\BOOKMARK [1][-]{chapter*.2}{Preface}{}% 1
\BOOKMARK [0][-]{chapter.1}{The Main Objects}{}% 2
\BOOKMARK [1][-]{section.1.1}{Torsion points on elliptic curves}{chapter.1}% 3
\BOOKMARK [2][-]{subsection.1.1.1}{The Tate module}{section.1.1}% 4
\BOOKMARK [1][-]{section.1.2}{Galois representations}{chapter.1}% 5
\BOOKMARK [1][-]{section.1.3}{Modular forms}{chapter.1}% 6
\BOOKMARK [1][-]{section.1.4}{Hecke operators}{chapter.1}% 7
\BOOKMARK [0][-]{chapter.2}{Modular Representations and Algebraic Curves}{}% 8
\BOOKMARK [1][-]{section.2.1}{Modular forms and Arithmetic}{chapter.2}% 9
\BOOKMARK [1][-]{section.2.2}{Characters}{chapter.2}% 10
\BOOKMARK [1][-]{section.2.3}{Parity conditions}{chapter.2}% 11
\BOOKMARK [1][-]{section.2.4}{Conjectures of Serre \(mod \040version\)}{chapter.2}% 12
\BOOKMARK [1][-]{section.2.5}{General remarks on mod p Galois representations}{chapter.2}% 13
\BOOKMARK [1][-]{section.2.6}{Serre's conjecture}{chapter.2}% 14
\BOOKMARK [1][-]{section.2.7}{Wiles's perspective}{chapter.2}% 15
\BOOKMARK [0][-]{chapter.3}{Modular Forms of Level 1}{}% 16
\BOOKMARK [1][-]{section.3.1}{The Definition}{chapter.3}% 17
\BOOKMARK [1][-]{section.3.2}{Some examples and conjectures}{chapter.3}% 18
\BOOKMARK [1][-]{section.3.3}{Modular forms as functions on lattices}{chapter.3}% 19
\BOOKMARK [1][-]{section.3.4}{Hecke operators}{chapter.3}% 20
\BOOKMARK [2][-]{subsection.3.4.1}{Relations Between Hecke Operators}{section.3.4}% 21
\BOOKMARK [1][-]{section.3.5}{Hecke operators directly on q-expansions}{chapter.3}% 22
\BOOKMARK [2][-]{subsection.3.5.1}{Explicit description of sublattices}{section.3.5}% 23
\BOOKMARK [2][-]{subsection.3.5.2}{Hecke operators on q-expansions}{section.3.5}% 24
\BOOKMARK [2][-]{subsection.3.5.3}{The Hecke algebra and eigenforms}{section.3.5}% 25
\BOOKMARK [2][-]{subsection.3.5.4}{Examples}{section.3.5}% 26
\BOOKMARK [1][-]{section.3.6}{Two Conjectures about Hecke operators on level 1 modular forms}{chapter.3}% 27
\BOOKMARK [2][-]{subsection.3.6.1}{Maeda's conjecture}{section.3.6}% 28
\BOOKMARK [2][-]{subsection.3.6.2}{The Gouvea-Mazur conjecture}{section.3.6}% 29
\BOOKMARK [1][-]{section.3.7}{An Algorithm for computing characteristic polynomials of Hecke operators}{chapter.3}% 30
\BOOKMARK [2][-]{subsection.3.7.1}{Review of basic facts about modular forms}{section.3.7}% 31
\BOOKMARK [2][-]{subsection.3.7.2}{The Naive approach}{section.3.7}% 32
\BOOKMARK [2][-]{subsection.3.7.3}{The Eigenform method}{section.3.7}% 33
\BOOKMARK [2][-]{subsection.3.7.4}{How to write down an eigenvector over an extension field}{section.3.7}% 34
\BOOKMARK [2][-]{subsection.3.7.5}{Simple example: weight 36, p=3}{section.3.7}% 35
\BOOKMARK [0][-]{chapter.4}{Duality, Rationality, and Integrality}{}% 36
\BOOKMARK [1][-]{section.4.1}{Modular forms for `39`42`"613A``45`47`"603ASL2\(Z\) and Eisenstein series}{chapter.4}% 37
\BOOKMARK [1][-]{section.4.2}{Pairings between Hecke algebras and modular forms}{chapter.4}% 38
