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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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Local Heights of CM Points

Local Heights of CM Points

(p.230) Chapter Eight Local Heights of CM Points
The Gross-Zagier Formula on Shimura Curves

Xinyi Yuan

Shou-Wu Zhang

Wei Zhang

Princeton University Press

This chapter computes the local heights and compares them with the derivatives computed before. It checks the theorem place by place and takes into account all the assumptions on the Schwartz function. According to the reduction of the Shimura curve, the situation is divided to the following four cases: archimedean case, supersingular case, superspecial case, and ordinary case. The treatments in different cases are similar in spirit, except that the fourth case is slightly different. The supersingular case is divided into two subcases: unramified case and ramified case. The chapter also describes local heights of CM points at any archimedean place v. The discussion covers the multiplicity function, the kernel function, unramified quadratic extension, ramified quadratic extension, ordinary components, supersingular components, and superspecial components.

Keywords:   local height, Schwartz function, Shimura curves, multiplicity function, kernel function, unramified quadratic extension, ramified quadratic extension, ordinary component, supersingular component, superspecial component

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