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The Gross-Zagier Formula on Shimura Curves
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The Gross-Zagier Formula on Shimura Curves

Xinyi Yuan, Shou-wu Zhang, and Wei Zhang

Abstract

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 i ... More

Keywords: incoherent quaternion algebra, Gross–Zagier formula, Shimura curves, L-series, incoherent automorphic representation, Waldspurger formula, Arakelov theory

Bibliographic Information

Print publication date: 2012 Print ISBN-13: 9780691155913
Published to Princeton Scholarship Online: October 2017 DOI:10.23943/princeton/9780691155913.001.0001

Authors

Affiliations are at time of print publication.

Xinyi Yuan, author
University of Califorinia, Berkeley

Shou-wu Zhang, author
Princeton University

Wei Zhang, author
Columbia University