The Gross-Zagier Formula on Shimura Curves (1)
The Gross-Zagier Formula on Shimura Curves (1)
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Abstract
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 Swinnerton-Dyer conjecture and Diophantine equations. The book begins with a conceptual formulation of the Gross–Zagier 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 L-series by means of Weil representations. The Gross–Zagier 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 Gross–Zagier formula is reduced to local formulas. This book will be of great use to students wishing to enter this area and to those already working in it.
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Front Matter
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One
Introduction and Statement of Main Results
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Two
Weil Representation and Waldspurger Formula
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Three
Mordell-Weil Groups and Generating Series
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Four
Trace of the Generating Series
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Five
Assumptions on the Schwartz Function
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Six
Derivative of the Analytic Kernel
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Seven
Decomposition of the Geometric Kernel
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Eight
Local Heights of CM Points
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End Matter
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