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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)
2.3. 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)
2.6. 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
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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