Analytic Properties of Automorphic L-Functions

Analytic Properties of Automorphic L-Functions

1st Edition - July 28, 1988

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  • Editors: J. Coates, S. Helgason
  • eBook ISBN: 9781483261034

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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.

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

Product details

  • No. of pages: 142
  • Language: English
  • Copyright: © Academic Press 1988
  • Published: July 28, 1988
  • Imprint: Academic Press
  • eBook ISBN: 9781483261034

About the Editors

J. Coates

S. Helgason

About the Authors

Stephen Gelbart

Freydoon Shahidi

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