Franklin

The Gross-Zagier Formula on Shimura Curves / Wei Zhang, Shou-wu Zhang, Xinyi Yuan.

Author/Creator:
Yuan, Xinyi author.
Publication:
Princeton, NJ : Princeton University Press, [2012]
Format/Description:
Book
1 online resource
Edition:
Course Book
Series:
Annals of Mathematics Studies ; 208
Contained In:
De Gruyter University Press Library.
Status/Location:
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Details

Subjects:
Arithmetical algebraic geometry.
Automorphic forms.
Quaternions.
Shimura varieties.
Language:
In English.
System Details:
Mode of access: Internet via World Wide Web.
text file PDF
Summary:
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.
Contents:
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
Notes:
Description based on online resource; title from PDF title page (publisher's Web site, viewed 08. Jul 2019)
Contributor:
Zhang, Shouwu, author.
Zhang, Wei, author.
De Gruyter.
ISBN:
9781400845644
OCLC:
979881796
Publisher Number:
10.1515/9781400845644 doi
Access Restriction:
Restricted for use by site license.