^{n}}

These are finite dimensional C-vector spaces whose dimensions are easily computed. Furthermore, they are generated by familiar elements (see Serre [24] or Lang [10].) The main tool is the formula

∑_{p∈D∪\{∞\}} \frac{1}{e(p)} ord_{p}(f) = \frac{k}{12}

where D is the fundamental domain for Γ and

e(p) = \begin{cases}1 & otherwise \\ 2 & if p = i \\ 3 & if p = ρ \end{cases}

One can alternatively define e(p) as follows. If p = τ and E = C/(Zτ + Z) then e(p) = \frac{1}{2}# Aut(E).

For k ≥ 4 we define the Eisenstein series G_{k} by

G_{k}(q) = \frac{1}{2}ζ(1 − k) + ∑_{n=1}^{∞} σ_{k−1}(n)q^{n},

then the map

τ ↦ ∑_{\begin{matrix}(m,n)≠(0,0) \\ m,n∈Z \end{matrix}} \frac{1}{(mτ + n)^{k}}

differs from G_{k} by a constant (no proof). Also, ζ(1 − k) ∈ Q and one may say, symbolically

at least, “ζ(1 − k) = ∑_{d=1}^{∞} d^{k−1} = σ_{k−1}(0).” The nth Bernoulli number B_{n} is defined by the equation

\frac{x}{e^{x} − 1} = ∑_{n=0}^{∞} \frac{B_{n}x^{n}}{n!} .

One can show that ζ(1−k) = −\frac{B_{k}}{k} so the constant coefficient of G_{k} is −\frac{B_{k}}{2k} which is rational.

4.2 Inner Product

In what follows we assume k ≥ 2 to avoid trivialities.. The Hecke operators T_{n} acts on the space M_{k}. Fix a subspace V ⊂ M_{k} which is stable under the action of the T_{n}. Let T(V) be the C-algebra generated by the endomorphism T_{n} acting on V and note that T(V) is actually a finite dimensional C-vector space since it is a subspace of End(V) and V is finite dimensional. Recall that T is commutative.

There is a bilinear form

T × V → C
⟨T, f⟩ ↦ a_{1}(f|T)

where f|T = ∑_{n=0}^{∞} a_{n}(f|T)q^{n}. We thus get maps

V → Hom(T, C) = T^{∗}
T → Hom(V, C) = V^{∗}.

Theorem 4.2.1. The above maps are isomorphisms.

Proof. It just remains to show each map is injective. Then since a finite dimensional vector space and its dual have the same dimension the result follows. First suppose f ↦ 0 ∈ Hom(T, C), then a_{1}(f|T) = 0 for all T ∈ T so, in particular, a_{n} = a_{1}(f|T_{n}) = 0 for all n ≥ 1. Thus f is a constant, but since k ≥ 2 this implies f = 0 (otherwise f wouldn’t transform correctly with respect to the action of the modular group).

Next suppose T ↦ 0 ∈ Hom(V, C), then a_{1}(f|T) = 0 for all f ∈ V. Substiting f|T_{n} for f and using the commutativity of T we have

a_{1}((f|T_{n})|T) = 0  for all f, n ≥ 1
a_{1}((f|T)|T_{n}) = 0  by commutativity
a_{n}(f|T) = 0  n ≥ 1
f|T = 0  since k ≥ 2, as above

Thus T = 0 which completes the proof.

Remark 4.2.2. The above isomorphisms are T-equivariant. Hom(T, C) is a T-module if we let T ∈ T act on φ ∈ Hom(T, C) by (T · φ)(T′) = φ(T T′). If α : V → Hom(T, C) is the above isomorphism (so α : f ↦ φ_{f} := (T′ ↦ a_{1}(f|T′))) then equivariance is the statement that α(T f) = T α(f). This follows since

α(T f)(T′) = φ_{T f}(T′) = a_{1}(T f|T′) = a_{1}(f|T′T)
= φ_{f}(T′T) = T φ(T′) = T α(f)(T′).

4.3 Eigenforms

We continue to assume that k ≥ 2.

Definition 4.3.1. A modular form f ∈ M_{k} is an eigenform for T if f|T_{n} = λ_{n}f for all n ≥ 1 and some complex numbers λ_{n}.

Let f be an eigenform, then a_{n}(f) = a_{1}(f|T_{n}) = λ_{n}a_{1}(f) so if a_{1}(f) = 0 then a_{n}(f) = 0 for all n ≥ 1 so since k ≥ 2 this would imply f = 0. Thus a_{1}(f) ≠ 0 and we may as well divide through by a_{1}(f) to obtain the normalized eigenform \frac{1}{a_{1}(f)}f. We thus assume that a_{1}(f) = 1, then the formula becomes a_{n}(f) = λ_{n} and so f|T_{n} = a_{n}(f)f, for all n ≥ 1.

Theorem 4.3.2. Let f ∈ V and let ψ be the image of f in Hom(T, C), thus ψ(T) = a_{1}(f|T). Then f is a normalized eigenform iff ψ is a

ring homomorphism.

Proof. First suppose f is a normalized eigenform so f|T_{n} = a_{n}(f)f. Then

ψ(T_{n}T_{m}) = a_{1}(f|T_{n}T_{m}) = a_{m}(f|T_{n})
= a_{m}(a_{n}(f)f) = a_{m}(f)a_{n}(f)
= ψ(T_{n})ψ(T_{m}),

so ψ is a homomorphism.

Conversely, assume ψ is a homomorphism. Then f|T_{n} = ∑ a_{m}(f|T_{n})q^{m}, so to show that f|T_{n} = a_{n}(f)f we must show that a_{m}(f|T_{n}) = a_{n}(f)a_{m}(f). Recall that ψ(T_{n}) = a_{1}(f|T_{n}) = a_{n}, thus

a_{n}(f)a_{m}(f) = a_{1}(f|T_{n})a_{1}(f|T_{m}) = ψ(T_{n})ψ(T_{m})
= ψ(T_{n}T_{m}) = a_{1}(f|T_{n}|T_{m})
= a_{m}(f|T_{n})

as desired.

18 CHAPTER 4. EMBEDDING HECKE OPERATORS IN THE DUAL

Rationality and Integrality Questions

5.1 Review

In the previous lecture we looked at subspaces V ⊂ M_{k} ⊂ C[[q]], (k ≥ 4), and considered the space T = T(V) = C[…, T_{n}, …] ⊂ End_{C} V of Hecke operators on V. We defined a pairing T × V → C by (T, f) ↦ a_{1}(f|T) and showed this pairing is nondegenerate and that it induces isomorphisms T ≅ Hom(V, C) and V ≅ Hom(T, C).

5.2 Integrality

Fix k ≥ 4 and let S = S_{k} be the space of weight k cusp forms with respect to the action of SL_{2}(Z). Let

S(Q) = S_{k} ∩ Q[[q]]
S(Z) = S_{k} ∩ Z[[q]].

Theorem 5.2.1. There is a C-basis of M_{k} consisting of forms with integral coefficients.

Proof. This is seen by exhibiting a basis. Recall that for all k ≥ 4

G_{k} = −\frac{b_{k}}{2k} + ∑_{k=1}^{∞} ∑_{d|k} d^{k−1}q^{n}

is the kth Eisenstein series which is a modular form of weight k and

E_{k} = −\frac{2k}{b_{k}} · G_{k} = 1 + ⋯

is its normalization. Since the Bernoulli numbers b_{2}, …, b_{8} have 1 as numerator (this isn’t always the case, b_{10} = \frac{5}{66}) we see that E_{4} and E_{6} have coefficients in Z and constant term 1. Furthermore one shows by dimension and independence arguments that the forms

{E_{4}^{a}E_{6}^{b}|4a + 6b = k}

form a basis for M_{k}.

5.3 Victor Miller’s Thesis

Let d = dim_{C} S_{k}, then Victor Miller showed in his thesis (see [11], ch. X, theorem 4.4) that there exists

f_{1}, …, f_{d} ∈ S_{k}(Z) such that a_{i}(f_{j}) = δ_{ij}

for 1 ≤ i, j ≤ d. The f_{i} clearly form a basis.

Proposition 5.3.1. Let R = Z[…, T_{n}, …] ⊂ End(S_{k}), then R = ⨁_{i=1}^{d} ZT_{i}.

Proof. To see that T_{1}, ⋯, T_{d} ∈ T = T(S_{k}) are linearly independent over C suppose ∑_{i=1}^{d} c_{i}T_{i} = 0, then

0 = a_{1}(f_{j}| ∑ c_{i}T_{i}) = ∑_{i} c_{i}a_{i}(f_{j}) = ∑_{i} c_{i}δ_{ij} = c_{j}.

From the isomorphism T ≅ Hom(S_{k}, C) we know that dim_{C} T = d, so we can write any T_{n} as a C-linear combination

T_{n} = ∑_{i=1}^{d} c_{n_{i}}T_{i}, c_{n_{i}} ∈ C.

But

Z ∋ a_{n}(f_{j}) = a_{1}(f_{j}|T_{n}) = ∑_{i=1}^{d} c_{n_{i}}a_{1}(f_{j}|T_{i}) = ∑_{i=1}^{d} c_{n_{i}}a_{i}(f_{j}) = c_{n_{j}}

so the c_{n_{i}} all lie in Z which completes the proof.

Thus R is an integral Hecke algebra of finite rank d over Z. We have a map

S(Z) × R → Z
(f, T) ↦ a_{1}(f|T)

which induces an embedding

S(Z) ↪ Hom(R, Z) ≅ Z^{d}.

Exercise 5.3.2. Prove that the map S(Z) ↪ Hom(R, Z) is in fact an isomorphism of Tmodules. [Hint: Show the cokernel is torsion free.]

5.4 Petersson Inner Product

The main theorem is

Theorem 5.4.1. The T_{n} ∈ T(S_{k}) are all diagonalizable over C.

To prove this we note that S_{k} supports a non-degenerate positive definite Hermitean inner product (the Petersson inner product)

(f, g) ↦ ⟨f, g⟩ ∈ C

such that ⟨f|T_{n}, g⟩ = ⟨f, g|T_{n}⟩. We need some background facts.

Definition 5.4.2. An operator T is normal if it commutes with its adjoint, thus T T^{∗} = T^{∗}T.

T_{n} is clearly normal since T_{n}^{∗} = T_{n},

Theorem 5.4.3. A normal operator is diagonalizable.

Thus each T_{n} is diagonalizable.

Theorem 5.4.4. A commuting family of semisimple (=diagonalizable) operators can be simultaneously diagonalized.

Since the T_{n} commute this implies S_{k} has a basis consisting of normalized eigenforms f. Their eigenvalues are real since

a_{n}(f)⟨f, f⟩ = ⟨a_{n}(f)f, f⟩ = ⟨f|T_{n}, f⟩
= ⟨f, a_{n}(f)f⟩ = \overline{a_{n}(f)}⟨f, f⟩.

Exercise 5.4.5. The coefficients a_{n} of the eigenforms are totally real algebraic integers. [Hint: The space S_{k} is stable under the action of Aut(C) on coefficients: if f = ∑_{n=1}^{∞} c_{n}q^{n} ∈ S_{k} and σ ∈ Aut(C) then σ(f) = ∑_{n=1}^{∞} σ(c_{n})q^{n} is again in S_{k} (check this by writing f in terms of a basis f_{1}, …, f_{d} ∈ S(Z)). Next use the fact that f is an eigenform iff σ(f) is an eigenform.]

Let

H = {x + iy : x, y ∈ R, and y > 0}

be the upper half plane. Then the volume form \frac{dx∧dy}{y^{2}} is invariant under the action of

GL_{2}^{+}(R) = {M ∈ GL_{2}(R)| det(M) > 0}.

If α = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}^{+}(R) then ( \begin{matrix}a & b \\ c & d \end{matrix} ) acts on H by

(\begin{matrix}a & b \\ c & d \end{matrix}) : z ↦ \frac{az + b}{cz + d}

and one has

Im(\frac{az + b}{cz + d}) = \frac{det(α)}{|cz + d|^{2}} y.

Differentiating \frac{az+b}{cz+d} gives

d(\frac{az + b}{cz + d}) = \frac{a(cz + d)dz − c(az + b)dz}{(cz + d)^{2}}

= \frac{(ad − bc)dz}{(cz + d)^{2}}

= \frac{det(α)}{(cz + d)^{2}} dz

Thus, under the action of α, dz ∧ d\overline{z} takes on a factor of

\frac{det(α)^{2}}{(cz + d)^{2}(c\overline{z} + d)^{2}} = (\frac{det(α)}{|cz + d|^{2}} )^{2}.

Definition 5.4.6. The Petersson inner product of forms f, g ∈ S_{k} is defined by

< f, g >= ∫_{Γ\H}(f(z)\overline{g(z)}y^{k})\frac{dx ∧ dy}{y^{2}} ,

where Γ = SL_{2}(Z).

Integrating over Γ\H can be taken to mean integrating over a fundamental domain for the action of H. Showing that the operators T_{n} are self-adjoint with respect to the Petersson inner product is a harder computation than Serre [24] might lead one to believe — it takes a bit of thought.

22 CHAPTER 5. RATIONALITY AND INTEGRALITY QUESTIONS

Modular Curves

6.1 Cusp Forms

Recall that if N is a positive integer we define the congruence subgroups Γ(N) ⊂ Γ_{1}(N) ⊂ Γ_{0}(N) by

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

Let Γ be one of the above subgroups. One can give a construction of the space S_{k}(Γ) of cusp forms of weight k for the action of Γ using the language of algebraic geometry. Let X_{Γ} = \overline{Γ\H^{∗}} be the compactifaction of the upper half plane (union the cusps) modulo the action of Γ. Then X_{Γ} can be given the structure of Riemann surface and S_{2}(Γ) = H^{0}(X_{Γ}, Ω^{1}) where Ω^{1} is the sheaf of differential 1-forms on X_{Γ}. This works since an element of H^{0}(X_{Γ}, Ω^{1}) is a differential form f(z)dz, holomorphic on H and the cusps, which is invariant with respect to the action of Γ. If γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ then

d(γ(z))/dz = (cz + d)^{−2}

so

f(γ(z))d(γ(z)) = f(z)dz

iff f satisfies the modular condition

f(γ(z)) = (cz + d)^{2}f(z).

There is a similar construction of S_{k} for k > 2.

6.2 Modular Curves

One knows that SL_{2}(Z) \H parameterizes isomorphism classes of elliptic curves. The other congruence subgroups also give rise to similar parameterizations. Thus Γ_{0}(N)\H parameterizes pairs (E, C) where E is an elliptic curve and C is a cyclic subgroup of order N, and Γ_{1}(N)\H parameterizes pairs (E, P) where E is an elliptic curve and P is a point of exact order N. Note that one can also give a point of exact order N by giving an injection Z/NZ ↪ E[N] or equivalently an injection μ μ μ_{N} ↪ E[N] where μ μ μ_{N} denotes the Nth roots of unity. Γ(N)\H parameterizes pairs (E, {α, β}) where {α, β} is a basis for E[N] ≅ (Z/NZ)^{2}.

The above quotients spaces are called moduli spaces for the moduli problem of determining equivalence classes of pairs (E+ extra structure).

6.3 Classifying Γ(N)-structures

Definition 6.3.1. Let S be an arbitrary scheme. An elliptic curve E/S is a proper smooth curve

\begin{aligned}& E \\ f & ↓ \\ & S \end{aligned}

with geometrically connected fibers all of genus one, give with a section “0”.

Loosely speaking, proper is a generalization of projective and smooth generalizes nonsingularity. See Hartshorne [7], chapter III, section 10, for the precise definitions.

Definition 6.3.2. Let S be any scheme and E/S an elliptic curve. A Γ(N)-structure on E/S is a group homomorphism

φ : (Z/NZ)^{2} → E[N](S)

whose image “generates” E[N](S).

A good reference is chapter 3 of Katz and Mazur [9].

Define a functor from the category of Q-schemes to the category of sets by sending a scheme S to the set of isomorphism classes of pairs

(E, Γ(N)-structure)

where E is an elliptic curve defined over S and isomorphisms (preserving the Γ(N)-structure) are taken over S. An isomorphism preserves the Γ(N)-structure if it takes the two distinguished generators to the two distinguished generators in the image (in the correct order).

Theorem 6.3.3. For N ≥ 4 the functor defined above is representable and the object representing it is the modular curve X corresponding to Γ(N).

What this means is that given a Q-scheme S, the set X(S) = Mor_{Q-schemes}(S, X) is isomorphic to the image of the functor’s value on S.

There is a natural way to map a pair (E, Γ(N)-structure) to an Nth root of unity. If P, Q are the distinguished basis of E[N] we send the pair (E, Γ(N)-structure) to

e_{N}(P, Q) ∈ μ μ μ_{N}

where e_{N} : E[N] × E[N] → μ μ μ_{N} is the Weil pairing. For the definition of this pairing see chapter III, section 8 of Silverman [33]. The Weil pairing is bilinear, alternating, nondegenerate, Galois invariant, and maps surjectively onto μ μ μ_{N}.

6.4 More on Integral Hecke Operators

We are considering the algebra of integral Hecke operators T = T_{Z} on the space of cusp forms S_{k}(C) with respect to the action of the full modular group SL_{2}(Z). Our goal is to see why T ≅ Z^{d} where d = dim_{C} S_{k}(C).

Suppose A ⊂ C is any subring of C and recall that

T_{A} = A[…, T_{n}, …] ⊂ End_{C} S_{k}.

We have a natural map

T_{A} ⊗_{A} C → T_{C}

but we do not yet know that it is an isomorphism.

6.5 Complex Conjugation

We have a conjugate linear map on functions

f(τ) ↦ \overline{f(−\overline{τ})}.

Since \overline{(e^{−2πi\overline{τ}})} = e^{2πiτ}, it follows that

∑_{n=1}^{∞} a_{n}q^{n} ↦ ∑_{n=1}^{∞} \overline{a_{n}}q^{n}

so it is reasonable to call this map “complex conjugation”. Furthermore, if we know that

S_{k}(C) = C ⊗_{Q} S_{k}(Q)

then it follows that complex conjugation takes S_{k}(C) into S_{k}(C). To see this note that if we have the above equality then every element of S_{k}(C) is a C-linear combination of elements of S_{k}(Q) and conversely, and it is clear that the set of such C-linear combinations is invariant under the action of complex conjugation.

6.6 Isomorphism in the Real Case

Proposition 6.6.1. T_{R} ⊗_{R} C ≅ T_{C}, as C-vector spaces.

Proof. Since S_{k}(R) = S_{k}(C) ∩ R[[q]] and since theorem 5.1 assures us that there is a Cbasis of S_{k}(C) consisting of forms with integral coefficients, we see that S_{k}(R) ≅ R^{d} where d = dim_{C} S_{k}(C). (Any element of S_{k}(R) is a C-linear combination of the integral basis, hence equating real and imaginary parts, an R-linear combination of the integral basis, and the integral basis stays independent over R.) By considering the explicit formula for the action of the Hecke operators T_{n} on S_{k} (see section 3) we see that T_{R} leaves S_{k}(R) invariant, thus

T_{R} = R[…, T_{d}, …] ⊂ End_{R} S_{k}(R).

In section 4 we defined a “perfect” pairing

T_{C} × S_{k}(C) → C

which allowed us to show that T_{C} ≅ S_{k}(C). By restricting to R we again get a perfect pairing so we see that T_{R} ≅ S_{k}(R) ≅ R^{d} which implies that

T_{R} ⊗_{R} C ⟶^{∼} T_{C}.

This also shows that S_{k}(C) ≅ C ⊗_{R} S_{k}(R) so we have complex conjugation over R.

6.7 The Eichler-Shimura Isomorphism

Our goal in this section is to outline a homological interpretation of S_{k}. For details see chapter 6 of Lang [11], the original paper of Shimura [32], or chapter VIII of Shimura [31].

How is S_{k}(C) sort of isomorphic to H^{1}(X_{Γ}, R)? Suppose k = 2 and Γ ⊂ SL_{2}(Z) is a congruence subgroup, let X_{Γ} = \overline{Γ\H} be the Riemann surface obtained by compactifying the upper half plane modulo the action of Γ. Then S_{k}(C) = H^{0}(X_{Γ}, Ω^{1}) so we have a pairing

H_{1}(X_{Γ}, Z) × S_{k}(C) → C

given by integration

(γ, ω) ↦ ∫_{γ} ω.

This gives an embedding

Z^{2d} ≅ H_{1}(X_{Γ}, Z) ↪ Hom_{C}(S_{k}(C), C) ≅ C^{d}

of a “lattice” in C^{d}. (We say “lattice” since there were some comments by Ribet that Z^{2d} isn’t a lattice because the rank might be too small since a subring of C^{d} having Z-rank 2d might not spans C^{d} over C). Passing to the quotient (and compactifying) gives a complex torus called the Jacobian of X_{Γ}. Again using the above pairing we get an embedding

C^{d} ≅ S_{k}(C) ↪ Hom(H_{1}(X_{Γ}, Z), C) ≅ C^{2d}

which, upon taking the real part, gives

S_{k}(C) → Hom(H_{1}(X_{Γ}, Z), R) ≅ H^{1}(X_{Γ}, R) ≅ H_{p}^{1}(Γ, R)

where H_{p}^{1}(Γ, R) denotes the parabolic group cohomology of Γ with respect to the trivial action. It is this result, that we may view S_{k}(C) as the cohomology group H_{p}^{1}(Γ, R), that was alluded to above.

Shimura generalized this for arbitrary k ≥ 2 so that

S_{k}(C) ≅ H_{p}^{1}(Γ, V_{k})

where V_{k} is a k − 1 dimensional R-vector space. The isomorphism is (approximately) the following: f ∈ S_{k}(C) is sent to the map

γ ↦ Re ∫_{τ_{0}}^{γτ_{0}} f(τ)τ^{i}dτ, i = 0, …, k − 2.

Let W = R ⊕ R, then Γ acts on W by

(\begin{matrix}a & b \\ c & d \end{matrix}) : (\begin{matrix}x \\ y \end{matrix}) ↦ (\begin{matrix}ax + by \\ cx + dy \end{matrix})

so Γ acts on

V_{k} = Sym^{k−2} W = W^{⊗k−2}/S_{k−2}

where S_{k−2} is the symmetric group on k − 2 symbols (note that dim V_{k} = k − 1). Let

L = H_{p}^{1}(Γ, Sym^{k−2}(Z ⊕ Z))

then under the isomorphism

S_{k}(C) ≅ H_{p}^{1}(Γ, R)

L is a sublattice of S_{k}(C) of Z-rank 2 which is T_{n}-stable for all n. Thus we have an embedding

T_{Z} = T ↪ End L

and so T_{R} ⊂ End_{R}(L ⊗ R) and T_{Z} ⊗_{Z} R ≅ T_{R} which has rank d.

6.8. THE PETTERSON INNER PRODUCT IS HECKE COMPATIBLE 27

6.8 The Petterson Inner Product is Hecke Compatible

Theorem 6.8.1. Let Γ = SL_{2}(Z), let f, g ∈ S_{k}(C), and let

⟨f, g⟩ = ∫_{Γ\H} f(τ)\overline{g(τ)}y^{k} \frac{dxdy}{y^{2}} .

Then this integral is well-defined and Hecke compatible, that is, ⟨f|T_{n}, g⟩ = ⟨f, g|T_{n}⟩ for all n. Proof. See Chapter 3 of Lang [11].

28 CHAPTER 6. MODULAR CURVES

Higher Weight Modular Forms

We are considering the spaces S_{k}(C), S_{k}(Q) and S_{k}(Z) which all have rank d. Each is acted upon by the Hecke algebra T. We defined a Hecke compatible inner product (the Peterrson product) and used it to show that

S_{k}(Z) ≅ Hom_{Z}(T, Z).

7.1 Definitions of T

“I may be asking you to explain something we have already discussed, but have we intrinsically defined the Hecke operators yet?”

– Saul Schliemer

The T_{n} are defined as operators on S_{k}(C) by defining their action on modular forms and noting from explicit formulas that S_{k}(C) is preserved. But the T_{n} can be thought of in other ways, for example, since

S_{k}(C) ≅ H_{p}^{1}(Γ, Sym^{k−2}(R ⊕ R))

one may give an explicit description of the action of T_{n} on H_{p}^{1}.

7.2 Double Cosets

Let p be a prime and

M_{p} = {α ∈ M_{2}(Z) : det(α) = p}.

Let F : L → C be a function on the free abelian group of lattices and recall that T_{p} acts on F by

(T_{p}F)(L) = p^{k−1} ∑_{\begin{matrix}L′⊂L \\ (L:L′)=p \end{matrix}} F (L′).

One can write M_{p} as a disjoint union of left cosets,

M_{p} = SL_{2}(Z) (\begin{matrix}p & 0 \\ 0 & 1 \end{matrix}) SL_{2}(Z) = ⋃_{\begin{matrix}ad=p \\ 0≤b<d \end{matrix}} (\begin{matrix}a & b \\ 0 & d \end{matrix}) SL_{2}(Z) .

Then ∑_{(L:L′)=p} L′ may be thought of as the sum of the images of L under the action of the left cosets of M_{p}. For a complete exposition, in greater generality, see Shimura [31], especially chapter 3.

7.3 More General Congruence Subgroups

Definition 7.3.1. A Dirichlet character mod N is a homomorphism

ε : (Z/nZ)^{∗} → C^{∗}

extended to Z/NZ by putting ε(m) = 0 if (m, N) ≠ 1.

Fix integers k ≥ 0 and N ≥ 1. In this section we consider the spaces

S_{k}(Γ_{1}(N), ϵ)

for Dirichlet characters ε mod N and explicitly describe the action of the Hecke operators

\begin{cases}T_{n}, & n ≥ 1 \\ ⟨d⟩, & d ∈ (Z/NZ)^{∗} \end{cases}

on these spaces. Remark 7.3.2. Let n be a positive integer. If (n, N) = 1, then the T_{n} behave like they do for SL_{2}(Z). In fact, the T_{n} and ⟨d⟩ commute and

(f|T_{n}, g) = (f, g|⟨n⟩^{−1}T_{n})
(f|⟨d⟩, g) = (f, g|⟨d⟩^{−1})

so the T_{n} (for n prime to N) and ⟨d⟩ are simultaneously diagonalizable. But if (n, N) ≠ 1 then T_{n} may not be diagonalizable.

Definition 7.3.3. Let

S_{k}(Γ_{1}(N)) = {f : f(γτ) = (cτ + d)^{k}f(τ) all γ ∈ Γ_{1}(N)}

where the f are assumed holomorphic on H ∪ {cusps}. For each Dirichlet character ε mod N let

S_{k}(Γ_{1}(N), ε) = \{f : f(γτ)(cτ + d)^{−k} = ε(d)f all γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N)\}.

When ε ≠ 0 and f ∈ S_{k}(Γ_{1}(N), ε) one calls ε the nebentypus of f.

Let d ∈ (Z/NZ)^{∗} and let f ∈ S_{k}(Γ_{1}(N)). Let γ = ( \begin{matrix}a & b \\ 0 & d \end{matrix} ) ∈ Γ_{1}(N) be a matrix whose lower right entry is congruent to d mod N. Then we define

f(τ)|⟨d⟩ = f(γτ)(cτ + d)^{−k}.

Since f|⟨d⟩ = ε(d)f, S_{k}(Γ_{1}(N), ε) is the ε(d) eigenspace of ⟨d⟩ and ⟨d⟩ is diagonalizable so one has a direct sum decomposition

S_{k}(Γ_{1}(N)) = ⨁_{ε:(Z/NZ)^{∗}→C^{∗}} S_{k}(Γ_{1}(N), ε).

If f ∈ S_{k}(Γ_{1}(N), ε) then

(\begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix}) ∈ Γ_{0}(N)

so

f(−τ)(−1)^{−k} = ε(−1)f(τ)

so that S_{k}(Γ_{1}(N), ε) = 0 unless ε(−1) = (−1)^{k}. Thus about half of the direct summands vanish.

7.4 Explicit Formulas

Let

f = ∑_{n=1}^{∞} a_{n}q^{n} ∈ S_{k}(Γ_{1}(N), ε)

and let p be a prime, then

f|T_{p} = \begin{cases}∑_{n=1}^{∞} a_{np}q^{n} + p^{k−1}ε(p) ∑_{n=1}^{∞} a_{n}q^{pn}, & p |̸N \\ ∑_{n=1}^{∞} a_{np}q^{n} + 0, & p|N \end{cases}

When p|N, T_{p} is often denoted U_{p} and called an Atkin-Lehner operator.

We have the relations

T_{m}T_{n} = T_{mn}, (m, n) = 1
T_{p^{k}} = \begin{cases}(T_{p})^{k}, & p|N \\ ?, & p |̸N \end{cases}

7.5 Old and New Forms

Warning: T_{p} is not necessarily diagonalizable if p|N. There is an example due to Shimura, to present it we must first introduce old and new forms.

Let M and N be positive integers such that M|N and let d|\frac{N}{M}. If f(τ) ∈ S_{k}(Γ_{1}(M)) then f(dτ) ∈ S_{k}(Γ_{1}(N)). We thus have a map S_{k}(Γ_{1}(M)) → S_{k}(Γ_{1}(N)) for each d|\frac{N}{M}. Combining these gives a map

φ_{M} : ⨁_{d| \frac{N}{M}} S_{k}(Γ_{1}(d)) → S_{k}(Γ_{1}(N)).

Definition 7.5.1. The old part of S_{k}(Γ_{1}(N)) is the subspace generated by the images of the φ_{M} for M|N, M ≠ N.

We remark that the new part of S_{k}(Γ_{1}(N)) is the orthogonal complement of the old part with respect to the Petersson inner product.

32 CHAPTER 7. HIGHER WEIGHT MODULAR FORMS

New Forms

Today we discuss how the Hecke operators T_{n} on S_{k}(Γ_{1}(N)) can fail to be diagonalizable. Let N be a positive integer and M a divisor of N. For each d|\frac{N}{M} we define a map

α_{d} : S_{k}(Γ_{1}(M)) → S_{k}(Γ_{1}(N)) : f(τ) ↦ f(dτ).

Note that when T_{p} acts on the image space S_{k}(Γ_{1}(N)) we will often denote it by U_{p}. We must check that f(dτ) ∈ S_{k}(Γ_{1}(N)). Define for γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ),

(f|[γ]_{k})(τ) = det(γ)^{k−1}(cz + d)^{−k}f(γ(τ)).

Thus f ∈ S_{k}(Γ_{1}(N)) iff f|[γ]_{k}(τ) = f(τ) (and f is holomorphic). Now let f(τ) ∈ Γ_{1}(M) and let ι_{d} = ( \begin{matrix}d & 0 \\ 0 & 1 \end{matrix} ). Then f|[ι_{d}]_{k}(τ) = d^{k−1}f(dτ) is a modular form on Γ_{1}(N) since ι_{d}^{−1}Γ_{1}(M)ι_{d} contains Γ_{1}(N) (check this directly by conjugating an element of Γ_{1}(N) by ι_{d}). Moreover if f is a cusp form then so is f|[ι_{d}]_{k}. If f ∈ S_{k}(Γ_{1}(M)) is nonzero, then as d varies over divisors of \frac{N}{M}, the various f(dτ) are linearly independent.

Suppose f ∈ S_{k}(Γ_{1}(M)) is a normalized eigenform for all of the Hecke operators T_{n} and ⟨n⟩, and p is a prime not dividiing M. Then

f|T_{p} = af and f|⟨p⟩ = ε(p)f.

Assume N = p^{r}M where r is an integer ≥ 1. Let

f_{i}(τ) = f(p^{i}τ),

so f_{0}, …, f_{r} are the images of f under the maps defined above and f = f_{0}. Consider the action of U_{p} on the f_{i}. From previous work we have

f|T_{p} = ∑_{n≥1} a_{np}q^{n} + ε(p)p^{k−1} ∑ a_{n}q^{pn}
= f_{0}|U_{p} + ε(p)p^{k−1}f_{1}

so

f_{0}|U_{p} = f|T_{p} − ε(p)p^{k−}f_{1} = af_{0} − ε(p)p^{k−1}f_{1}.

Also

f_{1}|U_{p} = (∑ a_{n}q^{pn})|U_{p} = ∑ a_{n}q^{n} = f_{0}.

More generally one can show that f_{i}|U_{p} = f_{i−1}.

U_{p} preserves the two dimensional vector space spanned by f_{0} and f_{1}. The matrix of U_{p} is

A = (\begin{matrix}a & 1 \\ −ε(p)p^{k−1} & 0 \end{matrix})

which has characteristic polynomial

χ_{A}(X) = X^{2} − aX + p^{k−1}ε(p).

8.1 Connection With Galois Representations

This leads to a striking connection with Galois representations. Let f be a modular form and E be the field generated over Q by the coefficients of f. Let ℓ be a prime and λ a prime lying over ℓ. Then one constructs a representation

ρ_{λ} : Gal(\overline{Q}/Q) → GL(2, E_{λ}).

If p |̸Nℓ, then ρ_{λ} is unramified at p, so there is a Frobenious element frob_{p} ∈ Gal(\overline{Q}/Q). One can show that

det(ρ_{λ}(frob_{p})) = p^{k−1}ε(p)
Tr(ρ_{λ}(frob_{p})) = a_{p} = a,

so the characteristic polynomial of ρ_{λ}(frob_{p}) ∈ GL_{2}(E_{λ}) is

X^{2} − a_{p}X + p^{k−1}ε(p).

8.2 Semisimplicity of U_{p}

Question 8.2.1. Is U_{p} semisimple on the span of f_{0} and f_{1}?

If the eigenvalues are distinct the answer is clearly yes. If the eigenvalues are the same, then X^{2} − aX + p^{k−1}ε(p) has discriminant zero, that is, 4ε(p)p^{k−1} = a^{2} so

a = 2p^{\frac{k−1}{2}} \sqrt{ε(p)}.

Is this possible? The answer is still unknown, although it is a curious fact that the Ramanujan conjectures (proved by Delign in 1973) imply that |a| ≤ 2p^{\frac{k−1}{2}} , so the above equality remains taunting.

When k = 2 Weil showed that ρ_{λ}(frob_{p}) is semisimple so if the eigenvalues of U_{p} are equal then ρ_{λ}(frob_{p}) is a scalar. But Edixhoven and Coleman [5] show that it is not a scalar by looking at the abelian variety attached to f.

8.3 Shimura’s Example of Nonsemisimple U_{p}

Let W be the space spanned by f_{0}, f_{1} and let V be the space spanned by f_{0}, f_{1}, f_{2}, f_{3}. U_{p} acts on V/W by \overline{f_{2}} ↦ 0 and \overline{f_{3}} ↦ \overline{f_{2}}. Thus the matrix of the action of U_{p} on V/W is ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) which is nonzero and nilpotent hence not semisimple. Since W is invariant under U_{p} this shows that U_{p} is not semisimple on V.

8.4 An Interesting Duality

Now suppose N = 1 thus Γ_{1}(N) = SL_{2}(Z). Because of the Petersson product all the T_{n} are diagonalizable, so S_{k} = S_{k}(Γ_{1}(1)) has a basis

f_{1}, …, f_{d}

of normalized eigenforms where d = dim S_{k}. Let T = T_{C}, then there is a canonical map

T_{C} ↪ C^{d} : T ↦ (λ_{1}, …, λ_{d})

where f_{i}|T = λ_{i}f_{i}. This map is clearly injective and we know by previous arguments that dim T_{C} = d so the map is an isomorphism of C-vector spaces.

The form

v = f_{1} + ⋯ + f_{n}

generates S_{k} as a T-module. Since v corresponds to the vector (1, …, 1) and T ≅ C^{d} acts on S_{k} ≅ C^{d} componentwise this is just the statement that C^{d} is generated by (1, …, 1) as a C^{d}-module – which is clear. Thus we have simultaneously:

1) S_{k} is free of rank 1 over T, and

2) S_{k} = Hom_{C}(T, C) as T-modules, thus

T ≅ Hom_{C}(T, C).

The isomorphism sends an element of T ∈ T to T v ∈ S_{k}. Since the identification S_{k} = Hom_{C}(T, C) was constructed using the Petersson product it is canonical and since the choice of a normalized eigenbasis f_{1}, …, f_{d} is canonical we see that the isomorphism T ≅ Hom_{C}(T, C) is canonical.

Proposition 8.4.1. v ∈ S_{k}(Q)

Proof. Let σ ∈ Gal(\overline{Q}/Q), then if f_{i} is a normalized eigenform so is σ(f_{i}) (from the explicit formula). Thus σ(f_{1} + ⋯ + f_{n}) = f_{1} + ⋯ + f_{n} for all σ as desired.

Now we consider the case for general N. Recall that we have defined maps

S_{k}(Γ_{1}(M)) → S_{k}(Γ_{1}(N))

for all M dividing N and all divisors d of \frac{N}{M}.

Definition 8.4.2. The old part of S_{k}(Γ_{1}(N)) is the space generated by all images of these maps with M|N but M ≠ N. The new part is the orthogonal complement of the old part with respect to the Petersson product.

