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

Analytic Properties of Automorphic L-Functions, Volume 6

1st Edition

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


Acknowledgements

Introduction

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

References

Index

Description

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.


Details

No. of pages:
142
Language:
English
Copyright:
© Academic Press 1988
Published:
Imprint:
Academic Press
eBook ISBN:
9781483261034

About the Editors

J. Coates Editor

S. Helgason Editor

About the Authors

Stephen Gelbart Author

Freydoon Shahidi Author