The Gross-Zagier Formula on Shimura Curves : : (AMS-184) / / Xinyi Yuan, Wei Zhang, Shou-wu Zhang.

• Superior document: Title is part of eBook package: De Gruyter Princeton Annals of Mathematics eBook-Package 1940-2020
• VerfasserIn: Yuan, Xinyi,; Zhang, Shou-wu,; Zhang, Wei,
• Place / Publishing House: Princeton, NJ : : Princeton University Press, , [2012]; ©2013
• Year of Publication: 2012
• Edition: Course Book
• Language: English
• Series: Annals of Mathematics Studies ; 184
• Physical Description: 1 online resource (272 p.)
• Other title: Frontmatter --; Contents --; Preface --; Chapter One. Introduction and Statement of Main Results --; Chapter Two. Weil Representation and Waldspurger Formula --; Chapter Three. Mordell-Weil Groups and Generating Series --; Chapter Four. Trace of the Generating Series --; Chapter Five. Assumptions on the Schwartz Function --; Chapter Six. Derivative of the Analytic Kernel --; Chapter Seven. Decomposition of the Geometric Kernel --; Chapter Eight. Local Heights of CM Points --; Bibliography --; Index
• Format: Mode of access: Internet via World Wide Web.
• ISBN: 9781400845644; 9783110494914; 9783110442502
• DOI: 10.1515/9781400845644?locatt=mode:legacy
• Access: restricted access
• Hierarchical level: Monograph
• Statement of Responsibility: Xinyi Yuan, Wei Zhang, Shou-wu Zhang.

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.