There is an algebraic definition of the new part. One defines certain trace maps

S_{k}(Γ_{1}(N)) → S_{k}(Γ_{1}(M))

for all M < N, M|N which are the adjoints to the above maps (w.r.t Petersson product). Then f is in the new part of S_{k}(Γ_{1}(N)) iff f is killed by all of these maps.

It can be shown that the T_{n} act semisimply on S_{k}(Γ_{1}(M))_{new} for all M ≥ 1. Thus S_{k}(Γ_{1}(M))_{new} has a basis of eigenforms. We have a natural map

⨁_{M|N} S_{k}(Γ_{1}(M))_{new} ↪ S_{k}(Γ_{1}(N)).

The image in S_{k}(Γ_{1}(N)) of an eigenform f for some S_{k}(Γ_{1}(M))_{new} is called a newform of level M_{f} = M. Note that a newform is not necessarily an eigenform for the Hecke operators acting on S_{k}(Γ_{1}(N)). Let

v = ∑_{f} f(q^{\frac{N}{M_{f}}}) ∈ S_{k}(Γ_{1}(N))

where the sum is taken over all newforms f of weight k and some level M|N. This generalizes the v constructed above when N = 1 and has many of the same good properties. For example, S_{k}(Γ_{1}(N)) is free of rank 1 over T with basis element v. The coefficients of v lie in Q, but to show this we need to know the new part of S_{k}(Γ_{1}(N)) is stable under the action of the Galois group of Q. This is not easy since the new part is defined in terms of the Petersson product which is an analytic construction. Serre circumvents this problem by giving an alternative definition in terms of trace maps going the other way.

8.5 Observations on T_{n}

Let T_{Q} = Q[⋯, T_{n}, ⋯] and Γ = Γ(1) = SL_{2}(Z). Let f_{1}, …, f_{d} be a basis of Γ consisting of normalized eigenforms.

Proposition 8.5.1. The coefficients of the f_{i} are totally real algebraic integers.

Proof. Gal(C/Q) acts on f_{i} by acting on the coefficients of its q-expansion. From the explicit formula in section 3.2 one sees that the set {f_{1}, …, f_{d}} is stable under the action of Gal(C/Q). For any i, a_{n}(f_{i}) is an eigenvalue of T_{n} since f_{i}|T_{n} = a_{n}(f_{i})f_{i}, and T_{n} is self-adjoint so a_{n}(f_{i}) must be real. Thus all conjugates of a_{n}(f_{i}) are real and there are only finitely many since a conjugate of a_{n}(f_{i}) must be a_{n}(f_{j}) for some j, 1 ≤ j ≤ d.

Proposition 8.5.2. The operators ⟨d⟩ on S_{k}(Γ_{1}(N)) lie in Z[…, T_{n}, …].

Proof. It is enough to show ⟨p⟩ ∈ Z[…, T_{n}, …] for There is a formula relating ⟨p⟩ and T_{p},

p^{k−1}⟨p⟩ = T_{p}^{2} − T_{p^{2}}.

By Dirichlet’s theorem on prime’s in arithmetic progression, see VIII.4 of Lang [10], there is another prime q congruent to p mod N. Since p^{k−1} and q^{k−1} are relatively prime there exist integers a and b so that ap^{k−1} + bq^{k−1} = 1. Then

⟨p⟩ = ⟨p⟩(ap^{k−1} + bq^{k−1}) = a(T_{p}^{2} − T_{p^{2}}) + b(T_{q}^{2} − T_{q^{2}}).

Let Σ be a set of representatives of {f_{1}, …, f_{d}}\Gal(C/Q). It is unknown whether or not #Σ can be larger than one, that is, whether the eigenforms are all conjugate under the action of Galois. Let K_{f} = Q(…, a_{n}(f), …) and defined a homomorphism of Q-algebras

T_{Q} → K_{f} : T_{n} ↦ λ where T_{n}f = λf

Taking the product over a set of representatives of the f_{i} yields a map

T_{Q} ⟶^{∼} ∏_{f∈Σ} K_{f}

which one can show is an isomorphism of Q-algebras. Example 8.5.3. Consider S_{2}(Γ_{0}(N)) with N prime, then

T_{Q} ≅ E_{1} × ⋯ E_{t}

with the E_{i} totally real fields. When N = 37, that T_{Q} ≅ Q × Q.

Some Explicit Genus Computations

9.1 Computing the Dimension of S_{k}(Γ)

Let k = 2 unless otherwise noted, and let Γ ⊂ SL_{2}(Z) be a congruence subgroup. Then

S_{2}(Γ) = H^{0}(X_{Γ}, Ω^{1})

where

X_{Γ} = (Γ\H) ∪ (Γ\P^{1}(Q)).

By definition dim H^{0}(X_{Γ}, Ω^{1}) is the genus of X_{Γ}.

Exercise 9.1.1. Prove that when Γ = SL_{2}(Z) then Γ\P^{1}(Q) is a point.

Since Γ ⊂ Γ(1) there is a covering

Γ\H ⟶ X_{Γ}
↓  ↓
Γ(1)\H ⟶ X_{Γ(1)} ⟶^{j} P^{1}(C)

which is only ramified at points above 0, 1728, ∞ (0 corresponds to i and ρ to 1728 under j).

Example 9.1.2. Suppose Γ = Γ_{0}(N), then the degree of the covering is the index (SL_{2}(Z) /{±1} : Γ_{0}(N)/{±1}). A point on Y_{Γ(1)} corresponds to an elliptic curve, whereas a points on Y_{0}(N) correspond to a pair consisting of an elliptic curve and a subgroup of order N.

9.2 Application of Riemann-Hurwitz

Now we compute the genus of X_{Γ} by applying the Riemann-Hurwitz formula. Intuitively the Euler charcteristic should be totally additive, that is, if A and B are disjoint spaces then

χ(A ∪ B) = χ(A) + χ(B).

Let X be a compact Riemann surface of genus g, then χ(X) = 2 − 2g. Since χ({point}) = 1 we should have that

χ(X − {p_{1}, …, p_{n}}) = χ(X) − nχ(1) = (2 − 2g) − n.

If we have an umramified covering X → Y of degree d then χ(X) = d · χ(Y). Consider the covering

X_{Γ} − {points over 0, 1728, ∞}
↓
X_{Γ(1)} − {0, 1728, ∞}

Since X_{Γ(1)} has genus 0, X_{Γ(1)} − {0, 1728, ∞} has Euler characteristic 2 − 3 = −1. If we let g = χ(X_{Γ}) then χ(X_{Γ} − {points over 0, 1728, ∞} = 2 − 2g − n_{0} − n_{1728} − n_{∞}, where n_{p} denotes the number of points lying over p. Thus −d = 2 − 2g − n_{0} − n_{1728} − n_{∞} whence

2g − 2 = d − n_{0} − n_{1728} − n_{∞}.

Suppose Γ = Γ_{0}(N) with N > 3, then n_{0} = d/3 and n_{1728} = d/2 (I’m not sure why). The degree d of the covering is equal to the number of unordered ordered basis of E[N], thus

d = # SL_{2}(Z/NZ)/2.

We still need to compute n_{∞}. SL_{2}(Z) acts on P^{1}(Q) if we view P^{1}(Q) as all pairs (a, b) of relatively prime integers and suppose ∞ corresponds to (1, 0). The stabilizer of (1, 0) is the sugroup \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z) : c = 0\} of upper triangular matrices. Since the points lying over ∞ are all conjugate by the Galois group of the covering (which is SL_{2}(Z/NZ)/{±1}),

number of cusps = \frac{order of SL_{2}(Z/NZ)/\{±1\}}{order of stabilizer of ∞}.

We thus have

2g(X(N)) − 2 = \frac{d}{6} − \frac{d}{N}

where \frac{d}{N} is the number of cusps.

9.3 Explicit Genus Computations

Let N > 3 and consider the modular curve X = X(N). There is a natural covering map X → X(1) ⟶^{j}C. Let d be the degree, then

2g − 2 = d − m_{0} − m_{1728} − m_{∞}

where g is the genus of X and m_{x} is the number of points lying over x. Since m_{0} is approximately \frac{d}{3} and m_{1728} is approximately \frac{d}{2},

2g − 2 = \frac{d}{6} − m_{∞} ± small correction factor.

9.4 The Genus of X(N)

Now we count the number of cusps of X(N), that is, the size of Γ(N)\P^{1}(Q). There is a surjective map from SL_{2}(Z) to P^{1}(Q) given by

(\begin{matrix}a & b \\ c & d \end{matrix}) ↦ (\begin{matrix}a & b \\ c & d \end{matrix})(\begin{matrix}1 \\ 0 \end{matrix}).

9.5. THE GENUS OF X_{0}(N) 39

Let U be the kernel, thus U is the stabilizer of ∞ = ( \begin{matrix}1 \\ 0 \end{matrix} ), so U = \{±( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) : a ∈ Z\}. Then the cusps of X(N) are the elements of

Γ(N)\(SL_{2}(Z) /U) = (Γ(N)\ SL_{2}(Z))/U = SL_{2}(Z/NZ)/U

which has order

\frac{# SL_{2}(Z/NZ)}{2N} = \frac{d}{N} .

Substituting this into the above formula gives

2g − 2 = \frac{d}{6} − \frac{d}{N} = \frac{d}{6N} (N − 6)

so

g = 1 + \frac{d}{12N} (N − 6).

When N is prime

d = \frac{1}{2}# SL_{2}(Z/NZ) = \frac{1}{2} · \frac{(N^{2} − 1)(N^{2} − N)}{N − 1}.

Thus when N = 5, d = 60 so g = 0, and when N = 7, d = 168 so g = 3.

9.5 The Genus of X_{0}(N)

Suppose N > 3 and N is prime. The covering map X_{0}(N) → X(1) is of degree N + 1 since a point of X_{0}(N) corresponds to an elliptic curve along with a subgroup of order N and there are N + 1 such subgroups because N is prime.

Exercise 9.5.1. X_{0}(N) has two cusps; they are the orbit of ∞ which is unramified and 0 which is ramified of order N.

Thus

2g − 2 = N + 1 − 2 − n_{1728} − n_{0}.

n_{0} is the number of pairs (E, C) (modulo isomorphism) such that E has j-invariant 0. So we consider E = C/Z[\frac{−1+i\sqrt{3}}{2}] which has endomorphism ring End(E) = Z[μ μ μ_{6}]. Now μ μ μ_{6}/±1 acts on the cyclic subgroups C so, letting ω be a primitive cube root of unity, we have

(E, C) ≅ (E, ωC) ≅ (E, ω^{2}C).

This might lead one to think that m_{0} is (N + 1)/3, but it may be bigger if, for example, C = ωC. Thus we must count those C so that ωC = C or ω^{2}C = C, that is, those C which are stable under O = Z[\frac{−1+i\sqrt{3}}{2}]. So we must compute the number of stable O/NOsubmodules of order N. This depends on the structure of O/NO:

O/NO = \begin{cases}F_{N} ⊕ F_{N} & if (\frac{−3}{N}) = 1 (N splits) \\ F_{N^{2}} & if (\frac{−3}{N}) = −1 (N stays inert) \end{cases}

Since O/NO = F_{N^{2}} is a field it has no submodules of order N, whereas F_{N} ⊕ F_{N} has two O/NO-submodules of order N, namely F_{N} ⊕ 0 and 0 ⊕ F_{N}. Thus

m_{0} = \begin{cases}\frac{N+1}{3} & if N ≡ 2 (mod 3) \\ \frac{N−1}{3} + 2 & if N ≡ 1 (mod 3) \end{cases}

Exercise 9.5.2. It is an exercise in elegance to write this as a single formula involving the quadratic symbol.

By similar reasoning one shows that

m_{1728} = \begin{cases}\frac{N+1}{2} & if N ≡ 3 (mod 4) \\ \frac{N−1}{2} + 2 & if N ≡ 1 (mod 4) \end{cases}

We can now compute the genus of X_{0}(N) for any prime N. For example, if N = 37 then 2g − 2 = 36 − (2 + 18) − (14) = 2 so g = 2. Similarly, X_{0}(13) has genus 0 and X_{0}(11) has genus 1. In general, X_{0}(N) has genus approximately N/12.

Serre constructed a nice formula for the above genus. Suppose N > 3 is a prime and write N = 12a + b with 0 ≤ b ≤ 11. Then Serre’s formula is

\begin{matrix}b & 1 & 5 & 7 & 11 \\ g & a − 1 & a & a & a + 1 \end{matrix}

9.6 Modular Forms mod p

Let N be a positive integer, let p be a prime and assume Γ is either Γ_{0}(N) or Γ_{1}(N).

Definition 9.6.1. Let M_{k}(Γ, Z) = M_{k}(Γ, C) ∩ Z[[q]], then

M_{k}(Γ, F_{p}) = M_{k}(Γ, Z) ⊗_{Z} F_{p}

is the space of modular forms mod p of weight k.

Suppose p = N, then one has Serre’s Equality:

M_{p+1}(SL_{2}(Z), F_{p}) = M_{2}(Γ_{0}(p), F_{p})

The map from the right hand side to the left hand side is accomplished via a certain normalized Eisenstein series. Recall that in SL_{2}(Z),

G_{k} = −\frac{B_{k}}{2k} + ∑_{n=1}^{∞}(∑_{d|n} d^{k−1})q^{n}

and

E_{k} = 1 − \frac{2k}{B_{k}} ∑_{n=1}^{∞}(∑_{d|n} d^{k−1})q^{n}.

One finds ord_{p}(−\frac{B_{k}}{2k}) using Kummer congruences. In particular, ord_{p}(B_{p−1}) = −1, so E_{p−1} ≡ 1 (mod p). Thus multiplication by E_{p−1} raises the level by p − 1 but does not change the q-expansion mod p. We thus get a map

M_{2}(Γ_{0}(p), F_{p}) → M_{p+1}(Γ_{0}(p), F_{p}).

The map

M_{p+1}(Γ_{0}(p), F_{p}) → M_{p+1}(SL_{2}(Z), F_{p})

is the trace map (which is dual to the natural inclusion going the other way) and is accomplished by averaging in order to get a form invariant under SL_{2}(Z).

The Field of Moduli

In this chapter we will study the field of definition of the modular curves X(N), X_{0}(N), and X_{1}(N).

The function field of X(1) = P_{Q}^{1} is Q(t). If E is an elliptic curve given by a Weierstrass equation y^{2} = 4x^{3} − g_{2}x − g_{3} then

j(E) = j(g_{2}, g_{3}) = \frac{1728g_{2}^{3}}{g_{2}^{3} − 27g_{3}^{2}} .

The j invariant determines the isomorphism class of E over C. Pick an elliptic curve E/Q(t) such that j(E) = t. In particular we could pick the elliptic curve with Weierstrass equation

y^{2} = 4x^{3} − \frac{27t}{t − 1728}x − \frac{27t}{t − 1728}.

Let E/k be an arbitrary elliptic curve and N a positive integer prime to char k. Then E[N](\overline{k}) ≅ (Z/NZ)^{2}. Let k(E[N]) be the field obtained by adjoining the coordinates of the N-torsion points of E. Consider the tower of fields \overline{k} ⊃ k(E[N]) ⊃ k. There is a Galois representation on the N torsion of E:

Gal(\overline{k}/k) ⟶^{ρ_{E,N}} Aut(E[N]) ≅ GL_{2}(Z/NZ)

and Gal(\overline{k}/k(E[N])) = ker(ρ_{E,N}). Thus the Galois group of the extension Q(t)(E[N]) over Q(t) is contained in GL_{2}(Z/NZ). Let X(N) be the curve corresponding to the function field Q(t)(E[N]) over Q. Since \overline{Q}∩Q(t)(E[N]) is contained in Q(μ μ μ_{N}), X(N) is defined over Q(μ μ μ_{N}).

Composing ρ_{E} with the natural map GL_{2}(Z/NZ) → GL_{2}(Z/NZ)/{±1} gives a map

\overline{ρ}_{E} : Gal(\overline{K}/K) → GL_{2}(Z/NZ)/\{±1\}.

Proposition 10.0.2. \overline{ρ}_{E} is surjective iff ρ_{E} is surjective.

Proof. If \overline{ρ}_{E} is surjective then either ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) or its negative lies in the image of ρ. Thus ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) lie in the image of ρ. Since \overline{ρ}_{E} is surjective this implies that ρ is surjective. The converse is trivial.

10.1 Digression on Moduli

X_{0}(N)(\overline{K}) is the set of \overline{K}-isomorpisms classes of pairs (E, C) where E/\overline{K} is an elliptic curve and C is a cyclic subgroup of order N. X_{0}(N)(K) is the set of isomorphism classes of pairs (E, C) such that for all σ ∈ Gal(\overline{K}/K), σ(E, C) = (E, C). There is a map

{k-isomorphism classes of pairs (E, C)/K} → X_{0}(N)(K)

which is “notoriously” non-injective. Deligne and Rapaport [13] prove the map is surjective. When N = 1 they observe that the map is surjective, then for N > 1 they show that certain obstructions vanish. A related question is

Question 10.1.1. If E/\overline{K} is isomorphic to all its Galois conjugates, is there E′/K which is isomorphic to E over \overline{K}?

10.2 When is ρ_{E} Surjective?

Proposition 10.2.1. Let E_{1} and E_{2} be elliptic curves defined over K with equal j-invariants, thus E_{1} ≅ E_{2} over \overline{K}. Assume E_{1} and E_{2} do not have complex multiplication over \overline{K}. Then ρ_{E_{1}} is surjective iff ρ_{E_{2}} is surjective.

Proof. Assume ρ_{E_{1}} is surjective. Since E_{1} has no complex multiplication over \overline{K}, Aut E_{1} = \{±1\}. Choose an isomorphism φ : E_{1} ⟶^{∼} E_{2} over \overline{K}. Then for σ ∈ Gal(\overline{K}/K) we have the diagram

E_{1} ⟶^{φ} E_{2}
=↓  ^{=}↓
σE_{1} ⟶^{σφ} ^{σ}E_{2}

Thus σφ = ±φ for all σ ∈ Gal(\overline{K}/K), so φ : E_{1}[N] → E_{2}[N] defines an equivalence \overline{ρ}_{E_{1}} ≅ \overline{ρ}_{E_{2}}. Since ρ_{E_{1}} is surjective this implies \overline{ρ}_{E_{2}} is surjective which, by the previous proposition, implies ρ_{E_{2}} is surjective.

Let K = C(j), with j transcendental over C. Let E/K be an elliptic curve with jinvariant j. Fix a positive integer N and let

ρ_{E} : Gal(\overline{K}/K) → GL_{2}(Z/NZ)

be the associated Galois representation. Then det ρ_{E} is the cyclotomic character which is trivial since C contains the Nth roots of unity. Thus the image of ρ_{E} lands inside of SL_{2}(Z/NZ). Our next theorem states that a generic elliptic curve has maximal possible Galois action on its division points.

Theorem 10.2.2. ρ_{E} : Gal(\overline{K}/K) → SL_{2}(Z/NZ) is surjective.

Igusa [8] found an algebraic proof of this theorem We will now make some comments on how an analytic proof goes.

Proof. Let C(j) = K = F_{1} be the field of modular functions for SL_{2}(Z). Suppose N ≥ 3 and let F_{N} be the field of mereomorphic functions for Γ(N). Then F_{N}/F_{1} is a Galois extension with Galois group SL_{2}(Z/NZ)/{±1}.

Let E be an elliptic curve over K with j-invariant j. We will show that Gal(F_{N}/F_{1}) = SL_{2}(Z/NZ)/{±1} acts transitively on the x-coordinates of the N torsion points of E. This will show that \overline{ρ}_{E} maps surjectively onto SL_{2}(Z/NZ)/\{±1\}. Then by proposition 10.0.2, ρ_{E} maps surjectively onto SL_{2}(Z/NZ), as claimed.

We will now construct the x-coordinates of E[N] as functions on H which are invariant under Γ(N). (Thus K(E[N]/{±1}) ⊂ F_{N}.)

Let τ ∈ H and let L_{τ} = Zτ + Z. Consider ℘(z, L_{τ}) which gives the x coordinate of C/L_{τ} in it standard form. Define, for each nonzero (r, s) ∈ ((Z/NZ)/{±1})^{2}, a function

f_{(r,s)} : H → C : τ ↦ \frac{g_{2}(τ)}{g_{3}(τ)}℘(\frac{rτ + s}{N}, L_{τ}).

First notice that for any α = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z),

f_{(r,s)}(ατ) = f_{(r,s)(\begin{matrix}a & b \\ c & d \end{matrix})}(τ).

Indeed, ℘ is homogenous of degree −2, g_{2} is modular of weight 4 and g_{3} is modular of weight 6, so

f_{(r,s)}(ατ) = \frac{g_{2}(ατ)}{g_{3}(ατ)}℘(\frac{rατ + s}{N})

= (cτ + d)^{−2} \frac{g_{2}(τ)}{g_{3}(τ)}℘(\frac{raτ + rb + csτ + sd}{N(cτ + d)})

= \frac{g_{2}(τ)}{g_{3}(τ)}℘(\frac{(ra + sc)τ + rb + sd}{N}) = f_{(r,s)α}(τ)

If τ ∈ H with g_{2}(τ), g_{3}(τ) ≠ 0 then the f_{(r,s)}(τ) are the x-coordinates of the nonzero N-division points of E_{j(τ)}. The various f_{(r,s)}(τ) are distinct. Thus SL_{2}(Z/NZ)/{±1} acts transitively on the f_{(r,s)}. The consequence is that the N^{2} − 1 nonzero division points of E_{j} have x-coordinates in \overline{F}_{N} equal to the f_{(r,s)} ∈ F_{N}.

10.3 Observations

Proposition 10.3.1. If E/Q(μ μ μ_{N})(t) is an elliptic curve with j(E) = t, then ρ_{E} has image SL_{2}(Z/NZ).

Proof. Since Q(μ μ μ_{N}) contains the Nth roots of unity Nth cyclotomic character is trivial hence the determinent of ρ_{E} is trivial. Thus the image of ρ_{E} lies in SL_{2}(Z/NZ). In the other direction, there is a natural inclusion

SL_{2}(Z/NZ) = Gal(C(t)(E[N])/C(t)) ↪ Gal(Q(μ μ μ_{N})(t)(E[N])/Q(μ μ μ_{N})(t)).

Proposition 10.3.2. If E/Q(t) is an elliptic curve with j(E) = t, then ρ_{E} has image GL_{2}(Z/NZ) and \overline{Q} ∩ Q(t)(E[N]) = Q(μ μ μ_{N}).

Proof. Since Q(t) contains no Nth roots of unity, the mod N cyclotomic character, and hence detρ_{E}, is surjective onto (Z/NZ)^{∗}. Since the image of ρ_{E} already contains SL_{2}(Z/NZ) it must equal GL_{2}(Z/NZ). For the second assertion consider the diagram

\overline{Q}  Q(t)(E[N])
| | SL_{2}
Q(μ μ μ_{N}) ⊂ Q(μ μ μ_{N})(t)
(Z/NZ)^{∗} | | GL_{2}/SL_{2} = (Z/NZ)^{∗}
Q ⊂  Q(t)

This gives a way to view X_{0}(N) as a projective algebraic curve over Q. Let K = Q(t) and let L = K(E[N]) = Q(μ μ μ_{N})(t). Then

H = \{( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} )\} ⊂ GL_{2}(Z/NZ) = Gal(L/K).

The fixed field L^{H} is an extension of Q(t) of transcendence degree 1 with field of constants \overline{Q} ∩ L^{H} = Q, i.e., a projective algebraic curve.

10.4 A Descent Problem

Consider the following exercise which may be approached in an honest or dishonest way.

Exercise 10.4.1. Suppose L/K is a finite Galois extension and G = Gal(L/K). Let E/L be an elliptic curve, assume Aut_{L} E = {±1}, and suppose that for all g ∈ G, there is an isomorphism ^{g}E ⟶^{∼} E over L. Show that there exists E_{0}/K such that E_{0} ≅ E over L.

Caution! The exercise is false as stated. Both the dishonet and honest approaches below work only if L is a separable closure of K. Now: can one construct a counterexample?

Discussion. First the hard, but “honest” way to look at this problem. For notions on descent see Serre [23]. By descent theory, to give E_{0} is the same as to give a family (λ_{g})_{g∈G}

of maps λ_{g} : ^{g}E ⟶^{∼} E such that λ_{gh} = λ_{g} ∘^{g}λ_{h} where ^{g}λ_{h} = g ∘ λ_{h} ∘ g^{−1}. Note that λ_{g} ∘^{g}λ_{h}

maps ^{gh}E → E. This is the natural condition to impose, because if f : E_{0} ⟶^{∼} E and we let λ_{g} = f ∘^{g}(f^{−1}) then λ_{gh} = λ_{g} ∘^{g}λ_{h}.

Using our hypothesis choose, for each g ∈ G, an isomorphism

λ_{g} : ^{g}E ⟶^{∼} E.

Define a map c by

c(g, h) = λ_{g} ∘^{g}λ_{h} ∘ λ_{gh}^{−1}.

Note that c(g, h) ∈ Aut E = {±1} so c defines an element of

H^{2}(G, \{±1\}) ⊂ H^{2}(Gal(\overline{L}/K), \{±1\}) = Br(K)[2].

Here Br(K)[2] denotes the 2-torsion of the Brauer group

Br(K) = H^{2}(Gal(\overline{L}/K), \overline{L}^{∗}).

This probably leads to an honest proof.

The dishonest approach is to note that g(j(E)) = j(E) for all g ∈ G since all conjugates of E are isomorphic and j(^{g}E) = g(j(E)). Thus j(E) ∈ K so we can define E_{0}/K by substituting j(E) into the universal elliptic curve formula (see III.1.4 of [33]). This gives an elliptic curve E_{0} defined over K but isomorphic to E over \overline{K}.

10.5 Second Look at the Descent Exercise

Last time we talked about the following problem. Suppose L/K is a Galois extension with char K = 0, and let E/L be an elliptic curve. Suppose that for all σ ∈ G = Gal(L/K), σE ≅ E over L. Conclude that there is an elliptic curve E_{0}/K such that E_{0} ≅ E over L. The conclusion may fail to hold if L is a finite extension of K, but the exercise is true when L = \overline{K}. First we give a descent argument which holds when L = \overline{K} and then give a counterexample to the more general statement.

For g, h ∈ G = Gal(L/K) we define an automorphism c(g, h) ∈ Aut E = {±1}. Choose for every g ∈ Gal(L/K) some isomorphism

λ_{g} : ^{g}E ⟶^{∼} E.

If the λ_{g} were to all satisfy the compatibility criterion λ_{gh} = λ_{g} ∘^{g}λ_{h} then by descent theory we could find a K-structure on E, that is a model for E defined over K and isomorphic to E over L. Define c(g, h) by c(g, h)λ_{gh} = λ_{g} ∘ ^{g}λ_{h} so c(g, h) measures how much the λ_{g} 10.6. ACTION OF GL_{2} 45

fail to satisfy the compatibility criterion. Since c(g, h) is a cocycle it defines an elements of H^{2}(G, \{±1\}). We want to know that this element is trivial. When L = \overline{K}, the map H^{2}(G, {±1}) → H^{2}(G, L^{∗}) is injective. To see this first consider the exact sequence

0 → \{±1\} → \overline{K}^{∗} ⟶^{2} \overline{K}^{∗} → 0

where 2 : \overline{K}^{∗} → \overline{K}^{∗} is the squaring map. Taking cohomology yields an exact sequence

H^{1}(G, \overline{K}^{∗}) → H^{2}(G, \{±1\}) → H^{2}(G, \overline{K}^{∗}).

By Hilbert’s theorem 90 ([25] Ch. X, Prop. 2), H^{1}(G, {±1}) = 0. Thus we have an exact sequence

0 → H^{2}(G, \{±1\}) → H^{2}(G, \overline{K}^{∗})[2] → 0.

Thus H^{2}(G, {±1}) naturally sits inside H^{2}(G, L^{∗}).

To finish Ribet does something with differentials and H^{0}(^{g}E, Ω^{1}) which I don’t understand.

The counterexample in the case when L/K is finite was provided by Kevin Buzzard (who said Coates gave it to him). Let L = Q(i), K = Q and E be the elliptic curve with Weirstrass equation iy^{2} = x^{3} + x + 1. Then E is isomorphic to its conjugate over L but one can show directly that E has no model over Q.

10.6 Action of GL_{2}

Let N > 3 be an integer and E/Q(j) an elliptic curve with j-invariant j(E) = j. Then there is a Galois extension

F_{N} = Q(j)(E[N]/{±1})
|
F_{1} = Q(j)

with Galois group GL_{2}(Z/NZ)/{±1}. Think of Q(j)(E[N]/{±1} as the field obtained from Q(j) by adjoining the x-coordinates of the N-torsion points of E. Note that this situation differs from the previous situation in that the base field C has been replaced by Q.

Consider

F = ⋃_{N} F_{N}

which corresponds to a projective system of modular curves . Let A_{f} be the ring of finite adèles, thus

A_{f} = Q̂ = Ẑ ⊗ Q ⊂ ∏_{p} Q_{p}.

A_{f} can be thought of as

{(x_{p}) : x_{p} ∈ Z_{p} for almost all p}.

The group GL_{2}(A_{f}) acts on F. To understand what this action is we first consider the subgroup GL_{2}(Ẑ) of GL_{2}(A_{f}).

It can be shown that

F = Q(f_{N,(r,s)} : (r, s) ∈ (Z/NZ)^{2} − {(0, 0)}, N ≥ 1)

where f_{N,(r,s)} is a function

f_{N,(r,s)} : H → C : τ ↦ \frac{g_{2}(τ)}{g_{3}(τ)}℘(\frac{rτ + s}{N}, L_{τ}).

We define the action of GL_{2}(Ẑ) on F as follows. Let g ∈ GL_{2}(Ẑ), then to give the action of g on f_{N,(r,s)} first map g into GL_{2}(Z/NZ) via the natural reduction map, then note that GL_{2}(Z/NZ) acts on f_{N,(r,s)} by

(\begin{matrix}a & b \\ c & d \end{matrix}) · f_{N,(r,s)} = f_{N,(r,s)(\begin{matrix}a & b \\ c & d \end{matrix})} = f_{N,(ra+sc,rb+sd)}.

Let E be an elliptic curve. Then the universal Tate module is

T (E) = \underrightarrow{lim} E[N] = ∏_{p} T_{p}(E).

There is an isomorphism α : Ẑ^{2} ⟶^{∼} T (E). Via right composition GL_{2}(Ẑ) acts on the collection of all such isomorphism α. So GL_{2}(Ẑ) acts naturally on pairs (E, α) but the action does nothing to E. One of the first important things we must do in understanding the construction of things like Shimura varieties is to free ourselves and allow GL_{2}(Ẑ) to act on the E’s as well.

Let

g = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}^{+}(Q)

(thus g has positive determinent). Let τ ∈ H and let E = E_{τ} be the elliptic curve determined by the lattice L_{τ} = Z + Zτ. Let

α_{τ} : L_{τ} = Z + Zτ ⟶^{∼} Z^{2}

be the isomorphism defined by τ ↦ (1, 0) and 1 ↦ (0, 1). Now view α = α_{τ} as a map

α : Z^{2} ⟶^{∼} H_{1}(E(C), Z).

Tensoring with Q then gives another map (also denoted α)

α : Q^{2} ⟶^{∼} H_{1}(E, Q).

Then α ∘ g is another isomorphism

Q^{2} ⟶^{α∘g} H_{1}(E, Q)

which induces an isomorphim Z^{2} ⟶^{∼} L′ ⊂ H_{1}(E, Q) where L′ is a lattice. There exists an elliptic curve E′/C and a map λ ∈ Hom(E′, E) ⊗ Q which induces a map (also denoted λ)

λ : H_{1}(E′, Z) ⟶^{∼} L′ ⊂ H_{1}(E, Q)

on homology groups.

Now we can define an action on pairs (E, α) by sending (E, α) to (E′, α′). Here α′ is the map α′ : Z^{2} → H_{1}(E^{1}, Z) given by the composition

Z^{2} ⟶^{αg} L′ ⟶^{λ^{−1}} H_{1}(E^{1}, Z).

In more concrete terms the action is

g : (E_{τ}, α_{τ}) ↦ (E′_{τ}, α′_{τ})

where τ′ = gτ = \frac{aτ+b}{cτ+d}. [[ CHECK THIS SOMETIME SOON!!]]

Hecke Operators as Correspondences

11.1 Some Philosophy

We are studying modular forms over C and more generally over subrings R of C. The Hecke algebras occur naturally as operators on various spaces of modular forms. We are aiming for an arithmetic perspective. One way is to study the arithmetic of cusp forms of weight 2 for congruence subgroups like Γ_{0}(N) or Γ_{1}(N). These cusp forms correspond to differentials on the modular curves X_{0}(N) and X_{1}(N). We have constructed models for each of these over Q.

When N′|N there is a natural map X(N) → X(N′). Thus we get a tower of curves and a corresponding tower of number fields.

\begin{matrix}⋯ & ⋯ \\ ↓ & ∪ \\ X(N) & F_{N} \\ ↓ & ∪ \\ X(N′) & F′_{N} \\ ↓ & ∪ \\ ⋯ & ⋯ \end{matrix}

Taking limits gives a curve X = \underleftarrow{lim} X(N) and a corresponding field F = \underrightarrow{lim} F_{N}.

There is an action of GL_{2}(A_{f}) on pairs (E, α). By A_{f} we mean the ring of finite adèles, which may be identified with the restricted product ∏_{p} Q_{p}. The subscript f, for ‘finite’, indicates that the infinite place is omitted. The full ring of adèles is A = A_{f} × R. If g ∈ GL_{2}(A_{f}), then g acts on pairs (E, α) where E is an elliptic curve and

α : Ẑ^{2} ⟶^{∼} T (E) = ∏_{p} T_{p}(E).

Note that

A_{f} = Ẑ ⊗_{Z} Q = ∏ Z_{p} ⊗_{Z} Q = Q̂,

and T (E) is free of rank 2 over Ẑ. Let

V (E) = T (E) ⊗_{Z} Q = ∏_{p} V_{p}(E)

where the product is restricted and V_{p}(E) = T_{p}(E) ⊗_{Z_{p}} Q_{p}. View g ∈ GL_{2}(A_{f}) as an automorphism of Q̂^{2}. Then α ∘ g sends Ẑ^{2} ⊂ Q̂^{2} to a lattice T′ ⊂ V (E). As a lemma, one shows that there is an elliptic curve E′ and a canonical map λ : E′ → E such that the induced map λ′ : V (E′) ⟶^{∼} V (E) is an isomorphism which sends T (E′) maps to T′ in V (E). Then g sends the pair (E, α) to (E′, λ′^{−1} ∘ α ∘ g).

11.2 Hecke Operators as Correspondences

Our goal is to think of Hecke operators (T_{n}, ⟨d⟩) as objects defined over Q. We will define the Hecke operators as correspondences.

Definition 11.2.1. Let C_{1} and C_{2} be curves, then a correspondence C_{1} →→ C_{2} is a curve C together with morphisms α : C → C_{1} and β : C → C_{2}.

\begin{aligned}& C \\ ↙^{α} & & ↘^{β} \\ C_{1} & & C_{2} \end{aligned}

Giving a correspondence C_{1} →→ C_{2} the dual correspondence C_{2} →→ C_{1} is obtained by looking at the diagram in a mirror

\begin{aligned}& C \\ ↙^{β} & & ↘^{α} \\ C_{2} & & C_{1} \end{aligned}

The simplest case to consider is the modular curve X_{0}(N) and Hecke operator T_{p}, where p |̸N. We view T_{p} as a correspondence X_{0}(N) →→ X_{0}(N), thus there is a curve C = X_{0}(pN) plus 2 maps α and β

X_{0}(pN)
↙^{α}  ↘^{β}
X_{0}(N)  X_{0}(N)

The maps α and β are degeneracy maps which forget data. To define them view X_{0}(N) as classifying pairs (E, C) where E is an elliptic curve and C ≅ Z/NZ is a cyclic subgroup of order N. Similarly X_{0}(pN) classifies pairs (E, G) where G = C ⊕ D ≅ Z/NZ ⊕ Z/pZ and D is cyclic of order p. Then

α : (E, G) ↦ (E, C)
β : (E, G) ↦ (E/D, (C + D)/D)

Now we now translate this into the language of complex analysis. The first map α corresponds to the map

Γ_{0}(pN)\H → Γ_{0}(N)\H

induced by the inclusion Γ_{0}(pN) ↪ Γ_{0}(N). The second map β is constructed by composing the map

Γ_{0}\H ⟶^{∼} (\begin{matrix}p & 0 \\ 0 & 1 \end{matrix})Γ_{0}(pN)(\begin{matrix}p & 0 \\ 0 & 1 \end{matrix})^{−1}\H

with the map to Γ_{0}(N)\H induced by the inclusion

Γ_{0}(N) ⊂ (\begin{matrix}p & 0 \\ 0 & 1 \end{matrix})Γ_{0}(pN)(\begin{matrix}p & 0 \\ 0 & 1 \end{matrix})^{−1}

The maps α and β induce maps

α^{∗}, β^{∗} : H^{0}(X_{0}(N), Ω^{1}) → H^{0}(X_{0}(pN), Ω^{1}).

