Ranks of Elliptic Curves and Random Matrix Theory

1 A concrete introduction to elliptic curves......Page 14
2 Variation of the rank in families of elliptic curves......Page 34
1 Introduction......Page 60
2 What is a family?......Page 62
3 Elliptic curve families......Page 67
4 Re ned modeling......Page 72
1 Introduction......Page 78
2 Bounding the average rank......Page 79
3 Nonvanishing results......Page 90
1 Introduction......Page 100
2 Random matrix calculations......Page 107
3 Discussion......Page 111
4 Acknowledgments......Page 112
Function elds and random matrices......Page 116
1 Function elds......Page 117
2......Page 130
3 Families of......Page 139
4 Further reading......Page 146
1 Introduction......Page 150
2 Weyl Integration Formula......Page 151
3 Schur functions as characters of the unitary group......Page 156
4 Frobenius-Schur duality & moments of traces......Page 159
5 Moments of characteristic polynomials......Page 162
6 Averages of ratios of characteristic polynomials......Page 164
7 Combinatorial de nition of Schur functions and some of its consequences......Page 167
The distribution of ranks in families of quadratic twists of elliptic curves......Page 178
1 Introduction......Page 184
2 The general method......Page 185
3 Rank......Page 186
4 Root numbers......Page 192
5 Rank......Page 193
1 Introduction......Page 196
2 Discussion......Page 197
3 Tamagawa numbers......Page 198
5 Acknowledgments......Page 199
1 Notations and introduction......Page 202
2 The example......Page 203
3 Generalization......Page 206
1 Introduction......Page 208
2 A model from Heegner points (largely due to Birch)......Page 210
3 Alternative ideas......Page 211
4 Data......Page 213
5 Conclusion......Page 218
6 Acknowledgments......Page 219
1 Introduction......Page 222
2 Moments of......Page 224
3 Vanishings of......Page 225
4 Evaluating the rst two terms of......Page 227
5 Conjecture for the rst two terms in......Page 230
6 Numerical Data......Page 231
Fudge Factors in the Birch and Swinnerton-Dyer Conjecture......Page 240
1 Introduction......Page 244
2 Numerical data......Page 245
4 Random matrix models......Page 248
5 Acknowledgments......Page 251
1 Introduction......Page 254
2 Special values and modular symbols......Page 256
3 Discretisation......Page 258
4 Unitary Random Matrices......Page 261
5 Numerical Evidence......Page 264
1......Page 267
2......Page 268
3......Page 269
4......Page 271
5......Page 276
1 Introduction......Page 280
2 Quaternion algebras and Brandt matrices......Page 281
3 Construction of......Page 283
4 Examples......Page 287
1 Introduction......Page 296
2 Quaternion algebras and Shimura correspondence......Page 297
4 On certain non-maximal orders and level......Page 299
5 An Algorithm for the Real Quadratic Twists......Page 300
6 Example: level......Page 301
7 Example: level......Page 304
8 Example: level......Page 312
9 Example: level......Page 314
1 Introduction......Page 322
2 Spherical polynomials and......Page 323
3 Two modular forms of weight......Page 325
Heuristics on class groups and on Tate-Shafarevich groups: The magic of the Cohen-Lenstra heuristics......Page 330
1 Analogy between number elds and elliptic curves de ned over......Page 331
2 Heuristics on class groups of quadratic number elds......Page 333
3 Heuristics on Tate-Shafarevich groups of elliptic curves......Page 339
4 Quadratic twist families......Page 344
A Note on the 2-Part of X for theCongruent Number Curves......Page 348
2-Descent Through the Ages......Page 352
Index......Page 364