Analytic Properties of Automorphic L-Functions

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