We can identify S_{2}(Γ_{0}(N)) with H^{0}(X_{0}(N), Ω^{1} by sending the cusp form f(τ) to the holomorphic differential f(τ)dτ. Now we get maps

α^{∗}, β^{∗} : S_{2}(Γ_{0}(N)) → S_{2}(Γ_{0}(pN)).

Exercise 11.2.2. Show that α^{∗}(f) = f thought of as a cusp form with respect to the smaller group Γ_{0}(pN). Then show that

β^{∗}(f) = p ∑_{n=1}^{∞} a_{n}q^{pN}.

(Ribet was unsure whether the factor of p should be there.)

11.3 Generalities on Correspondences

Let X, Y, and C be curves and let α and β be nonconstant holomorphic maps so that we have the correspondence

\begin{aligned}& C \\ ↙^{α} & & ↘^{β} \\ X & & Y \end{aligned}

Then we have maps

H^{0}(X, Ω^{1}) ⟶^{α^{∗}} H^{0}(C, Ω^{1}) ⟶^{β_{∗}} H^{0}(Y, Ω^{1}).

The composition β_{∗} ∘ α^{∗} is a map

H^{0}(X, Ω^{1}) → H^{0}(Y, Ω^{1}).

Switching the roles of X and Y gives a map

H^{0}(Y, Ω^{1}) → H^{0}(X, Ω^{1}).

In this context we can identify T_{p} by viewing it as the map

β_{∗} ∘ α^{∗} : H^{0}(X_{0}(N), Ω^{1}) → H^{0}(X_{0}(N), |Omega^{1})

and using the fact that S_{2}(Γ_{0}(N)) = H^{0}(X_{0}(N), Ω^{1}). One should recover the explicit representation

T_{p} : ∑ a_{n}q^{n} ↦ ∑ a_{np}q^{n} + p ∑ a_{n}q^{np}.

Now lets think more generally about correspondences. Suppose φ : X → Y is a map of curves. Let Γ ⊂ X × Y be the graph of φ. This gives a stupid correspondence

\begin{aligned}& Γ \\ ↙^{α} & & ↘^{β} \\ X & & Y \end{aligned}

We can reconstruct φ since φ(x) = β(α^{−1}(x)).

More generally suppose α : Γ → X has degree d ≥ 1. View α^{−1}(x) as a divisor on Γ. Then β(α^{−1}) is a divisor on Y. We thus get a map

Div^{n} X ⟶^{β∘α^{−1}} Div^{dn} Y.

In particular, when d = 0, there we get a map Div^{0} X → Div^{0} Y. Now apply this to the special case of T_{p}. T_{p} is the correspondence

X_{0}(pN)
↙^{α}  ↘^{β}
X_{0}(N)  X_{0}(N)

and the induced map is

(E, C) ↦^{α^{∗}} ∑_{D∈E[p]}(E, C ⊕ D) ↦^{β} ∑_{D∈E[p]}(E/D, (C + D)/D)

Thus we have a map

{ divisors on X_{0}(N) } ⟶ {divisors on X_{0}(N)}

This strongly resembles the original definition of T_{p} as a correspondence of lattices. But there was a factor of p^{k−1} then which annoyed Ogus. It disappears here though, and this is related to the ‘p’ in the exercise.

11.4 Jacobians of Curves

Let X be a curve of genus g over a field k. There is an important association

{ curves X/k } ⟶ { Jacobians Jac(X) = J(X) of curves }

between curves and their associated Jacobians.

Definition 11.4.1. Let X be a curve, then the Jacobian of X is an abelian variety of dimension g whose underlying group is isomorphic to the group of divisors of degree 0 on X modulo linear equivalence.

There are many constructions of the Jacobian of a curve. We first consider the Albanese construction. Recall that over C an abelian variety is just C^{g}/L where L is a lattice (and hence a free Z-module of rank 2g). There is an embedding

H_{1}(X, Z) ↪ H^{0}(X, Ω^{1})^{∗}
γ ↦ ∫_{γ} •

Then we realize J(X) as a quotient

J(X) = H^{0}(X, Ω^{1})^{∗}/H_{1}(X, Z).

In this construction J(X) is covariantly associated to X. There are other constructions in which J(X) is contravariantly associated to X. If F is a correspondence X →→ Y, then F induces a map J(X) → J(Y) and also a map J(Y) → J(X). If X = Y so that X and Y are the same, one can often be confused which duality should be used. Fortunately, for T_{p} when p is prime to N it does not matter. But it matters a lot if p|N since then we have noncommuting confusable operators and, “this has resulted in lots of mistakes in the literature.”

11.5 More on Hecke Operators

Our goal is to move things down to Q from C or \overline{Q}. In doing this we want to understand T_{n} (or T_{p}), that is, how they act on the associated Jacobians and how they can be viewed as correspondences. In characteristic p the formulas of Eichler-Shimura will play an important role.

We consider T_{p} as a correspondence on X_{1}(N) or X_{0}(N). To avoid confusion we will mainly consider T_{p} on X_{0}(N) with p |̸N. Thus assume, unless otherwise stated, that p |̸N. Remember that T_{p} was defined to be the correspondence

X_{0}(pN)
↙^{α}  ↘^{β}
X_{0}(N)  X_{0}(N)

Think of X_{0}(pN) as consisting of pairs (\underline{E}, D) where D is a cyclic subgroup of E of order p and \underline{E} is the enhanced elliptic curve consisting of an elliptic curve E along with a cyclic subgroup of order N. The degeneracy map α forgets the subgroup D and the degeneracy map β divides by it. By contravariant functoriality we have a commutative diagram

H^{0}(X_{0}(N), Ω^{1}) ⟶^{T_{p}^{∗}=α_{∗}∘β^{∗}} H^{0}(X_{0}(N), Ω^{1})
||  ||
S_{2}(Γ_{0}(N))  ⟶^{T_{p}}  S_{2}(Γ_{0}(N))

Our convention to define T_{p}^{∗} as α_{∗}∘β^{∗} instead of β_{∗}∘α^{∗} was completely psychological because there is a canonical duality relating the two. We chose the way we did because of the analogy with the case of a morphism φ : Y → X with graph Γ which induces a correspondence

\begin{aligned}& Γ \\ ↙^{π_{1}} & & ↘^{π_{2}} \\ Y & & X \end{aligned}

Since the morphism φ induces a map on global sections in the other direction

H^{0}(X, Ω^{1})Γ(X) ⟶^{φ^{∗}} Γ(Y) = H^{0}(Y, Ω^{1})

it is psychologically natural for more general correspondence such as T_{p} to map from the right to the left.

The morphisms α and β in the definition of T_{p} are defined over Q. This can be seen using the general theory of representable functors [How?]. Thus since T_{p} is defined over Q most of the algebraic geometric objects we will construct related to T_{p} will be defined over Q.

11.6 Hecke Operators acting on Jacobians

The Jacobian J(X_{0}(N)) = J_{0}(N) is an abelian variety defined over Q. There are both covariant and contravariant formations of J_{0}(N). Thus a map α : X_{0}(pN) → X_{0}(N) induces maps

\begin{matrix}J_{0}(pN) & J_{0}(pN) \\ α^{∗}↑ & ↓α_{∗} \\ J_{0}(N) & J_{0}(N) \end{matrix}

Note that α_{∗} ∘ α^{∗} : J_{0}(N) → J_{0}(N) is just multiplication by deg(α) = p + 1, since there are p+1 subgroups of order p in \underline{E}. (At least when p |̸N, when p|N there are only p subgroups.)

We will often refer to T_{p} as ξ_{p} as Shimura does in his book. There are two possible ways to define T_{p} = ξ_{p} as an endomorphism of J_{0}(N). We could either define T_{p} as β_{∗} ∘ α^{∗} or equivalently as α_{∗} ∘ β^{∗} (assuming still that p |̸N).

11.6.1 The Albanese Map

There is a way to map the curve X_{0}(N) into its Jacobian since the underlying group structure of J_{0}(N) is

J_{0}(N) = { divisors of degree 0 on X_{0}(N) } ; { principal divisors }

Once we have chosen a rational point, say ∞, on X_{0}(N) we obtain the Albanese map

θ : X_{0}(N) → J_{0}(N) : x ↦ x − ∞

which sends a point x to the divisor x−∞. The map θ gives us a way to pullback differentials on J_{0}(N). Let Cot J_{0}(N) denote the cotangent space of J_{0}(N) (or the space of regular differentials). The diagram

Cot J_{0}(N) ⟵^{ξ_{p}^{∗}} Cot J_{0}(N)
θ^{∗}↓≀  ≀↓θ^{∗}
H^{0}(X_{0}(N), Ω^{1}) ⟵^{T_{p}^{∗}} H^{0}(X_{0}(N), Ω^{1})

may be taken to give a definition of ξ_{p} since there is a unique endomorphism ξ_{p} : J_{0}(N) → J_{0}(N) inducing a map ξ_{p}^{∗} which makes the diagram commute.

Now suppose Γ is a correspondence X →→ Y so we have a diagram

\begin{aligned}& Γ \\ ↙^{α} & & ↘^{β} \\ X & & Y \end{aligned}

For example, think of Γ as the graph of a morphism φ : X → Y. Γ should naturally induce a map

H^{0}(Y, Ω^{1}) ⟶ H^{0}(X, Ω^{1}).

Taking Jacobians we see that the composition

J(X) ⟶^{α^{∗}} J(Γ) ⟶^{β_{∗}} J(Y)

gives a map β_{∗} ∘ α^{∗} : J(X) → J(Y). On cotangent spaces this induces a map

α^{∗} ∘ β_{∗} : H^{0}(Y, Ω^{1}) → H^{0}(X, Ω^{1}).

Now, after choice of a rational point, the map X → J(X) induces a map Cot J(X) → H^{0}(X, Ω^{1}). This is in fact independent of the choice of rational point since differentials on J(X) are invariant under translation.

The map J(X) → J(Y) is preferred in the literature. It is said to be induced by the Albanese functoriality of the Jacobian. We could have just as easily defined a map from J(Y) → J(X). To see this let

ψ = β_{∗} ∘ α^{∗} : J(X) → J(Y).

Dualizing induces a map ψ^{∨} = α_{∗} ∘ β^{∗}

J(X)^{∨} ⟵^{ψ^{∨}} J(Y)^{∨}
≀|  ≀|
J(X)  J(Y)

where we have used the fact that Jacobians are canonically autodual. This canonical duality is discussed in Mumford’s [17] and [18] and in Milne’s article in [30].

11.6.2 The Hecke Algebra

We now have ξ_{p} = T_{p} ∈ End J_{0}(N) for every prime p. If p|N, then we must decide between α_{∗} ∘ β^{∗} and β_{∗} ∘ α^{∗}. The usual choice is the one which induces the usual T_{p} on cusp forms. If you don’t like your choice you can get out of it with Atkin-Lehrer operators.

Let

T = Z[…, T_{p}, …] ⊂ End J_{0}(N)

then T is the same as T_{Z} ⊂ End(S_{2}(Γ_{0}(N))). To see this first note that there is a map T → T_{Z} which is not a prior injective, but which is injective because elements of End J_{0}(N) are completely determined by their action on Cot J_{0}(N).

11.7 The Eichler-Shimura Relation: Part I

J_{0}(N) is an abelian variety defined over Q so it is natural to ask for those primes p for which J_{0}(N) have good reduction. In the 1950’s Igusa managed to show that J_{0}(N) has good reduction for all p |̸N. He did this by first viewing J_{0}(N) as a scheme over spec Q, then “spreading things out” to make an abelian scheme over spec Z[1/N]. He did this by taking the Jacobian of the normalization of X_{0}(N) (which is defined over Z[1/N]) in P_{Z[1/N]}.

The Eichler-Shimura formula is a formula for T_{p} in characteristic p (or equivalently for endomorphisms ξ_{p} ∈ End(J_{0}(N)/F_{p})) for all p for which J_{0}(N) has good reduction at p. Suppose now that p |̸N, then X_{0}(N)/F_{p} has the same sort of properties as X_{0}(N)/Q. In particular it classifies enhanced elliptic curves \underline{E} where \underline{E} = (E, C) is an elliptic curve over F_{p} along with a cyclic subgroup of order N.

Next we ask what happens to the map ξ_{p} : X_{0}(N) → X_{0}(N)/Q in reduction. Thus consider the correspondence

X_{0}(Np)
↙^{α}  ↘^{β}
X_{0}(N)  X_{0}(N)

X_{0}(N) has good reduction at p, but X_{0}(Np) may not. Fortunately Deligne and Rapaport showed that X_{0}(Np) has relatively benign reduction at p. The first step is to form a model of X_{0}(Np)/F_{p}. Over F_{p}, X_{0}(Np) can be thought of as two copies of X_{0}(N) glued by pairing off the supersingular points (thus corresponding supersingular points are glued). The set of supersingular points

Σ ⊂ X_{0}(N)(\overline{F}_{p})

is the set of points represented by supersingular elliptic curves. There are only finitely many of these, in fact g + 1 where g is the genus of X_{0}(N).

Next we must understand the following constellation of diagrams

X_{0}(N)  X_{0}(N)
|  ↘  ↙  |
|  X_{0}(Np)  |
↓ ↙^{α}  ↘^{β}  ↓
X_{0}(N)  X_{0}(N)

11.8 The Eichler-Shimura Relation: Part II

Suppose p is a prime, p |̸N. Consider the correspondence T_{p} : X_{0}(N) →→ X_{0}(N) which takes an enhanced elliptic curve \underline{E} to the sum ∑_{#D=p} \underline{E}/D of quotients of \underline{E} by subgroups of order p. This is the correspondence

X_{0}(pN)
↙^{α}  ↘^{β}
X_{0}(N)  X_{0}(N)

From this one gets ξ_{p} = T_{p} : J_{0}(N) → J_{0}(N) by functoriality. (Henceforth we will denote Shimura’s ξ_{p} by T_{p}.) There are many ways to think of J_{0}(N). The cotangent space Cot J_{0}(N) of J_{0}(N) is the space of holomorphic (or translation invariant) differentials on J_{0}(N), which is isomorphic to S_{2}(Γ_{0}(N)). This gives a connection between our geometric definition of T_{p} and the definition, presented at the beginning of the course, of T_{p} as an operator on a space of cusp forms.

The Eichler-Shimura relation takes place in End(J_{0}(N)_{F_{p}}). Since X_{0}(N) reduces “nicely” in characteristic p, we can apply the Jacobian construction to X_{0}(N)_{F_{p}}. Furthermore the natural reduction map

End(J_{0}(N)) ↪ End(J_{0}(N))_{F_{p}}

is injective. [[Why is this true? It seems at first glance to be the sort of thing that might be false, so why is it true?]] Let F : X_{0}(N)_{F_{p}} → X_{0}(N)_{F_{p}} be the Frobenius map in characteristic p. Thus, if K = K(X_{0}(N)) is the function field of the nonsingular curve X_{0}(N), then F is induced by the p-th power map K → K : a ↦ a^{p}. By both functorialities F induces maps on the Jacobian of X_{0}(N)_{F_{p}}.

⟶^{Ver}

J_{0}(N)_{F_{p}}  J_{0}(N)_{F_{p}}

⟵^{Frob}

Note that Ver ∘ Frob = Frob ∘ Ver = p since p is the degree of F. Since the Albanese functoriality is given by pullback of differentials,

Frob = F_{∗} and Ver = Frob^{∨} = F^{∗}.

Theorem 11.8.1 (Eichler-Shimura). The formula of Eichler-Shimura is

T_{p} = Frob + Ver .

Proof. View X_{0}(pN)_{F_{p}} as two copies of X_{0}(N)_{F_{p}} glued along corresponding supersingular points. Now consider the diagram of schemes over F_{p}.

X_{0}(N)_{F_{p}}  X_{0}(N)_{F_{p}}
↘^{r}  ↙^{s}
X_{0}(pN)_{F_{p}}
↙^{α}  ↘^{β}
X_{0}(N)_{F_{p}}  X_{0}(N)_{F_{p}}

We must define r and s. How can we define a map X_{0}(N)_{F_{p}} → X_{0}(pN)_{F_{p}} in characteristic p? A point of X_{0}(N)_{F_{p}} is an enhanced elliptic curve \underline{E} = (E, C) consisting of an elliptic curve (not necessarily defined over F_{p}) along with a cyclic subgroup of order N. The map r sends \underline{E} to \underline{E} plus an isogeny of degree p,

E ⟶^{frob} E^{(p)}

where E^{(p)} is the curve obtained from E by hitting all defining equations by Frobenious, that is, by p-th powering the coefficients of the defining equations for E. We must introduce E^{(p)} since E might not be defined over F_{p} so Frobenious wouldn’t be an endomorphism of E. Thus r is the map

r : \underline{E} ↦ (\underline{E}, E ⟶^{frob} E^{(p)})

and similarly we define s to be the map

s : \underline{E} ↦ (\underline{E}, E ⟵^{ver} E^{(p)}

where ver is the dual of frob. We may view α as the map sending an isogeny E → E′ to the curve E, and similarly we may view β as the map sending an isogeny E′ → E to the curve E′. Viewing elements of X_{0}(N) as isogenies of degree N is equivalent to our previous view of X_{0}(N) as consisting of pairs (E, C) where C is a cyclic subgroup of order N, but the isogeny viewpoint is better suited to the purpose at hand. Thus

α : (E → E′) ↦ E
β : (E′ → E) ↦ E′

It now follows immediately that α ∘ r = id and β ∘ s = id. Note also that α ∘ s = β ∘ r = F is the map E ↦ E^{(p)}.

At least intuitively we may view X_{0}(pN)_{F_{p}} as the disjoint union of two copies of X_{0}(N)_{F_{p}}. Then

\begin{aligned}& X_{0}(pN)_{F_{p}} \\ ↙^{α} & & ↘^{β} \\ X_{0}(N)_{F_{p}} & & X_{0}(N)_{F_{p}} \end{aligned} = \begin{aligned}& X_{0}(N)_{F_{p}} \\ ↙^{id=α∘r} & & ↘^{F =β∘r} \\ X_{0}(N)_{F_{p}} & & X_{0}(N)_{F_{p}} \end{aligned}

+ \begin{aligned}& X_{0}(N)_{F_{p}} \\ ↙^{F =β∘r} & & ↘^{id=α∘r} \\ X_{0}(N)_{F_{p}} & & X_{0}(N)_{F_{p}} \end{aligned}

[[For some reason I guess this is supposed to prove that T_{p} = Frob + Ver. Why?]]

11.9 Applications

We have the relation T_{p} = Frob + Ver in End(J_{0}(N)_{F_{p}}). How can we apply this formula? Consider the ℓ-adic Tate module

Tate_{ℓ}(J_{0}(N)) = \underleftarrow{lim} J_{0}(N)[ℓ^{ν}] ⊗_{Z_{ℓ}} Q_{ℓ}

which is a vector space of dimension 2g(X_{0}(N)) over Q_{ℓ}. When ℓ ≠ p the image of Tate_{ℓ}(J_{0}(N)) under reduction mod p is basically the same as the original module in characteristic 0. Denote also by Tate_{ℓ}(J_{0}(N)) the image of Tate_{ℓ}(J_{0}(N)) under reduction mod p. On Tate_{ℓ}(J_{0}(N)) we have linear operators Frob_{p}, Ver_{p} and T_{p} which satisfy T_{p} = Frob_{p} + Ver_{p} and

Frob_{p} ∘ Ver_{p} = Ver_{p} ∘ Frob_{p} = p = multiplication by p.

Since p is invertible (because p is prime to ℓ),

T_{p} = Frob_{p} +p(F rob_{p})^{−1}

and thus

Frob_{p}^{2} −T_{p} Frob_{p} +p = 0.

This is a beautiful quadratic relation and so we should be able to get something out of it. We will come back to this shortly, but first we consider the various objects acting on the ℓ-adic Tate module. Tate_{ℓ}(J_{0}(N)) is acted upon in a natural way by

1. Gal(\overline{Q}/Q)

2. End_{Q}(J_{0}(N)) ⊗_{Z_{ℓ}} Q_{ℓ} (which acts by functoriality)

These actions commute with each other since endomorphisms defined over Q aren’t effected by the action of the Galois group of Q. Reducing mod p we also have the commuting actions

3. Gal(\overline{F}_{p}/F_{p})

4. End_{F_{p}}(J_{0}(N)) ⊗_{Z_{ℓ}} Q_{ℓ}

Note that a decomposition group group D_{p} ⊂ Gal(\overline{Q}/Q) acts (after modding out by the corresponding inertia group) in the same way as Gal(\overline{F}_{p}/F_{p}) and the action is unramified, thus 4 is a special case of 1. [[This could be wrong, because I’m perpetually confused about this idea.]]

The Frobenius elements Frob ∈ Gal(\overline{F}_{p}/F_{p}) and Frob ∈ End_{F_{p}}(J_{0}(N)) ⊗_{Z_{ℓ}} Q_{ℓ} induce the same operator on Tate modules. In other words, on the Tate module the morphism Frob_{p} on the Jacobian acts in the same way as φ_{p} ∈ Gal(\overline{F}_{p}/F_{p}). Note that while φ is in a quotient of a decomposition group one often takes a lift to get an element φ_{p} ∈ Gal(\overline{Q}/Q).

On Tate_{ℓ}(J_{0}(N)) we have a quadratic relationship

φ_{p}^{2} − T_{p}φ_{p} + p = 0.

This plays a role when one tries to separate out pieces of the Jacobian. Let

R = Z[…, T_{p}, …] ⊂ End J_{0}(N)

where we only adjoin those T_{p} with p |̸N. Think of R as a reduced Hecke algebra, in particular, R is a subring of T. Then

R ⊗ Q = ∏_{i=1}^{r} E_{i}

where the E_{i} are totally real number fields. The factors E_{i} are in bijection with the Galois conjugacy classes of weight 2 newforms f on Γ_{0}(M) (for some M|N). Galois acts on newforms by hitting their coefficients. The bijection is the map

f ↦ Q(coefficients of f) = E_{i}

Observe that the map is the same if we replace f by one of its conjugates. [[In order to see this don’t we need to know that Q(coeffics of f) is Galois? Do we know this?]] This decomposition is a decomposition of a subring

R ⊗ Q ⊂ End(J_{0}(N)) ⊗ Q =^{def} End(J_{0}(N) ⊗ Q).

Thus it induces a direct product decomposition of J_{0}(N), so J_{0}(N) gets divided up into subvarieties which correspond to conjugacy classes of newforms. A newform of exact level N corresponds to an abelian variety of dimension N.

The relationship

φ_{p}^{2} − T_{p}φ_{p} + p = 0

reminds one of the Caley-Hamilton theorem in which the coefficients would be the trace and determinent. Thus it seems reasonable to expect that

Tr φ_{p} = T_{p} and det φ_{p} = p.

This is is true but it does not follow formally from the given quadratic relation. It is proved in a fairly direct manner using the Weil pairing.

11.10 More on Eichler-Shimura

Remark 11.10.1 (Useful References). At the time of this writing (March, 1996) several useful papers on last year’s Fermat conference could be obtained via ftp from

                gauss.math.brown.edu:/dist/FermatConference

Viewing T_{p} as an element of End_{F_{p}}(J_{0}(N)_{F_{p}}) we have the Eichler-Shimura relation

T_{p} = Frob + Ver .

Let T be the algebra of Hecke operators viewed as a subring of End_{Q}(J_{0}(N)). Then since S_{2}(Γ_{0}(N)) is the tangent space to J_{0}(N)

T ⊂ End_{Q}(J_{0}(N)) ⊂ End_{C}(J_{0}(N))
⊂ End_{C}(Hom(S_{2}(Γ_{0}(N)), C).

[I don’t understand how this is the tangent space!] Thus the T constructed before is really the same ring of Hecke operators as we obtained via the modular forms construction.

Let X = Tate_{ℓ}(J_{0}(N)) and suppose p is a prime different from ℓ. If g is the genus of X_{0}(N) then X is a Q_{ℓ}-vector space of dimension 2g. Since the ℓ-torsion of J_{0}(N)_{F_{p}} is the reduction mod p of the ℓ-torsion of J_{0}(N) the operator T_{p} acts on X. Similarly if φ_{p} ∈ Gal(\overline{Q}/Q) is a Frobenius element then there is an associated map Frob_{p} on X. The Eichler-Shimura relation lifts to characteristic 0 to give [[how? why? this isn’t clear.]

T_{p} = φ_{p} + pφ_{p}^{−1}.

58 CHAPTER 11. HECKE OPERATORS AS CORRESPONDENCES

Abelian Varieties from Modular Forms

In this section we will describe how to break J_{0}(N) up into subvarieties A_{f} corresponding to cusp forms f. This was originally done by Shimura, see Theorem 7.14 of [31]. Let N be a positive integer and let T be the algebra of Hecke operators on J_{0}(N). Let T_{0} = Z[…, T_{n}, …: (n, N) = 1] ⊂ T be the subalgebra of T obtained by adjoining to Z only those Hecke operators T_{n} with n relatively prime to N. If f = ∑ a_{n}q^{n}, let E_{f} = Q(a_{1}, a_{2}, …). Then as in the previous section

T_{0} ⊗ Q = ∏_{f} E_{f},

where f runs over a set of representatives of the space of newforms of weight 2 and level dividing N modulo conjugation by Gal(\overline{Q}/Q). We call the conjugates of a newform its companions. Each f in the product T_{0} ⊗Q contains a projection π_{f} = (0, …, 0, 1, 0, …, 0) ∈ ∏ E_{f} where the 1 is in the fth position. Since T_{0} ⊂ T and T is free over Z, these projectors lie in T ⊗ Q. Since T ⊗ Q is commutative and the π_{f} are mutually orthogonal idempotents whose sum is (1, 1, …, 1) we see that T ⊗ Q breaks into a product of algebras

T ⊗ Q = ∏ 𝒜 _{f}.

(The map is x ↦ ∑ xπ_{f}.) Exercise 12.0.2. Show that dim_{E_{f}} 𝒜 _{f} is the number of divisors of \frac{N}{N(f)} where N(f) is the level of the newform f.

Let’s examine a particular case of this exercise. Let q be a prime, suppose f is a newform of level N(f), and N = N(f)q with q |̸N(f). We show that

𝒜 _{f} = E_{f}[U]/(U^{2} − a_{q}U + q)

and hence dim_{E_{f}} 𝒜 _{f} = 2 which, as expected, is the number of divisors of N/N(f) = q. The first step is to view 𝒜 _{f} as the space of operators generated by the T_{p} acting on the cusp forms f(dτ) for d|(N/N(f)). In our special case T_{p} acts as the scalar a_{p} when p |̸N. When q|N, T_{q} acts as the operator often called U_{q}. Recall that U_{q} corresponds to the matrix ( \begin{matrix}a_{q} & 1 \\ −q & 0 \end{matrix} ) with respect to a suitable basis (see section 9). Thus U_{q} satisfies the relation U_{q}^{2} − a_{q}U + q. Exercise 12.0.3. Write 𝒜 _{f} in general as

E_{f}[…, U_{q}, …]/I

where I is an ideal and the U_{q} correspond to primes dividing N/N(f). [Hint: See Lemma 4.4 of [4] for the answer.]

One can define an abelian subvariety A_{f} of J_{0}(N) as follows. As above let π_{f} be the fth projector in T_{0} ⊗ Q = ∏_{g} E_{g}. We would like to define A_{f} to be the image of J_{0}(N) under π_{f}. But π_{f} isn’t an endomorphism of J_{0}(N). But π_{f} ∈ End(J_{0}(N)) ⊗ Q so there exists n such that nπ_{f} ∈ End(J_{0}(N)). It thus makes sense to let A_{f} = nπ_{f}(J_{0}(N)). This is independent, up to isogeny, of the choice of n.

Due to the fact that A_{f} is only defined up to isogeny it is natural to consider the category of abelian varieties up to isogeny (whose objects are abelian varieties and morphisms are isogenies over Q). In this category J_{0}(N) is decomposes into subvarieties A_{f} since we have an isogeny

∏_{f} A_{f} → J_{0}(N).

Over C we can view this decomposition in yet another way. Think of J_{0}(N)_{C} as

J_{0}(N) = Hom(S_{2}(Γ_{0}(N)), C)/H_{1}(X_{0}(N), Z).

T acts on J_{0}(N) via the action of T on S_{2}(Γ_{0}(N)). This allows us to view J_{0}(N) naturally as V/L where V is a complex vector space and L a lattice in V. The Hecke algebra T breaks V apart into a direct sum V = ⨁_{f} V_{f}. Fortunately, since the Hecke operators act on V_{f} and L in a compatible way, V_{f} ∩ L is a lattice in V_{f}. Thus A_{f} = V_{f}/(V_{f} ∩ L) is an abelian variety.

Suppose A_{f} and A_{g} are not companions, thus f is not congruent to g modulo the actions of Gal(\overline{Q}/Q). What is #(A_{f} ∩ A_{g})? The general answer isn’t known but experimental and theoretical evidence indicates that this number is linked to congruences between f and g.

It is important to emphasize the case when f is a newform of level exactly N in which case 𝒜 _{f} = E_{f} [[why?]]. Fix such an f, let E = E_{f} and let

d = dim A_{f} = dim_{Q} E_{f} = [E_{f} : Q].

Let ℓ be a prime and consider Tate_{ℓ} the Q_{ℓ}-adic Tate module of A_{f}. Recall that

Tate_{ℓ} = \underleftarrow{lim}_{ν}A_{f}[ℓ^{ν}].

Note that Tate_{ℓ} ≅ Q_{ℓ}^{d}. There is a natural action of E ⊗_{Q} Q_{ℓ} on Tate_{ℓ}. By algebraic number theory

E ⊗_{Q} Q_{ℓ} = ∏_{λ|ℓ} E_{λ}

where λ runs through primes of E lying over ℓ and E_{λ} denotes the completion of E with respect to the absolute value induced by λ. Thus Tate_{ℓ} decomposes as a product

Tate_{ℓ} = ∏_{λ|ℓ} Tate_{λ}

where Tate_{λ} is an E_{λ} vector space. The following lemma will prove useful.

Lemma 12.0.4. Let the notation be as above. Then for all λ

dim_{E_{λ}} Tate_{λ} = 2.

Proof. Write A_{f} = V_{f}/L_{f}. The dimension of L_{f} ⊗ Q as a vector space over E_{f} is dim V_{f}/d [[UNFINISHED]] 12.1. COMPUTING THE DETERMINENT OF ρ_{λ} 61

Now consider Tate_{λ} which is an E_{λ} vector space of dimension 2. Since the Hecke operators are defined over Q it follows that Gal(\overline{Q}/Q) acts on Tate_{ℓ} in a way compatible with the action of E ⊗_{Q} Q_{ℓ}. We thus get a map

ρ_{ℓ} : Gal(\overline{Q}/Q) → Aut_{E⊗_{Q}Q_{ℓ}} Tate_{ℓ} = GL_{2}(E ⊗_{Q} Q_{ℓ}) = ∏_{λ} GL_{2}(E_{λ}).

Thus ρ_{ℓ} is the direct sum of maps ρ_{λ} where

ρ_{λ} : Gal(\overline{Q}/Q) → End_{E_{λ}}(Tate_{λ})

gives the action of Gal(\overline{Q}/Q) on Tate_{λ}. If p |̸ℓN then ρ_{λ} is unramified at p. In this case it makes sense to consider ρ_{λ}(φ_{p}) where φ_{p} is a Frobenius element at p. Then ρ_{λ}(φ_{p}) has a well-defined trace and determinent or equivalently a well-defined characteristic polynomial Φ(X).

Theorem 12.0.5. Suppose p |̸ℓN. Let Φ(X) be the characteristic polynomial of ρ_{λ}(φ_{p}). Then

Φ(X) = X^{2} − a_{p}X + p

where a_{p} is the pth coefficient of the modular form f (thus a_{p} is the image of T_{p} in E_{f}).

Proof. By the Cayley-Hamilton theorem

ρ_{λ}(φ)^{2} − Tr(ρ_{λ}(φ))ρ_{λ}(φ) + det(ρ_{λ}(φ)) = 0.

Using the Weil pairing one can show that Tr(ρ_{λ}(φ)) = a_{p}. We prove that det(ρ_{λ}(φ)) = p. This is a consequence of the fact that

det(ρ_{λ}) = χ_{ℓ}

where χ_{ℓ} is the ℓth cyclotomic character

χ_{ℓ} : Gal(\overline{Q}/Q) → Z_{ℓ}^{∗} ⊂ E_{λ}^{∗}

[[Apparently one continues from here by arguing that the Frobenii are dense in some product and so Tchebetarov density implies something.]]

The simplest case is when A_{f} is an elliptic curve. In [33] it is shown that det(ρ_{ℓ}) = χ_{ℓ}. This is proved using the Weil pairing. [[I couldn’t find where in silverman, but I didn’t look hard.]] More generally we define a Weil pairing

Tate A_{f} × Tate A_{f}^{∨} → E_{ℓ}(1).

Note that A_{f} is not necessarily self-dual as was the case when g = 1. Note also that the pairing maps into E_{ℓ}(1) instead of Q_{ℓ}(1) = μ μ μ_{ℓ}.

12.1 Computing the Determinent of ρ_{λ}

Let f = ∑ a_{n}q^{n} be a newform of weight 2 and level exactly N. Thus f is a common eigenform for the Hecke operators acting on S_{2}(Γ_{0}(N)) and f is orthogonal to the forms of lower weight. Then f gives rise to an abelian subvariety A = A_{f} ⊂ J_{0}(N). There is a canonical action of T ⊗ Q on A. Since A is the image of J_{0}(N) under the projection map π_{f}, the action of T ⊗ Q factors through the quotient E_{f} = Q(…, a_{n}, …). The dimension of A is equal to the degree [E_{f} : Q]. The module

Tate_{ℓ}(A) = \underleftarrow{lim} A[ℓ^{ν}]

is free of rank 2 over

E_{f} ⊗ Q_{ℓ} = ∏_{λ|ℓ} E_{λ}.

Thus we obtain a decomposition

Tate_{ℓ}(A) = ∏_{λ|ℓ} Tate_{λ}(A).

For each λ this gives a map

ρ_{λ} : Gal(\overline{Q}/Q) → Aut_{E_{λ}} Tate_{λ} = GL(2, E_{λ}).

If p |̸ℓN we consider ρ_{λ}(Frob_{p}) where Frob_{p} ∈ Gal(\overline{Q}/Q) is a Frobenius element for p (note the change in notation, before we called the Frobenius element φ_{p}). We next discuss why ρ_{λ}(Frob_{p}) satisfies X^{2} − a_{p}X + p = 0.

Theorem 12.1.1. Let the notation be as above. Then Tr(ρ_{λ}(Frob_{p})) = a_{p} and det(ρ_{λ}(Frob_{p})) = p.

The main step is to show that det ρ_{λ} = χ_{ℓ}. In order to do this we must first introduce the Weil pairing on Tate_{ℓ} A. We need a little background on abelian varieties.

12.2 Duality and Polarizations

Let A be an abelian variety. A divisor on A is an integral linear combination ∑ m_{i}D_{i} of codimension 1 algebraic subvarieties D_{i}. If f is a rational function on A then we define the divisor of f to be

(f) = ∑(ord_{D} f)D.

(See Hartshorne [7] II.6 for the definition of ord_{D} f.)

Two divisors D_{1} and D_{2} are linearly equivalent (denoted D_{1} ∼ D_{2}) if D_{1} − D_{2} = (f) for some rational function f. We denote by Pic(A) the group of divisors modulo linear equivalence. Two divisors D_{1} and D_{2} are algebraically equivalent if D_{1} is linearly equivalent to a translate of D_{2} (i.e. D_{1} ∼ D_{2} + a for some a ∈ A). The subgroup of Pic(A) of divisors algebraically equivalent to zero is denoted Pic^{0}(A). It is a fact that Pic^{0}(A) is an abelian variety called the dual of A and denoted A^{∨}. In the more general arithmetic setting Pic^{0}(A) is replaced by isomorphism classes of certain invertible sheaves of degree 0. A polarization of A is an isogeny A → Pic^{0}(A) of the form

a ↦ t_{a}^{∗}L ⊗ L^{−1}

where t_{a}^{∗} is translation by A and L is an ample invertible sheaf. For further details sees Milne’s article [16] or Murty’s book [19]. [[I’ve given two different definitions. Thus I’ve corned myself and should probably rewrite all of the above as soon as I have a better idea of what is going on from Milne’s point of view.]]

