The Gross-Zagier Formula on Shimura Curves
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Book Overview
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.
Format:Paperback
Language:English
ISBN:0691155925
ISBN13:9780691155920
Release Date:December 2012
Publisher:Princeton University Press
Length:272 Pages
Weight:0.90 lbs.
Dimensions:0.5" x 6.1" x 9.1"
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