Lectures On Automorphic $l$-functions (fields Institute Monographs)

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. Cover......Page 1 Title page......Page 4 Contents......Page 6 Preface......Page 12 Lectures on ��-functions, converse theorems, and functoriality for ����_{��}, by James W. Cogdell......Page 14 Preface......Page 16 Lecture 1. Modular forms and their ��-functions......Page 18 Lecture 2. Automorphic forms......Page 26 Lecture 3. Automorphic representations......Page 34 Lecture 4. Fourier expansions and multiplicity one theorems......Page 42 Lecture 5. Eulerian integral representations......Page 50 Lecture 6. Local ��-functions: The non-Archimedean case......Page 58 Lecture 7. The unramified calculation......Page 64 Lecture 8. Local ��-functions: The Archimedean case......Page 72 Lecture 9. Global ��-functions......Page 78 Lecture 10. Converse theorems......Page 86 Lecture 11. Functoriality......Page 94 Lecture 12. Functoriality for the classical groups......Page 100 Lecture 13. Functoriality for the classical groups, II......Page 104 Automorphic ��-functions, by Henry H. Kim......Page 110 Introduction......Page 112 Chevalley groups and their properties......Page 114 Cuspidal representations......Page 126 ��-groups and automorphic ��-functions......Page 128 Induced representations......Page 132 Eisenstein series and constant terms......Page 142 ��-functions in the constant terms......Page 150 Meromorphic continuation of ��-functions......Page 158 Generic representations and their Whittaker models......Page 160 Local coefficients and non-constant terms......Page 166 Local Langlands correspondence......Page 174 Local ��-functions and functional equations......Page 178 Normalization of intertwining operators......Page 184 Holomorphy and bounded in vertical strips......Page 190 Langlands functoriality conjecture......Page 194 Converse theorem of Cogdell and Piatetski-Shapiro......Page 196 Functoriality of the symmetric cube......Page 200 Functoriality of the symmetric fourth......Page 206 Bibliography......Page 212 Applications of symmetric power ��-functions, by M. Ram Murty......Page 216 Preface......Page 218 Lecture 1. The Sato-Tate conjecture......Page 220 Lecture 2. Maass wave forms......Page 226 Lecture 3. The Rankin-Selberg method......Page 232 Lecture 4. Oscillations of Fourier coefficients of cusp forms......Page 240 Lecture 5. Poincaré series......Page 250 Lecture 6. Kloosterman sums and Selberg’s conjecture......Page 256 Lecture 7. Refined estimates for Fourier coefficients of cusp forms......Page 260 Lecture 8. Twisting and averaging of ��-series......Page 266 Lecture 9. The Kim-Sarnak theorem......Page 270 Lecture 10. Introduction to Artin ��-functions......Page 278 Lecture 11. Zeros and poles of Artin ��-functions......Page 284 Lecture 12. The Langlands-Tunnell theorem......Page 288 Bibliography......Page 294 Back Cover......Page 298 Cover 1 Title page 4 Contents 6 Preface 12 Lectures on L-functions, converse theorems, and functoriality for GL_{n}, by James W. Cogdell 14 Preface 16 Lecture 1. Modular forms and their L-functions 18 Lecture 2. Automorphic forms 26 Lecture 3. Automorphic representations 34 Lecture 4. Fourier expansions and multiplicity one theorems 42 Lecture 5. Eulerian integral representations 50 Lecture 6. Local L-functions: The non-Archimedean case 58 Lecture 7. The unramified calculation 64 Lecture 8. Local L-functions: The Archimedean case 72 Lecture 9. Global L-functions 78 Lecture 10. Converse theorems 86 Lecture 11. Functoriality 94 Lecture 12. Functoriality for the classical groups 100 Lecture 13. Functoriality for the classical groups, II 104 Automorphic L-functions, by Henry H. Kim 110 Introduction 112 Chevalley groups and their properties 114 Cuspidal representations 126 L-groups and automorphic L-functions 128 Induced representations 132 Eisenstein series and constant terms 142 L-functions in the constant terms 150 Meromorphic continuation of L-functions 158 Generic representations and their Whittaker models 160 Local coefficients and non-constant terms 166 Local Langlands correspondence 174 Local L-functions and functional equations 178 Normalization of intertwining operators 184 Holomorphy and bounded in vertical strips 190 Langlands functoriality conjecture 194 Converse theorem of Cogdell and Piatetski-Shapiro 196 Functoriality of the symmetric cube 200 Functoriality of the symmetric fourth 206 Bibliography 212 Applications of symmetric power L-functions, by M. Ram Murty 216 Preface 218 Lecture 1. The Sato-Tate conjecture 220 Lecture 2. Maass wave forms 226 Lecture 3. The Rankin-Selberg method 232 Lecture 4. Oscillations of Fourier coefficients of cusp forms 240 Lecture 5. Poincaré series 250 Lecture 6. Kloosterman sums and Selberg’s conjecture 256 Lecture 7. Refined estimates for Fourier coefficients of cusp forms 260 Lecture 8. Twisting and averaging of L-series 266 Lecture 9. The Kim-Sarnak theorem 270 Lecture 10. Introduction to Artin L-functions 278 Lecture 11. Zeros and poles of Artin L-functions 284 Lecture 12. The Langlands-Tunnell theorem 288 Bibliography 294 Back Cover 298 A series of lectures from a spring 2003 graduate course at the Fields Institute introduce Langlands functoriality conjecture and its consequences in number theory and representation theory, focusing on how automorphic L-functions play a crucial role in the theory. They cover converse theorems and functionality for GL(n), automorphic L-functions, and applications of symmetric power L-functions. They are not indexed. Annotation : 2004 Book News, Inc., Portland, OR (booknews.com) James W. Cogdell, Henry H. Kim, M. Ram Murty. Includes Bibliographical References (p. 281-283).