The Gross-Zagier Formula on Shimura Curves

Preface vii

1 Introduction and Statement of Main Results 1
1.1 Gross-Zagier formula on modular curves . . . . . . . . . . . . . 1
1.2 Shimura curves and abelian varieties . . . . . . . . . . . . . . . 2
1.3 CM points and Gross-Zagier formula . . . . . . . . . . . . . . . 6
1.4 Waldspurger formula . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Plan of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6 Notation and terminology . . . . . . . . . . . . . . . . . . . . . 20

2 Weil Representation and Waldspurger Formula 28
2.1 Weil representation . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Shimizu lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3 Integral representations of the L-function . . . . . . . . . . . . 40
2.4 Proof of Waldspurger formula . . . . . . . . . . . . . . . . . . . 43
2.5 Incoherent Eisenstein series . . . . . . . . . . . . . . . . . . . . 44

3 Mordell-Weil Groups and Generating Series 58
3.1 Basics on Shimura curves . . . . . . . . . . . . . . . . . . . . . 58
3.2 Abelian varieties parametrized by Shimura curves . . . . . . . . 68
3.3 Main theorem in terms of projectors . . . . . . . . . . . . . . . 83
3.4 The generating series . . . . . . . . . . . . . . . . . . . . . . . . 91
3.5 Geometric kernel . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.6 Analytic kernel and kernel identity . . . . . . . . . . . . . . . . 100

4 Trace of the Generating Series 106
4.1 Discrete series at innite places . . . . . . . . . . . . . . . . . . 106
4.2 Modularity of the generating series . . . . . . . . . . . . . . . . 110
4.3 Degree of the generating series . . . . . . . . . . . . . . . . . . 117
4.4 The trace identity . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.5 Pull-back formula: compact case . . . . . . . . . . . . . . . . . 128
4.6 Pull-back formula: non-compact case . . . . . . . . . . . . . . . 138
4.7 Interpretation: non-compact case . . . . . . . . . . . . . . . . . 153

5 Assumptions on the Schwartz Function 171
5.1 Restating the kernel identity . . . . . . . . . . . . . . . . . . . 171
5.2 The assumptions and basic properties . . . . . . . . . . . . . . 174
5.3 Degenerate Schwartz functions I . . . . . . . . . . . . . . . . . 178
5.4 Degenerate Schwartz functions II . . . . . . . . . . . . . . . . . 181

6 Derivative of the Analytic Kernel 184
6.1 Decomposition of the derivative . . . . . . . . . . . . . . . . . . 184
6.2 Non-archimedean components . . . . . . . . . . . . . . . . . . . 191
6.3 Archimedean components . . . . . . . . . . . . . . . . . . . . . 196
6.4 Holomorphic projection . . . . . . . . . . . . . . . . . . . . . . 197
6.5 Holomorphic kernel function . . . . . . . . . . . . . . . . . . . . 202

7 Decomposition of the Geometric Kernel 206
7.1 Néron-Tate height . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.2 Decomposition of the height series . . . . . . . . . . . . . . . . 216
7.3 Vanishing of the contribution of the Hodge classes . . . . . . . 219
7.4 The goal of the next chapter . . . . . . . . . . . . . . . . . . . . 223

8 Local Heights of CM Points 230
8.1 Archimedean case . . . . . . . . . . . . . . . . . . . . . . . . . . 230
8.2 Supersingular case . . . . . . . . . . . . . . . . . . . . . . . . . 233
8.3 Superspecial case . . . . . . . . . . . . . . . . . . . . . . . . . . 239
8.4 Ordinary case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
8.5 The j -part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

Bibliography 251
Index 255