Analytic Properties of Automorphic L-Functions
Published on 14. July 2014
142 pages
978-1-4832-6103-4 (ISBN)
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.
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Stephen S. Gelbart | Fereidoon Shahidi
Analytic Properties of Automorphic L-functions
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01/1988
Academic Press
€45.19
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Content
AcknowledgementsIntroductionChapter 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, p) 1.6. Global Results for L(s, p) 1.7. Description of the L-Function Machine §2. 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) §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, 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) = 1Chapter 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,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 WordsReferencesIndex