\BOOKMARK [1][-]{section.4.3}{Eigenforms}{chapter.4}% 39
\BOOKMARK [1][-]{section.4.4}{Integrality}{chapter.4}% 40
\BOOKMARK [1][-]{section.4.5}{A Result from Victor Miller's thesis}{chapter.4}% 41
\BOOKMARK [1][-]{section.4.6}{The Petersson inner product}{chapter.4}% 42
\BOOKMARK [0][-]{chapter.5}{Analytic Theory of Modular Curves}{}% 43
\BOOKMARK [1][-]{section.5.1}{The Modular group}{chapter.5}% 44
\BOOKMARK [2][-]{subsection.5.1.1}{The Upper half plane}{section.5.1}% 45
\BOOKMARK [1][-]{section.5.2}{Points on modular curves parameterize elliptic curves with extra structure}{chapter.5}% 46
\BOOKMARK [1][-]{section.5.3}{The Genus of X\(N\)}{chapter.5}% 47
\BOOKMARK [0][-]{chapter.6}{Modular Curves}{}% 48
\BOOKMARK [1][-]{section.6.1}{Cusp Forms}{chapter.6}% 49
\BOOKMARK [1][-]{section.6.2}{Modular curves}{chapter.6}% 50
\BOOKMARK [1][-]{section.6.3}{Classifying \(N\)-structures}{chapter.6}% 51
\BOOKMARK [1][-]{section.6.4}{More on integral Hecke operators}{chapter.6}% 52
\BOOKMARK [1][-]{section.6.5}{Complex conjugation}{chapter.6}% 53
\BOOKMARK [1][-]{section.6.6}{Isomorphism in the real case}{chapter.6}% 54
\BOOKMARK [1][-]{section.6.7}{The Eichler-Shimura isomorphism}{chapter.6}% 55
\BOOKMARK [0][-]{chapter.7}{Modular Symbols}{}% 56
\BOOKMARK [1][-]{section.7.1}{Modular symbols}{chapter.7}% 57
\BOOKMARK [1][-]{section.7.2}{Manin symbols}{chapter.7}% 58
\BOOKMARK [2][-]{subsection.7.2.1}{Using continued fractions to obtain surjectivity}{section.7.2}% 59
\BOOKMARK [2][-]{subsection.7.2.2}{Triangulating X\(G\) to obtain injectivity}{section.7.2}% 60
\BOOKMARK [1][-]{section.7.3}{Hecke operators}{chapter.7}% 61
\BOOKMARK [1][-]{section.7.4}{Modular symbols and rational homology}{chapter.7}% 62
\BOOKMARK [1][-]{section.7.5}{Special values of L-functions}{chapter.7}% 63
\BOOKMARK [0][-]{chapter.8}{Modular Forms of Higher Level}{}% 64
\BOOKMARK [1][-]{section.8.1}{Modular Forms on 1\(N\)}{chapter.8}% 65
\BOOKMARK [1][-]{section.8.2}{Diamond bracket and Hecke operators}{chapter.8}% 66
\BOOKMARK [2][-]{subsection.8.2.1}{Diamond bracket operators}{section.8.2}% 67
\BOOKMARK [2][-]{subsection.8.2.2}{Hecke operators on q-expansions}{section.8.2}% 68
\BOOKMARK [1][-]{section.8.3}{Old and new subspaces}{chapter.8}% 69
\BOOKMARK [0][-]{chapter.9}{Newforms and Euler Products}{}% 70
\BOOKMARK [1][-]{section.9.1}{Atkin-Lehner-Li theory}{chapter.9}% 71
\BOOKMARK [1][-]{section.9.2}{The Up operator}{chapter.9}% 72
\BOOKMARK [2][-]{subsection.9.2.1}{A Connection with Galois representations}{section.9.2}% 73
\BOOKMARK [2][-]{subsection.9.2.2}{When is Up semisimple?}{section.9.2}% 74
\BOOKMARK [2][-]{subsection.9.2.3}{An Example of non-semisimple Up}{section.9.2}% 75
\BOOKMARK [1][-]{section.9.3}{The Cusp forms are free of rank 1 over TC}{chapter.9}% 76
\BOOKMARK [2][-]{subsection.9.3.1}{Level 1}{section.9.3}% 77
\BOOKMARK [2][-]{subsection.9.3.2}{General level}{section.9.3}% 78
\BOOKMARK [1][-]{section.9.4}{Decomposing the anemic Hecke algebra}{chapter.9}% 79
