Cover -- Title -- Copyright -- Contents -- Preface -- 1 Introduction and Statement of Main Results -- 1.1 Gross-Zagier formula on modular curves -- 1.2 Shimura curves and abelian varieties -- 1.3 CM points and Gross-Zagier formula -- 1.4 Waldspurger formula -- 1.5 Plan of the proof -- 1.6 Notation and terminology -- 2 Weil Representation and Waldspurger Formula -- 2.1 Weil representation -- 2.2 Shimizu lifting -- 2.3 Integral representations of the L-function -- 2.4 Proof of Waldspurger formula -- 2.5 Incoherent Eisenstein series -- 3 Mordell-Weil Groups and Generating Series -- 3.1 Basics on Shimura curves -- 3.2 Abelian varieties parametrized by Shimura curves -- 3.3 Main theorem in terms of projectors -- 3.4 The generating series -- 3.5 Geometric kernel -- 3.6 Analytic kernel and kernel identity -- 4 Trace of the Generating Series -- 4.1 Discrete series at infinite places -- 4.2 Modularity of the generating series -- 4.3 Degree of the generating series -- 4.4 The trace identity -- 4.5 Pull-back formula: compact case -- 4.6 Pull-back formula: non-compact case -- 4.7 Interpretation: non-compact case -- 5 Assumptions on the Schwartz Function -- 5.1 Restating the kernel identity -- 5.2 The assumptions and basic properties -- 5.3 Degenerate Schwartz functions I -- 5.4 Degenerate Schwartz functions II -- 6 Derivative of the Analytic Kernel -- 6.1 Decomposition of the derivative -- 6.2 Non-archimedean components -- 6.3 Archimedean components -- 6.4 Holomorphic projection -- 6.5 Holomorphic kernel function -- 7 Decomposition of the Geometric Kernel -- 7.1 Néron-Tate height -- 7.2 Decomposition of the height series -- 7.3 Vanishing of the contribution of the Hodge classes -- 7.4 The goal of the next chapter -- 8 Local Heights of CM Points -- 8.1 Archimedean case -- 8.2 Supersingular case -- 8.3 Superspecial case -- 8.4 Ordinary case -- 8.5 The j-part.

Bibliography -- Index.