The Gross-Zagier Formula on Shimura Curves

Xinyi Yuan; Shou-wu Zhang; Wei Zhang

Xinyi Yuan, Shou-wu Zhang and Wei Zhang

The Gross-Zagier Formula on Shimura Curves

(AMS-184)

In: Annals of Mathematics Studies, 184

Princeton University Press | 2013

https://doi.org/10.1515/9781400845644

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About this book

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.

The Gross-Zagier Formula on Shimura Curves will be of great use to students wishing to enter this area and to those already working in it.

Author information

Xinyi Yuan is assistant professor of mathematics at Princeton University. Shou-wu Zhang is professor of mathematics at Princeton University and Columbia University. Wei Zhang is assistant professor of mathematics at Columbia University.

Frontmatter Free access PDF PDF	i
Contents Free access PDF PDF	v
Preface Access restricted Content is available PDF PDF	vii
Chapter One. Introduction and Statement of Main Results Access restricted Content is available PDF PDF	1
Chapter Two. Weil Representation and Waldspurger Formula Access restricted Content is available PDF PDF	28
Chapter Three. Mordell-Weil Groups and Generating Series Access restricted Content is available PDF PDF	58
Chapter Four. Trace of the Generating Series Access restricted Content is available PDF PDF	106
Chapter Five. Assumptions on the Schwartz Function Access restricted Content is available PDF PDF	171
Chapter Six. Derivative of the Analytic Kernel Access restricted Content is available PDF PDF	184
Chapter Seven. Decomposition of the Geometric Kernel Access restricted Content is available PDF PDF	206
Chapter Eight. Local Heights of CM Points Access restricted Content is available PDF PDF	230
Bibliography Access restricted Content is available PDF PDF	251
Index Access restricted Content is available PDF PDF	255

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The Gross-Zagier Formula on Shimura Curves

(AMS-184)

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