The Gross-Zagier Formula on Shimura Curves: (Ams-184) (Paperback) (ISBN-13: 9780691155920)

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: 0691155925
ISBN-13: 9780691155920
Publisher: Princeton University Press
Language: English
Published: 12/02/2012
Pages: 272
Format: Paperback
Weight: 0.85lbs
Size: 9.10h x 6.10w x 0.50d