The Gross-Zagier Formula on Shimura Curves
eBook - ePub

The Gross-Zagier Formula on Shimura Curves

  1. 272 pages
  2. English
  3. ePUB (mobile friendly)
  4. Available on iOS & Android
eBook - ePub

The Gross-Zagier Formula on Shimura Curves

About this book

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.

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Yes, you can access The Gross-Zagier Formula on Shimura Curves by Xinyi Yuan,Shou-wu Zhang,Wei Zhang,Shou-Wu Zhang in PDF and/or ePUB format, as well as other popular books in Matematica & Teoria dei numeri. We have over one million books available in our catalogue for you to explore.

Table of contents

  1. Cover
  2. Title
  3. Copyright
  4. Contents
  5. Preface
  6. 1 Introduction and Statement of Main Results
  7. 2 Weil Representation and Waldspurger Formula
  8. 3 Mordell–Weil Groups and Generating Series
  9. 4 Trace of the Generating Series
  10. 5 Assumptions on the Schwartz Function
  11. 6 Derivative of the Analytic Kernel
  12. 7 Decomposition of the Geometric Kernel
  13. 8 Local Heights of CM Points
  14. Bibliography
  15. Index