Eisenstein Series and Automorphic L-Functions

✍ Scribed by Freydoon Shahidi

• Publisher: American Mathematical Society
• Year: 2010
• Tongue: English
• Leaves: 218
• Series: Colloquium Publications Volume no. 58
• Category: Library

✦ Synopsis

This book presents a treatment of the theory of L-functions developed by means of the theory of Eisenstein series and their Fourier coefficients, a theory which is usually referred to as the Langlands–Shahidi method. The information gathered from this method, when combined with the converse theorems of Cogdell and Piatetski-Shapiro, has been quite sufficient in establishing a number of new cases of Langlands functoriality conjecture; at present, some of these cases cannot be obtained by any other method. These results have led to far-reaching new estimates for Hecke eigenvalues of Maass forms, as well as definitive solutions to certain problems in analytic and algebraic number theory.

This book gives a detailed treatment of important parts of this theory, including a rather complete proof of Casselman–Shalika's formula for unramified Whittaker functions as well as a general treatment of the theory of intertwining operators. It also covers in some detail the global aspects of the method as well as some of its applications to group representations and harmonic analysis.

This book is addressed to graduate students and researchers who are interested in the Langlands program in automorphic forms and its connections with number theory.

✦ Table of Contents

Cover
Title page
Contents
Introduction
Reductive groups
Satake isomorphisms
Generic representations
Intertwining operators
Local coefficients
Eisenstein series
Fourier coefficients of Eisenstein series
Functional equations
Further properties of 𝐿–functions
Applications to functoriality
Appendices: Tables of Dynkin diagrams
Bibliography
Index
Back Cover

✦ Subjects

eisenstein series, langlands, automorphic L-functions