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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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Derivative of the Analytic Kernel

Derivative of the Analytic Kernel

(p.184) Chapter Six Derivative of the Analytic Kernel
The Gross-Zagier Formula on Shimura Curves

Xinyi Yuan

Shou-Wu Zhang

Wei Zhang

Princeton University Press

This chapter computes the derivative of the analytic kernel. It first decomposes the kernel function into a sum of infinitely many local terms indexed by places v of Fnonsplit in E. Each local term is a period integral of some kernel function. The chapter then considers the v-part for non-archimedean v. An explicit formula is given in the unramified case, and an approximation is presented in the ramified case assuming the Schwartz function is degenerate. An explicit result of the v-part for archimedean v is also introduced. The chapter proceeds by reviewing a general formula of holomorphic projection, and estimates the growth of the kernel function in order to apply the formula. It also computes the holomorphic projection of the analytic kernel function and concludes with a discussion of the holomorphic kernel function.

Keywords:   analytic kernel, kernel function, Schwartz function, holomorphic projection, analytic kernel function, holomorphic kernel function

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