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
- 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
- Title: The Gross-Zagier Formula on Shimura Curves
- Release date: November 2012
- Publisher(s): Princeton University Press
- ISBN: 9781400845644