The Gross-Zagier Formula on Shimura Curves: (AMS-184) (Annals of Mathematics Studies, 184) - Softcover

Yuan, Xinyi; Zhang, Shou-wu; Zhang, Wei

Softcover

ISBN 10: 0691155925 ISBN 13: 9780691155920

Publisher: Princeton University Press, 2012

Synopsis

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.

About this title

About the Author:

Xinyi Yuan is assistant professor of mathematics at Princeton University. Shou-wu Zhang is professor of mathematics at Princeton University and Columbia University. Wei Zhang is assistant professor of mathematics at Columbia University.

About this edition

• Publisher: Princeton University Press
• Publication date: 2012
• ISBN 10: 0691155925
• ISBN 13: 9780691155920
• Binding: Paperback
• Number of pages: 272