Lectures on Automorphic L-Functions

✍ Scribed by James W. Cogdell, Henry Hyeongsin Kim, Maruti Ram Murty

• Publisher: Amer Mathematical Society
• Year: 2004
• Tongue: English
• Leaves: 298
• Series: Fields Institute Monographs
• Category: Library

✦ Synopsis

This book provides a comprehensive account of the crucial role automorphic $L$-functions play in number theory and in the Langlands program, especially the Langlands functoriality conjecture. There has been a recent major development in the Langlands functoriality conjecture by the use of automorphic $L$-functions, namely, by combining converse theorems of Cogdell and Piatetski-Shapiro with the Langlands-Shahidi method. This book provides a step-by-step introduction to these developments and explains how the Langlands functoriality conjecture implies solutions to several outstanding conjectures in number theory, such as the Ramanujan conjecture, Sato-Tate conjecture, and Artin's conjecture. It would be ideal for an introductory course in the Langlands program.

✦ Table of Contents

Cover
Title page
Contents
Preface
Lectures on 𝐿-functions, converse theorems, and functoriality for 𝐺𝐿_{𝑛}, by James W. Cogdell
Preface
Lecture 1. Modular forms and their 𝐿-functions
Lecture 2. Automorphic forms
Lecture 3. Automorphic representations
Lecture 4. Fourier expansions and multiplicity one theorems
Lecture 5. Eulerian integral representations
Lecture 6. Local 𝐿-functions: The non-Archimedean case
Lecture 7. The unramified calculation
Lecture 8. Local 𝐿-functions: The Archimedean case
Lecture 9. Global 𝐿-functions
Lecture 10. Converse theorems
Lecture 11. Functoriality
Lecture 12. Functoriality for the classical groups
Lecture 13. Functoriality for the classical groups, II
Automorphic 𝐿-functions, by Henry H. Kim
Introduction
Chevalley groups and their properties
Cuspidal representations
𝐿-groups and automorphic 𝐿-functions
Induced representations
Eisenstein series and constant terms
𝐿-functions in the constant terms
Meromorphic continuation of 𝐿-functions
Generic representations and their Whittaker models
Local coefficients and non-constant terms
Local Langlands correspondence
Local 𝐿-functions and functional equations
Normalization of intertwining operators
Holomorphy and bounded in vertical strips
Langlands functoriality conjecture
Converse theorem of Cogdell and Piatetski-Shapiro
Functoriality of the symmetric cube
Functoriality of the symmetric fourth
Bibliography
Applications of symmetric power 𝐿-functions, by M. Ram Murty
Preface
Lecture 1. The Sato-Tate conjecture
Lecture 2. Maass wave forms
Lecture 3. The Rankin-Selberg method
Lecture 4. Oscillations of Fourier coefficients of cusp forms
Lecture 5. Poincaré series
Lecture 6. Kloosterman sums and Selberg’s conjecture
Lecture 7. Refined estimates for Fourier coefficients of cusp forms
Lecture 8. Twisting and averaging of 𝐿-series
Lecture 9. The Kim-Sarnak theorem
Lecture 10. Introduction to Artin 𝐿-functions
Lecture 11. Zeros and poles of Artin 𝐿-functions
Lecture 12. The Langlands-Tunnell theorem
Bibliography
Back Cover