Analytic Properties of Automorphic L-Functions

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.

Inhalt

Acknowledgements

Introduction

Chapter I. First Steps (1965-1970)

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, p)

1.6. Global Results for L(s, p)

1.7. Description of the L-Function Machine

Eisenstein Series and Langlands' Euler Products

2.1. The Example of L(s, )

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)

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

Eisenstein Series and Generic Representations

2.1. Whittaker Models: General Notions

2.2. Whittaker Models for I(s, pv)

2.3. Fourier Coefficients of Eisenstein Series

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

2.5. Examples

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

Chapter III. Recent Developments (1982- )

Explicit Construction of Zeta-Integrals 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)

LanglandsTheory Completed

2.1. Range of Applicability of the Method

2.2. A Uniform Line of Convergence for LS(s,p,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