Jump to ContentJump to Main Navigation
The Gross-Zagier Formula on Shimura Curves$
Users without a subscription are not able to see the full content.

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

Show Summary Details
Page of

PRINTED FROM PRINCETON SCHOLARSHIP ONLINE (www.princeton.universitypressscholarship.com). (c) Copyright Princeton University Press, 2018. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in HSO for personal use (for details see www.princeton.universitypressscholarship.com/page/privacy-policy).date: 16 January 2019

Weil Representation and Waldspurger Formula

Weil Representation and Waldspurger Formula

(p.28) Chapter Two Weil Representation and Waldspurger Formula
The Gross-Zagier Formula on Shimura Curves

Xinyi Yuan

Shou-Wu Zhang

Wei Zhang

Princeton University Press

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

Princeton Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.

Please, subscribe or login to access full text content.

If you think you should have access to this title, please contact your librarian.

To troubleshoot, please check our FAQs , and if you can't find the answer there, please contact us.