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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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Assumptions on the Schwartz Function

Assumptions on the Schwartz Function

(p.171) Chapter Five Assumptions on the Schwartz Function
The Gross-Zagier Formula on Shimura Curves

Xinyi Yuan

Shou-Wu Zhang

Wei Zhang

Princeton University Press

This chapter introduces two classes of degenerate Schwartz functions which significantly simplify the computations and arguments of both the analytic kernel and the geometric kernel functions. It first restates the kernel identity in terms of un-normalized kernel functions before stating the assumptions on the Schwartz function and claiming that these assumptions can be “added” to the kernel identity without losing the generality. It then considers some simple properties of the assumptions and proceeds by discussing the two classes of degenerate Schwartz functions. In the first case, a non-archimedean local field and a non-degenerate quadratic space are described. In the second case, since all the data are unramified, the lemma can be verified by explicit computations.

Keywords:   degenerate Schwartz function, analytic kernel, geometric kernel, kernel identity, un-normalized kernel function, non-archimedean local field, non-degenerate quadratic space

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