Modular Forms, Hecke Operators, and Modular Abelian Varieties Kenneth A. Ribet William A. Stein December 9, 2003 ii Contents Preface . 1 1 The Main objects 3 1.1 Torsion points . 3 1.1.1 The Tate module . 3 1.2 Galois representations . 4 1.3 Modular forms . 5 1.4 Hecke operators . 5 2 Modular representations and algebraic curves 7 2.1 Arithmetic of modular forms . 7 2.2 Characters . 8 2.3 Parity Conditions . 9 2.4 Conjectures of Serre (mod ` version) . 9 2.5 General remarks on mod p Galois representations . 9 2.6 Serre's Conjecture . 10 2.7 Wiles's Perspective . 10 3 Modular Forms of Level 1 13 3.1 The De¯nition . 13 3.2 Some Examples and Conjectures . 14 3.3 Modular Forms as Functions on Lattices . 15 3.4 Hecke Operators . 17 3.5 Hecke Operators Directly on q-expansions . 17 3.5.1 Explicit Description of Sublattices . 18 3.5.2 Hecke operators on q-expansions . 19 3.5.3 The Hecke Algebra and Eigenforms . 20 3.5.4 Examples . 20 3.6 Two Conjectures about Hecke Operators on Level 1 Modular Forms 21 iv Contents 3.6.1 Maeda's Conjecture . 21 3.6.2 The Gouvea-Mazur Conjecture . 22 3.7 A Modular Algorithm for Computing Characteristic Polynomials of Hecke Operators . 23 3.7.1 Review of Basic Facts About Modular Forms . 23 3.7.2 The Naive Approach . 24 3.7.3 The Eigenform Method . 24 3.7.4 How to Write Down an Eigenvector over an Extension Field 26 3.7.5 Simple Example: Weight 36, p = 3 . 26 4 Analytic theory of modular curves 29 4.1 The Modular group . 29 4.1.1 The Upper half plane . 29 4.1.2 Fundamental domain for the modular group . 30 4.1.3 Conjugating an element of the upper half plane into the fun- damental domain . 30 4.1.4 Writing an element in terms of generators . 30 4.1.5 Generators for the modular group . 30 4.2 Congruence subgroups . 30 4.2.1 De¯nition . 30 4.2.2 Fundamental domains for congruence subgroups . 30 4.2.3 Coset representatives . 30 4.2.4 Generators for congruence subgroups . 30 4.3 Modular curves . 30 4.3.1 The upper half plane is a disk . 30 4.3.2 The upper half plane union the cusps . 30 4.3.3 The Poincar¶e metric . 30 4.3.4 Fuchsian groups and Riemann surfaces . 30 4.3.5 Riemann surfaces attached to congruence subgroups . 30 4.4 Points on modular curves parameterize elliptic curves with extra structure . 31 4.5 The Genus of X(N) . 33 5 Modular Symbols 37 5.1 Modular symbols . 37 5.2 Manin symbols . 38 5.2.1 Using continued fractions to obtain surjectivity . 39 5.2.2 Triangulating X(G) to obtain injectivity . 40 5.3 Hecke Operators . 44 5.4 Modular Symbols and Rational Homology . 45 5.5 Special Values of L-functions . 46 6 Modular Forms of Higher Level 49 6.1 Modular Forms on ¡1(N) . 49 6.2 The Diamond Bracket and Hecke Operators . 50 6.2.1 Diamond Bracket Operators . 50 6.2.2 Hecke Operators on q-expansions . 52 6.3 Old and New Subspaces . 52 7 Newforms and Euler Products 55 Contents v 7.1 Atkin, Lehner, Li Theory . 55 7.2 The Up Operator . 59 7.2.1 A Connection with Galois Representations . 60 7.2.2 When is Up Semisimple? . 60 7.2.3 An Example of Non-semisimple Up . 