Electronic ISBN:  9781470431471 
Product Code:  FIM/20.E 
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Book DetailsFields Institute MonographsVolume: 20; 2004; 283 ppMSC: Primary 11; 22;
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 PiatetskiShapiro with the LanglandsShahidi method. This book provides a stepbystep 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, SatoTate conjecture, and Artin's conjecture. It would be ideal for an introductory course in the Langlands program.
ReadershipGraduate students and research mathematicians interested in representation theory and number theory.

Table of Contents

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 nonArchimedean 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

Chapter 1. Chevalley groups and their properties

Chapter 2. Cuspidal representations

Chapter 3. $L$groups and automorphic $L$functions

Chapter 4. Induced representations

Chapter 5. Eisenstein series and constant terms

Chapter 6. $L$functions in the constant terms

Chapter 7. Meromorphic continuation of $L$functions

Chapter 8. Generic representations and their Whittaker models

Chapter 9. Local coefficients and nonconstant terms

Chapter 10. Local Langlands correspondence

Chapter 11. Local $L$functions and functional equations

Chapter 12. Normalization of intertwining operators

Chapter 13. Holomorphy and bounded in vertical strips

Chapter 14. Langlands functoriality conjecture

Chapter 15. Converse theorem of Cogdell and PiatetskiShapiro

Chapter 16. Functoriality of the symmetric cube

Chapter 17. Functoriality of the symmetric fourth

Bibliography

Applications of symmetric power $L$functions, by M. Ram Murty

Preface

Lecture 1. The SatoTate conjecture

Lecture 2. Maass wave forms

Lecture 3. The RankinSelberg 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 KimSarnak theorem

Lecture 10. Introduction to Artin $L$functions

Lecture 11. Zeros and poles of Artin $L$functions

Lecture 12. The LanglandsTunnell theorem

Bibliography


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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 PiatetskiShapiro with the LanglandsShahidi method. This book provides a stepbystep 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, SatoTate conjecture, and Artin's conjecture. It would be ideal for an introductory course in the Langlands program.
Graduate students and research mathematicians interested in representation theory and number theory.

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 nonArchimedean 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

Chapter 1. Chevalley groups and their properties

Chapter 2. Cuspidal representations

Chapter 3. $L$groups and automorphic $L$functions

Chapter 4. Induced representations

Chapter 5. Eisenstein series and constant terms

Chapter 6. $L$functions in the constant terms

Chapter 7. Meromorphic continuation of $L$functions

Chapter 8. Generic representations and their Whittaker models

Chapter 9. Local coefficients and nonconstant terms

Chapter 10. Local Langlands correspondence

Chapter 11. Local $L$functions and functional equations

Chapter 12. Normalization of intertwining operators

Chapter 13. Holomorphy and bounded in vertical strips

Chapter 14. Langlands functoriality conjecture

Chapter 15. Converse theorem of Cogdell and PiatetskiShapiro

Chapter 16. Functoriality of the symmetric cube

Chapter 17. Functoriality of the symmetric fourth

Bibliography

Applications of symmetric power $L$functions, by M. Ram Murty

Preface

Lecture 1. The SatoTate conjecture

Lecture 2. Maass wave forms

Lecture 3. The RankinSelberg 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 KimSarnak theorem

Lecture 10. Introduction to Artin $L$functions

Lecture 11. Zeros and poles of Artin $L$functions

Lecture 12. The LanglandsTunnell theorem

Bibliography