\BOOKMARK [0][-]{chapter.10}{Some Explicit Genus Computations}{}% 80
\BOOKMARK [1][-]{section.10.1}{Computing the dimension of S2\(\)}{chapter.10}% 81
\BOOKMARK [1][-]{section.10.2}{Application of Riemann-Hurwitz}{chapter.10}% 82
\BOOKMARK [1][-]{section.10.3}{The Genus of X\(N\)}{chapter.10}% 83
\BOOKMARK [1][-]{section.10.4}{The Genus of X0\(N\), for N prime}{chapter.10}% 84
\BOOKMARK [1][-]{section.10.5}{Modular forms mod p}{chapter.10}% 85
\BOOKMARK [0][-]{chapter.11}{The Field of Moduli}{}% 86
\BOOKMARK [1][-]{section.11.1}{Algebraic definition of X\(N\)}{chapter.11}% 87
\BOOKMARK [1][-]{section.11.2}{Digression on moduli}{chapter.11}% 88
\BOOKMARK [1][-]{section.11.3}{When is E surjective?}{chapter.11}% 89
\BOOKMARK [1][-]{section.11.4}{Observations}{chapter.11}% 90
\BOOKMARK [1][-]{section.11.5}{A descent problem}{chapter.11}% 91
\BOOKMARK [1][-]{section.11.6}{Second look at the descent exercise}{chapter.11}% 92
\BOOKMARK [1][-]{section.11.7}{Action of `39`42`"613A``45`47`"603AGL2}{chapter.11}% 93
\BOOKMARK [0][-]{chapter.12}{Hecke Operators as Correspondences}{}% 94
\BOOKMARK [1][-]{section.12.1}{The Definition}{chapter.12}% 95
\BOOKMARK [1][-]{section.12.2}{Maps induced by correspondences}{chapter.12}% 96
\BOOKMARK [1][-]{section.12.3}{Induced maps on Jacobians of curves}{chapter.12}% 97
\BOOKMARK [1][-]{section.12.4}{More on Hecke operators}{chapter.12}% 98
\BOOKMARK [1][-]{section.12.5}{Hecke operators acting on Jacobians}{chapter.12}% 99
\BOOKMARK [2][-]{subsection.12.5.1}{The Albanese Map}{section.12.5}% 100
\BOOKMARK [2][-]{subsection.12.5.2}{The Hecke algebra}{section.12.5}% 101
\BOOKMARK [1][-]{section.12.6}{The Eichler-Shimura relation}{chapter.12}% 102
\BOOKMARK [1][-]{section.12.7}{Applications of the Eichler-Shimura relation}{chapter.12}% 103
\BOOKMARK [2][-]{subsection.12.7.1}{The Characteristic polynomial of Frobenius}{section.12.7}% 104
\BOOKMARK [2][-]{subsection.12.7.2}{The Cardinality of J0\(N\)\(Fp\)}{section.12.7}% 105
\BOOKMARK [0][-]{chapter.13}{Abelian Varieties}{}% 106
\BOOKMARK [1][-]{section.13.1}{Abelian varieties}{chapter.13}% 107
\BOOKMARK [1][-]{section.13.2}{Complex tori}{chapter.13}% 108
\BOOKMARK [2][-]{subsection.13.2.1}{Homomorphisms}{section.13.2}% 109
\BOOKMARK [2][-]{subsection.13.2.2}{Isogenies}{section.13.2}% 110
\BOOKMARK [2][-]{subsection.13.2.3}{Endomorphisms}{section.13.2}% 111
\BOOKMARK [1][-]{section.13.3}{Abelian varieties as complex tori}{chapter.13}% 112
\BOOKMARK [2][-]{subsection.13.3.1}{Hermitian and Riemann forms}{section.13.3}% 113
\BOOKMARK [2][-]{subsection.13.3.2}{Complements, quotients, and semisimplicity of the endomorphism algebra}{section.13.3}% 114
\BOOKMARK [2][-]{subsection.13.3.3}{Theta functions}{section.13.3}% 115
\BOOKMARK [1][-]{section.13.4}{A Summary of duality and polarizations}{chapter.13}% 116
\BOOKMARK [2][-]{subsection.13.4.1}{Sheaves}{section.13.4}% 117
