The Gross-Zagier Formula on Shimura Curves: (Ams-184) (Hardcover) (ISBN-13: 9780691155913)

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.

Author: Xinyi Yuan, Shou-Wu Zhang, Wei Zhang
ISBN-10: 0691155917
ISBN-13: 9780691155913
Publisher: Princeton University Press
Language: English
Published: 12/02/2012
Pages: 272
Format: Hardcover
Weight: 1.22lbs
Size: 9.21h x 6.14w x 0.63d