Since A ⊂ J_{0}(N), autoduality gives an isomorphism J_{0}(N) ⟶^{∼} J_{0}(N)^{∨} so we obtain a canonical polarization φ : A → A^{∨}. The diagram is

J_{0}(N) ⟶^{autoduality} J_{0}(N)^{∨}
∪  ↓
A ⟶^{polarization} A^{∨}

12.3 The Weil Pairing

Let A = A_{f} be the abelian variety connected to the modular form f. One can define a bilinear pairing

A[n] × A^{∨}[n] → μ μ μ_{n}

which is a natural generalization of the Weil pairing for an elliptic curve. See §16 of Milne’s arcticle [16]. The canonical polarization of A is an isogeny φ : A → A^{∨}. Combining this with the pairing gives a map

A[n] × A[n] → μ μ μ_{n}

which is alternating. But it is not a perfect pairing. Fortunately, upon taking limits, we obtain a perfect pairing

(·, ·) : Tate_{ℓ}(A) × Tate_{ℓ}(A) → Q_{ℓ}(1) = \underleftarrow{lim}μ μ μ_{ℓ^{ν}} ⊗_{Z_{ℓ}} Q_{ℓ}.

If t ∈ T_{0} ⊗ Q_{ℓ} then (ta, a′) = (a, ta′).

Using the polarization φ we can construct a map End(A) → End(A) ⊗ Q by

t ↦ t^{∗} = φ^{−1}t^{∨}φ.

The map t ↦ t^{∗} is called the Rosati involution and has the property that (ta, a′) = (a, t^{∗}a′) for a, a′ ∈ Tate_{ℓ}(A).

Now restrict the form

(·, ·) : Tate_{ℓ}(A) × Tate_{ℓ}(A) → Q_{ℓ}(1)

to Tate_{λ}(A) obtaining

⟨·, ·⟩ : Tate_{λ}(A) × Tate_{λ}(A) → Q_{ℓ}(1).

The form ⟨·, ·⟩ is alternating and nondegenerate. The nondegeneracy follows from the nondegeneracy of (·, ·) and the fact that (Tate_{λ}, Tate_{λ′}) = 0 when λ ≠ λ′. The pairing ⟨·, ·⟩ is Galois equivariant in the following sense. If σ ∈ Gal(\overline{Q}/Q) and x, y ∈ Tate_{λ} then

⟨σx, σy⟩ = σ⟨x, y⟩ = χ_{ℓ}(σ)⟨x, y⟩.

Note that σ acts on Q_{ℓ}(1) by multiplication by χ_{ℓ}(σ).

12.4 The Fancy Proof

There are two proofs of the theorem, a fancy proof and a concrete proof. We first present the fancy proof. The pairing ⟨·, ·⟩ is a alternating and bilinear form so it defines a map

⟨·, ·⟩ : Λ_{E_{λ}}^{2} Tate_{λ} → Q_{ℓ}(1).

It is not a priori true that we can take the wedge product over E_{λ} instead of Q_{ℓ}, but we can because ⟨tx, y⟩ = ⟨x, ty⟩ for any t ∈ E_{λ}. Let D = Λ^{2} Tate_{λ} and note that dim_{E_{λ}} D = 1.

There is a trace map which gives an identification

Hom_{Q_{ℓ}}(D, Q_{ℓ}(1)) ≅ Hom_{E_{λ}}(D, E_{λ}(1))

Thus ⟨·, ·⟩ is an isomorphism D ≅ E_{λ}(1). [[I don’t see this implication right now.]] It now follows that det ρ_{λ} = χ_{ℓ}.

12.5 The Concrete Proof

Now we consider the concrete proof. Suppose σ ∈ Gal(\overline{Q}/Q), then we must show that det(σ) = χ_{ℓ}(σ). Choose vectors x, y ∈ Tate_{λ} which are linearly independent over E_{λ}. Let a, b, c, d ∈ E_{λ} be defined by σ(x) = ax + cy and σ(y) = bx + dy. Let δ = ad − bc − χ_{ℓ}(σ), thus we want to show that δ = 0. We have

χ_{ℓ}(σ)⟨x, y⟩ = ⟨σx, σy⟩
= ⟨ax + cy, bx + dy⟩
= ⟨ax, bx⟩ + ⟨ax, dy⟩ + ⟨cy, bx⟩ + ⟨cy, dy⟩
= ⟨ax, dy⟩ + ⟨cy, bx⟩
= ⟨adx, y⟩ − ⟨bcx, y⟩ = ⟨(ad − bc)x, y⟩

To see that ⟨ax, bx⟩ = 0 we noted that

⟨ax, bx⟩ = ⟨abx, x⟩ = −⟨x, abx⟩ = −⟨ax, bx⟩.

[[Of course it looks like the next step is too use nondegeneracy to conclude that χ_{ℓ}(σ) = ad − bc, but it is not clear to me how to safely do this. My problem is that the expression (ad − bc)⟨x, y⟩ does not really make sense unless we know that ad − bc ∈ Q_{λ}(1) and, even then, why is this equal to the last expression? I clearly lack some crucial bit of knowledge about Q_{λ}(1)]]

12.6 The Construction for X_{1}(N)

Let f = ∑ a_{n}q^{n} be a newform for Γ_{1}(N), so f|T_{n} = a_{n}f for all n ≥ 1 and f|⟨d⟩ = ε(d)f for all d ∈ (Z/NZ)^{∗}. In the case of X_{1}(N), E_{f} is either totally real or it can be a CM field. If it is a CM field then it has a canonical complex conjugation \overline{·} : a_{n} ↦ ε(n)^{−1}a_{n}. The analogous formulas are

Tr(ρ_{ℓ}(Frob_{p})) = a_{p}
det(ρ_{ℓ}(Frob_{p})) = ε(p)p

There are some details concerning signs which we must consider. First, for every p we can define T_{p} as a correspondence on X_{1}(N) with α and β defined in an analogous way to the way they were defined in the case of X_{0}(N). But now we must choose if the corresponding endomorphism of J_{1}(N) should be (T_{p})_{∗} or (T_{p})^{∗}. The second choice concerns what happens in the Eichler-Shimura relation. Recall that X_{1}(N) classifies pairs (E, p) where E is an elliptic curve and p is a point of order N. Equivalently, X_{1}(N) classifies pairs consisting of an elliptic curve and an embedding Z/NZ ↪ E. But this is a bad way to look at this.

12.6. THE CONSTRUCTION FOR X_{1}(N) 65

A better way is to view X_{1}(N) as classifying pairs consisting of an elliptic curve and an embedding μ μ μ_{N} ↪ E because the associated Tate curve has a copy of μ μ μ_{N} naturally sitting in it. When passing to J_{1}(N) we take the Picard functorality such that if T_{p} : J_{1}(N) → J_{1}(N) then (T_{p}^{∨})^{∗} on H^{0}(X_{1}(N), Ω^{1}) = S_{2}(Γ_{1}(N)) is the classical T_{p}.

66 CHAPTER 12. ABELIAN VARIETIES FROM MODULAR FORMS

The Gorenstein Property

Suppose f = ∑ a_{n}q^{n} is a newform of exact level N and weight 2 for the congruence subgroup Γ_{0}(N). Let E = Q(…, a_{n}, …) and let λ be a place of E lieing over the prime ℓ of Q. The action of Gal(\overline{Q}/Q) on the ℓ-adic Tate module of the abelian variety A_{f} associated to f gives rise to a representation

ρ_{λ} : Gal(\overline{Q}/Q) → GL_{2}(E_{λ})

with det ρ_{λ} = χ_{ℓ} and Tr ρ_{λ}(Frob_{p}) = a_{p} for p |̸ℓN. We ask, how can we reduce this representation module λ?

Lemma 13.0.1. Let O be the ring of integers of E_{λ}. Then ρ_{λ} is equivalent to a representation which takes values in GL_{2}(O).

Proof. View GL_{2}(E_{λ}) as the group of automorphisms of a 2 dimensional E_{λ}-vector space V. A lattice is a free O-module of rank 2 such that L ⊗ E_{λ} ≅ V. It suffices to find an O-lattice L in V which is invariant under the action (via ρ_{λ}) of Gal(\overline{Q}/Q). For then the matrices of Gal(\overline{Q}/Q) with respect to a basis of V consisting of vectors from L will have coefficients in O. Choose any lattice L_{0} ⊂ V. Since L_{0} is discrete and Gal(\overline{Q}/Q) is compact the set of lattices gL_{0} with g ∈ Gal(\overline{Q}/Q) is finite. Let L = ∑ gL_{0} be the sum of the finitely many conjugates of L_{0}, then L is Galois invariant.

It now makes sense to let \overline{ρ}_{λ} = ρ_{λ} mod λ. This is expressed by the diagram

Gal(\overline{Q}/Q) ⟶^{ρ_{λ}} GL_{2}(O)
↘  ↓
GL_{2}(O/λ)

The drawback to this approach is that \overline{ρ}_{λ} is not intrinsic because the definition depends on the choice of L_{0}. In fact, it is possible to obtain nonisomorphic representations \overline{ρ}_{λ} by different choices of Gal(\overline{Q}/Q) → GL_{2}(O). By nonisomorphic we mean nonisomorphic over O (the representations are isomorphic over E_{λ}).

The good definition is to let \overline{ρ}_{λ} be the semisimplification of the reduction of ρ_{λ} modulo λ. By the semisimplification of a representation we mean the direct sum of the Jordan-Hölder factors in a filtration of V. Suppose σ : Gal(\overline{Q}/Q) → GL_{2}(F) is a mod ℓ Galois representation. If g ∈ Gal(\overline{Q}/Q) then the characteristic polynomial of the action of g in the mod λ representation is the same as the characteristic polynomial of g in the semisimplification. The representation \overline{ρ}_{λ} is semisimple (since it is a semisimplification) and det \overline{ρ}_{λ} = χ_{ℓ} mod ℓ is the mod ℓ cyclotomic character, so for example χ_{ℓ}(Frob_{p}) = p mod ℓ. Furthermore, Tr(\overline{ρ}_{λ}(Frob_{p})) = a_{p} mod λ. This shows that the characteristic polynomials corresponding to \overline{ρ}_{λ} are independent of our choice of an isomorphic representation with values in GL_{2}(O). Thus the following theorem implies that \overline{ρ}_{λ} only depends on f and λ.

Theorem 13.0.2 (Brauer-Nesbit). Suppose ρ_{1}, ρ_{2} : G → GL(V) are two finite dimensional semisimple representations of a group G over a finite field k. Assume furthermore that for every g ∈ G the characteristic polynomial of ρ_{1}(g) is the same as the characteristic polynomial of ρ_{2}(g). Then ρ_{1} and ρ_{2} are equivalent.

The proof can be found in section 30.16 of [3].

The Hecke operators T_{n} act on the space S_{2}(Γ_{0}(N)). Let T = Z[…, T_{n}, …] ⊂ End(S_{2}(Γ_{0}(N))) and recall that T is a commutative ring which is a Z-module of rank equal to dim_{C} S_{2}(Γ_{0}(N)). Let m be a maximal ideal of T and let k = T/m = F_{ℓ^{ν}} be the residue field.

Proposition 13.0.3. There is a (unique) semisimple representation

ρ_{m} : Gal(\overline{Q}/Q) → GL_{2}(k)

such that ρ_{m} is unramified outside ℓN and

Tr(ρ_{m}(Frob_{p})) = T_{p} mod m
det(ρ_{m}(Frob_{p})) = p mod m

Proof. It is enough to prove the assertion with T replaced by

T_{0} = Z[{…, T_{n}, …: (n, N) = 1}].

Indeed, a maximal ideal of T pulls back to a maximal ideal of T_{0} [[why is this true in this case?]] and k_{0} = T_{0}/m_{0} ⊂ k. Now

T_{0} ⊂ T_{0} ⊗ Q = ∏_{i=1}^{t} E_{i}

with the E_{i} number fields. Let O_{E_{i}} be the ring of integers of E_{i} and let O = ∏ O_{E_{i}}. Then

m_{0} ⊂ T_{0} ⊂ ∏ O_{E_{i}}.

By the going-up theorem there is a maximal ideal λ ⊂ ∏ O_{E_{i}} lieing over m_{0}, and k_{0} ⊂ O/λ. Using the construction outlined above (the field E is the only E_{i} for which E_{i} ∩ λ ≠ 0) we can make a representation

\overline{ρ}_{λ} : Gal(\overline{Q}/Q) → GL_{2}(O/λ)

with det \overline{ρ}_{λ} = χ_{ℓ} and Tr \overline{ρ}_{λ} = T_{p} mod m (because of the way we have set things up T_{p} plays the role of a_{p}). Thus this representation has the required properties, but it takes values in GL_{2}(O/λ) instead of k_{0}. Since the characteristic polynomial of every \overline{ρ}_{λ}(g) for g ∈ Gal(\overline{Q}/Q) has coefficients in the subfield k_{0} ⊂ O/λ there is a model for ρ_{λ} over k_{0}. This is the extra bit of information we need to complete the proof. It is a fact about finite fields and is somehow related to the fact that the Brauer group of a finite field is trivial. [[It would be great to have a reference for this fact about finite fields because, judging from the audience, it is not an extremely well-known result.]]

Let us look at this construction in another way. Write T_{0} ⊗ Q = E_{1} × ⋯ × E_{t} and recall that E_{i} corresponds to a newform of level M|N (one can obtain E_{i} by adjoining the coefficients of some newform of level M|N to Q). Also, J_{0}(N) is isogenous to a product A_{1} × ⋯ × A_{t}. Consider one of the factors, say E_{1}, and suppose, to fix ideas, that it corresponds to a newform of exact level N. Since Tate_{ℓ} A_{1} is free of rank 2 over E_{1} ⊗_{Q} Q_{ℓ}, we obtain a 2-dimensional representation ρ_{λ}. Reducing mod λ and semisimplifying gives the representation constructed in the above proposition. But it is also possible that one of the fields E_{i} corresponds to a newform f of level M properly dividing N. Then the associated abelian variety A_{i} has too large of dimension. So [[somehow?!]] we repeat the whole construction with J_{0}(N) to get a 2 dimensional representation.

Let A = A_{1} be as above. Then T acts on A_{1} which we view as an abelian subvariety of J_{0}(N). Let \overline{T} be the quotient of T by the kernel of the map T → End A_{1}. Although \overline{T} sits in O_{1} which is the ring of integers of a field, \overline{T} might not be integrally closed in O_{1}. Consider the usual Z_{ℓ}-adic Tate module Tate_{ℓ}(A_{1}) ≅ Z_{ℓ}^{2 dim A} (dim A = [E_{1} : Q]). This Tate module has a faithful action of \overline{T} ⊗ Z_{ℓ}. By the theory of semilocal rings this ring decomposes as a product

\overline{T} ⊗ Z_{ℓ} = ∏_{m|ℓ} \overline{T}_{m}.

Thus Tate_{ℓ} decomposes as a product ∏_{m} Tate_{m}(A_{1}). It would be nice if Tate_{ℓ} were free of rank 2 over \overline{T} ⊗ Z_{ℓ} but this is not known to be true in general, although it has often been verified in special cases. For this to be true we must have that, for all m|ℓ, Tate_{m} is free of rank 2 over \overline{T}_{m}.

Next we put things in a finite context instead of a projective limit context. Let J = J_{0}(N), then by Albanese or Picard functoriality T ⊂ End J. Let m ⊂ T be a maximal ideal. Let

J[m] = \{t ∈ J(\overline{Q}) : xt = 0 for all x ∈ m\}.

Note that J[m] ⊂ J[ℓ] where ℓ is the rational prime lying in m. Now J[ℓ] is an F_{ℓ} vector space of rank 2g where g is the genus of X_{0}(N). Although it is true that J[ℓ] is a T/ℓ-module it is not convenient to work with T/ℓ since it might not be a product of field and there can be some unpleasant ramification. It is more convenient to work with J[m] since T/m is a field. Thus, via another optic, J[m] is a k[G]-module where k = T/m and G = Gal(\overline{Q}/Q). The naive hope is that J[m] is a model for ρ_{m}, at least when ρ_{m} is irreducible. This does not quite work, but we do have the following

Theorem 13.0.4. J[m] is a model for ρ_{m} when ℓ ≠ 2 and ℓ |̸N.

13.1 The Gorenstein Property

Let

T = Z[…, T_{n}, …] ⊂ End J_{0}(N)

and let m ⊂ T be a maximal ideal. Let ℓ ∈ m be the prime of Z over which m lies. We have constructed a representation

ρ_{m} : Gal(\overline{Q}/Q) → GL_{2}(T/m)

which is semisimple by construction. It is unramified outside ℓN and for any prime p |̸ℓN,

Tr(ρ_{m}(Frob_{p})) = T_{p} mod m
det(ρ_{m}(Frob_{p})) = p mod m

We will usually be interested in the case when ρ_{m} is irreducible. Let T_{m} = \underleftarrow{lim} T/m^{i}T denote the completion of T at m. Note that T ⊗_{Z} Z_{ℓ} = ∏_{m|ℓ} T_{m}. Our goal is to prove that if m |̸2N and ρ_{m} is irreducible, then T_{m} is Gorenstein.

Definition 13.1.1. Let O be a complete discrete valuation ring. Let T be a local O-algebra which as a module if finite and free over O. Then T is a Gorenstein O-algebra if there is an isomorphism of T -modules T ⟶^{∼} Hom_{O}(T, O).

Thus T_{m} is Gorenstein if Hom_{Z_{ℓ}}(T_{m}, Z_{ℓ}) ⟶^{∼} T_{m}. as T_{m} modules. Intuitively this means T_{m} is “autodual”.

Theorem 13.1.2. Let J = J_{0}(N), let the representation ρ_{m} be the as constructed in Proposition ?? and assume ρ_{m} is irreducible and m |̸2N. Let

J[m] = {a : xa = 0 for all x ∈ m} ⊂ J[ℓ].

Then dim_{T/m} J[m] = 2 and J[m] as a Galois module is a 2-dimensional representation giving rise to ρ_{m}.

[[Does giving rise mean the char poly’s are all the same or that the semisimplifications are isomorphic?]] An easy argument shows that the theorem implies T_{m} is Gorenstein.

We first consider the structure of W = J[m]. Suppose the two dimensional representation

ρ_{m} : Gal(\overline{Q}/Q) → Aut_{T/m} V

constructed before is irreducible. Consider the semisimplification W^{s.s.} of W, thus W^{s.s.} is the direct sum of its Jordan-Hölder factors as a Gal(\overline{Q}/Q)-module. Mazur proved the following theorem.

Theorem 13.1.3. There is some integer t ≥ 0 so that

W^{s.s.} ≅ V × ⋯ × V = V^{t}.

If ρ_{m} is in fact absolutely irreducible then it is a result of Boston, Lenstra, and Ribet [2] that W ≅ V × ⋯ × V. A representation is absolutely irreducible if it is irreducible over the algebraic closure. It can be shown that if ℓ ≠ 2 and ρ_{m} is irreducible then ρ_{m} must be absolutely irreducible.

The construction of W is nice and gentle whereas the construction of V is accomplished via brute force.

Proof. (Mazur) We want to compare V with W. Let d = dim W. Let

W^{∗} = Hom_{T/m}(W, T/m(1))

where T/m(1) = T/m ⊗_{Z} μ μ μ_{ℓ}. We need to show that

W^{s.s.} ⊕ W^{∗s.s.} ≅ V × ⋯ × V = V^{d}

as representations of Gal(\overline{Q}/Q). Note that each side is a semisimple module of dimension 2d. To obtain the isomorphism we show that the two representations have the same characteristic polynomials so they are isomorphic.

We want to show that the characteristic polynomial of Frob_{p} is the same for both W^{s.s.} ⊕ W^{s.s.} and V × ⋯ × V. The characteristic polynomial of Frob_{p} on V is X^{2} − T_{p}X + p = (X−r)(X−pr^{−1}) where r lies in a suitable algebraic closure. It follows that the characteristic polynomial of Frob_{p} on V^{d} is (x − r)^{d}(x − pr^{−1})^{d}. On W the characteristic polynomial of Frob_{p} is (X − α_{1}) ⋯ (X − α_{d}) where α_{i} is either r or pr^{−1}. This is because Eichler-Shimura implies Frob_{p} must satisfy Frob_{p}^{2} −T_{p} Frob_{p} +p = 0. [[I don’t see this implication.]] On W^{∗} the characteristic polynomial of Frob_{p} is (X − pα_{1}^{−1}) ⋯ (X − pα_{d}^{−1}). [[This is somehow tied up with the definition of W^{∗} and I can’t quite understand it.]] Thus on W ⊕ W^{∗}, the characteristic polynomial of Frob_{p} is

∏_{i=1}^{d}(X − α_{i})(X − pα_{i}^{−1}) = ∏_{i=1}^{d}(X − r)(X − pr^{−1}) = (X − r)^{d}(X − pr^{−1})^{d}.

Therefore the characteristic polynomial of Frob_{p} on W ⊕W^{∗} is the same as the characteristic polynomial of Frob_{p} on V × ⋯ × V. The point is that although the α_{i} could all a priori be r or pr^{−1}, by adding in W^{∗} everything pairs off correctly. [[I don’t understand why we only have to check that the two representations agree on Frob_{p}. There are lots of other elements in Gal(\overline{Q}/Q), right?]]

We next show that J[m] ≠ 0. This does not follow from the theorem proved above because it does not rule out the possibility that t = 0 and hence W^{s.s.} ≅ 0. Suppose J[m] = 0, then we will show that J[m^{i}] = 0 for all i ≥ 1. We consider the ℓ-divisible group J_{m} = ∪_{i}J[m^{i}]. To get a better feel for what is going on, temporarily forget about m and just consider the Tate module corresponding to ℓ.

It is standard to consider the Tate module

Tate_{ℓ} J = \underleftarrow{lim} J[ℓ^{i}] ≅ Z_{ℓ}^{2 dim J}.

It is completely equivalent to consider

J_{ℓ} := ∪_{i=1}^{∞}J[ℓ^{i}] ⟶^{∼} (Q_{ℓ}/Z_{ℓ})^{2 dim J}.

Note that since Q_{ℓ}/Z_{ℓ} is not a ring the last isomorphism must be viewed as an isomorphism of abelian groups. In [14] Mazur called Tate_{ℓ} J ≅ Hom(Q_{ℓ}/Z_{ℓ}, J_{ℓ}) the covariant Tate module. Call

Tate_{ℓ}^{∗} J := Hom(J_{ℓ}, Q_{ℓ}/Z_{ℓ}) ≅ Hom_{Z_{ℓ}}(Tate_{ℓ} J, Z_{ℓ})

the contravariant Tate module. [[Why are the last two isomorphic?]] The covariant and contravariant Tate modules are related by a Weil pairing J[ℓ^{i}] × J[ℓ^{i}] → μ μ μ_{ℓ^{i}}. Taking projective limits we obtain a pairing

⟨·, ·⟩ : Tate_{ℓ} J × Tate_{ℓ} J → Z_{ℓ}(1) = \underleftarrow{lim}μ μ μ_{ℓ^{i}}.

This gives a map

Tate_{ℓ} J → Hom(Tate_{ℓ} J, Z_{ℓ}(1)) = (Tate_{ℓ}^{∗} J)(1)

where (Tate_{ℓ}^{∗} J)(1) = (Tate_{ℓ}^{∗} J) ⊗ Z_{ℓ}(1). Z_{ℓ}(1) is a Z_{ℓ}-module where

∑ a_{i}ℓ^{i} · ζ = ζ^{∑ a_{i}ℓ^{i}}

[[This should probably be said long ago.]] This pairing is not a pairing of T-modules, since if t ∈ T then ⟨tx, y⟩ = ⟨x, t^{∨}y⟩. It is more convenient to use an adapted pairing defined as follows. Let w = w_{V} ∈ End J_{0}(N) be the Atkin-Lehner involution so that t^{∨} = wtw. Define a new T-compatible pairing by [x, y] := ⟨x, wy⟩. Then

[tx, y] = ⟨tx, wy⟩ = ⟨x, t^{∨}wy⟩ = ⟨x, wty⟩ = [x, ty].

The pairing [·, ·] defines an isomorphism of T ⊗ Z_{ℓ}-modules

Tate_{ℓ} J ⟶^{∼} Hom_{Z_{ℓ}}(Tate_{ℓ} J, Z_{ℓ}(1)).

Since Z_{ℓ}(1) is a free module of rank 1 over Z_{ℓ} a suitable choice of basis gives an isomorphism of T ⊗ Z_{ℓ}-modules

(Tate_{ℓ}^{∗} J)(1) ≅ Hom_{Z_{ℓ}}(Tate_{ℓ} J, Z_{ℓ}(1)) ≅ Hom_{Z_{ℓ}}(Tate_{ℓ} J, Z_{ℓ}) = Tate_{ℓ}^{∗} J.

Thus Tate_{ℓ} J ⟶^{∼} Tate_{ℓ}^{∗} J.

Proof. (That J[m] ≠ 0.) The point is that the contravariant Tate module Hom(J_{ℓ}, Q_{ℓ}/Z_{ℓ}) is the Pontrjagin dual of T_{ℓ}. How does this relate to Tate_{m} J? Since T ⊗ Z_{ℓ} ≅ ∏_{m|ℓ} T_{m}, Tate_{ℓ} J ≅ ∏_{m|ℓ} Tate_{m} J so we can define Tate_{m}^{∗} J := Hom_{Z_{ℓ}}(Tate_{m} J, Z_{ℓ}). Weil proved that Tate_{m} J ≅ T ate_{m}^{∗}J is nonzero. View Tate_{m}^{∗} J as being dual to J_{m} in the sense of Pontrjagin duality and so (Tate_{m}^{∗} J)/(m Tate_{m}^{∗} J) is dual to J[m]. If J[m] = 0 then this quotient is 0, so Nakayama’s lemma would imply that Tate_{m}^{∗} J = 0. This would contradicts Weil’s assertion. Therefore J[m] ≠ 0.

13.2 Proof the Gorenstein Property

We are considering the situation with respect to J_{0}(N) although we could consider J_{1}(N). Let T ⊂ End J_{0}(N) be the Hecke algebra and let m ⊂ T be a maximal ideal. Let ℓ be the characteristic of the residue class field T/m. Let T_{m} = \underleftarrow{lim} T/m^{i}T. Then T ⊗_{Z} Z_{ℓ} = ∏_{m|ℓ} T_{m}. [[I want to put a good reference for this Atiyah-Macdonald like fact here.]] Each T_{m} acts on Tate_{ℓ} J_{0}(N) so we obtain a product decomposition

Tate_{ℓ} J_{0}(N) = ∏_{m|ℓ} Tate_{m} J_{0}(N).

We have the following two facts:

1. Tate_{m} J_{0}(N) ≠ 0

2. Tate_{ℓ} J_{0}(N) is T ⊗ Z_{ℓ}-autodual and each Tate_{m} J_{0}(N) is Z_{ℓ}-autodual.

[[autoduality for which dual? I think it is the linear dual since this is used.]]

Let W = J[m], then the action of Gal(\overline{Q}/Q) on W gives a representation of Gal(\overline{Q}/Q) over the field T/m. We compared W with a certain two dimensional representation ρ_{m} : Gal(\overline{Q}/Q) → V over T/m. Assume unless otherwise stated that V is irreducible as a Gal(\overline{Q}/Q)-module. Let Tate_{ℓ} = Tate_{ℓ} J_{0}(N) and Tate_{m} = Tate_{m} J_{0}(N). A formal argument due to Mazur showed that

W^{s.s.} ≅ V × ⋯ × V = V^{⊕t}.

We have not yet determined t but we would like to show that t = 1.

Definition 13.2.1. The Pontrjagin dual of a module M is the module M^{∧} := Hom(M, Q/Z) where M is viewed as an abelian group (if M is topological, only take those homomorphisms whose kernel is compact). The linear dual of a module M over a ring R is the module M^{∗} = Hom_{R}(M, R).

Exercise 13.2.2. Note that (Q_{ℓ}/Z_{ℓ})^{∧} = Z_{ℓ} and Z_{ℓ}^{∧} = Q_{ℓ}/Z_{ℓ}.

Solution. We can think of Q_{ℓ}/Z_{ℓ} as

{ ∑_{n=−k}^{−1} a_{n}ℓ^{n} : k > 0 and 0 ≤ a_{i} < ℓ}.

Let (b_{i}) ∈ Z_{ℓ} so b_{i} ∈ Z/ℓ^{i}Z and b_{i+1} ≡ b_{i} (mod ℓ^{i}). Define a map Q_{ℓ}/Z_{ℓ} → Q/Z by 1/ℓ^{i} ↦ b_{i}/ℓ^{i}. To check that this is well-defined it suffices to check that 1/ℓ^{i} maps to the same place as ℓ · 1/ℓ^{i+1}. Now 1/ℓ^{i} ↦ b_{i}/ℓ^{i} and

ℓ · 1/ℓ^{i+1} ↦ ℓ · b_{i+1}/ℓ^{i+1} = b_{i+1}/ℓ^{i}.

So we just need to check that

b_{i+1}/ℓ^{i} ≡ b_{i}/ℓ^{i} (mod Z).

This is just the assertion that (b_{i+1} − b_{i})/ℓ^{i} ∈ Z which is true since b_{i+1} ≡ b_{i} (mod ℓ^{i}).

Proposition 13.2.3. Let the notation be as above, then t > 0.

Proof. The idea is to use Nakayama’s lemma to show that if t = 0 and hence W = 0 then Tate_{m} = 0 which is clearly false. But the relation between W and Tate_{m} is rather convoluted. In fact J[ℓ^{∞}] is the Pontrjagin dual of Tate_{ℓ}^{∗}, that is,

J[ℓ^{∞}]^{∧} = Tate_{ℓ}^{∗} = Hom_{Z_{ℓ}}(Tate_{ℓ}, Z_{ℓ})

and

(Tate_{ℓ}^{∗})^{∧} = Hom(Tate_{ℓ}^{∗}, Q_{ℓ}/Z_{ℓ}) = J[ℓ^{∞}].

[[First: Why are they dual? Second: Why are we homing into Q_{ℓ}/Z_{ℓ} instead of Q/Z?]] Looking at the m-adic part shows that

J[m^{∞}] = Hom(Tate_{m}^{∗}, Q_{ℓ}/Z_{ℓ})

and hence

J[m] = Hom(Tate_{m}^{∗} /m Tate_{m}^{∗}, Z/ℓZ).

Thus if J[m] = 0 then Nakayama’s lemma implies Tate_{m}^{∗} = 0. By autoduality this implies Tate_{m} = 0.

We have two goals. The first is to show that t = 1, i.e., that J[m] is 2-dimensional over T/m. The second is to prove that T_{m} is Gorenstein, i.e., that T_{m} ≅ Hom_{Z_{ℓ}}(T_{m}, Z_{ℓ}). This is one of the main theorems in the subject. We are assuming throughout that ρ_{m} is irreducible and ℓ |̸2N. Loosely speaking the condition that ℓ |̸2N means that J[m] has good reduction at ℓ and that J[m] can be understood just by understanding J[m] in characteristic ℓ. We want to prove that T_{m} is Gorenstein because this property plays an essential role in proving that T_{m} is a local complete intersection.

Example 13.2.4. K. Buzzard presented the following example of a non-Gorenstein ring. Let

T = {(a, b, c, d) ∈ Z_{p}^{4} : a ≡ b ≡ c ≡ d (mod p)}.

Then T is local but not Gorenstein.

For now we temporarily postpone the proof of the first goal and instead show that the first goal implies the second.

Theorem 13.2.5. Suppose J[m] is two dimensional over T/m (thus t = 1). Then T_{m} is Gorenstein. Proof. We have seen before that

J[m] = Hom_{Z/ℓZ}(Tate_{m}^{∗} /m Tate_{m}^{∗}, Z/ℓZ)
= Hom_{T/m}(Tate_{m}^{∗} /m Tate_{m}^{∗}, T/m).

Thus the dual of Tate_{m}^{∗} /m Tate_{m}^{∗} is two dimensional over T/m and hence Tate_{m}^{∗} /m Tate_{m}^{∗} itself is two dimensional over T/m. By Nakayama’s lemma and autoduality of Tate_{m} this implies Tate_{m} is generated by 2 elements over T_{m}. There is a surjection

T_{m} × T_{m} ⟶ Tate_{m} .

In fact it is true that rank_{Z_{ℓ}} Tate_{m} = 2 rank_{Z_{ℓ}} T_{m}. We temporarily postpone the proof of this claim. Assuming this claim and using the fact that a surjection between Z_{ℓ}-modules of the same rank is an isomorphism implies that Tate_{m} ≅ T_{m} × T_{m}. Now T_{m} is a direct summand of the free Z_{ℓ} module Tate_{m} so T_{m} is projective. A projective module over a local ring is free. Thus T_{m} is free of rank 1 and hence autodual (Gorenstein). [[This argument is an alternative to Mazur’s – it seems too easy... maybe I am missing something.]]

We return to the claim that

rank_{Z_{ℓ}} Tate_{m} = 2 rank_{Z_{ℓ}} T_{m}.

This is equivalent to the assertion that

dim_{Q_{ℓ}} Tate_{m} ⊗_{Z_{ℓ}}Q_{ℓ} = 2 dim_{Q_{ℓ}} T_{m} ⊗_{Z_{ℓ}} Q_{ℓ}.

The module Tate_{ℓ} J_{0}(N) is the projective limit of the ℓ-power torsion on the Jacobian

J(C) = \frac{Hom_{C}(S_{2}(Γ_{0}(N), C), C)}{H_{1}(X_{0}(N), Z)}.

Let L = H_{1}(X_{0}(N), Z) be the lattice. Then L is a T-module and Tate_{ℓ} = L ⊗_{Z} Z_{ℓ} (since L/ℓ^{i}L ≅ (\frac{1}{ℓ^{i}} L)/L). Tensoring with R gives

L ⊗_{Z} R = Hom_{C}(S_{2}(Γ_{0}(N), C), C)
= Hom_{R}(S_{2}(Γ_{0}(N), R), C)
= Hom_{R}(S_{2}(Γ_{0}(N), R), R) ⊗_{R} C = (T ⊗_{Z} R) ⊗_{R} C

Thus L ⊗_{Z} R is free of rank 2 over T ⊗_{Z} R and L ⊗_{Z} C is free of rank 2 over T ⊗_{Z} C. Next choose an embedding Q_{ℓ} ↪ C. Now Tate_{ℓ} is a module over T ⊗ Z_{ℓ} = ∏_{m|ℓ} T_{m} so we have a decomposition Tate_{ℓ} = ∏_{m|ℓ} Tate_{m}. Since

Tate_{ℓ} ⊗_{Z_{ℓ}}Q_{ℓ} = ∏_{m} Tate_{m} ⊗_{Z_{ℓ}}Q_{ℓ}

we can tensor with C to see that

Tate_{ℓ} ⊗_{Z_{ℓ}}C = ∏_{m} Tate_{m} ⊗_{Z_{ℓ}}C.

But Tate_{ℓ} ⊗_{Z_{ℓ}}C = L⊗_{Z}C is free of rank 2 over T⊗C. Therefore the product ∏_{m} Tate_{m} ⊗_{Z_{ℓ}}C is free of rank 2 over T ⊗ C. Since

T ⊗ C = (T ⊗_{Z} Z_{ℓ}) ⊗_{Z_{ℓ}} C = ∏_{m}(T_{m} ⊗_{Z_{ℓ}} C)

we conclude that for each m, Tate_{m} ⊗_{Z_{ℓ}}C is free of rank 2 over T_{m} ⊗_{Z_{ℓ}} C. This implies

dim_{C} Tate_{m} ⊗_{Z_{ℓ}}C = 2 dim_{C} T_{m} ⊗_{Z_{ℓ}} C

which completes the proof.

13.2.1 Vague Comments

Ogus commented that this same proof shows that T ⊗_{Z} C is Gorenstein. Then he said that something called “faithfully flat descent” could then show that T ⊗_{Z} Q is Gorenstein.

We have given the classical argument of Mazur that T_{m} is Gorenstein, but we still haven’t shown that J[m] has dimension 2. This will be accomplished next time using Dieudonné modules.

13.3 Finite Flat Group Schemes

Definition 13.3.1. Let S be a scheme. Then a group scheme over S is a group object in the category of S-schemes.

Thus a group scheme over S is a scheme G/S equipped with S-morphisms m : G×G → G, inv : G → G and a section 1_{G} : S → G satisfying the usual group axioms.

Suppose G is a group scheme over the finite field F_{q}. If R is an F_{q}-algebra then G(R) = Mor(spec R, G) is a group. It is the group of R-valued points of G.

We consider several standard examples of group schemes.

Example 13.3.2. The multiplicative group scheme G_{m} is G_{m} = spec Z[x, \frac{1}{x}] with morphisms. [[give maps, etc.]] The additive group scheme is spec Z[x]...

