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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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Weil Representation and Waldspurger Formula

Weil Representation and Waldspurger Formula

Chapter:
(p.28) Chapter Two Weil Representation and Waldspurger Formula
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/9780691155913.003.0002

This chapter reviews some basic results on Weil representations, theta liftings and Eisenstein series. In particular, it introduces a proof of the Waldspurger formula. The theory of Weil representation is applied to an integral representation of the Rankin–Selberg L-function and to a proof of Waldspurger's central value formula. The chapter mostly follows Waldspurger's treatment with some modifications including Kudla's construction of incoherent Eisenstein series. It first describes the classical theory of Weil representation for an orthogonal space over a local field before discussing theta functions, the Siegel–Weil formula, and normalized local Shimizu lifting. The main result is an integral formula for the L-series using a kernel function. The Waldspurger formula is a direct consequence of the Siegel–Weil formula. After presenting the proof of Waldspurger formula, the chapter lists some computational results on three types of incoherent Eisenstein series.

Keywords:   theta liftings, Weil representation, Eisenstein series, Waldspurger formula, Rankin–Selberg L-function, incoherent Eisenstein series, orthogonal space, theta function, Siegel–Weil formula, Shimizu lifting

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