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# The Gross-Zagier Formula on Shimura CurvesXinyi Yuan, Shou-wu Zhang & Wei Zhang

 TABLE OF CONTENTS:Preface vii 1 Introduction and Statement of Main Results 11.1 Gross-Zagier formula on modular curves . . . . . . . . . . . . . 11.2 Shimura curves and abelian varieties . . . . . . . . . . . . . . . 21.3 CM points and Gross-Zagier formula . . . . . . . . . . . . . . . 61.4 Waldspurger formula . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Plan of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . 121.6 Notation and terminology . . . . . . . . . . . . . . . . . . . . . 20 2 Weil Representation and Waldspurger Formula 282.1 Weil representation . . . . . . . . . . . . . . . . . . . . . . . . . 282.2 Shimizu lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.3 Integral representations of the L-function . . . . . . . . . . . . 402.4 Proof of Waldspurger formula . . . . . . . . . . . . . . . . . . . 432.5 Incoherent Eisenstein series . . . . . . . . . . . . . . . . . . . . 44 3 Mordell-Weil Groups and Generating Series 583.1 Basics on Shimura curves . . . . . . . . . . . . . . . . . . . . . 583.2 Abelian varieties parametrized by Shimura curves . . . . . . . . 683.3 Main theorem in terms of projectors . . . . . . . . . . . . . . . 833.4 The generating series . . . . . . . . . . . . . . . . . . . . . . . . 913.5 Geometric kernel . . . . . . . . . . . . . . . . . . . . . . . . . . 973.6 Analytic kernel and kernel identity . . . . . . . . . . . . . . . . 100 4 Trace of the Generating Series 1064.1 Discrete series at infinite places . . . . . . . . . . . . . . . . . . 1064.2 Modularity of the generating series . . . . . . . . . . . . . . . . 1104.3 Degree of the generating series . . . . . . . . . . . . . . . . . . 1174.4 The trace identity . . . . . . . . . . . . . . . . . . . . . . . . . 1224.5 Pull-back formula: compact case . . . . . . . . . . . . . . . . . 1284.6 Pull-back formula: non-compact case . . . . . . . . . . . . . . . 1384.7 Interpretation: non-compact case . . . . . . . . . . . . . . . . . 153 5 Assumptions on the Schwartz Function 1715.1 Restating the kernel identity . . . . . . . . . . . . . . . . . . . 1715.2 The assumptions and basic properties . . . . . . . . . . . . . . 1745.3 Degenerate Schwartz functions I . . . . . . . . . . . . . . . . . 1785.4 Degenerate Schwartz functions II . . . . . . . . . . . . . . . . . 181 6 Derivative of the Analytic Kernel 1846.1 Decomposition of the derivative . . . . . . . . . . . . . . . . . . 1846.2 Non-archimedean components . . . . . . . . . . . . . . . . . . . 1916.3 Archimedean components . . . . . . . . . . . . . . . . . . . . . 1966.4 Holomorphic projection . . . . . . . . . . . . . . . . . . . . . . 1976.5 Holomorphic kernel function . . . . . . . . . . . . . . . . . . . . 202 7 Decomposition of the Geometric Kernel 2067.1 Néron-Tate height . . . . . . . . . . . . . . . . . . . . . . . . . 2077.2 Decomposition of the height series . . . . . . . . . . . . . . . . 2167.3 Vanishing of the contribution of the Hodge classes . . . . . . . 2197.4 The goal of the next chapter . . . . . . . . . . . . . . . . . . . . 223 8 Local Heights of CM Points 2308.1 Archimedean case . . . . . . . . . . . . . . . . . . . . . . . . . . 2308.2 Supersingular case . . . . . . . . . . . . . . . . . . . . . . . . . 2338.3 Superspecial case . . . . . . . . . . . . . . . . . . . . . . . . . . 2398.4 Ordinary case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2448.5 The j -part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Bibliography 251Index 255 Return to Book DescriptionFile created: 4/21/2017 Questions and comments to: webmaster@press.princeton.eduPrinceton University Press