Example 13.3.3. The group scheme μ μ μ_{p} is the kernel of the morphism G_{m} → G_{m} induced by x ↦ x^{p}. Thus μ μ μ_{p} = spec Z[x]/(x^{p} − 1) and so for any F_{q}-algebra R we have that μ μ μ_{p}(R) = {r ∈ R : r^{p} = 1}. The group scheme α_{p} is the kernel of the morphism G_{a} → G_{a} induced by [[what!! what is alphap?? it should be the additive group scheme of order p, no?]].

Let A be a finite algebra over F_{p} and suppose G = spec A affine group scheme (over F_{p}). Then the order of G is defined to be the dimension of A as an F_{p} vector space.

Example 13.3.4. Let E/F_{p} be an elliptic curve. Then G = E[p] is a group scheme of order p^{2} [[why is this true?]]. This is wonderful because this is the order that E[p] should have in analogy with the characteristic 0 situation. When we just look at points we have

#G(\overline{F_{p}}) = \begin{cases}1 & supersingular \\ p & ordinary \end{cases} .

13.4 Reformulation of V = W problem

Let J = J_{0}(N) be the Jacobian of X_{0}(N). Then J is defined over Q and has good reduction at all primes not dividing N. Assume ℓ is a prime not dividing N. J[ℓ] extends to a finite flat group scheme over Z[\frac{1}{N}]. This is a nontrivial result of Grothendieck (SGA 7I, LNM 288). Since ℓ |̸N, J[ℓ] gives rise to a group scheme over F_{ℓ}.

We have “forcefully” constructed a Galois representation ρ_{m} : G → V of dimension 2 over T/m. Our goal is to show that this is isomorphic to the naturally defined Galois representation W = J[m]. So far we know that

0 ⊂ V ⊂ W ⊂ J[ℓ].

Our assumptions are that ℓ |̸N, V is irreducible, and ℓ ≠ 2.

Let \underline{J} be J thought of as a scheme over Z_{ℓ}. Grothendieck showed that \underline{J} is the spectrum of a free finite Z_{ℓ}-module. Raynaud (1974) showed that if ℓ ≠ 2 then essentially everything about \underline{J}[ℓ] can be seen in terms of J[ℓ](\overline{Q}_{ℓ}). He goes on to construct group scheme \underline{V} and \underline{W} over Z_{ℓ} such that

\underline{V} ⊂ \underline{W} ⊂ \underline{J}[ℓ].

Our goal is to prove that the inclusion V ↪ W of Galois modules is an isomorphism. Raynaud showed noted that the category of finite flat group schemes over Z_{ℓ} is an abelian category so the cokernel \underline{Q} = \underline{W} /\underline{V} is defined. Furthermore, V = W iff \underline{Q} = 0. Since \underline{Q} is flat \underline{Q}_{F_{ℓ}} has the same dimension over F_{ℓ} as \underline{Q}_{Q_{ℓ}} has over Q_{ℓ}. Passing to characteristic ℓ yields an exact sequence

0 → \underline{V}_{F_{ℓ}} → \underline{W}_{F_{ℓ}} → \underline{Q}_{F_{ℓ}} → 0.

Thus V ↪ W is an isomorphism iff \underline{V}_{F_{ℓ}} ↪ \underline{W}_{F_{ℓ}} is an isomorphism. Since \underline{V}, \underline{W}, and \underline{Q} have an action of k = T/m that are k-vector space schemes. This leads us to Dieudonné theory.

13.5 Dieudonné Theory

Let G/k be a finite k-vector space scheme where k is a finite field of order q. Suppose G has order q^{n} so G is locally the spectrum of a rank n algebra over k. The Dieudonné functor contravariantly associates to G an n dimensional k-vector space D(G). Let Frob : G → G be the morphism induced by the pth power map on the underlying rings and let Ver be the dual of Frob. Let φ = D(Frob) and ν = D(Ver), then it is a property of the functor D that φ ∘ ν = ν ∘ φ = 0. The functor D is a fully faithful functor. Example 13.5.1. Let k = F_{p}. If G is either μ μ μ_{p}, α_{p}, or Z/pZ then D(G) is a one-dimensional vector space over F. In the case of α_{p}, φ = ν = 0. For μ μ μ_{p}, φ = 0 and ν ≠ 1 and for Z/pZ, φ ≠ 1 and ν = 0. [[The latter two could be reversed!]]

Let G^{∨} = Hom(G,μ μ μ_{p}) denote the Cartier dual of the scheme G. Then

D(G^{∨}) = Hom_{k}(D(G), k)

(φ and ν are switched.) Example 13.5.2. Let A be an abelian variety over F_{ℓ} and let G = A[ℓ]. Then G is an F_{ℓ}-vector space scheme of order ℓ^{2g}. Thus D(G) is a 2g-dimensional F_{ℓ}-vector space and furthermore D(G) = H_{DR}^{1}(A/F_{ℓ}). The Hodge filtration on H_{DR}^{1} of the abelian variety A gives rise to a diagram

Hom(H^{0}(A^{∨}, Ω^{1}), F_{ℓ}) = T an(A^{∨})
||
0 → H^{0}(A, Ω^{1}) → D(G) → H_{DR}^{1}(A/F_{ℓ}) → H^{1}(A, O) → 0
||  ||
D(A_{/F_{ℓ}}[ℓ])  → D(A_{/F_{ℓ}}[Ver]) → 0

There is an exact sequence

0 → W_{F_{ℓ}} → J_{F_{ℓ}}[ℓ] ⟶^{m} J_{F_{ℓ}}[ℓ]

so because D is an exact functor the sequence

D(J_{F_{ℓ}}[ℓ]) ⟶^{m} D(J_{F_{ℓ}}[ℓ]) → D(W_{F_{ℓ}}) → 0

is exact. Following Fontaine we consider

D(W_{F_{ℓ}}[Ver]) = H^{1}(J, O)/mH^{1}(J, O).

13.6 The Proof: Part II

[[We all just returned from the Washington D.C. conference and will now resume the proof.]]

Let J_{0}(N) be the Jacobian of X_{0}(N). Let m ⊂ T be a maximal ideal and suppose m|ℓ. Assume that ℓ ≠ 2 and ℓ |̸N. The assumption that ℓ ≠ 2 is necessary for Raynaud’s theory and we assume that ℓ |̸N so that our group schemes will have good reduction. We attach to m a 2 dimensional semisimple representation ρ_{m} : Gal(\overline{Q}/Q) → V and only consider the case that ρ_{m} is irreducible.

The m-torsion of the Jacobian, W = J_{0}(\overline{Q})[m], is naturally a Galois module. We have shown that W ≠ 0. By (Boston-Lenstra-Ribet) W ≅ V × ⋯ × V (the number of fractions is not determined). We proved that W^{s.s.} ≅ V × ⋯ × V. Choose an inclusion V ↪ W and let Q = W/V be the cokernel.

Theorem 13.6.1. Q = 0 so dim_{T/m} W = 2

To prove the theorem we introduce the “machine” of finite flat group schemes over Z_{ℓ}. For example, W extends to a finite flat group scheme W_{Z_{ℓ}} which is defined to be the Zariski closure of W in J_{Z_{ℓ}}[ℓ]. Passing to group schemes yields an exact sequence

0 → V_{Z_{ℓ}} → W_{Z_{ℓ}} → Q_{Z_{ℓ}} → 0.

Reducing mod ℓ then yields an exact sequence of F_{ℓ}-group schemes

0 → V_{F_{ℓ}} → W_{F_{ℓ}} → Q_{F_{ℓ}} → 0.

The point is that Q = 0 iff Q_{Z_{ℓ}} = 0 iff Q_{F_{ℓ}} = 0.

Next we introduced the exact contravariant Dieudonné functor

D : ( Groups Schemes /F_{ℓ} ) ⟶ ( Linear Algebra ).

D sends a group scheme G to a T/m vector space equipped with two endomorphisms φ = Frob and ν = Ver. Applying D gives an exact sequence of T/m-vector spaces

0 → D(Q) → D(W) → D(V) → 0

where everything is now viewed over F_{ℓ}.

Lemma 13.6.2. D(W [Ver]) = (H^{0}(X_{0}(N)_{F_{ℓ}}, Ω^{1})[m])^{∗}

Proof. We have the diagram

D(J_{F_{ℓ}}[ℓ]) =  H_{DR}^{1}(J_{F_{ℓ}})
↓  ↓
D(J_{F_{ℓ}}[Ver]) =  H^{1}(J_{F_{ℓ}}, O_{J_{F_{ℓ}}})
↓  ↓
D(W [Ver]) = H^{1}(J_{F_{ℓ}}, O_{J})/mH^{1}(J_{F_{ℓ}}, O_{J})

Furthermore we have the identifications

H^{1}(J_{F_{ℓ}}, O_{J}) = Tan(J_{F_{ℓ}}^{∨}) = Cot(J_{F_{ℓ}}^{∨})^{∗}
= H^{0}(J^{∨}, Ω^{1})^{∗} = H^{0}(X_{0}(N), Ω^{1})^{∗}

For the last identification we must have J^{∨} = Alb(X_{0}(N)). Finally

D(W [Ver]) = H^{1}(J, O_{J})/mH^{1}(J, O_{J})
= (H^{0}(X_{0}(N)_{F_{ℓ}}, Ω^{1})[m])^{∗}.

Lemma 13.6.3. H^{0}(X_{0}(N)_{F_{ℓ}}, Ω^{1})[m] has T/m dimension ≤ 1.

Proof. Let S = H^{0}(X_{0}(N)_{F_{ℓ}}, Ω^{1})[m]. Then S ↪ F_{ℓ}[[q]]. We defined the Hecke operators T_{n} on S via the identification S ≅ H^{1}(J, O_{J}) so that they act on S ⊂ F_{ℓ}[[q]] in the standard way. Let T(S) be the subalgebra of End(S) generated by the images of the T_{n} in End(S). (T(S) is not a subring of T.) There is a perfect pairing

T(S) × S ⟶ F_{ℓ}
(T, f) ↦ a_{1}(f|T)

Thus dim_{F_{ℓ}} T(S) = dim_{F_{ℓ}} S and so dim_{T(S)} S ≤ 1. Since m acts trivially on S there is a surjection T/m ⟶ T(S). Thus

dim_{T/m} S ≤ dim_{T(S)} S ≤ 1

as desired.

An application of the above lemma shows that D(W [Ver]) has T/m dimension ≤ 1.

Lemma 13.6.4. D(W [Ver]) ≅ D(V [Ver])

Proof. Consider the following diagram.

0  0  0
↓  ↓  ↓
0 →  D(Q)  →  D(W)  →  D(V)  → 0
↓ Ver  ↓ Ver  ↓ Ver
0 →  D(Q)  →  D(W)  →  D(V)  → 0
↓  ↓  ↓
0 → D(Q)/ Ver D(Q) → D(W)/ Ver D(W) ⟶^{?∼?} D(V)/ Ver D(V) → 0
↓  ↓  ↓
0  0  0

D(V) has dimension 2 so since Ver ∘ Frob = Frob ∘ Ver = 0 and Ver, Frob are both nonzero they must each have rank 1 (in the sense of undergraduate linear algebra). Since D is exact, D(V [Ver]) = D(V)/ Ver D(V) and D(W [Ver]) = D(W)/ Ver D(W). By the previous lemma dim D(W)/ Ver D(W) = 1. Thus D(W)/ Ver D(W) → D(V)/ Ver D(V) is a map of 1 dimensional vector spaces so to show that it is an isomorphism we just need to show that it is surjective. This follows from the commutativity of the square

D(W)  →  D(V)  → 0
↓  ↓
D(W)/ Ver D(W) → D(V)/ Ver D(V)

↓ 0

Suppose for the moment that we admit (Boston-Lenstra-Ribet). Then

W = V × ⋯ × V = V^{⊕t}

so

D(W [Ver]) ≅ D(V [Ver])^{⊕t}

and hence t = 1.

Alternatively we can avoid the use of Boston-Lenstra-Ribet. Suppose Q ≠ 0. Then there is an injection V ↪ Q. [[I can’t see this without using B-L-R. It isn’t obvious to me from 0 → V → W → Q → 0.]] Thus over F_{ℓ}, V [Ver] ↪ Q[Ver]. Since V [Ver] ≠ 0 this implies Q[Ver] ≠ 0. Thus D(Q)/ Ver D(Q) = D(Q[Ver]) ≠ 0. But the bottom row of the above diagram implies D(Q)/ Ver D(Q) = 0 so Q = 0.

13.7 Key Result of Boston-Lenstra-Ribet

Let G be a group (i.e., G = Gal(\overline{Q}/Q)), let k be a field (i.e., k = T/m), and let V be a two dimensional k-representation of G given by

ρ : k[G] → End_{k}(V) = M_{2}(k).

The key hypothesis is that V is absolutely irreducible, i.e., that ρ is surjective. For each g ∈ G consider

p_{g} = g^{2} − g Tr ρ(g) + det ρ(g) ∈ k[G].

By the Cayley-Hamilton theorem ρ(p_{g}) = 0. Let J be the two-sided ideal of k[G] generated by all p_{g} such that g ∈ G. Since J ⊂ ker ρ , ρ induces a map

σ : k[G]/J → End_{k}(V).

Theorem 13.7.1 (Boston-Lenstra-Ribet). If σ is surjective then σ is an isomorphism.

In particular if V is absolutely irreducible then σ is surjective. The theorem can be false when dim V > 2.

Suppose W is a second representation of G given by μ : k[G] → End(W) and that μ(J) = {0} ⊂ End(W). Then W is a module over k[G]/J = End(V). But End(V) is a semisimple ring so any End(V) module is a direct sum of simple End(V) modules. The only simple End(V) module is V. Thus W ≅ V^{⊕^{n}} for some n.

80 CHAPTER 13. THE GORENSTEIN PROPERTY

Local Properties of ρ_{λ}

Let f be a newform of weight 2 on Γ_{0}(N). To f we have associated an abelian variety A = A_{f} furnished with an action of E = Q(…, a_{n}(f), …). Let O_{E} be the ring of integers of E and λ ⊂ O_{E} a prime. Then we obtain a λ-adic representation

ρ_{λ} : Gal(\overline{Q}/Q) → GL_{2}(E_{λ})

on the Tate module Tate_{ℓ} A = \underleftarrow{lim} A[λ^{i}]. We will study the local properties of ρ_{λ} at various primes p.

14.1 Definitions

To view ρ_{λ} locally at p we restrict to the decomposition group D_{p} = Gal(\overline{Q}/Q) at p. Recall the definition of D_{p}. Let K be a finite extension of Q and let w be a prime of K lying over p. Then the decomposition group at w is defined to be

D_{w} = {σ ∈ Gal(K/Q) : σw = w}.

Proposition 14.1.1. D_{w} ≅ Gal(K_{w}/Q_{p})

Proof. Define a map Gal(K_{w}/Q_{p}) → D_{w} by σ ↦ σ_{|K}. Since σ_{|K} fixes Q this restriction is an element of Gal(K/Q). Since wO_{K_{w}} is the unique maximal ideal of O_{K_{w}} and σ induces an automorphism of O_{K_{w}}, it follows that σ(wO_{K_{w}}) = wO_{K_{w}}. Thus σ_{|K}(w) = w so σ_{|K} ∈ D_{w}. The map σ ↦ σ_{|K} is bijective because K is dense in K_{w}.

Let

D_{p} = \underleftarrow{lim}_{w|p}D_{w} = \underleftarrow{lim}_{w|p} Gal(K_{w}/Q_{p}) = Gal(\overline{Q}_{p}/Q_{p}).

For each w|p let the inertia group I_{w} be the kernel of the map from D_{w} into Gal(O_{K}/w, Z/pZ). Let I_{p} = \underleftarrow{lim}_{w|p}I_{w}.

14.2 Local Properties when p |̸N

Next we study local properties of ρ_{λ} at primes p |̸N. Thus suppose p |̸N and p ≠ ℓ = char(O_{E}/λ). Let D_{p} = Gal(\overline{Q_{p}}/Q_{p}). Then

1) ρ_{λ}|D_{p} is unramified (i.e., ρ_{λ}(I_{p}) = {1}) thus ρ_{λ}|D_{p} factors through D_{p}/I_{p} so ρ_{λ}(Frob_{p}) is defined.

82 CHAPTER 14. LOCAL PROPERTIES OF ρ_{λ}

2) Tr(ρ_{λ}(Frob_{p})) = a_{p}(f)

2+) We can describe ρ_{λ}|D_{p} up to isomorphism. It is the unique semisimple representation satisfying 1) and 2).

14.3 Weil-Deligne Groups

Notice that everything is sort of independent of λ. Using Weil-Deligne groups we can summarize all of these λ-adic representations in terms of data which makes λ disappear.

We have an exact sequence

1 → I_{p} → Gal(\overline{Q}_{p}/Q_{p}) → Gal(\overline{F}_{p}/F_{p}) → 0.

Since Gal(\overline{F}_{p}/F_{p}) = Ẑ there is an injection Z ↪ Gal(\overline{F}_{p}/F_{p}). Define the Weil group W (\overline{Q}_{p}/Q_{p}) ⊂ Gal(\overline{Q}_{p}/Q_{p}) to be the set of elements of Gal(\overline{Q}_{p}/Q_{p}) mapping to Z ⊂ Gal(\overline{F}_{p}/F_{p}). W fits into the exact sequence

1 → I_{p} → W (\overline{Q}_{p}/Q_{p}) → Z → 1.

There is a standard way in which the newform f gives rise to a representation of W. Factor the polynomial x^{2} − a_{p}(f)x + p as a product (x − r)(x − r′) with r, r′ ∈ C. Define maps α, β by

α : Z → C^{∗} : 1 ↦ r
β : Z → C^{∗} : 1 ↦ r′

Combining α and β and the map W (\overline{Q}_{p}/Q_{p}) → Z yields a map

α ⊕ β : W (\overline{Q}_{p}/Q_{p}) → GL_{2}(C)

σ ↦ (\begin{matrix}α(σ) & 0 \\ 0 & β(σ) \end{matrix}).

Moreover α ⊕ β gives rise via some construction to all the λ-adic representations ρ_{λ}.

14.4 Local Properties when p|N

Suppose p|N but p ≠ ℓ. Carayol was able to generalize 1) in his thesis which builds upon the work of Langlands and Deligne in the direction of Deligne-Rapaport and Katz-Mazur. The idea is that the abelian variety A has a conductor which is a positive integer divisible by those primes of bad reduction. The conductor of A satisfies cond(A) = M^{g} where g = dim A and M is the reduced conductor of A.

Theorem 14.4.1 (Carayol). M = N.

How can we generalize 2) or 2+)? For each p dividing N there is a representation σ_{p} of WD(\overline{Q}_{p}/Q_{p}) over C such that σ_{p} gives rise to ρ_{λ}|D_{p} for all λ |̸p. Here WD(\overline{Q}_{p}/Q_{p}) is the Weil-Deligne group which is Deligne’s generalization of the Weil group. WD(\overline{Q}_{p}/Q_{p}) is an extension of W (\overline{Q}_{p}/Q_{p}). What is σ_{p} supposed to be? The point is that σ_{p} is determined by f. Thus every f gives rise to a family (σ_{p})_{p prime}. To really think about σ_{p} we must think about modular forms in an adelic context instead of viewing them as holomorphic functions on the complex upper halfplane.

If p^{2}|N and f = ∑ a_{n}q^{n} is a newform of level p^{2} then it is a classical fact that a_{p} = 0. But the study of ρ_{λ}|D_{p} is rich and “corresponds to a rather innocuous looking crystal”.

14.5 Definition of the Reduced Conductor

We now define the reduced conductor. Let λ be a prime of E and p a prime of Q such that λ |̸p. We want to define some integer e(p) so that p^{e(p)} is the p-part of the reduced conductor. We will not define e(p) but what we will do is define an integer e(p, λ) which is the p part of the conductor. e(p, λ) is independent of λ but this will not be proved here.

Consider

ρ_{λ}|D_{p} : Gal(\overline{Q}_{p}/Q_{p}) → Aut_{E_{λ}} V.

Let V^{I} ⊂ V be the inertia invariants of V, i.e.,

V^{I} = {v ∈ V : ρ_{λ}(σ)(v) = v for all σ ∈ I}.

Since ρ_{λ} is unramified at p iff ρ_{λ}(I_{p}) = {1} we comment that ρ_{λ} is unramified at p iff V^{I} = V. Let

e(p, λ) = dim V/V^{I} + δ(p, λ)

where δ(p, λ) is the Swan conductor. By working with finite representations we define δ(p, λ) as follows. Choose a D_{p}-stable lattice L in V by first choosing an arbitrary one then taking the sum of its finitely many conjugates. Let \overline{V} = L/λL which is a 2 dimensional vector space over k = O_{E}/λ. Let G be the quotient of Gal(\overline{Q}_{p}/Q_{p}) by the kernel of the map Gal(\overline{Q}_{p}/Q_{p}) → Aut_{K} \overline{V}. Thus G = Gal(K/Q_{p}) for some finite extension K/Q_{p}. The extension K is finite over Q_{p} since G ⊂ Aut_{k} \overline{V} and Aut_{k} \overline{V} is a 2 × 2 matrix ring over a finite field. The corresponding diagram is

Gal(\overline{Q}_{p}/Q_{p}) →  Aut_{k} \overline{V}
↘  ∪
G = Gal(K/Q_{p})

Consider in G the sequence of “higher ramification groups”

G = G_{−1} ⊃ G_{0} ⊃ G_{1} ⊃ · · · .

Here G_{0} is the inertia group of K/Q_{p} and G_{1} is the p-sylow subgroup of G_{0} [[the usages of “the” in this sentence makes me nervous.]] Let G_{i} = {g ∈ G_{0} : ord(gπ − π) ≥ i + 1} where π is some kind of uniformizing parameter [[I missed this – what is π?]] Let

δ(p, λ) = ∑_{i=1}^{∞} \frac{1}{(G_{0} : G_{i})} dim(\overline{V} /\overline{V}^{G_{i}}).

It is a theorem that δ(p, λ) is an integer and does not depend on λ.

If to start with we only had \overline{V} and not V we could define

cond(\overline{V}) = ∑_{i=0}^{∞} \frac{1}{(G_{0} : G_{i})} dim(\overline{V} /\overline{V}^{G_{i}}).

Then

e(p, λ) = cond(\overline{V}) + (dim \overline{V}^{I} − dim V^{I}).

A reference for much of this material is Serre’s Local Factors of L-functions of λ-adic Representations.

14.6 Introduction

Our goal is to study local properties of the λ-adic representations ρ_{λ} arising from a weight 2 newform of level N on Γ_{0}(N). There is a theorem of Carayol which states roughly that if p ≠ ℓ then ρ_{λ}|D_{p} is predictable from “the component at p of f”. To understand this theorem we must understand what is meant by “the component at p of f”. If p^{2} |̸N this component is easy to determine but if p^{2}|N it is harder. One reason is that when p^{2}|N then a_{p}(f) = 0. [[this should be easy to see so there should be an argument here.]] If a_{p}(f) = 0 and ρ_{λ} : Gal(\overline{Q}/Q) → Aut(V_{λ}) then V_{λ}^{I_{p}} = 0. This means that there is no ramification going on at p. See Casselman, “On representations of GL(2) and the arithmetic of modular curves”, Antwerp II.

14.7 Adelic Representations Associated to Modular Forms

Let R be a subring of A^{2} = R^{2} × (Ẑ ⊗_{Z} Q)^{2}, suppose that R ≅ Q^{2} and that R ⊗_{Q} A ≅ A^{2}. Let L = R ∩ (R^{2} × Ẑ^{2}). Then the natural map L ⊗ Ẑ → Z^{2} is an isomorphism. [[Is the isomorphism implied by the definition of L or is it part of the requirement for L to actually form an adelic lattice?]] L is called an adelic lattice.

The space of modular forms S_{2}(Γ_{0}(N)) is isomorphic to a certain space of functions on G(A) = GL(2, A). See Borel-Jacquet [[Corvallis?]] or Diamond-Taylor, Inventiones Mathematica, 115 (1994) [[what is title?]] We will describe this isomorphism.

Write A = R × A_{f} = A_{∞} × A^{∞} where A_{f} = ∏ Q_{p} (restricted product) is the ring of finite adeles. A_{∞} denotes the adeles with respect to the place ∞ so A_{∞} = R, and A^{∞} denotes the adeles away from the place ∞ so A^{∞} = A_{f}.

S_{2}(Γ_{0}(N)) is isomorphic to the set of functions φ : G(A) → C which satisfy

0) φ is left invariant by G(Q), i.e., φ(x) = φ(gx) for all g ∈ G(Q) and all x ∈ G(A),

1) φ(xu^{∞}) = φ(x) for all x ∈ G(A) and all u^{∞} ∈ U^{∞},

2) φ(xu_{∞}) = φ(x)j(u_{∞}, i)^{−k} det(u_{∞}) for all x ∈ G(A) and u_{∞} ∈ U_{∞},

3) holomorphy, cuspidal, and growth conditions.

U^{∞} is the adelic version of Γ_{0}(N). Thus U^{∞} is the compact open subgroup

U^{∞} = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(Ẑ) : c ≡ 0 (mod N)\} ⊂ G(A_{f}).

(For Γ_{1}(N) the condition is that c ≡ 0 (mod N) and d ≡ 1 (mod N) but a is not restricted.)

Next we describe U_{∞} ⊂ GL_{2}(R). GL_{2}(R) operates on H^{±} = C − R by z ↦ \frac{az+b}{cz+d}. Let U_{∞} be the stabilizer of i.

The third condition involves the automorphy factor j defined by

j((\begin{matrix}a & b \\ c & d \end{matrix}), z) = cz + d.

To explain the holomorphy condition 3) we define, for any g ∈ G(A_{f}) a map

α_{g,φ} : H^{±} → C
hi ↦ φ(gh)j(h, i)^{k}(det(h))^{−1}.

Here h ∈ G(A_{f}) so hi ∈ H^{±}. There may be several different h ∈ G(A_{f}) which give the same hi ∈ H^{±} so it must be checked that α_{g,φ} is well-defined. Suppose hi = h′i, then h^{−1}h′i = i. Thus h^{−1}h′ ∈ U_{∞}, so by 2),

φ(gh^{−1}h′) = φ(g)j(h^{−1}h′, i)^{−k} det(h^{−1}h′).

Thus

φ(gh^{−1}h′) det(h′)^{−1} = φ(g)j(h^{−1}h′, i)^{−k} det(h)^{−1}.

Substituting gh for g yields

φ(gh′)(det(h′))^{−1} = φ(gh)j(h^{−1}h′, i)^{−k}(det(h))^{−1}.

[[Why does not the automorphy factor work out right?? :(]] The holomorphy condition is that the family of maps α_{g,φ} are all holomorphic.

The cuspidal condition is that for all g ∈ G(A), the integral

∫_{u∈A/Q} φ(( \begin{matrix}1 & u \\ 0 & 1 \end{matrix} ), g)du = 0

vanishes. Since A/Q is compact it has a Haar measure defined modulo k^{∗} which induces du. Although the integral is not well-defined the vanishing or non-vanishing of the integral is.

We can now describe the isomorphism between S_{k}(Γ_{0}(N)) and the space of such functions φ on G(A).

{ space of φ satisfying 0-3} → S_{k}(Γ_{0}(N))
φ ↦ f

where f is the restriction to H = H^{+} of the function

hi ↦ φ(h)j(h, i)^{k}(det h)^{−1}.

Now we can associate to a newform f a representation of G(A). We can weaken condition 1) to get

1-) φ(xu^{∞}) = φ(x) for all x ∈ G(A) and all u^{∞} ∈ U^{∞} where U^{∞} is some compact open subgroup of G(A_{f}).

[[can the U^{∞} vary for each x ∈ G(A) or are they fixed throughout?]] Let S be the space of all functions satisfying all conditions except 1-) replaces 1). This space has a left action of G(A):

(g ∗ φ)(x) = φ(xg)

If f is a newform corresponding to some φ via the above isomorphism then via this action f gives rise to an infinite dimensional representation π of G(A). In fact we obtain, for each prime p, a representation π_{p} of GL(2, Q_{p}). The representation space is ∑_{g∈GL_{2}(Q_{p})} C · g ∗ φ. Our immediate goal is to understand π_{p} for as many p as possible. [[spherical representations have something to do with this. are the π_{p} spherical reps?]]

We are studying local properties of the λ-adic representations ρ_{λ} associated to a newform f of weight 2, level N and character ε : (Z/NZ)^{∗} → C^{∗} (with cond(ε)|N). Let ℓ ∈ Z be the prime over which λ lies. We look at ρ_{λ} locally at p, p ≠ ℓ.

As we saw last time f gives rise to an irreducible representation of GL(2, A). An irreducible representation of GL(2, A) gives rise to a family of representations (π_{v}) where v is a prime or ∞ and π_{v} is an irreducible representation of GL(2, Q_{v}). This is because Q_{v} ⊂ A so GL(2, Q_{v}) ⊂ GL(2, A).

Carayol proved that if p ≠ ℓ then ρ_{λ}|_{D_{p}} depends, up to isomorphism, only on π_{p}. The most interesting case of the proof of this theorem is the case when p^{2} |̸N. Everything necessary to obtain a proof in this case was available in the work of Langlands [1972].

To get an idea of what is going on we will first consider the case when p |̸N. The characteristic polynomial of Frobenious (at least psychologically) under the representation ρ_{λ}|_{D_{p}} is

x^{2} − a_{p}x + pε(p) = (x − r)(x − s).

Because of Weil’s proof of the Riemann hypothesis for abelian varieties [over finite fields?] one knows that |r| = |s| = \sqrt{p}. Since ρ_{λ} arises from the action of Galois on an abelian variety which has good reduction at p (since p |̸N) it follows that ρ_{λ}|_{D_{p}} is unramified. [[Is this in Serre-Tate, 1968?]] We also know that ρ_{λ}(Frob_{p}) has characteristic polynomial

x^{2} − a_{p}x + pε(p) ∈ E[x].

In this situation one also knows that ρ_{λ}(Frob_{p}) is semisimple [[proof: Ribet nodded at Coleman who smiled at nodded back.]] Thus

ρ_{λ} ∼ (\begin{matrix}r & 0 \\ 0 & s \end{matrix}).

In this situation what is the representation π_{p} of GL(2, Q_{p})? There are two characters

α, β : Q_{p}^{∗} → C^{∗}

(called “Grössencharacters of type (a,0)”) such that

1. α and β are unramified in the sense that

α|_{Z_{p}^{∗}} = β|_{Z_{p}^{∗}} = 1.

This is a reasonable condition since under some sort of local class field theory Q_{p}^{∗} embeds as a dense subgroup of Gal(\overline{Q_{p}}/Q_{p}) and under this embedding the inertia subgroup I ⊂ Gal(\overline{Q_{p}}/Q_{p}) corresponds to Z_{p}^{∗}. [[This could be wrong. Also, is I ∩Q_{p}^{∗} = Z_{p}^{∗}?]]

2. α(p^{−1}) = r, β(p^{−1}) = s.

In the 1950’s Weil found a way under which α and β correspond to continuous characters α_{λ}, β_{λ} on Gal(\overline{Q_{p}}/Q_{p}) with values in \overline{E}_{λ}^{∗} such that

1. α_{λ} and β_{λ} are unramified.

2. α_{λ}(Frob_{p}) = r and β_{λ}(Frob_{p}) = s.

One has that ρ_{λ} = α_{λ} ⊕ β_{λ}. See Serre-Tate [29]. [[Why see this?]] Define a character Θ on the Borel subgroup

B = (\begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix}) ⊂ GL(2, Q_{p})

by

Θ(\begin{matrix}x & y \\ 0 & z \end{matrix}) = α(x)β(z) ∈ C^{∗}.

14.8. MORE LOCAL PROPERTIES OF THE ρ_{λ}. 87

Then

π_{p} = Ind_{B}^{GL(2,Q_{p})} := C[GL(2, Q_{p})] ⊗_{C[B]} C.

We call this induced representation π_{p} the unramified principal series representation associated to α and β and write π_{p} = PS(α, β). People say π_{p} is spherical in the sense that there is a vector in the representation space invariant under the maximal compact subgroup of GL(2, Q_{p}). [[Ribet was slightly unsure about the correct definition of spherical.]] [[For some mysterious reason]] since π_{p} = PS(α, β) it follows that α, β and hence ρ_{λ}|D_{p} is completely determined by π_{p}. [[This is the point and i don’t see this.]]

Next we consider the more difficult case when p||N (p divides N exactly). There are two cases to consider

a ε is ramified at p (p| cond(ε))

b ε is unramified at p (p |̸ cond(ε)).

14.8 More Local Properties of the ρ_{λ}.

Let f be a newform of level N. Then f corresponds to a representation π = ⊗π_{v}. If λ|ℓ and ℓ ≠ p then ρ_{λ}|D_{p} corresponds to π_{p} under the Langlands correspondence. The details of this correspondence were figured out by Philip Kutzko, but Carayol completed it in the exceptional case (to be defined later). There are 3 cases to consider.

a) p |̸N

b) p||N

c) p^{2}|N

We considered the first two cases last time. The third case is different because the same sort of analysis as we applied to the first two cases no longer works in the sense that we no longer know what π_{p} looks like.

14.8.1 Possibilities for π_{p}

Case 1 (principal series) In this case, π_{p} = PS(α, β). Here α, β : Q_{p}^{∗} → C^{∗} are unramified characters. Q_{p}^{∗} corresponds to a dense subgroup of the abelian Galois group of Q_{p} under the correspondence elucidated in Serre’s “Local Classfield Theory” (in Cassels and Frohlich).

Q_{p}^{∗} → Gal(\overline{Q_{p}}/Q_{p})^{ab}
↓  ↓
Z → Ẑ = Gal(\overline{F_{p}}/F_{p})

Under this correspondence α and β correspond to Galois representations α_{λ} and β_{λ} and ρ_{λ}|D_{p} ∼ α_{λ} ⊕ β_{λ}.

Case 2 (special) In this case, π_{p} is the special automorphic representation corresponding to the Galois representation κ ⊗ st where st is the Steinberg representation which arises somehow from a split-multiplicative reduction elliptic curve and κ is a Dirichlet character. In this case

ρ_{λ}|D_{p} = κ ⊗ (\begin{matrix}χ_{ℓ} & ∗ \\ 0 & 1 \end{matrix}).

Case 3 (cuspidal) Case 3 occurs when π_{p} does not fall into either of the previous cases. Such a π_{p} is called cuspidal or super-cuspidal. Some of these π_{p} come from the following recipe. Fix an algebraic closure \overline{Q_{p}} of Q_{p}. Let K be a quadratic extension of Q_{p}. Let ψ : K^{∗} → C^{∗} be a Grössencharacter. Then ψ gives rise to a character

ψ_{λ} : Gal(\overline{Q_{p}}/K) → \overline{E}_{λ}^{∗}

which induces ρ_{λ}|D_{p}. That is,

ρ_{λ}|D_{p} = Ind_{Gal(\overline{Q_{p}}/K)}^{Gal(\overline{Q_{p}}/Q_{p})} ψ_{λ} : Gal(\overline{Q_{p}}/Q_{p}) → GL(2, \overline{E}_{λ}).

This representation is irreducible iff ψ is not invariant under the canonical conjugation of K/Q_{p}. The pair ψ, K gives rise (via the construction of Jacquet-Langlands) to a representation of π_{ψ,K} of GL(2, Q_{p}). Those representations which do not come from this recipe and which do not fall into case 1 or case 2 above are called extraordinary. They can only occur when p ≤ 2.

When can the various cases occur?

Case 1) occurs, e.g, if p |̸N, and also if p||N and ε (the character of f) is ramified at p. Case 2) occurs if p||N and ε is unramified at p.

14.8.2 The case ℓ = p

We consider ρ_{λ}|D_{p} where λ|p and p = ℓ. Write f = ∑ a_{n}q^{n}. The case when λ |̸a_{p} is called the ordinary case. This case is very similar to the case for an ordinary elliptic curve. In other words,

ρ_{λ}|D_{p} = (\begin{matrix}α & ∗ \\ 0 & β \end{matrix}).

[[Ribet mentioned that there is a paper on this by Mazur and Wiles in the American Journal of Mathematics. Look the reference up.]] An important point is that β is unramified so it makes sense to consider β(Frob_{p}) = a_{p} ∈ E_{λ}^{∗}. Since αβ is the determinent, αβ = χ_{p}^{k−1}ε = χ_{p}ε (after setting k = 2 to fix ideas), this gives some description of what is going on. The obvious question to ask is whether or not * is nontrivial. That is, is ρ_{λ}|D_{p} semisimple or not?

When f has weight 2, then f gives rise to an abelian variety A = A_{f}. Then ρ_{λ} is defined by looking at the action of Galois on the λ-adic division points on A. If none of the λ lying over p divide a_{p} then A is ordinary at p. A stronger statement is that the p-divisible group A[p^{∞}] has good ordinary reduction.

