The Gross-Zagier Formula on Shimura Curves

Xinyi Yuan, Shou-wu Zhang, Wei Zhang, Shou-Wu Zhang

• Publisher: Princeton University Press
• Year: 2012
• Print ISBN:
  9780691155920
  9780691155913
• eBook ISBN:
  9781400845644
• Topic: Mathematics
• Subtopic: Number Theory
• Index: Mathematics

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.

Table of contents

• Cover
• Title
• Copyright
• Contents
• Preface
• 1 Introduction and Statement of Main Results
• 2 Weil Representation and Waldspurger Formula
• 3 Mordell–Weil Groups and Generating Series
• 4 Trace of the Generating Series
• 5 Assumptions on the Schwartz Function
• 6 Derivative of the Analytic Kernel
• 7 Decomposition of the Geometric Kernel
• 8 Local Heights of CM Points
• Bibliography
• Index