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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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Trace of the Generating Series

Trace of the Generating Series

Chapter:
(p.106) Chapter Four Trace of the 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/9780691155913.003.0004

This chapter proves the theorem that asserts the modularity of the generating series and the theorem dealing with abelian varieties parametrized by Shimura curves. Before presenting the proofs, the chapter considers the new space of Schwartz functions and constructs theta series and Eisenstein series from such functions. It proceeds by discussing discrete series at infinite places, modularity of the generating series, degree of the generating series, and the trace identity. It also presents the pull-back formula for the compact and non-compact cases. In particular, it describes CM cycles on the Shimura curve, pull-back as cycles, degree of the pull-back, and some coset identities.

Keywords:   modularity, generating series, abelian varieties, Shimura curves, Schwartz function, theta series, Eisenstein series, discrete series, trace identity, pull-back formula

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