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The Gross-Zagier Formula on Shimura Curves$
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Xinyi Yuan, Shou-wu Zhang, and Wei Zhang

Print publication date: 2012

Print ISBN-13: 9780691155913

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691155913.001.0001

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Introduction and Statement of Main Results

Introduction and Statement of Main Results

(p.1) Chapter One Introduction and Statement of Main Results
The Gross-Zagier Formula on Shimura Curves

Xinyi Yuan

Shou-Wu Zhang

Wei Zhang

Princeton University Press

This chapter states the main result of this book regarding Shimura curves and abelian varieties as well as the main idea of the proof of a complete Gross–Zagier formula on quaternionic Shimura curves over totally real fields. It begins with a discussion of the original formula proved by Benedict Gross and Don Zagier, which relates the Néeron–Tate heights of Heegner points on X⁰(N) to the central derivatives of some Rankin–Selberg L-functions under the Heegner condition. In particular, it considers the Gross–Zagier formula on modular curves and abelian varieties parametrized by Shimura curves. It then decribes CM points and the Waldspurger formula before concluding with an outline of our proof, along with the notation and terminology.

Keywords:   abelian varieties, Gross–Zagier formula, Shimura curves, Benedict Gross, Don Zagier, Néeron–Tate height, Heegner point, Rankin–Selberg L-function, modular curve, Waldspurger formula

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