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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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Mordell-Weil Groups and Generating Series

Mordell-Weil Groups and Generating Series

Chapter:
(p.58) Chapter Three Mordell-Weil Groups and Generating Series
Source:
The Gross-Zagier Formula on Shimura Curves
Author(s):

Xinyi Yuan

Shou-Wu Zhang

Wei Zhang

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691155913.003.0003

This chapter deals with Mordell–Weil groups and generating series. It first provides an overview of the basics on Shimura curves and abelian varieties parametrized by Shimura curves before introducing a theorem, which is an identity between the analytic kernel and the geometric kernel. It then defines the generating series and uses it to describe the geometric kernel. It also presents a theorem, which is an identity formulated in terms of projectors, and reviews some basic notations and results on Shimura curves. Other topics covered include the Eichler–Shimura theory for abelian varieties parametrized by Shimura curves, normalization of the geometric kernel, and the analytic kernel function. The chapter concludes with an analysis of the kernel identity implied in the first theorem.

Keywords:   abelian varieties, Mordell–Weil group, generating series, Shimura curve, analytic kernel, geometric kernel, projector, Eichler–Shimura theory, kernel identity

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