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The Gross-Zagier Formula on Shimura Curves Book

The Gross-Zagier Formula on Shimura Curves
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The Gross-Zagier Formula on Shimura Curves, This comprehensive account of the Gross-Zagier formula on Shimura curves over totally real fields relates the heights of Heegner points on abelian varieties to the derivatives of L-series. The formula will have new applications for the Birch and Swinnerto, The Gross-Zagier Formula on Shimura Curves
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  • The Gross-Zagier Formula on Shimura Curves
  • Written by author Xinyi Yuan
  • Published by Princeton University Press, 11/11/2012
  • This comprehensive account of the Gross-Zagier formula on Shimura curves over totally real fields relates the heights of Heegner points on abelian varieties to the derivatives of L-series. The formula will have new applications for the Birch and Swinnerto
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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 NĂ©ron-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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The Gross-Zagier Formula on Shimura Curves, This comprehensive account of the Gross-Zagier formula on Shimura curves over totally real fields relates the heights of Heegner points on abelian varieties to the derivatives of L-series. The formula will have new applications for the Birch and Swinnerto, The Gross-Zagier Formula on Shimura Curves

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The Gross-Zagier Formula on Shimura Curves, This comprehensive account of the Gross-Zagier formula on Shimura curves over totally real fields relates the heights of Heegner points on abelian varieties to the derivatives of L-series. The formula will have new applications for the Birch and Swinnerto, The Gross-Zagier Formula on Shimura Curves

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The Gross-Zagier Formula on Shimura Curves, This comprehensive account of the Gross-Zagier formula on Shimura curves over totally real fields relates the heights of Heegner points on abelian varieties to the derivatives of L-series. The formula will have new applications for the Birch and Swinnerto, The Gross-Zagier Formula on Shimura Curves

The Gross-Zagier Formula on Shimura Curves

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