One simple case is when f has CM. By this we mean that there is a character κ ≠ 1 of order 2 such that a_{n} = κ(n)a_{n} for all n prime to cond(κ). [[This is not a typo, I do not mean \overline{a_{n}} = κ(n)a_{n}. Coleman said that a_{n} = κ(n)a_{n} is just a funny way to say that half of the a_{n} are 0.]] It is easy to prove that f has CM iff the ρ_{λ} become abelian on some open subgroup of Gal(\overline{Q}/Q) of finite index. Ribet explains this in his article in [21]. If f has CM then since the representation ρ_{λ} is almost abelian one can show that * is trivial. Ribet said he does not know whether the converse is true. Note that f has CM iff A_{f}/\overline{Q} has CM. If all λ|p are ordinary (i.e., they do not divide a_{p}) and if * is trivial for every ρ_{λ}|D_{p} it is easy to show using [26] that A has CM.

Next we will say something more about representations which appear to be ordinary. Consider the situation in which f has weight 2 and p exactly divides the level N of f. Suppose furthermore that the character ε of f is unramified at p. Then π_{p} is a special representation. The λ-adic representations for ℓ ≠ p are (up to characters of finite order) like representations attached to some Tate curve. The situation is similar when ℓ = p since

ρ_{λ}|D_{p} = (\begin{matrix}α & ∗ \\ 0 & β \end{matrix}).

As in the case of a Tate curve α/β = χ_{p}. Up to a quadratic character we know the situation since αβ = χ_{p}ε. Also β is still unramified and β(Frob_{p}) = a_{p} is a unit. We know that a_{p}^{2} = ε(p) is a root of unity. When k = 2 the case of a spherical representation mimics what happens for ordinary reduction. The upper right hand entry * is never trivial in this case [[because of something to do with extensions and Kummer theory]].

Ribet said he knows nothing about the situation when k > 2. If p||N and ε is unramified at p then π_{p} is special so ρ_{λ}, ℓ ≠ p are again of the form ( \begin{matrix}α & ∗ \\ 0 & β \end{matrix} ). So we still know everything up to a quadratic character. But if ℓ = p some characters are Hodge-Tate [[what does that mean?]] so by a theorem of Faltings they can not be bizarre powers of the cyclotomic character. The multiplicative case like the ordinary case is very special to the case k = 2. Wiles uses this heavily in his proof. If k is arbitrary then a_{p}^{2} = ε(p)p^{\frac{k}{2} −1} so a_{p} is not a unit for k > 2 so there is no invariant line. So the representation is probably irreducible in the case k > 2, interestingly enough. [[Echos of “yeah”, “strange” and “very strange” are heard throughout the room.]]

14.8.3 Tate Curves

K. Buzzard told me something about Tate curves. Suppose E/Q_{p} is an elliptic curve with multiplicative reduction at p and that j ∈ Q_{p} is the j-invariant of E. Using a formula which can be found in Chapter V of [34] one finds q = q(j) with |q| < 1. The Tate curve is E(\overline{Q}_{p}) = \overline{Q}_{p}/q^{Z}. The p torsion on the Tate curve is \{ζ_{p}^{a}(q^{1/p})^{b} : 0 ≤ a, b ≤ p − 1\}. Galois acts by ζ_{p} ↦ ζ_{p}^{a} and q^{1/p} ↦ ζ_{p}^{a}q^{1/p}. Thus the associated Galois representation is ( \begin{matrix}χ_{p} & ∗ \\ 0 & 1 \end{matrix} ).

90 CHAPTER 14. LOCAL PROPERTIES OF ρ_{λ}

The Weight and Serre’s Conjectures

15.1 Introduction

The next topic is a close study of the reduction of these representations mod ℓ. Starting with some eigenform f we make a representation ρ_{λ} then reducing mod λ gives a representation \overline{ρ}_{λ}. The motivating question is

Question 15.1.1. Which possible eigenforms of various weight and level might give rise to some \overline{ρ}_{λ}.

We might as well assume \overline{ρ}_{λ} is irreducible although interesting work has been done in the reducible case. The main trends in the subject are “lowering” and “raising”. The result after much works is that certain of Serre’s conjectures are valid in the sense that if ρ_{λ} arises from a modular form then the level and weight of the modular form is what Serre expected. If ρ_{λ} “looks like” it comes from a certain space then it does.

The question can also be viewed via the opposite optic. What is the most bizarre kind of f that could give rise to ρ_{λ}? Using certain properties of cond(ρ_{λ}) one can obtain some sort of control on the possible weights and levels. This is used in the work of Wiles.

15.2 Review of the λ-adic case

A newform f of level N gives rise to a system (ρ_{λ,f}) of λ-adic representations. We try to describe what goes on with ρ_{λ}|D_{p}. We know everything, in principal at least, when p ≠ ℓ and λ|ℓ because of Carayol’s theorem. When p = ℓ the situation is more complicated. The simplest cases are when

1. p |̸N and (a_{p}, N) = 1, this is the ordinary good reduction case.

2. p||N, the weight of f is 2, and the character ε associated to f is unramified at p.

These two cases are parallel. But if in case 2 the weight is strictly greater than 2 then nothing is known.

15.3 Serre’s conjecture 0

Next we talk about the situation for mod ℓ representations. The basic reference is Serre’s 1987 article [27].

Conjecture 15.3.1 (Serre’s Conjecture 0). Suppose

ρ : Gal(\overline{Q}/Q) → GL(2, F)

is a modular representation with F a finite field. Assume ρ is irreducible and odd, i.e., that det(ρ)(c) = −1 where c is complex conjugation. Then Serre conjectures that ρ is modular.

Formally this means that ρ is isomorphic to a reduction of some ρ_{f,λ} where f is a newform on Γ_{1}(N) for some N and λ is a prime of O = O_{E_{f}}. Fix an algebraic closure \overline{F} of F. Then there is an embedding of O/λ into \overline{F} and a natural inclusion of F into \overline{F}.

\begin{aligned}O/λ & ↪ & \overline{F} \\ & & ∪ \\ & & F \end{aligned}

To say that ρ is isomorphic to the reduction of ρ_{f,λ} we mean that \overline{ρ}_{λ,f} and ρ are isomorphic over \overline{F}.

15.3.1 Problems

One problem is that ρ lives in a slightly smaller field that O/λ would suggest.

Example 15.3.2. Let f be the unique newform [?] of level 23 weight 2 and trivial character. The coefficients of f live in O = O_{Q(\sqrt{5})} = Z[\frac{1+\sqrt{5}}{2}]. For example, a_{2} satisfies x^{2} − x − 1 = 0. Take λ = 2. Then O/λ ≅ F_{4}. Thus \overline{ρ}_{λ,f} lives over F_{4}. If we just consider those a_{p} with p ≠ 2, 23 one can show that these elements all actually live in Z[\sqrt{5}] which maps mod λ to F_{2} ⊂ F_{4}. Because of this it can be shown that \overline{ρ}_{λ,f} has a model ρ over F_{2}. This is because the traces and dets of Frobenius at primes where Frobenius is defined (i.e., p ≠ 2, 23) take values in F_{2}. Geometrically, GL_{2}(F_{2}) ≅ S_{3} so the representation ρ we get is one in which Galois acts as S_{3} on 3 points of X_{0}(23). To see that ρ is modular we might need to pass to an algebraic closure. Starting with ρ : Gal(\overline{Q}/Q) → GL_{2}(F_{2}) we want to say ρ ≅ \overline{ρ}_{λ,f}. But this isomorphism only takes place over F_{4}.

At this point K. Buzzard says, “Maybe there is some better modular form so that the associated representation actually takes values in F_{2} and is isomorphic to ρ. That would be a stronger conjecture.” Ribet responds that he has never thought about that question. Next Caskel asks if the determinent of any ρ_{f,λ} is a cyclotomic character. The answer is no unless f is a modular form with respect to Γ_{0}(N). Here f is a modular form with respect to Γ_{1}(N).

15.4 Serre’s conjecture 1

The next conjecture in Serre’s paper [27] is now a theorem when ℓ is odd.

Theorem 15.4.1. Suppose ℓ is odd. If the mod ℓ representation ρ is irreducible and modular then ρ arises from a newform f of some specific weight k(ρ) and level N(ρ).

Here N(ρ) is a sort of conductor of ρ defined as before. It is called the “Artin conductor”. It is a product ∏_{p≠ℓ} p^{e(p)} where e(p) is a sum

∑_{i=0}^{∞} \frac{1}{(G_{0} : G_{i})} dim(V/V^{G_{i}}).

Here V is the representation space of ρ. The factor \frac{1}{(G_{0}:G_{i})} depends only on G_{0} = ρ|I_{p}. [[Should I really say G_{0} = ρ(I_{p}) the image of I_{p} sitting inside Aut(V)?]] There is, in particular, a term corresponding to i = 0 which is dim(V/V^{I_{p}}). Thus

e(p) = dim(V/V^{I_{p}}) + δ(p).

Since dim V ≤ 2 it follows that dim(V/V^{I_{p}}) ≤ 2. This dimension is 0 if ρ is unramified at p. The term δ(p) is called the Swan conductor. It is 0 if ρ is tamely ramified at p. There is a relationship between this conductor and something involving f. Suppose ρ ≅ \overline{ρ}_{λ,f} where f is a newform of level N. It turns out that

ord_{p}(N) = dim(V_{λ}/V_{λ}^{I_{p}}) + δ(p) = e(p) + error term

where the error term is dim(V^{I_{p}}) − dim(V_{λ}^{I_{p}}). [[Cryptic statement: The point is that when reducing mod λ more things can become invariant.]] The thing to keep in mind is that if f gives rise to ρ then the power of p in the level of f is constrained since it is given by certain formulas only depending on e(p) and an error term which has magnitude at most 2. The Artin conductor of ρ is some minimal prime to p level which can give rise to ρ. Serre conjectured that there is such an f which gives rise to ρ.

The weight k(ρ) depends only on ρ|I_{ℓ}. Thus k(ρ) is some analog for e(ℓ). It is kind of hard to know where to begin to explain everything. Why is Serre’s conjecture actually a theorem?

15.4.1 Key background points

One idea is that every modular representation ρ must come from an f of level prime to ℓ. Imagine f is a modular form of level ℓ. Consider the corresponding mod ℓ representation. It is unramified outside ℓ so N(ℓ) = 1. The conjecture then implies this mod ℓ representation comes from a level 1 modular form (i.e., a modular form on SL_{2}(Z)) of higher weight. This is a classical result.

There is a relationship between mod ℓ forms on SL_{2}(Z) of weight ℓ + 1 and mod ℓ forms on Γ_{0}(ℓ) of weight 2. The dimensions of these spaces are the same over F_{ℓ} or over C. There is a map from one space to the other. If \overline{f} is a modular form of weight 2 on Γ_{0}(ℓ) multiplication by a certain form gives a form of weight ℓ + 1 on Γ_{0}(ℓ). Then averaging (or taking the trace) gives a modular form SL_{2}(Z). The whole process is, in some sense, just the identity map. It was discovered through an appropriate calculation of Serre, 1972, Antwerp. That the formula works was proved (via explicit computations) in the Berkeley Ph.D. thesis of C. Queen [20]. Katz thinks it is easy to prove the formula from the right magical moduli point of view. [[Ribet says he has never seen this.]] Ribet gave a concrete proof for ℓ > 2 in [22]. It would be nice if someone would construct a clear proof for the case ℓ = 2. [[I am confused about what Ribet is talking about. What formula?]]

We now allude to the proof of the theorem (Serre’s conjecture 1) when ℓ > 2. Start with a representation ρ. It comes from some (possibly terrible) modular from f.

Step 1. Make the level prime to ℓ via some concrete argument.

Step 2. Try to compare N(ρ) and N(f). These are two prime to ℓ numbers. What happens if p|\frac{N(f)}{N(ρ)}? Carayol separated things into appropriate cases. In each case replace f by a better form so the ratio has a smaller power of p in it. This is “level lowering”. Eventually we get to the case N(ρ) = N(f). Using a paper of Edixhoven [6] one shows that the weight k of f can be made equal to k(ρ).

What is k(ρ)? We give some of the background which leads to Serre’s numerical recipe. Start with a modular form f of low weight k. Try to describe what happens if we take the representation attached to f and reduce mod λ then restrict to obtain \overline{ρ}_{λ,f}|I_{ℓ}. Assume first that we are in the following situation:

• the level N = N(f) is prime to ℓ,

• λ|ℓ,

• the weight k is such that 2 ≤ k ≤ ℓ + 1.

There are other cases to consider but we look at this one first. Recall that det(ρ) = χ_{ℓ}^{k−1}ε where cond(ε)|N. Therefore

det(ρ)|I_{ℓ} = χ_{ℓ}^{k−1}

so we see immediately that det(ρ) gives k mod ℓ − 1. This permits the possibility that k = ℓ + 1 but in fact ρ comes from another form of weight 2, since 2 ≡ ℓ + 1 (mod ℓ − 1). But this is only a possibility when 2 different weights could give some ρ. So we should make the following agreement: if there is a form of weight 2 giving ρ then maybe it is better not to look at f. [[There is laughter throughout the room.]]

Next suppose the weight is between 2 and ℓ and we are in the ordinary case so a_{ℓ}(f) ≢ 0 (mod λ). Then ρ|I_{ℓ} = ( \begin{matrix}χ_{ℓ}^{k−1} & ∗ \\ 0 & 1 \end{matrix} ) exactly as before. [[Ogus says, “* is never 0”. Ribet says, “No, no, no. Oh wait, it’s my theorem that if this representation were a direct sum of 2 characters, i.e., ∗ = 0, then one can find a weight 2 form giving ρ.]] This can be described using Kummer theory but “I do not think about it this way.”

15.5 The weight and fundamental characters

Let

ρ : Gal(\overline{Q}/Q) → GL(2, F_{ℓ^{ν}})

be an irreducible modular mod ℓ representation. Let D = D_{p} = Gal(\overline{Q}_{p}/Q_{p}) be the decomposition group at p. Today we will see how studying the local properties of ρ help us to understand the following theorem.

Theorem 15.5.1. If ℓ > 2 and ρ is modular then ρ is modular of a certain weight k(ρ) and level N(ρ).

Since ρ is modular, ρ ∼ \overline{ρ}_{λ,f} for some modular form f. The theorem asserts that we can actually take f ∈ S_{k(ρ)}(Γ_{1}(N(ρ))). As we have seen before N(ρ) is the prime to ℓ Artin conductor of ρ.

Today we attack the question of what k(ρ) should be. To do this we restrict to the inertia group I_{p}. Before doing this we introduce a certain pair of characters. First we prove a lemma. Let σ be the semisimplification of ρ|_{D}. Thus σ is either a direct sum of two characters or ρ|_{D} depending on whether or not ρ|_{D} is irreducible.

Lemma 15.5.2. The representation σ is tame, i.e, σ is trivial on the ℓ-Sylow subgroup of I_{ℓ}.

Proof. Let P be the ℓ-Sylow subgroup of I_{ℓ}. Since I_{ℓ} is normal in D and P is unique it follows that P is normal in D. Let W = F_{ℓ^{ν}} × F_{ℓ^{ν}} be the representation space of σ. Then

W^{P} = {w ∈ W : σ(τ)w = w for all τ ∈ P }

is a subspace of W invariant under the action of D. To see this let α ∈ D and suppose w ∈ W^{P}. Since P is normal in D, α^{−1}τα = τ′ for some τ′ ∈ P. Therefore

σ(α)^{−1}σ(τ)σ(α)w = σ(τ′)w = w

so σ(τ)σ(α)w = σ(α)w hence σ(α)w ∈ W^{P}.

But W^{P} ≠ 0. To see this write W as a disjoint union of its orbits under the action of P. Since P is an ℓ-Sylow group and W is finite we see that the size of each orbit is either 1 or a positive power of ℓ. Now {0} is a singleton orbit, W has ℓ-power order, and all non-singleton orbits have order a positive power of ℓ so there must be at least ℓ − 1 other singleton orbits. Each of these other singleton orbits gives a nonzero element of W^{P}.

If W^{P} = W then P acts trivially so we are done. If W^{P} ≠ W then W^{P} is a one dimensional subspace invariant under D so by semisimplicity σ is a diagonal representation. Suppose τ ∈ P then τ has order ℓ^{n} for some n. Write

σ(τ) = ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} )

then α^{ℓ^{n}} = 1 and β^{ℓ^{n}} = 1. Since α, β ∈ F_{ℓ^{ν}} it follows that α = β = 1.

The lemma implies σ|_{I} factors through the tame quotient I_{t} = I/P. We now describe I_{t} explicitly. Let Q_{p}^{tame} be the maximal tame extension of Q_{p} and let K = Q_{p}^{ur} be the maximal unramified extension. Since P is the part of inertia fixing Q_{p}^{tame},

I_{t} = I/P = Gal(Q_{p}^{tame}/Q_{p}^{ur}).

For each n prime to p there is a tower of fields

Q_{p}^{tame}
|
K(p^{\frac{1}{n}})
|
K = Q_{p}^{ur}
|
Q_{p}

By Kummer theory Gal(K(p^{\frac{1}{n}})/K) = μ_{n}(K) where μ_{n}(K) denotes the group of nth roots of unity in K. Thus for each n prime to p we obtain by restriction a map I_{t} → μ_{n}(K). They are compatible so passing to the limit we obtain a map

I_{t} → lim_{←n} μ_{n}(K) = ∏_{r≠p} lim_{←a} μ_{r^{a}}(K) = ∏_{r≠p} Z_{r}(1).

Viewed more cleverly mod p we obtain a map

I_{t} → lim_{←n} μ_{n}(\overline{F}_{p}) = lim_{←n} F_{p^{n}}^{∗}.

We now fix attention on n = p^{i}−1. The point is, that for every such i we have a standard map

I_{t} → F_{p^{i}}^{∗}.

We call this map the fundamental character of level i. This is unnatural because we had to take an embedding F_{ℓ^{ν}} ↪ \overline{F}_{ℓ}. Instead Serre begins with some disembodied field F of order p^{i}. There are i different maps F_{p^{i}} → F corresponding to the i automorphisms of F_{p^{i}}. Restricting these maps to F_{p^{i}}^{∗} and composing with the fundamental character I_{t} → F_{p^{i}}^{∗} defined above gives Serre’s fundamental characters of I_{t} with values in F. We have thus defined some characters on I_{t} which can be viewed as standard.

Recall that we have a mod ℓ^{ν} representation ρ which gives rise to

σ : I_{t} → GL(2, F_{ℓ^{ν}}).

Since the elements of I_{t} have order prime to the characteristic ℓ this representation is semisimple so it can be diagonalized upon passing to an algebraic closure of F_{ℓ^{ν}}. Thus σ corresponds to a pair of characters

α, β : I_{t} → F_{ℓ^{2ν}}^{∗}.

These characters have some stability properties since σ is the restriction of a homomorphism from the full decomposition group. Consider the tower of fields

K(p^{\frac{1}{n}}) ⊃ K = Q_{p}^{ur} ⊃ Q_{p}.

Let G = Gal(K(p^{\frac{1}{n}})/Q_{p}). Recall that Gal(K(p^{\frac{1}{n}})/K) = μ_{n}(K) and Gal(K/Q_{p}) is generated by Frob_{p}. What happens is that if h ∈ μ_{n}(K) and g ∈ G is such that g restricts to Frob_{p} then we have a conjugation formula: ghg^{−1} = h^{p}. Applying this reasoning to the σ situation with h ∈ I_{t} and g restricting to Frob_{p} we find that

σ(ghg^{−1}) = σ(h^{p}) = σ(h)^{p}.

Thus the pair of characters {α, β} is stable under pth powering, i.e., {α, β} = {α^{p}, β^{p}}. The point is that

σ(g)σ(h)σ(g^{−1}) = σ(ghg^{−1}) = σ(h)^{p}.

Thus the representation h ↦ σ(h)^{p} is isomorphic to h ↦ σ(h) via conjugation by σ(g). Thus α, β as a pair are the same as α^{p}, β^{p} as a pair since they came from isomorphic representations.

What does this mean? One possibility is that α = α^{p} and β = β^{p}. This means α, β take values in F_{p}^{∗}. The other possibility is that α^{p} = β and β^{p} = α. Then α^{p^{2}} = α and β^{p^{2}} = β so α, β take values in F_{p^{2}}^{∗}. The first situation in which α and β take values in F_{p}^{∗} is called the level 1 situation. The second situation is called the level 2 situation.

We will play a carnival game, “guess your weight.” First we consider the level 2 case. Our strategy is to try and express α and β in terms of the two fundamental characters of level 2.

First some background. Before stating the general result we talk about a special case. Suppose E is an elliptic curve over Q and p is a prime (2 is allowed). Assume E has good supersingular reduction at p. Then there is a representation

Gal(\overline{Q}/Q) → Aut(E[p])

which may or may not be irreducible. It gives rise via restriction to two characters α, β : I_{t} → F_{p}^{∗}. Serre [Groupes de Galois attachés aux points d’ordre fini des courbes elliptiques sur un corps de nombres, 1972] proves that α, β are the two fundamental characters of level 1 and that I_{t} acts tamely and irreducibly. He also obtained a map

I_{t} → F_{p^{2}}^{∗} ⊂ GL(2, F_{p})

where F_{p^{2}}^{∗} sits inside GL(2, F_{p}) via the action of the multiplicative group of a field on itself after choice of a basis. The action of I_{t} on F_{p^{2}}^{∗} is through the fundamental character. This was the world’s first view of fundamental characters.

Serre next asked Fontaine the corresponding question for more general weight. The first published proof appears in [6]. Suppose f is a newform of weight k such that 2 ≤ k ≤ p and that the level of f is prime to p = ℓ. Take \overline{ρ}_{f,λ} where E_{f} = Q(coefficients of f) and λ is a prime of O_{E_{f}} dividing p. Suppose we are in the supersingular case so a_{p} ≡ 0 mod λ. Semisimplifying and restricting as before gives a pair of characters α, β : I_{t} → F_{p^{ν}}^{∗}.

Theorem 15.5.3. Under the above hypothesis, {α, β} equals {ψ^{k−1}, (ψ′)^{k−1}} where ψ and ψ′ are the two fundamental characters of level 2, i.e., ψ = (ψ′)^{p} and ψ′ = ψ^{p}.

Edixhoven’s method is to reduce from weight k to weight 2. Given a modular form mod p on Γ_{1}(N) of weight k and character ε he shows that it is enough to look at a corresponding modular form on Γ_{1}(Np) of weight 2 and character εω^{k−2} where ω is a Teichmuller character on (Z/pZ)^{∗}. This is a method which helps us understand what happens at primes p where a certain abelian variety does not have good reduction but has potentially good reduction. In this direction Ogus suggests reading the paper [Faltings and Jordan, Crystalline cohomology and GL(2, Q), Israel J. Math. 90 (1995), no. 1-3, 1–66.]

What can we guess about the arbitrary situation? The situation is as follows. We begin with a representation ρ which gives rise to a representation σ which in turn gives rise to a pair of fundamental characters ψ^{a} and (ψ′)^{a} = ψ^{pa}. (Here a should be thought of as a number mod p^{2} − 1.) The condition that we are not in level 1 means that a is not divisible by p + 1 since

ψψ′ = ψ^{p+1} : I_{t} → F_{p}^{∗}

is the unique fundamental character of level 1 (namely the mod p cyclotomic character χ).

We next do a Euclidean division. Take 0 ≤ a < p^{2} − 1 and write a = qp + r. Then pa = qp^{2} + rp, but we are working mod p^{2} − 1 so this becomes pa = q + rp. What are the possible values for q and r? By the Euclidean algorithm 0 ≤ r ≤ p − 1 and 0 ≤ q ≤ p − 1. Now what are the constraints on r and q together? The main constraint is that r ≠ q, since if r = q then a is a multiple of p + 1. So we can assume that 0 ≤ r < q ≤ p − 1. We know that

{α, β} = {(ψψ′)^{r}(ψ′)^{q−r}, (ψψ′)^{r}ψ^{q−r}}.

For example, if α = ψ^{a} then

α = ψ^{qp+r} = (ψ′)^{q}ψ^{r} = (ψψ′)^{r}(ψ′)^{q−r}.

Since ψψ′ = χ we can view {α, β} as a pair of characters (ψ′)^{q−r} and ψ^{q−r} which has been multiplied as a pair by χ^{r}.

What would happen if for example r = 0? In this case

{α, β} = {(ψ′)^{k−1}, ψ^{k−1}}

where k = q + 1. So when r = 0 we guess the weight to be k(ρ) = q + 1.

What happens more generally? Suppose f is a modular form thought of mod p where λ|p. If f = ∑ a_{n}q^{n} gives rise to ρ then we might well ask what modular form gives rise to ρ ⊗ χ. In [Lecture Notes in Math, no. 601] we find that θf = ∑ na_{n}q^{n} is a form giving rise to ρ ⊗ χ. If k is the weight of f, then θf has weight k + p + 1. More generally, since χ^{r} appears in the pair

{ψ^{q−r}, (ψ′)^{q−r}} · χ^{r}

we obtain

k(ρ) = q − r + 1 + (p + 1)r = q + pr + 1.

But be careful, the weight k does not have to go up. In fact, it goes up p − 1 times then ends up where it started. This gives rise to the theory of theta cycles which we will study in the next lecture.

15.6 The weight in Serre’s conjectures on modular representations

Let

ρ : Gal(\overline{Q}/Q) → GL(2, F_{p^{ν}})

be a mod p representation. Let ℓ = p and assume that ρ is supersingular and of level 2 at ℓ.

Let ψ and ψ′ be the fundamental characters of level 2. Let I = I_{ℓ} be the inertia group at ℓ. After extension of scalars ρ|_{I} has the form

ρ|_{I} ∼ (\begin{matrix}ψ^{a}ψ′^{b} & 0 \\ 0 & ψ′^{a}ψ^{b} \end{matrix}).

We may assume 0 ≤ a < b ≤ p − 1. Since ψ′ = ψ^{p} we also know that ψ^{a}ψ′^{b} = ψ^{a+pb} and ψ′^{a}ψ^{b} = ψ^{b+pa}.

The conjectured weight is k(ρ) = b + pa + 1. Since χ = ψψ′ we see that

ρ|_{I} ∼ χ^{a}(\begin{matrix}ψ′^{b−a} & 0 \\ 0 & ψ^{b−a} \end{matrix}).

The weight of ψ′^{b−a} ⊕ ψ^{b−a} is b − a + 1. Twisting a representation by χ raises the weight by p + 1 (at least when the weight is in the right range). This is because twisting by χ corresponds to applying the θ operator to the corresponding modular form. Thus

k(ρ) = a(p + 1) + b − a + 1 = b + pa + 1.

15.6.1 θ-series

The theory of the θ operator was first developed by Serre and Swinnerton–Dyer, and later jazzed up by Katz [Lecture Notes in Math, Vol. 601, Springer-Verlag]. There is a notion of modular forms mod p and of q-expansion which gives a map

α : ⨁_{k≥0} M_{k}(Γ_{1}(N), F_{p}) → F_{p}[[q]].

This map is far from injective. The kernel is generated by 1 element, a certain Eisenstein series E_{p}.

Suppose f ∈ F_{p}[[q]] is in the image of α. If f ≠ 0 let w(f) denote the smallest k so that f comes from some M_{k}. If f does not coming from any single M_{k} do not define w(f).

Define an operator θ on F_{p}[[q]] by

θ(∑ a_{n}q^{n}) = q \frac{d}{dq} (∑ a_{n}q^{n}) = ∑ na_{n}q^{n}.

Work of Serre and Swinnerton-Dyer shows that θ preserves the image of α.

15.6. THE WEIGHT IN SERRE’S CONJECTURES ON MODULAR REPRESENTATIONS99

Theorem 15.6.1. Suppose f ≠ 0 is a mod p modular form as above. If w(f) ≢ 0 mod p then w(θf) = w(f) + p + 1.

The theta series associated to f is

w(f), w(θf), w(θ^{2}f), …

This series is periodic because Fermat’s Little Theorem implies θ^{p}f = θf and so w(θ^{p}f) = w(θf).

Tate asked the question: What are the possible θ-cycles? That is, what possible sequences w(f), w(θf), … can occur? The question was basically answered by work of Edixhoven [Invent. Math. 109, 563-594 (1992)] and Jochnowitz [Trans. A.M.S. vol. 270, 1982 (253-267)].

Suppose f is an eigenform with 2 ≤ w(f) ≤ p. Suppose f is supersingular, i.e., a_{p}(f) = 0. Since f is an eigenform the a_{n} are multiplicative (a_{nm} = a_{n}a_{m} for (n, m) = 1) and there is a recurrence among prime power coefficients so a_{n}(f) = 0 whenever p|n. Thus θ^{p−1}f = f hence w(θ^{p−1}f) = w(f) = k.

If we start with f and apply θ successively what happens to the weight? First suppose k = 2. The θ-cycle is

2, 2 + (p + 1), 2 + 2(p + 1), …, 2 + (p − 2)(p + 1).

This is because we can apply θ thus raising the weight by p + 1 so long as the weight is not a multiple of p. Only the last term in the above sequence is divisible by p. There are p − 1 terms so this the full θ-cycle.

Next suppose k = p. The θ-cycle is

p, 3, 3 + (p + 1), …, 3 + (p + 1)(p − 3).

The last term is divisible by p, no earlier term is, and there are p − 1 terms so this is the full θ-cycle. We know that the second term must be 3 since it is the only number so that the θ-cycle works out right, i.e., so that the (p − 1)st term is divisible by p but no earlier term is. If we would have tried 2 instead of 3 we would have obtained the sequence

p, 2, 2 + (p + 1), …, 2 + (p + 1)(p − 3), 2 + (p + 1)(p − 2).

This sequence has one too many terms.

Now we consider the general case. Let k be a weight such that 2 < k < p. The θ-cycle is

k, k + (p + 1), …, k + (p + 1)(p − k), k′, k′ + p + 1, …, k′ + (p + 1)(k − 3).

The first p − k + 1 terms of the sequence are obtained by adding p + 1 successively until obtaining a term which is divisible by p. Applying θ to a form of weight k + (p + 1)(p − k) causes the weight to drop to some k′. It can be proved that there is at most one drop in the sequence.

How can we guess k′? It must be such that k′ + (p + 1)(k − 3) is divisible by p. Thus the correct answer is

k′ = p + 3 − k.

15.6.2 Edixhoven’s paper

Suppose that ρ is an irreducible mod ℓ = p representation of level 2. To avoid problems in the exceptional case assume ℓ ≥ 3. Edixhoven proves that if ρ is modular in the sense that ρ comes from an eigenform in S_{k}(Γ_{1}(N)) with (N, p) = 1, then ρ comes from an eigenform in S_{k(ρ)}(Γ_{1}(N)).

What are the elements of the proof?

1. The behavior of ρ|_{I_{p}} when 2 ≤ w(f) ≤ p + 1,

2. the fact that every modular representation ρ coming from a cusp form f has the form ρ_{f} ⊗ χ^{i} where i ∈ Z/(p − 1)Z, and

3. the fact that w(f) satisfies 2 ≤ w(f) ≤ p + 1.

Serre knew that 2 ≤ w(f) ≤ p+1 but never published a proof. How did Serre talk about his result before Edixhoven? The eigenforms f correspond to systems of eigenvalues in \overline{F}_{p} of the Hecke operators T_{r}, r |̸N, r |̸p. Eigenforms of weight at most p + 1 give, up to twist, all systems of eigenvalues. A possible proof uses the construction of ρ_{f} in terms of certain étale cohomology groups.

15.7 The extra assumption

[[The mysterious slogan of the day which Ribet put up this time is “will soon admit”.]]

Let ρ : G = Gal(\overline{Q}/Q) → GL(2, F_{ℓ^{ν}}) be a Galois representation . Assume that ρ is irreducible, ρ ≅ \overline{ρ}_{λ,f}, and ℓ > 2. Then the theorem is

Theorem 15.7.1. Under the above hypothesis, ρ comes from a modular form f of weight k(ρ), level N(ρ), and under a certain extra assumption, character ε of order prime to ℓ.

The extra assumption alluded to in the theorem is the following. Extra Assumption.

Not all of the following are true.

1. ℓ = 3,

2. ρ| Gal(\overline{Q}/Q(\sqrt{−3})) is abelian, and

3. det(ρ) is not a power of the mod 3 cyclotomic character χ.

Example 15.7.2. If det(ρ) = χ is the mod 3 cyclotomic character, for example when ρ comes from the Galois representation on the 3-torsion of an elliptic curve, then the extra assumption holds so the theorem applies. [[as long as the additional assumption hold?]]

Let ρ be a representation as above. Then

det ρ : G → F_{ℓ^{ν}}

is very likely ramified at ℓ. Let χ be the mod ℓ cyclotomic character. As an exercise one can show that det ρ = χ^{i}θ for some i and some θ where θ : G → F_{ℓ^{ν}} is unramified at ℓ. Using the Kronecker-Weber theorem and class-field theory one can prove the following two facts.

1. One can write θ as a Dirichlet character (Z/MZ)^{∗} → F_{ℓ^{ν}}^{∗}. [[Does this mean that θ factors through the mod M cyclotomic character?]] 2. In fact M|N(ρ), since the conductor of the determinent of a representation should

divide the conductor of the representation. [[?? My notes have the opposite remark?]]

Thus we can view θ as a map (Z/N(ρ)Z)^{∗} → F_{ℓ^{ν}}^{∗}.

Let H = ker θ ⊂ (Z/N(ρ)Z)^{∗}. Define a congruence subgroup Γ_{H}(N) by

Γ_{H}(N) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ_{0}(N) : a, d ∈ H\}.

We have the following theorem.

Theorem 15.7.3. Suppose ℓ > 2 and ρ satisfies the above assumptions including the extra assumption. Then ρ arises from a form f in S_{k(ρ)}(Γ_{H}(N(ρ))).

In particular, the theorem is true if ρ comes from the ℓ-torsion representation on an elliptic curve since det = χ so the extra assumption is satisfied.

One might ask if the extra assumption is really necessary. At first Serre did not think that it was. But he was surprised to discover the following example which shows that the extra assumption can not be eliminated.

Example 15.7.4. (Serre) The space S_{2}(Γ_{1}(13)) is 2 dimensional. Let ℓ = 3. There is a newform f ∈ S_{2}(Γ_{1}(13)) of character ε : (Z/13Z)^{∗} → C^{∗}. This character must be trivial on {±1} and has order 6. Using [[cite Ribet’s LNM 601 article]] one shows that

Q(…, a_{n}, …) ⊃ Q(…, ε(d), …) = Q(\sqrt{−3}).

Furthermore, since S_{2}(Γ_{1}(13)) has dimension 2 there can be at most 2 embeddings of Q(…, a_{n}, …) into C (each embedding gives an independent newform). Thus one actually has that Q(\sqrt{−3}) = Q(…, a_{n}, …).

Let λ = \sqrt{3} and let \overline{ρ}_{λ,f} : G → GL(2, F_{3}) be the associated Galois representation . Then det(\overline{ρ}_{λ,f}) = χθ where χ is the mod 3 cyclotomic character and θ ≡ ε (mod 3) [[I am not really sure what this congruence means]]. In particular, θ has order 2. Thus H = ker(θ) ⊂ (Z/13Z)^{∗} is exactly the squares in (Z/13Z)^{∗}.

The conclusion of the theorem can not hold since S_{2}(Γ_{H}(13)) = 0. This is because any form would have to have a character whose order is at most two since it must be trivial on H, but S_{2}(Γ_{H}(13)) ⊂ S_{2}(Γ_{1}(13)) and S_{2}(Γ_{1}(13)) consists of forms whose character is of order 6. [[There are a lot of compatibilities to check here which I have not checked.]]

In this example

• ℓ = 3,

• det ρ is not a power of χ, and

• \overline{ρ}_{λ,f}| Gal(\overline{Q}/Q(\sqrt{−3})) is abelian.

A good way to see the last assertion is to look at the following formula:

f ⊗ ε^{−1} = \overline{f}

(up to an Euler factor at 13) in the sense that

ρ_{f,λ} ⊗ ε^{−1} = ρ\overline{f}.

Now reduce mod 3 to obtain

\overline{ρ}_{λ,f} ⊗ ε^{−1} ≅ \overline{ρ}_{λ,f}

since f ≡ \overline{f} (mod \sqrt{−3}) (since 3 is ramified). Thus \overline{ρ}_{λ,f} is isomorphic to a twist of itself by a complex character thus by elementary facts \overline{ρ}_{λ,f} is reducible and abelian over the field corresponding to its kernel. [[So why is the Galois group of Q(\sqrt{−3}) in the kernel of \overline{ρ}_{λ,f}?]] [[Ribet suddenly gets very excited and notes that \overline{ρ}_{λ,f} is also abelian on the Galois groups of Q(\sqrt{13}) and Q(\sqrt{39}) by the same argument.]]

There are two main ingredients in this theorem. The first is companion forms.