61 7.3 The Cusp Forms are Free of Rank One over TC . 61 7.3.1 Level 1 . 61 7.3.2 General Level . 62 7.4 Decomposing the Anemic Hecke Algebra . 63 8 Hecke operators as correspondences 65 8.1 The De¯nition . 65 8.2 Maps induced by correspondences . 67 8.3 Induced maps on Jacobians of curves . 68 8.4 More on Hecke operators . 69 8.5 Hecke operators acting on Jacobians . 69 8.5.1 The Albanese Map . 70 8.5.2 The Hecke Algebra . 71 8.6 The Eichler-Shimura Relation . 72 8.7 Applications of the Eichler-Shimura Relation . 75 8.7.1 The Characteristic Polynomial of Frobenius . 75 8.7.2 The Cardinality of J0(N)(Fp) . 77 9 Abelian Varieties 79 9.1 Abelian Varieties . 79 9.2 Complex Tori . 80 9.2.1 Homomorphisms . 80 9.2.2 Isogenies . 82 9.2.3 Endomorphisms . 83 9.3 Abelian Varieties as Complex Tori . 84 9.3.1 Hermitian and Riemann Forms . 84 9.3.2 Complements, Quotients, and Semisimplicity of the Endo- morphism Algebra . 85 9.3.3 Theta Functions . 87 9.4 A Summary of Duality and Polarizations . 88 9.4.1 Sheaves . 88 9.4.2 The Picard Group . 88 9.4.3 The Dual as a Complex Torus . 88 9.4.4 The N¶eron-Several Group and Polarizations . 89 9.4.5 The Dual is Functorial . 89 9.5 Jacobians of Curves . 90 9.5.1 Divisors on Curves and Linear Equivalence . 90 9.5.2 Algebraic De¯nition of the Jacobian . 91 9.5.3 The Abel-Jacobi Theorem . 92 9.5.4 Every abelian variety is a quotient of a Jacobian . 94 9.6 N¶eron Models . 95 9.6.1 What are N¶eron Models? . 96 9.6.2 The Birch and Swinnerton-Dyer Conjecture and N¶eron Models 97 9.6.3 Functorial Properties of Neron Models . 99 vi Contents 10 Abelian Varieties Attached to Modular Forms 101 10.1 Decomposition of the Hecke Algebra . 101 10.1.1 The Dimension of Lf . 102 10.2 Decomposition of J1(N) . 103 10.2.1 Aside: Intersections and Congruences . 104 10.3 Galois Representations Attached to Af . 104 10.3.1 The Weil Pairing . 106 10.3.2 The Determinant . 107 10.4 Remarks About the Modular Polarization . 108 11 Modularity of Abelian Varieties 111 11.1 Modularity Over Q . 111 11.2 Modularity of Elliptic Curves over Q . 114 12 L-functions 115 12.1 L-functions Attached to Modular Forms . 115 12.1.1 Analytic Continuation and Functional Equation . 116 12.1.2 A Conjecture About Nonvanishing of L(f; k=2) . 117 12.1.3 Euler Products . 118 12.1.4 Visualizing L-function . 119 13 The Birch and Swinnerton-Dyer Conjecture 121 13.1 The Rank Conjecture . 121 13.2 Re¯ned Rank Zero Conjecture . 125 13.2.1 The Number of Real Components . 125 13.2.2 The Manin Index . 125 13.2.3 The Real Volume ­A . 127 13.2.4 The Period Mapping . 127 13.2.5 Manin-Drinfeld Theorem . 128 13.2.6 The Period Lattice . 128 13.2.7 The Special Value L(A; 1) . 128 13.2.8 Rationality of L(A; 1)=­A . 129 13.3 General Re¯ned Conjecture . 131 13.4 The Conjecture for Non-Modular Abelian Varieties . 132 13.5 Visibility of Shafarevich-Tate Groups . 133 13.5.1 De¯nitions . 133 13.5.2 Every Element of H1(K; A) is Visible Somewhere . 134 13.5.3 Visibility in the Context of Modularity . 135 13.5.4 Future Directions . ..