Based on a workshop on Special Values of Rankin L-series held at the MSRI in December 2001, this volume presents thirteen articles written by leading contributors on the history of the Gross-Zagier formula and recent developments. Topics include the theory of complex multiplication, automorphic forms, the Rankin-Selberg method, arithmetic intersection theory, and Iwasawa theory.
Author(s): Henri Darmon, Shou-wu Zhang
Series: Mathematical Sciences Research Institute Publications
Publisher: Cambridge University Press
Year: 2004
Language: English
Pages: 381
Half-title......Page 3
Series-title......Page 6
Title......Page 7
Copyright......Page 8
Contents......Page 9
Preface......Page 10
1. Prologue: The Opportune Arrival of Heegner Points......Page 15
2. Prehistory......Page 16
3. Heegner......Page 18
4. Simplification and Generalisation......Page 19
5. 1982......Page 22
References......Page 23
Gross to Birch: March 1, 1982......Page 25
Birch to Gross: May 6, 1982......Page 27
Gross to Birch: May 14, 1982......Page 28
Birch to Gross: around September 6, 1982......Page 31
Gross to Birch: September 17, 1982......Page 35
Gross to Birch: December 1, 1982......Page 36
Birch to Gross: December 27, 1982......Page 37
1. Introduction......Page 39
2. The Deuring–Heilbronn Phenomenon......Page 41
3. Existence of L-functions of Elliptic Curves with Triple Zeros......Page 42
4. Solution of the Class Number One Problem......Page 43
References......Page 49
Heegner Points and Representation Theory......Page 51
1. Heegner Points on X0(N)......Page 52
2. Rankin L-Series and a Height Formula......Page 53
3. Starting from the L-Function......Page 54
4. Local Representation Theory......Page 55
5. Unitary Similitudes......Page 56
7. Inner Forms......Page 57
8. Langlands Parameters......Page 58
9. Local epsilon Factors......Page 59
11. Local Test Vectors......Page 60
12. An Explicit Local Formula......Page 62
13. Ad grave accent lic Groups......Page 64
14. A Special Case......Page 65
15. Automorphic Representations......Page 66
17. Global Test Vectors......Page 67
18. An Explicit Global Formula......Page 69
20. Shimura Varieties......Page 72
21. Nearby Quaternion Algebras......Page 74
22. The Global Representation......Page 75
23. The Global Linear Form......Page 76
References......Page 78
1. Introduction......Page 81
2. Some Properties of Abelian Schemes and Modular Curves......Page 84
3. The Serre–Tate Theorem and the Grothendieck Existence Theorem......Page 90
4. Computing Naive Intersection Numbers......Page 95
5. Intersection Formula Via Hom Groups......Page 99
6. Supersingular Cases with r A(m) = 0......Page 102
7. Application of a Construction of Serre......Page 112
8. Intersection Theory Via Meromorphic Tensors......Page 123
9. Self-Intersection Formula and Application to Global Height Pairings......Page 131
10. Quaternionic Explications......Page 144
Appendix by W. R. Mann: Elimination of Quaternionic Sums......Page 153
References......Page 176
1. Introduction......Page 179
2. Notation and Hypotheses......Page 180
3. Atkin–Lehner Theory on GL2......Page 183
4. Quaternion Algebras and the Jacquet–Langlands Correspondence......Page 184
5. The Work of Waldspurger......Page 185
6. Test Vectors: The Work of Gross and Prasad......Page 186
7. The Work of Gross and Zhang......Page 189
References......Page 203
1. Introduction and Notation......Page 205
2. Automorphic Forms......Page 209
3. Weights and Levels......Page 211
4. Automorphic L-Series......Page 213
5. Rankin–Selberg L-Series......Page 214
6. The Odd Case......Page 215
7. The Even Case......Page 218
8. The Idea of Gross and Zagier......Page 219
9. Calculus on Arithmetic Surfaces......Page 222
10. Decomposition of Heights......Page 224
11. Construction of the Kernels......Page 226
12. Geometric Pairing......Page 230
13. Local Gross–Zagier Formula......Page 233
14. Gross–Zagier Formula in Level ND......Page 235
15. Green’s Functions of Heegner Points......Page 237
16. Spectral Decomposition......Page 240
17. Lowering Levels......Page 242
18. Continuous Spectrum......Page 248
19. Periods of Eisenstein Series......Page 250
References......Page 254
Special Cycles and Derivatives of Eisenstein Series......Page 257
1. Shimura Varieties of Orthogonal Type......Page 258
2. Algebraic Cycles......Page 259
3. Modular Generating Functions......Page 261
4. Connections with Values of Eisenstein Series......Page 265
5. Integral Models and Cycles......Page 267
6. Connections with Derivatives of Eisenstein Series......Page 269
III. Derivatives of L-Series......Page 272
8. Connections with Derivatives of L-Functions......Page 273
Appendix: Shimura Curves......Page 276
References......Page 280
Faltings Heights and the Derivative of Zagier’s Eisenstein Series......Page 285
1. The Chowla–Selberg Formula......Page 286
2. Bost’s L21 -Arithmetic Divisors and Intersection Theory......Page 288
3. The Main Result......Page 290
4. Construction of the Green’s Function xi(m; v)......Page 294
5. The proof of Theorem 3.2......Page 296
Acknowledgment......Page 297
References......Page 298
Elliptic Curves and Analogies Between Number Fields and Function Fields......Page 299
1. Introduction......Page 300
2. Review of the Birch and Swinnerton-Dyer Conjecture over Function Fields......Page 301
3. Function Field Analogues of the Gross–Zagier Theorem......Page 303
4. Ranks over Function Fields......Page 314
5. Rank Bounds......Page 318
6. Ranks over Number Fields......Page 320
7. Algebraic Rank Bounds......Page 321
8. Arithmetic and Geometric Bounds I: Cyclotomic Fields......Page 323
9. Arithmetic and Geometric Bounds II: Function Fields over Number Fields......Page 324
References......Page 326
Heegner Points and Elliptic Curves of Large Rank over Function Fields......Page 331
References......Page 336
Introduction......Page 337
1. Classical Heegner Points......Page 339
2. Heegner Points and p-adic Integration......Page 344
3. Forms on Tp multiplication H......Page 355
4. Complex Periods and Heegner Points......Page 358
5. p-adic Periods and Stark–Heegner Points......Page 363
6. Heegner Points and Integration on Hp multiplication Hq......Page 368
7. Periods of Hilbert Modular Forms......Page 375
References......Page 379
