Lectures on Automorphic $L$-functions ( Fields Institute Monographs )

Publication series ：Fields Institute Monographs

Author： James W. Cogdell;Henry H. Kim;M. Ram Murty  

Publisher： American Mathematical Society‎

Publication year： 2009

E-ISBN: 9781470431471

P-ISBN(Hardback):  9780821848005

Subject： O156.2 theory of algebraic numbers

Keyword： 暂无分类

Language： ENG

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Lectures on Automorphic $L$-functions

Description

This book provides a comprehensive account of how automorphic $L$-functions play a crucial role in the Langlands program, especially, the Langlands functoriality conjecture, and in number theory. Recently there has been a 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 introduces the reader to these developments step by step, 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. This book would be ideal for an introductory course in the Langlands program.

Chapter

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

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