15.7.1 Companion Forms

The key papers dealing with companion forms are Gross and Coleman-Voloch. The context is the following. Suppose we are interested in the recipe for k(ρ). There are two cases to consider, the level 1 case and the level 2 case. We have already discussed the level 2 case. In the level 1 case what happens? If we semisimplify and restrict to inertia we obtain a direct sum of two representations. There are two possibilities for ρ|I:

ρ|I = (\begin{matrix}χ^{α} & ∗ \\ 0 & χ^{β} \end{matrix}) or ρ|I = (\begin{matrix}χ^{α} & 0 \\ 0 & χ^{β} \end{matrix}).

In the second case it is easy to guess k(ρ). The thing to do is to look at the exponents and normalize as best you can. Since α and β are only defined mod p − 1 we may, after relabeling if necessary, assume that 0 ≤ α ≤ β ≤ p − 2. Factor out χ^{α} to obtain

χ^{α} ⊗ (\begin{matrix}1 & 0 \\ 0 & χ^{β−α} \end{matrix}).

Next (secretly) recall that if f is ordinary of weight k then f gives rise to the representation

(\begin{matrix}χ^{k−1} & ∗ \\ 0 & χ \end{matrix})

(here * can be trivial). [[Something must be wrong here because nothing makes sense in the following argument unless the lower right entry of this matrix is 1 instead of χ.]] If f is of weight β − α + 1 twisting α times by q \frac{d}{dq} gives the desired representation. Thus the weight k(ρ) is

(p + 1)α + β − α + 1 = β + pα + 1.

There is one caveat. Since Serre was terrified of weight 1 forms, if α = β = 0 then Serre defines k(ρ) = p instead of k(ρ) = 1. There is no distinction between Serre’s formula and the above formula when the weight is > 1 since, as Katz proved, any form of weight > 1 lifts to a characteristic 0 form. But this is not true for weight 1. To understand the weight 1 forms we must use sophisticated tools of algebraic geometry.

Suppose f is of weight k, 2 ≤ k ≤ p + 1. Let \overline{ρ}_{λ,f} be the associated representation where λ is some prime dividing ℓ = p. Then

\overline{ρ}_{λ,f}|I_{p} = (\begin{matrix}χ^{k−1} & ∗ \\ 0 & 1 \end{matrix}).

[[I really wonder why this should be so in such brave generality? It seems like things were not like this just a minute ago.]] To introduce the idea of a companion form suppose that somehow by chance ∗ = 0 so we have ( \begin{matrix}χ^{k−1} & 0 \\ 0 & 1 \end{matrix} ). Take ρ and twist by χ^{1−k} then

ρ ⊗ χ^{1−k}|I_{p} = ρ ⊗ χ^{p−k}|I_{p} = (\begin{matrix}1 & 0 \\ 0 & χ^{p−k} \end{matrix}).

What is the minimum weight giving rise to this representation. The level is 1 [[why??]] the representation is semisimple and reducible and α = 0, β = p − k, so the natural weight is p − k + 1. Thus the conjecture predicts that there should exist another form g of weight p + 1 − k such that ρ_{g} = ρ_{f} ⊗ χ^{p−k}. Here g is a companion form of f. Thus g = θ^{p−k}f where θ = q \frac{d}{dq} is the θ-operator we defined when studying θ-series. This conjecture was proved by Gross when k ≠ 2, p. It was proved by Coleman and Voloch when k = p. This is how we know about the weight. Next time we will talk about the character.

[[What is the other main ingredient in the proof.]]

15.8 The exceptional level 1 case

Let

ρ : G = Gal(\overline{Q}/Q) → GL(2, F_{ℓ^{ν}})

be a Galois representation with ℓ > 2. Assume that ρ is irreducible and modular. Then ρ comes from a modular form in S_{k(ρ)}(Γ_{1}(Np)). Here p = ℓ [I think] and 2 ≤ k(ρ) ≤ p^{2} − 1.

We still must talk about the exceptional case in level 1. In this situation

ρ|I_{ℓ} = (\begin{matrix}χ^{α} & ∗ \\ 0 & χ^{β} \end{matrix}) = χ^{β} ⊗ (\begin{matrix}χ & ∗ \\ 0 & 1 \end{matrix})

since α = β + 1. [[I do not remember why.]] Define a representation σ of Gal(\overline{Q}/Q) by

σ := ρ ⊗ χ^{−β} ≅ (\begin{matrix}χ & ∗ \\ 0 & 1 \end{matrix}).

Fortunately we know enough to say something about the weight k(σ). In fact,

k(σ) = \begin{cases}2 & σ if is finite at ℓ \\ ℓ + 1 & otherwise \end{cases}

This is enough to determine k(ρ) since

k(ρ) = k(χ^{β} ⊗ σ) = (p + 1)β + k(σ).

We have not yet said what we mean for σ to be finite at ℓ.

Definition 15.8.1. σ|D_{ℓ} is finite if σ|D_{ℓ} is equivalent to a representation defined by G(\overline{Q}_{ℓ}), where G is a finite flat F_{ℓ^{ν}} vector space scheme over Z_{ℓ}.

This definition is not terribly enlightening. To help fathom what is going on we consider the following special case. Suppose E/Q is an elliptic curve with semistable (=good or multiplicative) reduction at ℓ. Then E[ℓ] defines a representation

σ : G → Aut E[ℓ] = GL(2, F_{ℓ}).

Let Δ_{E} be the discriminant of E, then σ is finite at ℓ iff

ord_{ℓ} Δ_{E} ≡ 0 (mod ℓ).

Thus when Δ_{E} satisfies this requirement, k(σ) = 2. Next suppose p ≠ ℓ and assume E has semistable reduction at p. Then the representation σ attached to E[ℓ] is unramified at p iff

ord_{p} Δ_{E} ≡ 0 (mod ℓ).

The reference for everything we have been discussing is [27].

The bridge is Kummer theory. Let E be an elliptic curve with multiplicative reduction at p. Suppose V = E[ℓ] = F_{ℓ} ⊕ F_{ℓ}. We have a map

τ : D = Gal(\overline{Q}_{ℓ}/Q_{ℓ}) → Aut V.

We treat the case p = ℓ and p ≠ ℓ at the same time. By the theory of Tate curves there is an exact sequence of D-modules

0 → X → V ⟶^{α} Y → 0.

[[I do not remember him explicitly saying that these maps should be D-modules maps but they must be for the twist ^{d}s to be a splitting.]] Each of these terms is a D module and X and Y are 1 dimensional. The action of D on Y is given by an unramified character ε of order dividing 2 (i.e., the order is 1 or 2). The action of D on X is given by χε.

Next we define an element of H^{1}(D, Hom(Y, X)). Choose a splitting s : Y → V. Thus s is a linear map (not necessarily a map of D-modules) [[otherwise all twists of a splittings would be trivial]] such that αs = 1. For each d ∈ D consider the twisting ^{d}s : Y → V defined by ^{d}s(y) = ds(d^{−1}y). Since

α(ds(d^{−1}y)) = d(α(s(d^{−1}y))) = d(d^{−1}y) = y

if follows that ^{d}s is again a splitting. Thus ^{d}s − s : Y → V followed by α : V → Y is 0. We therefore have that ^{d}s − s ∈ Hom(Y, X). The map d ↦ ^{d}s defines an element of H^{1}(D, Hom(Y, X)).

We have isomorphisms

H^{1}(D, Hom(Y, X)) ≅ H^{1}(D, μ_{ℓ}) ≅ Q_{p}^{∗}/Q_{p}^{∗ℓ}.

The last isomorphism follows from Kummer theory. [[I do not know where the first one comes from.]] Thus τ defines an element of Q_{p}^{∗}/Q_{p}^{∗ℓ}. Serre proved that τ is finite iff the corresponding element of Q_{p}^{∗}/Q_{p}^{∗ℓ} defined by τ is in the image of Z_{p}^{∗}.

If E/Q_{p} is an elliptic curve with multiplicative reduction then there exists a Tate parameter q ∈ Q_{p}^{∗} with Val_{p}(q) > 0 such that

E ≅ E_{q} := G_{m}/q^{Z}

over the unique quadratic unramified extension of Q_{p}. The kernel of multiplication by p gives rise to an exact sequence as above. [[Work out the details sometime!]] Furthermore, the element of Q_{p}^{∗}/Q_{p}^{∗^{ℓ}} defined by the representation comming from E[ℓ] is just the image of q. In this context there is a formula for Δ_{E} which is

Δ_{E} = q ∏_{n=1}^{∞}(1 − q^{n})^{24}.

Note that the product factor is a unit in Q_{p}. Thus Val_{p} Δ_{E} = Val_{p} q.

[[At this moment Ogus asks if Serre uses Raynaud’s result. Ribet first says no, but then changes his mind and says “Yes, he must be able to explicitly describe finite flat vector space schemes.” Then Ogus asks what is wrong with weight 1. Ribet replies that Serre didn’t know how to define forms in weight 1. But L. Merel says something about Serre being frustrated because he couldn’t compute in weight 1.]]

Fermat’s Last Theorem

16.1 The application to Fermat

“As part of this parcel, I can sketch the application to Fermat.”

Suppose that semistable elliptic curves over Q are modular. Then FLT is true. Why? “As I have explained so many times … ” Suppose ℓ > 5 and

a^{ℓ} + b^{ℓ} + c^{ℓ} = 0

with abc ≠ 0, all relatively prime, and such that A = a^{ℓ} ≡ −1 (mod 4), B = b^{ℓ} is even and C = c^{ℓ} ≡ 1 (mod 4). Then we consider the elliptic curve

E : y^{2} = x(x − A)(x − B).

The minimal discriminant is

Δ_{E} = \frac{(ABC)^{2}}{2^{8}}

as discussed in Serre’s [27] and Diamond-Kramer [[cite Math Research Letters, 1995]]. The conductor N_{E} is equal to the product of the primes dividing ABC (so in particular N_{E} is square-free). Furthermore, E is semistable – the only hard place to check is at 2. Diamond-Kramer checks this explicitly.

Here is how to get Fermat’s theorem. View E[ℓ] as a G = Gal(\overline{Q}/Q)-module. The idea is to show that this representation must come from a modular form of weight 2 and level 2. This will be a contradiction since there are no modular forms of weight 2 and level 2. But to apply the level and weight theorem we need to know that E[ℓ] is irreducible. The proof of this is due to Mazur.

Let ρ : G → Aut E[ℓ] be the Galois representation on the ℓ torsion of E. Since E is semistable for p ≠ ℓ,

ρ|I_{p} ≅ (\begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix}).

Assume ρ is reducible. Then ρ has an invariant subspace so

ρ ≅ (\begin{matrix}α & ∗ \\ 0 & β \end{matrix}).

Then from the form of ρ|I_{p} we see that the characters α and β could be ramified only at ℓ. Thus α = χ^{i} and β = χ^{1−i} where χ is the mod ℓ cyclotomic character. The exponents are i and 1 − i since αβ is the determinent which is χ. [[Why is χ supposed to be the only possible unramified character? probably since whatever the character is, it is a product of χ times something else, and the other factor is ramified.]]

What happens to ρ at ℓ, i.e., what is ρ|I_{ℓ}? There are only two possibilities. Either ρ|I_{ℓ} is the direct sum of the two fundamental characters or it is the sum of the trivial character with χ. The second possibility must be the one which occurs. [[I do not understand why... something about “characters globally are determined by local information.”]] So either i = 0 or i = 1. If i = 0,

ρ = (\begin{matrix}1 & ∗ \\ 0 & χ \end{matrix}).

This means that there is an element of E[ℓ] whose subspace is left invariant under the action of Galois. Thus E has a point of order ℓ rational over Q. If i = 1 then

ρ = (\begin{matrix}χ & ∗ \\ 0 & 1 \end{matrix}).

Therefore μ_{ℓ} ↪ E[ℓ]. Divide to obtain E′ = E/μ_{ℓ}. The representation on E[ℓ]/μ_{ℓ} is constant (this is basic linear algebra) so the resulting representation on E′[ℓ] has the form ( \begin{matrix}1 & ∗ \\ 0 & χ \end{matrix} ) so E′ has a rational point of order ℓ.

Now E[2] is a trivial Galois module since it contains 3 obvious rational points, namely (0, 0), (A, 0), and (B, 0). Thus the group structure on the curve E (or E′) (which we constructed from a counterexample to Fermat) contains

Z/2Z ⊕ Z/2Z ⊕ Z/ℓZ.

In Mazur’s paper [[“rational isogenies of prime degree”]] he proves that ℓ ≤ 3. Since we assumed that ℓ > 5, this is a contradiction. Notice that we have not just proved FLT. We have demonstrated the irreducibility of the Galois representation on the ℓ torsion of the elliptic curve E arising from a hypothetical counterexample to FLT.

We now have a representation

ρ : Gal(\overline{Q}/Q) → GL(2, F_{ℓ}) = Aut E[ℓ]

which is irreducible and modular of weight 2 and level N = N_{E} (the conductor of E). Because ρ is irreducible we conclude that ρ is modular of weight k(ρ) and level N(ρ). Furthermore

ord_{ℓ} Δ_{E} = ord_{ℓ}(ABC)^{2} = 2ℓ ord_{ℓ} abc ≡ 0 (mod ℓ)

so k(ρ) = 2.

We can also prove that N(ρ) = 2. Clearly N(ρ)|N_{E}. This is because N(ρ) computed locally at p ≠ ℓ divides the power of p in the conductor of the ℓ-adic representation for E at p. [[I do not understand this.]] Since ρ is only ramified at 2 or maybe ℓ, N(ρ) must be a power of 2. For p ≠ ℓ, ρ is ramified at p iff ord_{p} Δ_{E} ≢ 0 (mod ℓ) which does happen when p = 2. Since N_{E} is square free this implies that N(ρ) = 2. But S_{2}(Γ_{1}(2)) = 0, which is the ultimate contradiction!

But how do we know semistable elliptic curves over Q are modular?

16.2 Modular Elliptic Curves

Theorem 16.2.1 (Theorem A). Every semistable elliptic curve over Q is modular.

There are several ways to define a modular elliptic curve.

Definition 16.2.2. Let E be an elliptic curve. Then E is modular if there is a prime ℓ > 2 such that the associated ℓ-adic Galois representation

ρ_{E,ℓ^{∞}} : G = Gal(\overline{Q}/Q) → GL(2, Z_{ℓ})

defined by the ℓ-power division points on E is modular (i.e., it is a ρ_{f,λ}).

Definition 16.2.3. Let E be an elliptic curve of conductor N_{E}. For each prime p not dividing N_{E} one associates a number a_{p} related [[in a simple way!]] to the number of points on the reduction of E modulo p. Then E is modular if there exists a cusp form f = ∑ b_{n}q^{n} which is an eigenform

for the Hecke operators such that a_{p} = b_{p} for almost all p. In the end one deduces that f can be chosen to have weight 2, trivial character, and level N_{E}.

Definition 16.2.4 (Shimura). An elliptic curve E is modular if there is nonconstant map X_{0}(N) → E for some N.

Theorem 16.2.5 (Theorem B. Wiles, Taylor-Wiles). Suppose E is a semistable elliptic curve over Q and suppose ℓ is an odd prime such that E[ℓ] is irreducible and modular, then ρ_{E,ℓ^{∞}} is modular and hence E is modular.

Now we sketch a proof that theorem B implies theorem A. First take ℓ = 3. If E[3] is irreducible then by work of Langlands-Tunnel we win. The idea is to take

ρ : G → GL(2, F_{3}) ↪ GL(2, Z[\sqrt{−2}]) ⊂ GL(2, C).

The point is that there are two maps Z[\sqrt{−2}] → F_{3} given by reduction modulo each of the primes lying over 3. Choosing one gives a map GL(2, Z[\sqrt{−2}]) → GL(2, F_{3}) which, for some amazing reason [[which is?]], has a section. Then ρ : G → GL(2, C) is a continuous representation with odd determinent which must still be irreducible. By Langlands-Tunnel we know that ρ is modular and in fact ρ comes from a weight 1 cusp form f of level a power of 3 times powers of most primes dividing cond(E). Reducing f modulo some prime of Z[\sqrt{−2}] lieing over 3 we obtain a mod 3 modular form which corresponds to ρ : G → E[3]. The proof of all this uses the immense base-change business in Langlands’ book. [[Ribet next says: “have to get 3’s out of the level! This jacks up the weight, and the level is still not square free. Then have to adjust the weight again.” I do not know what the point of this is.]]

Kevin Buzzard asked a question relating to how one knows the hypothesis needed for the theorem on weights and levels applies in our situation. To answer this suppose ℓ = 3 or 5. Form the associated representation ρ : G → GL(2, F_{ℓ}) coming from E[ℓ] and assume it is irreducible, modular and semistable.

Definition 16.2.6. A mod ℓ Galois representation is semistable if for all p ≠ ℓ, the inertia group I_{p} acts unipotently and the conjectured weight is 2 or ℓ + 1.

Note that det ρ = χ.

Lemma 16.2.7. Under the above assumptions ℓ divides the order of the image of ρ.

Proof. If not, then ρ is finite at all primes p, since for primes p ≠ ℓ inertia acts trivially [[some other argument for ℓ]]. Inertia acts trivially because if the order of the image of ρ is prime to ℓ then ρ acts diagonally. For if not then since ρ|I_{p} is unipotent (hence all eigenvalues are 1), in a suitable basis something like ( \begin{matrix}1 & ψ \\ 0 & 1 \end{matrix} ) is in the image of ρ and has order ℓ, a contradiction. Because of finiteness k(ρ) = 2 and N(ρ) = 1 which is a contradiction since there are no forms of weight 2 and level 1.

Next we consider what happens if E[3] is reducible. There are two cases to consider. First suppose E[5] is also reducible. Then E contains a rational subgroup of order 15. We can check by hand that all such curves are modular. The key result that is that X_{0}(15)(Q) is finite.

The second possibility is that E[5] is irreducible. In this case we first find a curve E′ which is semistable over Q such that

• E′[5] ≅ E[5] (this is easy to do because of the lucky coincidence that X_{0}(5) has genus 0)

• E′[3] is irreducible

Next we discover that E′ is modular since E′[3] is irreducible. This implies E′[5] is modular hence E[5] is modular. Theorem B then implies E is modular.

Deformations

17.1 Introduction

For the rest of the semester let ℓ be an odd prime. Let ρ : G → GL(2, F_{ℓ^{ν}}) be such that • ρ is modular

• ρ is irreducible

• ρ is semistable Definition 17.1.1. The representation ρ is semistable if

• for all p ≠ ℓ,

ρ|I_{p} ≅ (\begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix}),

(∗ is typically trivial since most primes are unramified.)

• k(ρ) = 2 or ℓ + 1. This means that there are 2 possibilities.

1. ρ is finite at D_{ℓ}.

2.

ρ|D_{ℓ} ≅ (\begin{matrix}α & ∗ \\ 0 & β \end{matrix})

where β is unramified. (Since det(ρ) = χ we can add that α|I_{ℓ} = χ.) If k(ρ) = 2 then possibility 1 occurs. If k(ρ) = ℓ + 1 then we are in case 2, but being in case 2 does not imply that k(ρ) = ℓ + 1. If ρ comes from an elliptic curve E/Q, then case 1 occurs if E has good reduction at ℓ whereas case 2 occurs if E has ordinary multiplicative reduction [[I am not sure about this last assertion because I missed it in class.]].

What is a deformation of ρ and when can we prove that it is modular?

Let A be a complete local Noetherian ring with maximal ideal m and residue field F_{ℓ^{ν}} (so A is furnished with a map A/m ⟶^{∼} F_{ℓ^{ν}}). Let

ρ̃_{A} : G → GL(2, A)

be a representation which is ramified at only finitely many primes. Assume ρ̃ = ρ̃_{A} lifts ρ, i.e., the reduction of ρ̃ mod m gives ρ. Theorem 17.1.2. Let the notation be as above, then ρ̃ is modular iff it satisfies (∗).

Neither of the terms in this theorem have been defined yet.

17.2 Condition (∗)

Let

ρ : G = Gal(\overline{Q}/Q) → GL(2, F_{ℓ^{ν}})

be a Galois representation.

The statement we wish to understand is

“All ρ̃ which satisfy (∗) are modular.”

Let A be a complete local Noetherian ring with residue field F_{ℓ^{ν}}. This means that we are given a map A/m ≅ F_{ℓ^{ν}}. Suppose ρ̃: G → GL(2, A) satisfies the following conditions:

• ρ̃ lifts ρ,

• det ρ̃ = χ̃,

• ρ̃ is ramified at only finitely many primes, and

• condition (∗).

What is condition (∗)? It the requirement that ρ̃ have the same qualitative properties as ρ locally at ℓ. There are two cases to consider.

Case 1. (arising from supersingular reduction at ℓ) Suppose ρ is finite and flat at ℓ. Then ρ|I_{ℓ} is given by the 2 fundamental characters

I_{ℓ} → F_{ℓ^{2}}^{∗}

of level 2 (instead of from powers of these characters because of the semistability assumption). Condition (∗) is that the lift of ρ is also constrained to be finite and flat. This means that for every n ≥ 1,

ρ̃|D_{ℓ} mod m^{n} : D_{ℓ} → GL(2, A/m^{n})

is finite and flat, i.e., it comes from a finite flat group scheme over Z_{ℓ} which is provided with an action of A/m^{n} as endomorphisms.

Case 2. (bad multiplicative reduction at ℓ or good ordinary reduction at ℓ) In case 2

ρ|D_{ℓ} ≅ (\begin{matrix}α & ∗ \\ 0 & β \end{matrix})

where β is unramified (equivalently α|I_{ℓ} = χ). [[In the case of good ordinary reduction β is given by the action of Galois on E[ℓ] in characteristic ℓ.]] Condition (∗) is the requirement that

ρ̃|D_{ℓ} ≅ (\begin{matrix}α̃ & ∗ \\ 0 & β̃ \end{matrix}),

where β̃ is unramified, α̃|I_{ℓ} = χ̃. It follows automatically that β̃ lifts β.

17.2.1 Finite flat representations

In general what does it mean for ρ to be finite and flat. It means that there exists a finite flat group scheme G over Z_{ℓ} such that G(\overline{Q}_{ℓ}) is the representation space of ρ. This definition is subtle.

Coleman asked if there is a way to reformulate the definition without mentioning group schemes. Ribet mentioned Hopf algebras but then stopped. Buzzard suggested some of the subtlety of the definition by claiming that in some situation χ is finite flat whereas χ^{2} is not. Ogus vaguely conjectured that Fontaine’s language is the way to understand this.

It is possible in case 2 above for ρ|D_{ℓ} to be finite flat without ρ̃ finite flat. The quintessential example is an elliptic curve E with supersingular reduction at ℓ such that ℓ| ord_{ℓ} Δ_{E}.

17.3 Classes of Liftings

Let Σ be a finite set of prime numbers. We characterize a class of liftings ρ̃ which depends on Σ. What does it mean for ρ̃ to be in the class of deformations corresponding to Σ?

17.3.1 The case p ≠ ℓ

First we talk about the case when p ≠ ℓ. If p ∈ Σ then there is no special condition on ρ̃|I_{p}. If p ∉ Σ one requires that ρ̃ is qualitatively the same as ρ. This means

1. If ρ is unramified at p (which is the usual case), then we just require that ρ̃ is unramified at p.

2. If ρ is ramified but unipotent at p so

ρ|I_{p} ≅ (\begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix})

we require that

ρ̃|I_{p} ≅ (\begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix}).

This situation can occur with an elliptic curve which has multiplicative reduction at p and for which ℓ |̸ ord_{p} Δ_{E}.

At this point I mention the prime example. Example 17.3.1. Suppose ρ̃ = ρ_{f,λ} is the λ-adic representation attached to f. What can we say about ord_{p} N(f)? We know that

ord_{p} N(f) = ord_{p} N(ρ) + dim(ρ)^{I_{p}} − dim(ρ̃)^{I_{p}}

where (ρ)^{I_{p}} means the inertia invariants in the representation space of ρ. If ρ is semistable then

ord_{p} N(ρ) + dim(ρ)^{I_{p}} = 2.

Since we are assuming ρ is semistable, ord_{p} N(f) ≤ 2. Furthermore, the condition p ∉ Σ is a way of saying

ord_{p} N(f) = ord_{p} N(ρ).

Thus the requirement that p ∉ Σ is that the error term dim(ρ)^{I_{p}} − dim(ρ̃)^{I_{p}} vanish.

Note that ord_{p} N(ρ̃) = ord_{p} N(f) by Carayol’s theorem. Thus ord_{p} N(f) is just a different way to write ord_{p} N(ρ̃).

Example 17.3.2. Imagine ρ is ramified at p and ord_{p} N(ρ) = 1. Then ord_{p} N(f) is either 1 or 2. The requirement that p ∉ Σ is that ord_{p} N(f) = 1.

17.3.2 The case p = ℓ

Next we talk about the case p = ℓ. There are two possibilities: either ℓ ∈ Σ or ℓ ∉ Σ. If ℓ ∈ Σ then we impose no further condition on ρ̃ (besides the already imposed condition (∗), semistability at ℓ, etc.). If ℓ ∉ Σ and ρ if finite and flat (which is not always the case) then we require ρ̃ to be finite and flat. If ℓ ∉ Σ and ρ is not finite flat then no further restriction (this is the Tate curve situation).

Suppose ρ̃ = ρ_{f,λ}, and ρ̃ belongs to the class defined by Σ. We want to guess (since there is no Carayol theorem) an integer N_{Σ} such that N(f)|N_{Σ}. What is N_{Σ}? It will be

N_{Σ} = ∏_{\begin{matrix}p≠ℓ \\ p∈Σ \end{matrix}} p^{2} · ∏_{\begin{matrix}p≠ℓ \\ p∉Σ \end{matrix}} p^{ord_{p} N(ρ)} · ℓ^{δ}.

Here ord_{p} N(ρ) is 1 iff ρ is ramified at p. The exponent δ is either 0 or 1. It is 1 iff k(ρ) = ℓ+1 or ℓ ∈ Σ and ρ is ordinary at ℓ, i.e.,

ρ|D_{ℓ} ≅ (\begin{matrix}α & ∗ \\ 0 & β \end{matrix}).

[[This definition could be completely wrong. It was definitely not presented clearly in class.]]

There is an exercise associated with this. It is to justify a priori the definition of δ. Suppose, for example, that ρ_{f,λ} satisfies (∗), then we want to show that ℓ^{2} |̸N(f).

Theorem 17.3.3. Every ρ̃ of class Σ comes from S_{2}(Γ_{0}(N_{Σ})).

Define an approximation T to the Hecke algebra by letting

T = Z[…, T_{n}, …] ⊂ End(S_{2}(Γ_{0}(N_{Σ})))

where we adjoin only those T_{n} for which (n, ℓN_{Σ}) = 1. For some reason there exists a map

T → F_{ℓ^{ν}} : T_{r} ↦ Tr ρ(Frob_{r}).

Why should such a map exist? The point is that we know by the theorem that ρ comes by reduction from a ρ_{f,λ} with f ∈ S_{2}(Γ_{0}(N_{Σ})).

But there is a wrinkle. [[I do not understand: He says: “Clearly N(ρ)|N_{Σ}. k(ρ) = 2 or ℓ + 1. If k(ρ) = ℓ + 1 then δ = 1 so ℓ|N_{Σ}. Have to trickily slip over to weight 2 in order to get f.” This does not make any sense to me.]]

17.4 Wiles’ Hecke algebra

Composing the map T → O_{E_{f}} with reduction mod λ from O_{E_{f}} to F_{ℓ^{ν}} we obtain a map T → F_{ℓ^{ν}}. Let m be the kernel. Then m is a maximal ideal of T. Wiles’ Hecke algebra is the completion T_{m} of T at m. [[For some reason]] there exists

ρ̃: G = Gal(\overline{Q}/Q) → GL(2, T_{m})

such that Tr(ρ̃(Frob_{r})) = T_{r}. What makes this useful is that ρ̃ is universal for lifts of type Σ. This means that given any lift τ of type Σ there exists a map

φ : GL(2, T_{m}) → GL(2, A)

such that τ = φρ̃.

Another key idea that the approximation T obtained by just adjoining those T_{n} with (n, ℓN_{Σ}) = 1 is, after completing at certain primes, actually equal to the whole Hecke algebra.

114 CHAPTER 17. DEFORMATIONS

The Hecke Algebra T_{Σ}

18.1 The Hecke Algebra

Throughout this lecture ℓ ≠ p and ℓ ≥ 3. We are studying the representation ρ : G → GL(2, F_{ℓ^{ν}}). This is an irreducible representation, ℓ is odd, ρ is semistable, and det ρ = χ. To single out certain classes of liftings we let Σ be a finite set of primes. Let A be a complete local Noetherian ring with residue filed F_{ℓ^{ν}}. We take liftings ρ̃: G → GL(2, A) such that ρ̃ reduces down to ρ, det ρ̃ = χ̃, and ρ̃ is “like” ρ away from Σ. For example, if ρ is unramified at p we also want ρ̃ unramified at p, etc.

Assume ρ̃ is modular. We guess the serious divisibility condition that N(f)|N_{Σ}. Recall that

N_{Σ} = ∏_{\begin{matrix}p≠ℓ \\ p∈Σ \end{matrix}} p^{2} · ∏_{\begin{matrix}p≠ℓ \\ p∉Σ \end{matrix}} p^{ord_{p} N(ρ)} · ℓ^{δ}.

To define δ consider two cases.

• level 1 case. Take δ = 1 if ℓ ∈ Σ or if ρ is not finite at ℓ. Take δ = 0 otherwise.

• level 2 case. This is the case when ρ|I_{ℓ} has order ℓ^{2} − 1. Take δ = 0.

A priori nobody seems to know how to prove that N(f)|N_{Σ} given only that ρ̃ is modular. In the end we will show that all modular ρ̃ in fact come from

S_{2}(Γ_{0}(N_{Σ})).

This can be regarded as a proof that N(ρ)|N_{Σ}.

Last time we tried to get things going by defining the anemic Hecke algebra

T = Z[…, T_{n}, …] ⊂ End(S_{2}(Γ_{0}(N_{Σ})))

where we only adjoin those T_{n} for which (n, ℓN_{Σ}) = 1. By some level lowering theorem there exists an f ∈ S_{w}(Γ_{0}(N_{Σ})) giving ρ̃ so we obtain a map T → F. The map sends T_{n} to the reduction modulo λ of its eigenvalue on f. (λ is a prime of the ring of integers of E_{f} lying over ℓ.) [[... something about needing an f of the right level = N(ρ).]]

Let m ⊂ T be the kernel of the above defined map T → F. Then m is a maximal ideal. Let T_{m} be the completion of T at m and note that

T_{m} ↪ T ⊗_{Z} Z_{ℓ}.

116 CHAPTER 18. THE HECKE ALGEBRA T_{Σ}

We need to know that T_{m} is Gorenstein. This is done by comparing T_{m} to some full Hecke ring. Thus

T ⊂ R = Z[…, T_{n}, …] ⊂ End(S_{2}(Γ_{0}(N_{Σ})))

where the full Hecke ring R is obtained by adjoining the T_{n} for all integers n. We should think of R as

R = T[T_{ℓ}, {U_{P} : p|N_{Σ}}].

Note that T_{ℓ} may or may not be a U_{ℓ} depending on if ℓ|N_{Σ}.

Lemma 18.1.1. If ℓ |̸N_{Σ} then R = T[U_{p}, …]. Thus if T_{ℓ} is not a U_{ℓ} then we do not need T_{ℓ}.

Proof. This lemma is an interesting thing and the proof goes as follows. Oooh. Sorry, this is not true. Ummm.... hmm.

The ring T[U_{p}, …] is clearly of finite index in R since: if q is a random prime number consider R ⊗_{Z} Q_{q} compared to T[U_{p}, …] ⊗_{Z} Q_{q}. [[I do not know how to do this argument. The lemma as stated above probably isn’t really true. The point is that the following lemma is the one we need and it is true.]]

Let T[U_{p}, …] be the ring obtained by adjoining to T just those U_{p} with p|N_{Σ}.

Lemma 18.1.2. If ℓ |̸N_{Σ} then the index (R : T[U_{p}, …]) is prime to ℓ. Note that we assume ℓ ≥ 3.

Proof. We must show that the map T[U_{p}, …] → R/ℓR is surjective. [[I thought about it for a minute and did not see why this suffices. Am I being stupid?]] Let A = F_{ℓ}[T_{n} : (n, ℓ) = 1] be the image of T[U_{p}, …] in R/ℓR so we have a diagram

T[U_{p}, …] ⟶ R/ℓR
↘  ∪
A

We must show that A = R/ℓR. There is a beautiful duality

R/ℓR × S_{2}(Γ_{0}(N_{Σ}); F_{ℓ}) → F_{ℓ} (perfect pairing).

Thus A^{⊥} = 0 iff A = R/ℓR.

Suppose 0 ≠ f ∈ A^{⊥}, then a_{n}(f) = 0 for all n such that (n, ℓ) = 1. Thus f = ∑ a_{nℓ}q^{nℓ}. Let θ = q \frac{d}{dq} be the theta operator. Since the characteristic is ℓ, θ(f) = 0. On the other hand w(f) = 2 and since ℓ ≥ 3, ℓ |̸2. Thus w(θf) = w(f) + ℓ + 1 = 2 + ℓ + 1 which is a contradiction since θf = 0 and w(0) = 0 ≠ 3 + ℓ. [[The weight is an integer not a number mod ℓ, right?]]

Example 18.1.3. The lemma only applies if ℓ ≥ 3. Suppose ℓ = 2 and consider S_{2}(Γ_{0}(23)). Then

T[U_{p}, …] = Z[\sqrt{5}] ⊂ R = Z[\frac{1 + \sqrt{5}}{2}]

so (R : T[U_{p}, …]) is not prime to 2.

18.2. THE MAXIMAL IDEAL IN R 117

Remark 18.1.4. If N = N_{Σ} then

rank_{Z} T = ∑_{M|N} S_{2}(Γ_{0}(M))^{new}

and

rank_{Z} R = dim S_{2}(Γ_{0}(N)).

There is an injection

⨁_{M|N} S_{2}(Γ_{0}(M))^{new} ↪ S_{2}(Γ_{0}(N))

but ⊕S_{2}(Γ_{0}(M))^{new} is typically much smaller than S_{2}(Γ_{0}(N)).

18.2 The maximal ideal in R

The plan is to find a special maximal ideal m_{R}

\begin{aligned}m_{R} & ⊂ & R \\ ∪ & & ∪ \\ m & ⊂ & T \end{aligned} .

Once we finally found the correct m and m_{R} then T_{m} ⟶^{∼} R_{m_{R}} will be an isomorphism.

In finding m_{R} we will not invoke some abstract going up theorem but we will “produce” m_{R} by some other process.

The ideal m was defined by a newform f level M|N_{Σ}. The coefficients of f lie in

O_{λ} = (O_{E_{f}})_{λ}

where E_{f} is the coefficient ring of f and λ is a prime lying over ℓ. Thus O_{λ} is an ℓ-adic integer ring. Composing the residue class map O_{λ} → \overline{F} with the eigenvalue map T → O_{λ} we obtain the map T → \overline{F}.

In order to obtain the correct m_{R} we will make a sequence of changes to f to make some good newform. [[Is the motivation for all this the fact that the lemma will not apply if ℓ|N_{Σ}?]]

18.2.1 Strip away certain Euler factors

Write f = ∑ a_{n}q^{n}. Replace f by

h = ∑_{certain n} a_{n}q^{n}

where the sum is over those n which are prime to each p ∈ Σ. What does this mean? If we think about the L-function L(f, s) = ∏_{p} L_{p}(f, s), then h has L-function

L(h, s) = ∏_{p∉Σ} L_{p}(f, s).

Furthermore making this change does not take us out of S_{2}(Γ_{0}(N_{Σ})), i.e., h ∈ S_{2}(Γ_{0}(N_{Σ})). [[He explained why but my notes are very incomplete. They say: Why. Because h = (f ⊗ε)⊗ ε, (ε Dirichlet character ramified at primes in N_{Σ}). Get a form of level lcm(∏_{p∈Σ} p^{2}, N(f))|N_{Σ}. Can strip and stay in space since N_{Σ} has correct squares built into it.]] Remark 18.2.1. Suppose that p||N_{Σ}. Then p ∉ Σ and p||N(ρ). The level of f is

N(f) = \begin{cases}N(ρ), & if k(ρ) = 2 \\ N(ρ)ℓ, & if k(ρ) = ℓ + 1 \end{cases} .

If p||N(ρ) then f|T_{p} = a_{p}(f)f. Thus h is already an eigenform for T_{p} unless p ∈ Σ in which the eigenvalue is 0.

[[I do not understand this remark. Why would the eigenvalue being 0 mean that T_{p} is not an eigenform? Furthermore, what is the point of this remark in the wider context of transforming f into a good newform.]]

