 | Yuan X. The Gross-Zagier formula on Shimura curves / X.Yuan, S.-W.Zhang, W.Zhang. - Princeton; Oxford: Princeton univ. press, 2013. - viii, 256 p. - (Annals of mathematics studies; N 184). - Bibliogr.: p.251-254. - Ind.: p.255-256. - Пер. загл.: Формула Гросса-Цагира на кривых Шимуры. - ISBN 978-0-691-15591-3
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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 infinite 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 Neron-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
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