\BOOKMARK [2][-]{subsection.13.4.2}{The Picard group}{section.13.4}% 118
\BOOKMARK [2][-]{subsection.13.4.3}{The Dual as a complex torus}{section.13.4}% 119
\BOOKMARK [2][-]{subsection.13.4.4}{The N\351ron-Severi group and polarizations}{section.13.4}% 120
\BOOKMARK [2][-]{subsection.13.4.5}{The Dual is functorial}{section.13.4}% 121
\BOOKMARK [1][-]{section.13.5}{Jacobians of curves}{chapter.13}% 122
\BOOKMARK [2][-]{subsection.13.5.1}{Divisors on curves and linear equivalence}{section.13.5}% 123
\BOOKMARK [2][-]{subsection.13.5.2}{Algebraic definition of the Jacobian}{section.13.5}% 124
\BOOKMARK [2][-]{subsection.13.5.3}{The Abel-Jacobi theorem}{section.13.5}% 125
\BOOKMARK [2][-]{subsection.13.5.4}{Every abelian variety is a quotient of a Jacobian}{section.13.5}% 126
\BOOKMARK [1][-]{section.13.6}{N\351ron models}{chapter.13}% 127
\BOOKMARK [2][-]{subsection.13.6.1}{What are N\351ron models?}{section.13.6}% 128
\BOOKMARK [2][-]{subsection.13.6.2}{The Birch and Swinnerton-Dyer conjecture and N\351ron models}{section.13.6}% 129
\BOOKMARK [2][-]{subsection.13.6.3}{Functorial properties of Neron models}{section.13.6}% 130
\BOOKMARK [0][-]{chapter.14}{Abelian Varieties Attached to Modular Forms}{}% 131
\BOOKMARK [1][-]{section.14.1}{Decomposition of the Hecke algebra}{chapter.14}% 132
\BOOKMARK [2][-]{subsection.14.1.1}{The Dimension of the algebras Lf}{section.14.1}% 133
\BOOKMARK [1][-]{section.14.2}{Decomposition of J1\(N\)}{chapter.14}% 134
\BOOKMARK [2][-]{subsection.14.2.1}{Aside: intersections and congruences}{section.14.2}% 135
\BOOKMARK [1][-]{section.14.3}{Galois representations attached to Af}{chapter.14}% 136
\BOOKMARK [2][-]{subsection.14.3.1}{The Weil pairing}{section.14.3}% 137
\BOOKMARK [2][-]{subsection.14.3.2}{The Determinant}{section.14.3}% 138
\BOOKMARK [1][-]{section.14.4}{Remarks about the modular polarization}{chapter.14}% 139
\BOOKMARK [0][-]{chapter.15}{Modularity of Abelian Varieties}{}% 140
\BOOKMARK [1][-]{section.15.1}{Modularity over Q}{chapter.15}% 141
\BOOKMARK [1][-]{section.15.2}{Modularity of elliptic curves over Q}{chapter.15}% 142
\BOOKMARK [1][-]{section.15.3}{Modularity of abelian varieties over Q}{chapter.15}% 143
\BOOKMARK [0][-]{chapter.16}{L-functions}{}% 144
\BOOKMARK [1][-]{section.16.1}{L-functions attached to modular forms}{chapter.16}% 145
\BOOKMARK [2][-]{subsection.16.1.1}{Analytic continuation and functional equations}{section.16.1}% 146
\BOOKMARK [2][-]{subsection.16.1.2}{A Conjecture about nonvanishing of L\(f,k/2\)}{section.16.1}% 147
\BOOKMARK [2][-]{subsection.16.1.3}{Euler products}{section.16.1}% 148
\BOOKMARK [2][-]{subsection.16.1.4}{Visualizing L-function}{section.16.1}% 149
\BOOKMARK [0][-]{chapter.17}{The Birch and Swinnerton-Dyer Conjecture}{}% 150
\BOOKMARK [1][-]{section.17.1}{The Rank conjecture}{chapter.17}% 151
\BOOKMARK [1][-]{section.17.2}{Refined rank zero conjecture}{chapter.17}% 152
\BOOKMARK [2][-]{subsection.17.2.1}{The Number of real components}{section.17.2}% 153