18.2.2 Make into an eigenform for U_{ℓ}

We perform this operation to f to obtain a form g then apply the above operation to get the ultimate h having the desired properties. Do this if ℓ|N_{Σ} but ℓ |̸N(f). This happens precisely if ℓ ∈ Σ and ρ is good and ordinary at ℓ. Then f|U_{ℓ} is just some random junk. Consider g = f +∗f(q^{ℓ}) where ∗ is some coefficient. We see that if ∗ is chosen correctly then g|U_{ℓ} = Cg for some constant C. There are 2 possible choices for ∗ which lead to 2 choices for C. Let a_{ℓ} = a_{ℓ}(f), then C can be either root of

X^{2} + a_{ℓ}X + ℓ = 0.

This equation has exactly one unit root in O_{λ}. The reason is because we are in the ordinary situation so a_{ℓ} is a unit. But ℓ is not a unit. The sum of the roots is a unit but the product is not. [[Even this is not clear to me right now. Definitely check this later.]] Make the choice of ∗ so that real root is C. Then we obtain a g such that

g|U_{ℓ} = (unit) · g.

Next apply the above procedure to strip g and end up with an h such that

• h|U_{ℓ} = (unit) · h,

• h|U_{p} = 0 for p ∈ Σ (p ≠ ℓ), and

• h|T_{p} = a_{p}(f) · h for p ∉ Σ.

Now take the form h. It gives a map R → O_{λ} which extends T → O_{λ}. Let m_{R} be the kernel of the map R → \overline{F} obtained by composing R → O_{λ} with O_{λ} → \overline{F}. A lot of further analysis shows that T_{m} → R_{m_{r}} is an isomorphism. We end up having to show separately that the map is injective and surjective.

[[In this whole lecture p ≠ ℓ.]]

[[Wiles’ notation: His T_{m_{R}} is my R and his T′ is my T_{m}.

18.3 The Galois Representation

We started with a representation ρ, chose a finite set of primes Σ and then made the completed Hecke algebra T_{m}. Our goal is to construct the universal deformation of ρ of type Σ. The universal deformation is a representation

ρ̃: G → GL(2, T_{m})

such that for all primes p with p |̸ℓN_{Σ}, ρ̃(Frob_{p}) has trace T_{p} ∈ T_{m} and determinent p. We now proceed with the construction of ρ̃. Let

T = Z[…, T_{n}, …], (n, ℓN_{Σ}) = 1

be the anemic Hecke algebra. Then T ⊗ Q decomposes as a product of fields

T ⊗ Q = ∏_{f} E_{f}

where the product is over a set of representatives for the Galois conjugacy classes of newforms of weight 2, trivial character, and level dividing N_{Σ}. Since T is integral (it is for example a finite rank Z-module), T ↪ ∏_{f} O_{f}. Since Z_{ℓ} is a flat Z-module,

T ⊗ Z_{ℓ} ↪ ∏_{f} O_{f} ⊗ Z_{ℓ} = ∏_{f,λ} O_{f,λ}

where the product is over a set of representatives f and all λ|ℓ.

T_{m} is a direct factor of T ⊗ Z_{ℓ}. [[This is definitely not the assertion that T_{m} is an O_{f,λ}. What exactly is it the assertion of really?]]

We can restrict the product to a certain finite set S and still obtain an injection

T_{m} ↪ ∏_{(f,λ)∈S} O_{f,λ}.

The finite set S consists of those (f, λ) such that the prime λ of O_{f} pulls back to m under the map T → O_{f} obtained by composing T → ∏_{f} O_{f} with the projection onto O_{f}. [[Why is this enough so that T_{m} still injects in?]] Restricting to a finite product is needed so that

[ ∏_{(f,λ)∈S} O_{f,λ} : T_{m}] < ∞.

Given f and λ there exists a representation

ρ_{f,λ} : G → GL(2, O_{f,λ}).

It is such that Tr ρ_{f,λ}(Frob_{p}) = a_{p} is the image of T_{p} under the inclusion

T ⊗ Z_{ℓ} ↪ ∏_{(f,λ)∈S} O_{f,λ}.

Put some of these ρ_{f,λ} together to create a new representation

ρ′ = ∏_{(f,λ)∈S} ρ_{f,λ} : G → GL(2, ∏ O_{f,λ}) ⊂ GL(2, T_{m} ⊗ Q).

The sought after universal deformation ρ̃ is a map making the following diagram commute

G ⟶^{ρ′} GL(2, T_{m} ⊗ Q)
ρ̃ ↘  ∪
GL(2, T_{m})

Theorem 18.3.1. ρ′ is equivalent to a representation taking values in GL(2, T_{m}).

One way to [[try to]] prove this theorem is by invoking a general theorem of Carayol. [[and then what? does this way work? why is it not a good way?]] But the right way to prove the theorem is Wiles’ way.

120 CHAPTER 18. THE HECKE ALGEBRA T_{Σ}

18.3.1 The structure of T_{m}

Just as an aside let us review the structure of T_{m}.

• T_{m} is local.

• T_{m} is not necessarily a discrete valuation ring.

• T_{m} ⊗ Q is a product of finite extensions of Q_{ℓ}.

• T_{m} is not necessarily a product of rings O_{f,λ}.

• T_{m} need not be integral. [[or can I say, “IS not integral.” ?]]

18.3.2 The philosophy in this picture

Choose c to be a complex conjugation in G = Gal(\overline{Q}/Q). Since ℓ is odd det(c) = −1 is a very strong condition which rigidifies the situation.

18.3.3 Massage ρ

Choose two 1-dimensional subspaces so that

ρ(c) = (\begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix})

with respect to any basis consisting of one vector from each subspace. For any σ ∈ G write

ρ(σ) = (\begin{matrix}a_{σ} & b_{σ} \\ c_{σ} & d_{σ} \end{matrix}).

Then a_{σ}d_{σ} and b_{σ}c_{σ} are somehow intrinsically defined. This is because

ρ(σc) = (\begin{matrix}−a_{σ} & ? \\ ? & d_{σ} \end{matrix})

so

a_{σ} = \frac{Tr(ρ(σ)) − Tr(ρ(σc))}{2}

and

d_{σ} = \frac{Tr(ρ(σ)) + Tr(ρ(σc))}{2}.

Since we know the determinent it follows that b_{σ}c_{σ} is also intrinsically known. [[The point is that we know certain things about these matrices in terms of their traces and determinants.]] Proposition 18.3.2. There exists g ∈ G such that b_{g}c_{g} ≠ 0. Proof. Since ρ is irreducible there exists σ_{1} such that b_{σ_{1}} ≠ 0 and there exists σ_{2} such that c_{σ_{2}} ≠ 0. If b_{σ_{2}} ≠ 0 or c_{σ_{1}} ≠ 0 then we are done. So the only problem case is when b_{σ_{1}} = 0 and c_{σ_{2}} = 0. Easy linear algebra shows that in this situation g = σ_{1}σ_{2} has the required property.

Now rigidify by choosing a basis so that b_{g} = 1. Doing this does not fix a basis because they are many ways to choose such a basis.

18.3.4 Massage ρ′

Choose a basis of (T_{m} ⊗ Q)^{2} so that

ρ′(c) = (\begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix}).

For any σ ∈ G write

ρ′(σ) = (\begin{matrix}a_{σ} & b_{σ} \\ c_{σ} & d_{σ} \end{matrix})

Using an argument as above shows that a_{σ}, d_{σ} ∈ T_{m} since the traces live in T_{m}. Furthermore b_{σ}c_{σ} ∈ T_{m} since the determinent is in T_{m}.

The key observation is that b_{σ}c_{σ} reduces mod m to give the previous b_{σ}c_{σ} ∈ F corresponding to ρ(σ). This is because the determinants and traces of ρ′ are lifts of the ones from ρ. Since m is the maximal ideal of a local ring and b_{g}c_{g} reduces mod m to something nonzero it follows that b_{g}c_{g} is a unit in T_{m}.

Choose a basis so that

ρ′(c) = (\begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix})

and also so that

ρ′(g) = (\begin{matrix}a_{g} & 1 \\ u & d_{g} \end{matrix}) ∈ GL(2, T_{m}).

Here u is a unit in T_{m}.

Proposition 18.3.3. Write

ρ′(σ) = (\begin{matrix}a_{σ} & b_{σ} \\ c_{σ} & d_{σ} \end{matrix})

with respect to the basis chosen above. Then a_{σ}, b_{σ}, c_{σ}, d_{σ} ∈ T_{m}.

Proof. We already know that a_{σ}, d_{σ} ∈ T_{m}. The question is how to show that b_{σ}, c_{σ} ∈ T_{m}. Since

ρ′(σg) = (\begin{matrix}a_{σ}a_{g} + b_{σ}u & ? \\ ? & c_{σ} + d_{σ}d_{g} \end{matrix})

we see that a_{σ}a_{g} + b_{σ}u ∈ T_{m}. Since a_{σ}a_{g} ∈ T_{m} it follows that b_{σ}u ∈ T_{m}. Since u is a unit in T_{m} this implies b_{σ} ∈ T_{m}. Similarly c_{σ} + d_{σ}d_{g} ∈ T_{m} so c_{σ} ∈ T_{m}.

As you now see, in this situation we can prove Carayol’s theorem with just some matrix computations. [[This is basically a field lowering representation theorem. The thing that makes it easy is that there exists something (namely c) with distinct eigenvalues which is rational over the residue field. Schur’s paper, models over smaller fields. “Schur’s method”.]]

18.3.5 Representations from modular forms mod ℓ

If you remember back in the 70’s people would take an f ∈ S_{2}(Γ_{0}(N); F) which is an eigenform for almost all the Hecke operators

T_{p}f = c_{p}f for almost all p, and c_{p} ∈ F.

The question is then: Can you find

ρ : G → GL(2, F)

such that

Tr(ρ(Frob_{p})) = c_{p} and det(ρ(Frob_{p})) = p

for all but finitely many p? The answer is yes. The idea is to find ρ by taking ρ_{f,λ} [[which was constructed by Shimura?]] for some f and reducing mod λ. The only special thing that we need is a lemma saying that the eigenvalues in characteristic ℓ lift to eigenvalues in characteristic 0.

18.3.6 Representations from modular forms mod ℓ^{n}

Serre and Deligne asked: “What happens mod ℓ^{n}?”

More precisely, let R be a local finite Artin ring such that ℓ^{n}R = 0 for some n. Take f ∈ S_{2}(Γ_{0}(N); R) satisfying the hypothesis

{r ∈ R : rf = 0} = {0}.

This is done to insure that certain eigenvalues are unique. Assume that for almost all p one has T_{p}f = c_{p}f with c_{p} ∈ R. The problem is to find ρ : G → GL(2, R) such that

Tr(ρ(Frob_{p})) = c_{p} and det(ρ(Frob_{p})) = p

for almost all p.

The big stumbling block is that ρ need not be the reduction of some ρ_{g,λ} for any g, λ. [[I couldn’t understand why – I wrote “can mix up f’s from characteristic 0 so can not get one which reduces correctly.”]]

Let T = Z[…, T_{p}, …] where we only adjoin those T_{p} for which f is an eigenvector [[I made this last part up, but it seems very reasonable]]. Then f obviously gives a rise to a map

T → R : T ↦ eigenvalue of T on f .

The strange hypothesis on f insures that the eigenvalue is unique. Indeed, suppose T f = af and T f = bf, then 0 = T f − T f = af − bf = (a − b)f so by the hypothesis a − b = 0 so a = b.

Since the pullback of the maximal ideal of R is a maximal ideal of T we get a map T_{m} → R for some m. [[I do not understand why we suddenly get this map and I do not know why the pullback of the maximal ideal is maximal.]]

Now the problem is solved. Take ρ′ : G → GL(2, T_{m}) with the sought after trace and determinent properties. Then let ρ be the map obtained by composing with the map GL(2, T_{m}) → GL(2, R).

18.4 ρ′ is of type Σ

Let ρ be modular irreducible and semistable mod ℓ representation with ℓ > 2. Let Σ be a finite set of primes. Then N(ρ)|N_{Σ}. We constructed the anemic Hecke algebra T which contains a certain maximal ideal m. We then consider the completion T_{m} of T at m. Next we constructed

ρ′ : G → GL(2, T_{m})

lifting ρ. Thus the diagram

G ⟶^{ρ′} GL(2, T_{m})
ρ ↘  ↓
GL(2, F)

18.5. ISOMORPHISM BETWEEN T_{M} AND R_{M_{R}} 123

commutes.

Some defining properties of ρ′ are

• det ρ′ = χ̃.

• Tr ρ′(Frob_{r}) = T_{r}. Since topologically T_{m} is generated by the Frob_{r} this is a tight condition.

• ρ′ is a lift of type Σ.

To say ρ′ is a lift of type Σ entails that ρ′ is unramified outside primes p|N_{Σ}. This is true because ρ′ is constructed from various ρ_{f,λ} with N(f)|N_{Σ}. If p ≠ ℓ and p|N(ρ) then p||N_{Σ}. Recall that

ρ|D_{p} ∼ (\begin{matrix}α & ∗ \\ 0 & β \end{matrix})

where α = βχ and α and β are unramified. For ρ′ to be a lift of type Σ we require that

ρ′|D_{p} ∼ (\begin{matrix}α̃ & ∗ \\ 0 & β̃ \end{matrix})

where α̃ and β̃ are unramified lifts and α̃ = β̃χ. [[I find it mighty odd that α is χ times an unramified character and yet α is not ramified! How can that be? Restricted to inertia β and α would be trivial but χ would not be.]] Is this true of ρ′? Yes since by a theorem of Langlands the factors

ρ_{f,λ}|D_{p} ∼ (\begin{matrix}α̃ & ∗ \\ 0 & β̃ \end{matrix}) .

[[Ribet said more about this but it does not form a cohesive whole in my mind. Here is what I have got. Since ρ_{f,λ} obviously ramified at p, p||N(f)|N_{Σ}. ρ_{f,λ}|D_{p} is like an elliptic curve with bad multiplicative reduction at p. That ρ_{f,λ}|D_{p} ∼ ( \begin{matrix}α̃ & ∗ \\ 0 & β̃ \end{matrix} ) really comes down to

Deligne-Rapaport. If write f = ∑ a_{n}q^{n}, then a_{p} ≠ 0 and β̃(Frob_{p}) = a_{p}, α̃(Frob_{p}) = pa_{p}. Thus a_{p}^{2} = 1 since α̃β̃ = χ. Thus a_{p} = ±1 and a_{p} mod λ = β(Frob_{p}) = ±1 independent of (f, λ). So we have these numbers a_{p} = a_{p}(f) = ±1, independent of λ. ]]

18.5 Isomorphism between T_{m} and R_{m_{R}}

Let T ⊂ R = Z[…, T_{n}, …] be the anemic Hecke algebra with maximal ideal m. The difference between T and R is that R contains all the Hecke operators whereas T only contains the T_{p} with p |̸ℓN_{Σ}. Wiles proved that the map T_{m} → R_{m_{R}} is an isomorphism. Which Hecke operators are going to hit the missing T_{p}? If we do the analysis in R_{m_{R}} we see that [[I think for p ≠ ℓ!]]

T_{p} = \begin{cases}±1, & for p|N_{Σ}, p ∉ Σ \\ 0, & for p ∈ Σ \end{cases} .

This takes care of everything except T_{ℓ}. In proving the surjectivity of T_{m} → R_{m_{R}} we are quite happy to know that T_{p} = ±1 or 0. Proving this is a bit unpalatable. It is described in [4].

Consider the commuting diagram

\begin{aligned}T_{m} & ↪ & ∏ O_{f,λ} \\ & ↘ & ↑ \\ & & R_{m_{R}} \end{aligned} .

The map R_{m_{R}} → ∏ O_{f,λ} is constructed by massaging f by stripping away certain Euler factors so as to obtain an eigenvector for all the Hecke operators. This diagram forces T_{m} → R_{m_{R}} to be injective.

[[Ogus: Is it clearly surjective on the residue field? Ribet: Yes. Ogus: OK, then we just need to prove it is ètale.]]

From the theory of the θ operator we already know two-thirds of the times that T contains T_{ℓ}.

Suppose ℓ|N_{Σ}. This entails that we are in the ordinary case, ρ is not finite at ℓ, or ℓ ∈ Σ. We did not prove in this situation that T_{ℓ} ∈ Z[…, T_{n}, …: (n, ℓ) = 1].

Using generators and relations and brute force one shows that R_{m_{R}} → ∏ O_{f,λ} is an injection. Then we can compare everything in ∏ O_{f,λ}. Now

ρ_{f,λ}|D_{ℓ} ∼ (\begin{matrix}α̃ & ∗ \\ 0 & β̃ \end{matrix})

and

β̃(Frob_{ℓ}) = T_{ℓ} ∈ ∏ O_{f,λ}.

Using arguments like last time one shows that β̃(Frob_{ℓ}) can be expressed in terms of the traces of various operators. This proves surjectivity in this case.

Ultimately we have T_{m} ≅ R_{m_{R}}. The virtue of T_{m} is that it is generated by traces. The virtue of R_{m_{R}} is that it is Gorenstein. We have seen this if ℓ |̸N_{Σ}. In fact it is Gorenstein even if ℓ||N_{Σ}. [[Ribet: As I stand here today I do not know how to prove this last assertion in exactly one case. Ogus: You mean there is another gap in Wiles’ proof. Ribet: No, it is just something I need to work out.]] When ℓ||N_{Σ} there are 2 cases. Either ρ is not finite at ℓ or it is. The case when ρ is not finite at ℓ was taken care of in [[cite a paper by Mazur-Ribet, Asterisque, reference?]]. A proof that R_{m_{R}} is Gorenstein when ρ is finite at ℓ (ℓ ∈ Σ) is not in the literature.

Now forget R_{m_{R}} and just think of T_{m} in both ways: trace generated and Gorenstein.

18.6 Deformations

Fix an absolutely irreducible modular mod ℓ representation ρ and a finite set of primes Σ. Consider the category C of complete local Noetherian W (F)-algebras A (with A/m = F). Here F = F_{ℓ^{ν}} and W (F) is the ring of Witt vectors, i.e., the ring of integers of an unramified extension of Q_{ℓ} of degree ν.

Define a functor F : C → Set by sending A in C to the set of equivalence classes of lifts

ρ̃: G → GL(2, A)

of ρ of type Σ. The equivalence relation is that ρ̃_{1} ∼ ρ̃_{2} iff there exists M ∈ GL(2, A) with

M ≡ (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) (mod m) such that ρ̃_{1} = M^{−1}ρ̃_{2}M.

Mazur proved [15] that F is representable.

Theorem 18.6.1. There exists a lift

ρ_{univ} : G → GL(2, R_{Σ})

of type Σ such that given any lift ρ̃ : G → GL(2, A) there exists a unique homomorphism φ : R_{Σ} → A such that φ ∘ ρ_{univ} ∼ ρ̃ in the sense of the above equivalence relation.

Lenstra figured out how to concretely construct R_{Σ}.

Back in the student days of Ribet and Ogus, Schlessinger wrote a widely quoted thesis which gives conditions under which a certain class of functors can be representable. Mazur checks these conditions in his paper.

[[Buzzard: What happened to Schlessinger anyways? Ribet: He ended up at University of North Carolina, Chapel Hill.]]

We have constructed ρ̃ = ρ′ : G → GL(2, T_{m}). By the theorem there exists a unique morphism φ : R_{Σ} → T_{m} such that ρ′ = φ ∘ ρ_{univ}.

Theorem 18.6.2. φ is an isomorphism thus ρ′ is the universal deformation and T_{m} is the universal deformation ring.

This will imply that any lift of type Σ is modular.

The morphism φ is surjective since

T_{p} = Tr ρ̃(Frob_{p}) = Tr φ ∘ ρ_{univ}(Frob_{p}) = φ(Tr(ρ_{univ}(Frob_{p}))).

We have two very abstractly defined local Noetherian rings. How would you prove they are isomorphic? Most people would be terrified by this question. Wiles dealt with it.

18.7 Wiles Main Conjecture

“We are like a train which is trying to reach Fermat’s Last Theorem. Of course it has not made all of its scheduled stops. But it is on its way.”

We have a representation ρ : G → GL(2, F). Take F = F_{ℓ} for our applications today. Then the ring of Witt vectors is W (F) = Z_{ℓ}. The Hecke algebra can be embedded as

T_{m} ⊂ ∏_{(g,μ)∈A} O_{g,μ}.

The Hecke algebra T_{m} has the following properties.

• The index of T_{m} in ∏ O_{g,μ} is finite.

• Gorenstein as a Z_{ℓ}-module, i.e., there exists an isomorphism Hom_{Z_{ℓ}}(T_{m}, Z_{ℓ}) ≅ T_{m}.

• T_{m} is generated by the T_{r} with r prime and (r, ℓN_{Σ}) = 1.

We have constructed a representation

ρ′ : G → GL(2, T_{m}).

Composing appropriately with the map T_{m} ↪ ∏ O_{g,μ} gives a map

G → ∏_{(g,μ)} GL(2, O_{g,μ}).

This is the product of representations ∏ ρ_{g,μ}. The triangle is

G  ⟶^{ρ′}  GL(2, T_{m})
∏ ρ_{g,μ} ↘  ↓
∏ GL(2, O_{g,μ})

Moreover, ρ′ is a deformation of ρ of type Σ so it lifts ρ and satisfies certain “nice as ρ” properties at primes p ∉ Σ.

Let

ρ_{univ} : G → GL(2, R_{Σ})

be the universal deformation of ρ of type Σ. Lenstra gave a very concrete paper [12] constructing this ρ_{univ}. Before his paper there was only Schlesinger’s thesis. By the definition of ρ_{univ} there exists a unique map φ : R_{Σ} → T_{m} = T_{Σ} such that φ ∘ ρ_{univ} = ρ′. By φ ∘ ρ_{univ} we mean the composition of ρ_{univ} with the map GL(2, R_{Σ}) → GL(2, T_{m}) induces by φ. As noted last time it is easy to see that φ is surjective.

Theorem 18.7.1 (Wiles main ‘conjecture’). φ is an isomorphism (for each Σ).

The theorem implies the following useful result.

Theorem 18.7.2. Suppose E is a semistable elliptic curve over Q and that for some ℓ > 2 the representation ρ = ρ_{ℓ,E} on E[ℓ] is irreducible and modular. Then E is modular.

Proof. The representation

ρ̃ = ρ_{ℓ^{∞},E} : G → GL(2, Z_{ℓ})

on the ℓ-power torsion E[ℓ^{∞}] = ∪E[ℓ^{n}] is a lift of

ρ : G → Aut(E[ℓ]) = GL(2, F_{ℓ}).

Furthermore, ρ̃ is a deformation of ρ of type Σ. Applying universality and using the theorem that R_{Σ} ≅ T_{Σ} we get a map

T_{Σ} → Z_{ℓ} : T_{r} ↦ a_{r} = a_{r}(E) = trace ρ̃(Frob_{r}).

The relevant diagram is

\begin{matrix}R_{Σ} \\ ↓ & ↘ \\ T_{Σ} & ⟶ & Z_{ℓ} \end{matrix}

where the map R_{Σ} → Z_{ℓ} is given by

trace ρ_{univ}(Frob_{r}) ↦ a_{r}.

Now the full Hecke algebra Z[…, T_{n}, …] embeds into the completion T_{Σ}. Composing this with the map T_{Σ} → Z_{ℓ} above we obtain a map

α : Z[…, T_{n}, …] → Z_{ℓ}.

Because of the duality between the Hecke algebra and modular forms there exists a modular form h ∈ S_{2}(Γ_{0}(N_{Σ}), Z_{ℓ}) corresponding to α. Since α is a homomorphism h is a normalized eigenform. Furthermore a_{r}(h) = a_{r}(E) ∈ Z for all primes r |̸ℓN_{Σ}. Since almost all coefficients of h are integral it follows that h is integral. Because we know a lot about eigenforms we can massage h to an eigenform in S_{2}(Γ_{0}(N_{E}), Z).

[[Some undigested comments follow.]]

• Once there is any connection between ρ and a modular form one can prove Taniyama-Shimura in as strong a form as desired. See the article Number theory as Gadfly.

• Take the abelian variety A_{h} attached to h. The λ-adic representation will have pieces with the same representations. Using Tate’s conjecture we see that E is isogenous to A_{h}. Use at some points Carayol’s theorem: If g is a form giving rising to the abelian variety A then the conductor of A is the same as the conductor of g.

• Tate proved that if two elliptic curves have isomorphic ρ_{ℓ^{∞}} for some ℓ then they are isogenous.

18.8 T_{Σ} is a complete intersection

Recall the construction of T_{Σ} = T_{m}. Let T be the anemic Hecke algebra. Then

T ⊗ Z_{ℓ} ↪ ∏ O_{g,μ}

where the product is over a complete set of representatives (for the action of Galois on eigenforms) g and primes μ lying over ℓ. We found a specific (f, λ) for Σ = ∅ such that \overline{ρ}_{λ,f} = ρ. The maximal ideal m was defined as follows. The form f induces a map T → O_{f,λ}. Taking the quotient of O_{f,λ} by its maximal ideal we obtain a map T → \overline{F}_{ℓ}. Then m is the kernel of this map. The diagram is

\begin{matrix}T \\ ↓ & ↘ \\ O_{f,λ} & ⟶ & \overline{F}_{ℓ} \end{matrix}

The map O_{f,λ} → \overline{F}_{ℓ} is

a_{r}(f) ↦ trace ρ(Frob_{r}).

To fix ideas we cheat and suppose O_{f,λ} = Z_{ℓ}. [[In Wiles’ optic this is OK since he can work this way then tensor everything at the end.]]

Now ρ_{f,λ} is a distinguished deformation of ρ [[“Distinguished” is not meant in a mathematical sense]]. The map f gives rise to a map T_{Σ} → O = Z_{ℓ} which we also denote by f

R_{Σ} ⟶^{φ} T_{Σ}
↘ ↓ f
O = Z_{ℓ}

18.9 The inequality #O/η ≤ #℘_{T}/℘_{T}^{2} ≤ ℘_{R}/℘_{R}^{2}

Let

ρ : G = Gal(\overline{Q}/Q) → GL(2, F_{ℓ})

be irreducible and modular with ℓ > 2. Let Σ be a finite set of primes. We assume there is a modular form f of weight 2 with coefficients in Z_{ℓ} which gives rise ρ. Let

ρ_{f,λ} : G → GL(2, Z_{ℓ})

be the representation coming from f, then ρ_{f,λ} reduces to ρ modulo ℓ.

Let R_{Σ} be the universal deformation ring, so every deformation of ρ of type Σ factors through R_{Σ} in an appropriate sense. Let T_{Σ} be the Hecke ring associated to Σ. It is a Z_{ℓ}-algebra which is free of finite rank. Furthermore

T_{Σ} ⊂ ∏_{(g,μ)∈A} O_{g,μ} = O_{f,μ} × ∏_{(g,μ)∈A−{(f,μ)}} O_{g,μ}

where A is as defined before. Define projections pr_{1} and pr_{2} onto the first and rest of the factors, respectively

pr_{1} : ∏_{(g,μ)∈A} O_{g,μ} → O = O_{f,λ}
pr_{2} : ∏_{(g,μ)∈A} O_{g,μ} → ∏_{(g,μ)≠(f,λ)} O_{g,μ}

Let φ : R_{Σ} → T_{Σ} be the map coming from the universal property of R_{Σ}. This map is surjective. The famous triangle which dominates all of the theory is

R_{Σ} ⟶^{φ} T_{Σ}
↓ pr_{1}
O = Z_{ℓ}

18.9.1 The definitions of the ideals

We now define two ideals. View T_{Σ} as sitting in the product ∏ O_{g,μ}.

1. The congruence ideal η ⊂ O is

η := O ∩ T_{Σ} = ker(pr_{2} : T_{Σ} → ∏_{(g,μ)≠(f,λ)} O_{g,μ})

2. The prime ideal ℘_{T} ⊂ T_{Σ} is

℘_{T} = ker(pr_{1} : T_{Σ} → O)

It is true that

#O/η ≤ #℘_{T} /℘_{T}^{2} .

The condition for equality is a theorem of Wiles. Theorem 18.9.1. T_{Σ} is a complete intersection iff #O/η = #℘_{T} /℘_{T}^{2} .

There is an analogous construction for

ψ = pr_{1} ∘ φ : R_{Σ} → O.

The diagram is

\begin{aligned}R_{Σ} & ⟶^{φ} & T_{Σ} \\ & ψ ↘ & ↓ pr_{1} \\ & & O \end{aligned} .

Let ℘_{R} be the kernel of ψ. From the commutativity of the above diagram we see that ψ maps ℘_{R} ⟶ ℘_{T}. Thus we have an induced map \overline{ψ} on “tangent spaces”

\overline{ψ} : ℘_{R}/℘_{R}^{2} ⟶ ℘_{T} /℘_{T}^{2} .

It follows that

#O/η ≤ #℘_{T} /℘_{T}^{2} ≤ #℘_{R}/℘_{R}^{2}.

There is an analogous theorem. Theorem 18.9.2. The above inequalities are all equalities iff

• φ : R_{Σ} → T_{Σ} is an isomorphism, and

• T_{Σ} is a complete intersection ring.

18.9. THE INEQUALITY #O/η ≤ #℘_{T} /℘_{T}^{2} ≤ ℘_{R}/℘_{R}^{2} 129

18.9.2 Aside: Selmer Groups

Let M be the set of matrices in M(2, Q_{ℓ}/Z_{ℓ}) which have trace 0. Then GL(2, Z_{ℓ}) operates on M by conjugation. Thus G acts on M via the representation ρ′ : G → GL(2, Z_{ℓ}). To ℘_{R}/℘_{R}^{2} there corresponds the Selmer group which is a subgroup of H^{1}(Gal(\overline{Q}/Q), M). The subgroup is

H_{Σ}^{1}(G, M) = Hom_{O}(℘_{R}/℘_{R}^{2}, Q_{ℓ}/Z_{ℓ}) ⊂ H^{1}(Gal(\overline{Q}/Q), M).

[[Since Flach’s thesis there has been a problem of trying to get an upper bound for the Selmer group. Wiles converted it into the above problem.]]

18.9.3 Outline of some proofs

We outline the key steps in the proof that #℘_{R}/℘_{R}^{2} ≤ #O/η.

Step 1: Σ = ∅

The key step is the minimal case when Σ = ∅. This is done in [35]. They claim to be proving the apparently weaker statement that T_{Σ} a complete intersection implies

#℘_{T} /℘_{T}^{2} = #O/η.

But in Wiles’ paper [36] he obtains the inequality

#℘_{R}/℘_{R}^{2} ≤ \frac{(#℘_{T} /℘_{T}^{2} )^{2}}{#O/η} .

Combining these two shows that

#℘_{R}/℘_{R}^{2} ≤ #O/η.

In an appendix to [35] Faltings proves directly that

#℘_{R}/℘_{R}^{2} ≤ #O/η.

[[At this point there were some remarks about why Wiles might have taken a circular route in his Annals paper. Ribet replied,

“As Serre says, it is sometimes better to leave out any psychological behavior related to how people did something but instead just report on what they did.”]]

Step 2: Passage from Σ = ∅ to σ general

The second step is the induction step in which we must understand what happens as Σ grows. Thus Σ is replaced by Σ′ = Σ ∪ {q} where q is some prime not in Σ.

We will use the following notation. The object attached to Σ′ will be denoted the same way as the object attached to Σ but with a ′. Thus (℘_{R}/℘_{R}^{2})′ denotes the Selmer group for Σ′.

The change in the Selmer group when Σ is replaced by Σ′ is completely governed by some local cohomology group. There is a constant c_{q} such that

#(℘_{R}/℘_{R}^{2})′ ≤ c_{q}#(℘_{R}/℘_{R}^{2}) ≤ c_{q}#O/η.

So we just need to know that

#O/η′ ≥ c_{q}#O/η,

i.e., that η′ is small as an ideal in O. We need a formula for the ratio of the two orders.

Let T be the anemic ring of Hecke operators on S_{2}(Γ_{0}(N_{Σ})) obtained by adjoining to Z all the Hecke operators T_{n} with n prime to ℓN_{Σ}. Let T′ be the anemic Hecke ring of Hecke operators on S_{2}(Γ_{0}(N_{Σ′})).

Since N_{Σ}|N_{Σ′} there is an inclusion

S_{2}(Γ_{0}(N_{Σ})) ↪ S_{2}(Γ_{0}(N_{Σ′})).

There is one subtlety, this injection is not equivariant for all of the Hecke operators. But this is no problem because T and T′ are anemic. So the inclusion induces a restriction map r : T′ → T.

We now introduce a relative version of η which is an ideal I ⊂ T. One way to think of I is as T ∩ T′ where T and T′ are both viewed as subrings of ∏ O_{g,μ}

T′  ⟶^{r}  T
∩  ∩
T′_{m′}  T_{m}
∩  ∩
∏_{(g,μ)} O_{g,μ} ↪ ∏_{more (g,μ)} O_{g,μ}

The definition Lenstra would give is that

I := r(Ann_{T′}(ker(r))).

The amazing formula is

η′ = η · f(I)

where f : T → O is the map induced by the modular form f. [[After introducing this definition Ogus was very curious about how deep it is, in particular, about whether its proof uses the Gorensteiness of T. Ribet said, “somehow I do not think this formula can possibly be profound.”]]

We pause with an aside to consider Wiles’ original definition of η. By duality the map f : T_{Σ} → Z_{ℓ} induces

f^{∨} : Hom_{Z_{ℓ}}(Z_{ℓ}, Z_{ℓ}) → Hom_{Z_{ℓ}}(T_{Σ}, Z_{ℓ}) ≅ T_{Σ}.

Because T_{Σ} is Gorenstein there is an isomorphism Hom_{Z_{ℓ}}(T_{Σ}, Z_{ℓ}) ≅ T_{Σ}. Now f^{∨}(id) ∈ T_{Σ} so f(f^{∨}(id)) ∈ O = Z_{ℓ}. Wiles let η = (f(f^{∨}(id))) be the ideal generated by f(f^{∨}(id)).

To finish step 2 we must show that #O/f(I) ≥ c_{q}, i.e., that “I is small”. [[I do not see how this actually finishes step 2, but it is reasonable that it should. How does this index relate to the index of f(I)η in O?]]

Let J = J_{0}(N_{Σ}) and J′ = J_{0}(N_{Σ′}). Since

S_{2}(Γ_{0}(N_{Σ})) ↪ S_{2}(Γ_{0}(N_{Σ′}))

functoriality of the Jacobian induces a map J ↪ J′. By autoduality we also obtain an injection J^{∨} ↪ J′ and J ∩ J^{∨} is a finite subgroup of J′. [[I definitely do not understand why J is not just equal to J^{⊥}. Where does the other embedding J^{⊥} ↪ J′ come from?]]

It can be seen that J ∩ J^{∨} = J[δ] for some δ ∈ T. It turns out that

Ann_{T}(J ∩ J^{∨}) = T ∩ δ End(J) ⊇ δT.

It is an observable fact that f(δT) is an ideal of O of norm c_{q}. The heart of the whole matter is to see that the inclusion

δT ⊆ T ∩ δ End(J)

is an equality after localization at m. To do this we have to know that Tate_{m}(J) ≅ T_{m}^{2}. This is equivalent to the Gorenstein business. With this in hand one can just check this equality.

[[Unfortunately we were now 10 minutes passed when the course should end. Realizing this Ribet brought everything to a close. As is the tradition at the end of a course the room erupted in applause.]] 132 CHAPTER 18. THE HECKE ALGEBRA T_{Σ}

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BIBLIOGRAPHY 135

Index

Γ(N)-structures, 23, 24

Albanese, 50, 52–54, 69

Brauer-Nesbit theorem, 68 Buzzard, 45, 73, 89, 92, 107, 111, 125

complete intersection, 73, 127–129 congruence subgroup, 2 cusp forms, 2

deformations, 109, 111, 118, 119, 124–127 Deligne, 3 descent, 44, 75

Eichler, 3 Eichler-Shimura, 25, 51, 53–55, 57, 64, 71 eigenforms, 17 Eisenstein series, 15, 19, 40, 98

finite flat representations, 111, 112 finite flat schemes, 75–77, 103, 104, 110,

111

Galois representations, 1, 7, 34 Gorenstein, 67, 69, 72 Grothendieck, 75, 76

Hecke operator, 2, 11

inner product, 16, 20, 27

Jacobian, 1, 26, 50–54, 56, 74, 75, 77, 130

Lenstra, 125, 126, 130

Mazur, 6, 70–72, 75, 105, 106 modular curves, 1, 23, 41, 45, 47, 84

newform, 31 non-Gorenstein, 73

Ogus, 50, 75, 94, 97, 104, 111, 124, 130 optic, 69, 91, 127

Selmer group, 129 semistable, 103, 105–109, 111, 115, 122,

126 Serre, 3, 5, 7–9, 15, 21, 36, 40, 44, 83, 86, 87, 91–96, 98, 100–102

Serre’s conjecture, 8

Shimura, 2, 3, 5, 26, 29, 31, 34, 52, 107 Shimura variety, 46

Tate curve, 65, 89, 104, 112