The Gross-Zagier Formula on Shimura Curves

Book Description

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. Cover
  2. Title
  3. Copyright
  4. Contents
  5. Preface
  6. 1 Introduction and Statement of Main Results
    1. 1.1 Gross-Zagier formula on modular curves
    2. 1.2 Shimura curves and abelian varieties
    3. 1.3 CM points and Gross-Zagier formula
    4. 1.4 Waldspurger formula
    5. 1.5 Plan of the proof
    6. 1.6 Notation and terminology
  7. 2 Weil Representation and Waldspurger Formula
    1. 2.1 Weil representation
    2. 2.2 Shimizu lifting
    3. 2.3 Integral representations of the L-function
    4. 2.4 Proof of Waldspurger formula
    5. 2.5 Incoherent Eisenstein series
  8. 3 Mordell–Weil Groups and Generating Series
    1. 3.1 Basics on Shimura curves
    2. 3.2 Abelian varieties parametrized by Shimura curves
    3. 3.3 Main theorem in terms of projectors
    4. 3.4 The generating series
    5. 3.5 Geometric kernel
    6. 3.6 Analytic kernel and kernel identity
  9. 4 Trace of the Generating Series
    1. 4.1 Discrete series at infinite places
    2. 4.2 Modularity of the generating series
    3. 4.3 Degree of the generating series
    4. 4.4 The trace identity
    5. 4.5 Pull-back formula: compact case
    6. 4.6 Pull-back formula: non-compact case
    7. 4.7 Interpretation: non-compact case
  10. 5 Assumptions on the Schwartz Function
    1. 5.1 Restating the kernel identity
    2. 5.2 The assumptions and basic properties
    3. 5.3 Degenerate Schwartz functions I
    4. 5.4 Degenerate Schwartz functions II
  11. 6 Derivative of the Analytic Kernel
    1. 6.1 Decomposition of the derivative
    2. 6.2 Non-archimedean components
    3. 6.3 Archimedean components
    4. 6.4 Holomorphic projection
    5. 6.5 Holomorphic kernel function
  12. 7 Decomposition of the Geometric Kernel
    1. 7.1 Neron-Tate height
    2. 7.2 Decomposition of the height series
    3. 7.3 Vanishing of the contribution of the Hodge classes
    4. 7.4 The goal of the next chapter
  13. 8 Local Heights of CM Points
    1. 8.1 Archimedean case
    2. 8.2 Supersingular case
    3. 8.3 Superspecial case
    4. 8.4 Ordinary case
    5. 8.5 The j-part
  14. Bibliography
  15. Index

Product Information

  • Title: The Gross-Zagier Formula on Shimura Curves
  • Author(s): Xinyi Yuan, Shou-wu Zhang, Wei Zhang
  • Release date: November 2012
  • Publisher(s): Princeton University Press
  • ISBN: 9781400845644