\BOOKMARK [2][-]{subsection.17.2.2}{The Manin index}{section.17.2}% 154
\BOOKMARK [2][-]{subsection.17.2.3}{The Real volume A}{section.17.2}% 155
\BOOKMARK [2][-]{subsection.17.2.4}{The Period mapping}{section.17.2}% 156
\BOOKMARK [2][-]{subsection.17.2.5}{The Manin-Drinfeld theorem}{section.17.2}% 157
\BOOKMARK [2][-]{subsection.17.2.6}{The Period lattice}{section.17.2}% 158
\BOOKMARK [2][-]{subsection.17.2.7}{The Special value L\(A,1\)}{section.17.2}% 159
\BOOKMARK [2][-]{subsection.17.2.8}{Rationality of L\(A,1\)/A}{section.17.2}% 160
\BOOKMARK [1][-]{section.17.3}{General refined conjecture}{chapter.17}% 161
\BOOKMARK [1][-]{section.17.4}{The Conjecture for non-modular abelian varieties}{chapter.17}% 162
\BOOKMARK [1][-]{section.17.5}{Visibility of Shafarevich-Tate groups}{chapter.17}% 163
\BOOKMARK [2][-]{subsection.17.5.1}{Definitions}{section.17.5}% 164
\BOOKMARK [2][-]{subsection.17.5.2}{Every element of `39`42`"613A``45`47`"603AH1\(K,A\) is visible somewhere}{section.17.5}% 165
\BOOKMARK [2][-]{subsection.17.5.3}{Visibility in the context of modularity}{section.17.5}% 166
\BOOKMARK [2][-]{subsection.17.5.4}{Future directions}{section.17.5}% 167
\BOOKMARK [1][-]{section.17.6}{Kolyvagin's Euler system of Heegner points}{chapter.17}% 168
\BOOKMARK [2][-]{subsection.17.6.1}{A Heegner point when N=11}{section.17.6}% 169
\BOOKMARK [2][-]{subsection.17.6.2}{Kolyvagin's Euler system for curves of rank at least 2}{section.17.6}% 170
\BOOKMARK [0][-]{chapter.18}{The Gorenstein Property for Hecke Algebras}{}% 171
\BOOKMARK [1][-]{section.18.1}{Mod \040representations associated to modular forms}{chapter.18}% 172
\BOOKMARK [1][-]{section.18.2}{The Gorenstein property}{chapter.18}% 173
\BOOKMARK [1][-]{section.18.3}{Proof of the Gorenstein property}{chapter.18}% 174
\BOOKMARK [2][-]{subsection.18.3.1}{Vague comments}{section.18.3}% 175
\BOOKMARK [1][-]{section.18.4}{Finite flat group schemes}{chapter.18}% 176
\BOOKMARK [1][-]{section.18.5}{Reformulation of V=W problem}{chapter.18}% 177
\BOOKMARK [1][-]{section.18.6}{Dieudonn\351 theory}{chapter.18}% 178
\BOOKMARK [1][-]{section.18.7}{The proof: part II}{chapter.18}% 179
\BOOKMARK [1][-]{section.18.8}{Key result of Boston-Lenstra-Ribet}{chapter.18}% 180
\BOOKMARK [0][-]{chapter.19}{Local Properties of }{}% 181
\BOOKMARK [1][-]{section.19.1}{Definitions}{chapter.19}% 182
\BOOKMARK [1][-]{section.19.2}{Local properties at primes pN}{chapter.19}% 183
\BOOKMARK [1][-]{section.19.3}{Weil-Deligne Groups}{chapter.19}% 184
\BOOKMARK [1][-]{section.19.4}{Local properties at primes pN}{chapter.19}% 185
\BOOKMARK [1][-]{section.19.5}{Definition of the reduced conductor}{chapter.19}% 186
\BOOKMARK [0][-]{chapter.20}{Adelic Representations}{}% 187
\BOOKMARK [1][-]{section.20.1}{Adelic representations associated to modular forms}{chapter.20}% 188
\BOOKMARK [1][-]{section.20.2}{More local properties of the .}{chapter.20}% 189
\BOOKMARK [2][-]{subsection.20.2.1}{Possibilities for p}{section.20.2}% 190
