**Fields Institute Monographs**

Volume: 20;
2004;
283 pp;
Softcover

MSC: Primary 11; 22;

Print ISBN: 978-0-8218-4800-5

Product Code: FIM/20.S

List Price: $92.00

Individual Member Price: $73.60

**Electronic ISBN: 978-1-4704-3147-1
Product Code: FIM/20.E**

List Price: $92.00

Individual Member Price: $73.60

#### Supplemental Materials

# Lectures on Automorphic $L$-functions

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*James W. Cogdell; Henry H. Kim; M. Ram Murty*

A co-publication of the AMS and Fields Institute

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.

Titles in this series are co-published with The Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

#### Table of Contents

# Table of Contents

## Lectures on Automorphic $L$-functions

- Cover Cover11
- Title page iii4
- Contents v6
- Preface xi12
- Lectures on 𝐿-functions, converse theorems, and functoriality for 𝐺𝐿_{𝑛}, by James W. Cogdell 114
- Preface 316
- Lecture 1. Modular forms and their 𝐿-functions 518
- Lecture 2. Automorphic forms 1326
- Lecture 3. Automorphic representations 2134
- Lecture 4. Fourier expansions and multiplicity one theorems 2942
- Lecture 5. Eulerian integral representations 3750
- Lecture 6. Local 𝐿-functions: The non-Archimedean case 4558
- Lecture 7. The unramified calculation 5164
- Lecture 8. Local 𝐿-functions: The Archimedean case 5972
- Lecture 9. Global 𝐿-functions 6578
- Lecture 10. Converse theorems 7386
- Lecture 11. Functoriality 8194
- Lecture 12. Functoriality for the classical groups 87100
- Lecture 13. Functoriality for the classical groups, II 91104
- Automorphic 𝐿-functions, by Henry H. Kim 97110
- Introduction 99112
- Chevalley groups and their properties 101114
- Cuspidal representations 113126
- 𝐿-groups and automorphic 𝐿-functions 115128
- Induced representations 119132
- Eisenstein series and constant terms 129142
- 𝐿-functions in the constant terms 137150
- Meromorphic continuation of 𝐿-functions 145158
- Generic representations and their Whittaker models 147160
- Local coefficients and non-constant terms 153166
- Local Langlands correspondence 161174
- Local 𝐿-functions and functional equations 165178
- Normalization of intertwining operators 171184
- Holomorphy and bounded in vertical strips 177190
- Langlands functoriality conjecture 181194
- Converse theorem of Cogdell and Piatetski-Shapiro 183196
- Functoriality of the symmetric cube 187200
- Functoriality of the symmetric fourth 193206
- Bibliography 199212
- Applications of symmetric power 𝐿-functions, by M. Ram Murty 203216
- Preface 205218
- Lecture 1. The Sato-Tate conjecture 207220
- Lecture 2. Maass wave forms 213226
- Lecture 3. The Rankin-Selberg method 219232
- Lecture 4. Oscillations of Fourier coefficients of cusp forms 227240
- Lecture 5. Poincaré series 237250
- Lecture 6. Kloosterman sums and Selberg’s conjecture 243256
- Lecture 7. Refined estimates for Fourier coefficients of cusp forms 247260
- Lecture 8. Twisting and averaging of 𝐿-series 253266
- Lecture 9. The Kim-Sarnak theorem 257270
- Lecture 10. Introduction to Artin 𝐿-functions 265278
- Lecture 11. Zeros and poles of Artin 𝐿-functions 271284
- Lecture 12. The Langlands-Tunnell theorem 275288
- Bibliography 281294
- Back Cover Back Cover1298

#### Readership

Graduate students and research mathematicians interested in representation theory and number theory.