Xinyi Yuan is assistant professor of mathematics at Princeton University. Shou-wu Zhang is professor of mathematics at Princeton University and Columbia University. Wei Zhang is assistant professor of mathematics at Columbia University.

• 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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Xinyi Yuan is assistant professor of mathematics at Princeton University. Shou-wu Zhang is professor of mathematics at Princeton University and Columbia University. Wei Zhang is assistant professor of mathematics at Columbia University.

• 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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Xinyi Yuan is assistant professor of mathematics at Princeton University. Shou-wu Zhang is professor of mathematics at Princeton University and Columbia University. Wei Zhang is assistant professor of mathematics at Columbia University.

• 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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