\BOOKMARK [2][-]{subsection.20.2.2}{The case =p}{section.20.2}% 191
\BOOKMARK [2][-]{subsection.20.2.3}{Tate curves}{section.20.2}% 192
\BOOKMARK [0][-]{chapter.21}{Serre's Conjecture}{}% 193
\BOOKMARK [1][-]{section.21.1}{The Family of -adic representations attached to a newform}{chapter.21}% 194
\BOOKMARK [1][-]{section.21.2}{Serre's Conjecture A}{chapter.21}% 195
\BOOKMARK [2][-]{subsection.21.2.1}{The Field of definition of }{section.21.2}% 196
\BOOKMARK [1][-]{section.21.3}{Serre's Conjecture B}{chapter.21}% 197
\BOOKMARK [1][-]{section.21.4}{The Level}{chapter.21}% 198
\BOOKMARK [2][-]{subsection.21.4.1}{Remark on the case N\(\)=1}{section.21.4}% 199
\BOOKMARK [2][-]{subsection.21.4.2}{Remark on the proof of Conjecture B}{section.21.4}% 200
\BOOKMARK [1][-]{section.21.5}{The Weight}{chapter.21}% 201
\BOOKMARK [2][-]{subsection.21.5.1}{The Weight modulo -1}{section.21.5}% 202
\BOOKMARK [2][-]{subsection.21.5.2}{Tameness at }{section.21.5}% 203
\BOOKMARK [2][-]{subsection.21.5.3}{Fundamental characters of the tame extension}{section.21.5}% 204
\BOOKMARK [2][-]{subsection.21.5.4}{The Pair of characters associated to }{section.21.5}% 205
\BOOKMARK [2][-]{subsection.21.5.5}{Recipe for the weight}{section.21.5}% 206
\BOOKMARK [2][-]{subsection.21.5.6}{The World's first view of fundamental characters}{section.21.5}% 207
\BOOKMARK [2][-]{subsection.21.5.7}{Fontaine's theorem}{section.21.5}% 208
\BOOKMARK [2][-]{subsection.21.5.8}{Guessing the weight \(level 2 case\)}{section.21.5}% 209
\BOOKMARK [2][-]{subsection.21.5.9}{-cycles}{section.21.5}% 210
\BOOKMARK [2][-]{subsection.21.5.10}{Edixhoven's paper}{section.21.5}% 211
\BOOKMARK [1][-]{section.21.6}{The Character}{chapter.21}% 212
\BOOKMARK [2][-]{subsection.21.6.1}{A Counterexample}{section.21.6}% 213
\BOOKMARK [1][-]{section.21.7}{The Weight revisited: level 1 case}{chapter.21}% 214
\BOOKMARK [2][-]{subsection.21.7.1}{Companion forms}{section.21.7}% 215
\BOOKMARK [2][-]{subsection.21.7.2}{The Weight: the remaining level 1 case}{section.21.7}% 216
\BOOKMARK [2][-]{subsection.21.7.3}{Finiteness}{section.21.7}% 217
\BOOKMARK [0][-]{chapter.22}{Fermat's Last Theorem}{}% 218
\BOOKMARK [1][-]{section.22.1}{The application to Fermat}{chapter.22}% 219
\BOOKMARK [1][-]{section.22.2}{Modular elliptic curves}{chapter.22}% 220
\BOOKMARK [0][-]{chapter.23}{Deformations}{}% 221
\BOOKMARK [1][-]{section.23.1}{Introduction}{chapter.23}% 222
\BOOKMARK [1][-]{section.23.2}{Condition \(*\)}{chapter.23}% 223
\BOOKMARK [2][-]{subsection.23.2.1}{Finite flat representations}{section.23.2}% 224
\BOOKMARK [1][-]{section.23.3}{Classes of liftings}{chapter.23}% 225
\BOOKMARK [2][-]{subsection.23.3.1}{The case p=}{section.23.3}% 226
\BOOKMARK [2][-]{subsection.23.3.2}{The case p=}{section.23.3}% 227
\BOOKMARK [1][-]{section.23.4}{Wiles's Hecke algebra}{chapter.23}% 228
\BOOKMARK [0][-]{chapter.24}{The Hecke Algebra T}{}% 229
