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Analytic Properties of Automorphic L-Functions - 1st Edition - ISBN: 9780122791758, 9781483261034

Analytic Properties of Automorphic L-Functions, Volume 6

1st Edition

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Authors: Stephen Gelbart Freydoon Shahidi
Editors: J. Coates S. Helgason
eBook ISBN: 9781483261034
Imprint: Academic Press
Published Date: 28th July 1988
Page Count: 142
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Table of Contents



Chapter I. First Steps (1965-1970)

§1. An Analysis of the Method of Jacquet-Langlands

1.1. Cuspidal Representations and L-Functions for GL(2)

1.2. Global Zeta-Integrals and their Factorization

1.3. The Local Zeta-Integrals

1.4. More Local Theory

1.5. Global Results for LS(s, π)

1.6. Global Results for L(s, π)

1.7. Description of the L-Function Machine

§2. Eisenstein Series and Langlands' Euler Products

2.1. The Example of L(s, χ)

2.2. L-Groups

2.3. Unramified Representations

2.4. The General Set-Up: Preliminary Definitions

2.5. The General Set-Up: Eisenstein Series, Constant Terms, and Langlands' "Euler Products"

Chapter II. Developments and Refinements (1970-1982)

§1. Zeta-Integrals for GL(n) and Related Groups

1.1. The Method of Tate-Godement-Jacquet

1.2. Jacquet's Theory for GL(2) x Gl(2) and the Method of Rankin-Selberg

Appendix to Section (1.2): Analysis and Reformulation of the Method of Rankin-Selberg-Jacquet for GL(2) x GL(2)

1.3. Shimura's Method

1.4. Hecke Theory for GL(n)

1.5. The Metaplectic Group

1.6. Symmetric Powers of L-functions

1.7. GL(n) X GL(m)

1.8. Additional Notes and References: L-Functions and the Lifting Problem

1.9. Concluding Remarks

§2. Eisenstein Series and Generic Representations

2.1. Whittaker Models: General Notions

2.2. Whittaker Models for I(s, πv)

2.3. Fourier Coefficients of Eisenstein Series

2.4. Local Coefficients and the Functional Equation for LS(s, π)

2.5. Examples

2.6. On the Non-Vanishing of L-Functions for Re(s) = 1

Chapter III. Recent Developments (1982- )

§1. Explicit Construction of Zeta-Integrals á la Piatetski-Shapiro

1.1. Origins of the Method of Piatetski-Shapiro and Rallis

1.2. The Construction of Piatetski-Shapiro and Rallis

1.3. Summing Up of the Method

1.4. Rankin Triple Products

1.5. L-Functions for G x GL(n)

§2. LanglandsTheory Completed

2.1. Range of Applicability of the Method

2.2. A Uniform Line of Convergence for LS(s,π,r)

2.3. Ramanujan-Type Estimates

2.4. Analytic Continuation of the Completed L-function

2.5. More Examples

2.6. On the Uniquenessof Local Factors

Last Words




Analytic Properties of Automorphic L-Functions is a three-chapter text that covers considerable research works on the automorphic L-functions attached by Langlands to reductive algebraic groups.

Chapter I focuses on the analysis of Jacquet-Langlands methods and the Einstein series and Langlands’ so-called “Euler products”. This chapter explains how local and global zeta-integrals are used to prove the analytic continuation and functional equations of the automorphic L-functions attached to GL(2). Chapter II deals with the developments and refinements of the zeta-inetgrals for GL(n). Chapter III describes the results for the L-functions L (s, ?, r), which are considered in the constant terms of Einstein series for some quasisplit reductive group.

This book will be of value to undergraduate and graduate mathematics students.


No. of pages:
© Academic Press 1988
28th July 1988
Academic Press
eBook ISBN:

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About the Authors

Stephen Gelbart

Freydoon Shahidi

About the Editors

J. Coates

S. Helgason