Lectures on modular forms and Hecke operators

Preface......Page 10
The Tate module......Page 12
Galois representations......Page 13
Modular forms......Page 14
Hecke operators......Page 15
Modular forms and Arithmetic......Page 16
Conjectures of Serre (mod  version)......Page 18
General remarks on mod p Galois representations......Page 19
Wiles's perspective......Page 20
The Definition......Page 22
Some examples and conjectures......Page 23
Modular forms as functions on lattices......Page 24
Hecke operators......Page 26
Relations Between Hecke Operators......Page 27
Hecke operators directly on q-expansions......Page 28
Explicit description of sublattices......Page 29
Hecke operators on q-expansions......Page 30
The Hecke algebra and eigenforms......Page 31
Examples......Page 32
The Gouvea-Mazur conjecture......Page 33
An Algorithm for computing characteristic polynomials of Hecke operators......Page 34
The Naive approach......Page 35
The Eigenform method......Page 36
How to write down an eigenvector over an extension field......Page 37
Simple example: weight 36, p=3......Page 38
Modular forms for `39`42`"613A``45`47`"603ASL2(Z) and Eisenstein series......Page 40
Pairings between Hecke algebras and modular forms......Page 41
Eigenforms......Page 42
A Result from Victor Miller's thesis......Page 43
The Petersson inner product......Page 44
The Upper half plane......Page 48
Points on modular curves parameterize elliptic curves with extra structure......Page 49
The Genus of X(N)......Page 52
Modular curves......Page 56
Classifying (N)-structures......Page 57
Isomorphism in the real case......Page 58
The Eichler-Shimura isomorphism......Page 59
Modular symbols......Page 62
Manin symbols......Page 63
Using continued fractions to obtain surjectivity......Page 64
Triangulating X(G) to obtain injectivity......Page 65
Hecke operators......Page 68
Modular symbols and rational homology......Page 69
Special values of L-functions......Page 70
Modular Forms on 1(N)......Page 74
Diamond bracket operators......Page 75
Old and new subspaces......Page 77
Atkin-Lehner-Li theory......Page 80
The Up operator......Page 84
A Connection with Galois representations......Page 85
Level 1......Page 86
General level......Page 87
Decomposing the anemic Hecke algebra......Page 89
Computing the dimension of S2()......Page 92
Application of Riemann-Hurwitz......Page 93
The Genus of X(N)......Page 94
Modular forms mod p......Page 95
Algebraic definition of X(N)......Page 98
Digression on moduli......Page 99
When is E surjective?......Page 100
Observations......Page 101
A descent problem......Page 102
Second look at the descent exercise......Page 103
Action of `39`42`"613A``45`47`"603AGL2......Page 104
The Definition......Page 106
Maps induced by correspondences......Page 108
More on Hecke operators......Page 109
Hecke operators acting on Jacobians......Page 110
The Albanese Map......Page 111
The Eichler-Shimura relation......Page 112
The Characteristic polynomial of Frobenius......Page 116
The Cardinality of J0(N)(Fp)......Page 118
Abelian varieties......Page 120
Homomorphisms......Page 121
Isogenies......Page 123
Abelian varieties as complex tori......Page 124
Hermitian and Riemann forms......Page 125
Complements, quotients, and semisimplicity of the endomorphism algebra......Page 126
A Summary of duality and polarizations......Page 128
The Dual as a complex torus......Page 129
Jacobians of curves......Page 130
Divisors on curves and linear equivalence......Page 131
Algebraic definition of the Jacobian......Page 132
The Abel-Jacobi theorem......Page 133
Every abelian variety is a quotient of a Jacobian......Page 134
What are Néron models?......Page 136
The Birch and Swinnerton-Dyer conjecture and Néron models......Page 138
Functorial properties of Neron models......Page 140
Decomposition of the Hecke algebra......Page 142
The Dimension of the algebras Lf......Page 143
Decomposition of J1(N)......Page 144
Aside: intersections and congruences......Page 145
Galois representations attached to Af......Page 146
The Weil pairing......Page 147
The Determinant......Page 149
Remarks about the modular polarization......Page 150
Modularity over Q......Page 152
Modularity of abelian varieties over Q......Page 155
L-functions attached to modular forms......Page 158
Analytic continuation and functional equations......Page 159
Euler products......Page 161
Visualizing L-function......Page 162
The Rank conjecture......Page 164
Refined rank zero conjecture......Page 166
The Manin index......Page 167
The Real volume A......Page 168
The Period lattice......Page 169
Rationality of L(A,1)/A......Page 170
The Conjecture for non-modular abelian varieties......Page 172
Visibility of Shafarevich-Tate groups......Page 173
Definitions......Page 174
Visibility in the context of modularity......Page 175
Future directions......Page 177
Kolyvagin's Euler system of Heegner points......Page 178
A Heegner point when N=11......Page 187
Kolyvagin's Euler system for curves of rank at least 2......Page 188
Mod  representations associated to modular forms......Page 190
The Gorenstein property......Page 193
Proof of the Gorenstein property......Page 195
Vague comments......Page 198
Reformulation of V=W problem......Page 199
Dieudonné theory......Page 200
The proof: part II......Page 201
Key result of Boston-Lenstra-Ribet......Page 203
Definitions......Page 206
Local properties at primes pN......Page 207
Definition of the reduced conductor......Page 208
Adelic representations associated to modular forms......Page 210
More local properties of the .......Page 213
Possibilities for p......Page 214
The case =p......Page 215
Tate curves......Page 216
Serre's Conjecture......Page 218
Serre's Conjecture A......Page 219
The Field of definition of......Page 220
The Level......Page 221
Remark on the case N()=1......Page 222
Remark on the proof of Conjecture B......Page 223
Tameness at......Page 224
Fundamental characters of the tame extension......Page 225
The Pair of characters associated to......Page 226
Recipe for the weight......Page 227
Guessing the weight (level 2 case)......Page 228
-cycles......Page 229
The Character......Page 231
A Counterexample......Page 233
Companion forms......Page 234
The Weight: the remaining level 1 case......Page 235
Finiteness......Page 236
The application to Fermat......Page 238
Modular elliptic curves......Page 240
Introduction......Page 242
Condition (*)......Page 243
The case p=......Page 244
The case p=......Page 245
Wiles's Hecke algebra......Page 246
The Hecke algebra......Page 248
Strip away certain Euler factors......Page 250
Make into an eigenform for U......Page 251
The Galois representation......Page 252
Massage......Page 253
Massage '......Page 254
Representations from modular forms mod n......Page 255
' is of type......Page 256
Isomorphism between Tm and RmR......Page 257
Deformations......Page 258
Wiles's main conjecture......Page 259
The Inequality #O/#T/T2# R/R2......Page 261
The Definitions of the ideals......Page 262
Outline of some proofs......Page 263
Computing with Modular Forms and Abelian Varieties......Page 266
The Modular Curve X0(389)......Page 268
Newforms of level 389......Page 269
Mordell-Weil ranks......Page 270
The Discriminant is divisible by p......Page 271
Congruences primes in Sp+1(0(1))......Page 272
The Shafarevich-Tate group......Page 273
The Field generated by points of small prime order on an elliptic curve......Page 274
References......Page 276