\BOOKMARK [1][-]{section.24.1}{The Hecke algebra}{chapter.24}% 230
\BOOKMARK [1][-]{section.24.2}{The Maximal ideal in R}{chapter.24}% 231
\BOOKMARK [2][-]{subsection.24.2.1}{Strip away certain Euler factors}{section.24.2}% 232
\BOOKMARK [2][-]{subsection.24.2.2}{Make into an eigenform for U}{section.24.2}% 233
\BOOKMARK [1][-]{section.24.3}{The Galois representation}{chapter.24}% 234
\BOOKMARK [2][-]{subsection.24.3.1}{The Structure of Tm}{section.24.3}% 235
\BOOKMARK [2][-]{subsection.24.3.2}{The Philosophy in this picture}{section.24.3}% 236
\BOOKMARK [2][-]{subsection.24.3.3}{Massage }{section.24.3}% 237
\BOOKMARK [2][-]{subsection.24.3.4}{Massage '}{section.24.3}% 238
\BOOKMARK [2][-]{subsection.24.3.5}{Representations from modular forms mod }{section.24.3}% 239
\BOOKMARK [2][-]{subsection.24.3.6}{Representations from modular forms mod n}{section.24.3}% 240
\BOOKMARK [1][-]{section.24.4}{' is of type }{chapter.24}% 241
\BOOKMARK [1][-]{section.24.5}{Isomorphism between Tm and RmR}{chapter.24}% 242
\BOOKMARK [1][-]{section.24.6}{Deformations}{chapter.24}% 243
\BOOKMARK [1][-]{section.24.7}{Wiles's main conjecture}{chapter.24}% 244
\BOOKMARK [1][-]{section.24.8}{T is a complete intersection}{chapter.24}% 245
\BOOKMARK [1][-]{section.24.9}{The Inequality \043O/\043T/T2\043 R/R2}{chapter.24}% 246
\BOOKMARK [2][-]{subsection.24.9.1}{The Definitions of the ideals}{section.24.9}% 247
\BOOKMARK [2][-]{subsection.24.9.2}{Aside: Selmer groups}{section.24.9}% 248
\BOOKMARK [2][-]{subsection.24.9.3}{Outline of some proofs}{section.24.9}% 249
\BOOKMARK [0][-]{chapter.25}{Computing with Modular Forms and Abelian Varieties}{}% 250
\BOOKMARK [0][-]{chapter.26}{The Modular Curve X0\(389\)}{}% 251
\BOOKMARK [1][-]{section.26.1}{Factors of J0\(389\)}{chapter.26}% 252
\BOOKMARK [2][-]{subsection.26.1.1}{Newforms of level 389}{section.26.1}% 253
\BOOKMARK [2][-]{subsection.26.1.2}{Isogeny structure}{section.26.1}% 254
\BOOKMARK [2][-]{subsection.26.1.3}{Mordell-Weil ranks}{section.26.1}% 255
\BOOKMARK [1][-]{section.26.2}{The Hecke algebra}{chapter.26}% 256
\BOOKMARK [2][-]{subsection.26.2.1}{The Discriminant is divisible by p}{section.26.2}% 257
\BOOKMARK [2][-]{subsection.26.2.2}{Congruences primes in Sp+1\(0\(1\)\)}{section.26.2}% 258
\BOOKMARK [1][-]{section.26.3}{Supersingular points in characteristic 389}{chapter.26}% 259
\BOOKMARK [2][-]{subsection.26.3.1}{The Supersingular j-invariants in characteristic 389}{section.26.3}% 260
\BOOKMARK [1][-]{section.26.4}{Miscellaneous}{chapter.26}% 261
\BOOKMARK [2][-]{subsection.26.4.1}{The Shafarevich-Tate group}{section.26.4}% 262
\BOOKMARK [2][-]{subsection.26.4.2}{Weierstrass points on X0+\(p\)}{section.26.4}% 263
\BOOKMARK [2][-]{subsection.26.4.3}{A Property of the plus part of the integral homology}{section.26.4}% 264
\BOOKMARK [2][-]{subsection.26.4.4}{The Field generated by points of small prime order on an elliptic curve}{section.26.4}% 265
\BOOKMARK [0][-]{chapter*.9}{References}{}% 266