The GrossZagier formula on Shimura curves
 Author/Creator
 Yuan, Xinyi, 1981
 Language
 English.
 Imprint
 Princeton : Princeton University Press, 2013.
 Physical description
 viii, 256 p. ; 25 cm.
 Series
 Annals of mathematics studies ; no. 184.
Access
Available online
 proquest.safaribooksonline.com Safari Books Online
 ebrary

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Unknown
QA1 .A665 NO.184

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QA1 .A665 NO.184
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Contributors
 Contributor
 Zhang, Shouwu.
 Zhang, Wei, 1981
Contents/Summary
 Bibliography
 Includes bibliographical references and index.
 Contents

 Preface vii 1 Introduction and Statement of Main Results 1 1.1 GrossZagier formula on modular curves ... 1 1.2 Shimura curves and abelian varieties ... 2 1.3 CM points and GrossZagier 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 Lfunction ... 40 2.4 Proof of Waldspurger formula ... 43 2.5 Incoherent Eisenstein series ... 44 3 MordellWeil 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 Pullback formula: compact case ... 128 4.6 Pullback formula: noncompact case ... 138 4.7 Interpretation: noncompact 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 Nonarchimedean 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 NeronTate 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.
 (source: Nielsen Book Data)
 Publisher's Summary
 This comprehensive account of the GrossZagier formula on Shimura curves over totally real fields relates the heights of Heegner points on abelian varieties to the derivatives of Lseries. The formula will have new applications for the Birch and SwinnertonDyer conjecture and Diophantine equations. The book begins with a conceptual formulation of the GrossZagier formula in terms of incoherent quaternion algebras and incoherent automorphic representations with rational coefficients attached naturally to abelian varieties parametrized by Shimura curves. This is followed by a complete proof of its coherent analogue: the Waldspurger formula, which relates the periods of integrals and the special values of Lseries by means of Weil representations. The GrossZagier formula is then reformulated in terms of incoherent Weil representations and Kudla's generating series. Using Arakelov theory and the modularity of Kudla's generating series, the proof of the GrossZagier formula is reduced to local formulas. "The GrossZagier Formula on Shimura Curves" will be of great use to students wishing to enter this area and to those already working in it.
(source: Nielsen Book Data)
Bibliographic information
 Publication date
 2013
 Responsibility
 Xinyi Yuan, Shouwu Zhang, and Wei Zhang.
 Series
 Annals of mathematics studies ; no. 184
 ISBN
 9780691155913
 0691155917
 9780691155920
 0691155925