Lectures on automorphic L-functions:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, R.I.
American Mathematical Society
2004
|
| Series: | Fields Institute monographs
20 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references |
| Physical Description: | XII, 283 Seiten |
| ISBN: | 0821835165 |
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MARC
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| 245 | 1 | 0 | |a Lectures on automorphic L-functions |c James M. Cogdell, Henry H. Kim, M. Ram Murty |
| 264 | 1 | |a Providence, R.I. |b American Mathematical Society |c 2004 | |
| 300 | |a XII, 283 Seiten | ||
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| 490 | 1 | |a Fields Institute monographs |v 20 | |
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Contents
Preface xi
Lectures on // functions, Converse Theorems, and
Functionality for GL(n)
James W. Cogdell
Preface 3
Lecture 1. Modular Forms and Their // functions 5
1. Examples 6
2. Growth estimates on cusp forms 7
3. The L function of a cusp form 8
4. The Euler product 10
5. References 12
Lecture 2. Automorphic Forms 13
1. Automorphic forms on GL2 13
2. Automorphic forms on GLn 16
3. Smooth automorphic forms 17
4. L2 automorphic forms 18
5. Cusp forms 18
6. References 19
Lecture 3. Automorphic Representations 21
1. (/sT fmite) automorphic representations 21
2. Smooth automorphic representations 24
3. L2 automorphic representations 25
4. Cuspidal representations 25
5. Connections with classical forms 26
6. References 27
Lecture 4. Fourier Expansions and Multiplicity One Theorems 29
1. The Fourier expansion of a cusp form 29
2. Whittaker models 31
3. Multiplicity one for GLn 33
4. Strong multiplicity ones for GLn 34
5. References 35
V
vi Contents
Lecture 5. Eulerian Integral Representations 37
1. GL2 x GLX 37
2. GLn x GLm with m n 38
3. GLn x GLn 41
4. Summary 43
5. References 43
Lecture 6. Local L functions: The Non Archimedean Case 45
1. Whittaker functions 45
2. The local L function (m n) 46
3. The local functional equation 48
4. The conductor of tt 49
5. Multiplicativity and stability of 7 factors 49
6. References 50
Lecture 7. The Unramified Calculation 51
1. Unramified representations 52
2. Unramified Whittaker functions 53
3. Calculating the integral 55
4. References 57
Lecture 8. Local L functions: The Archimedean Case 59
1. The arithmetic Langlands classification 59
2. The L functions 59
3. The integrals (m n) 61
4. Is the L factor correct? 62
5. References 64
Lecture 9. Global L functions 65
1. Convergence 65
2. Meromorphic continuation 66
3. Poles of L functions 67
4. The global functional equation 67
5. Boundedness in vertical strips 68
6. Summary 69
7. Strong Multiplicity One revisited 69
8. Generalized Strong Multiplicity One 70
9. References 70
Lecture 10. Converse Theorems 73
1. Converse Theorems for GLn 73
2. Inverting the integral representation 74
3. Proof of Theorem 10.1 (i) 77
4. Proof of Theorem 10.1 (ii) 77
5. Theorem 10.2 and beyond 78
6. A useful variant 79
7. Conjectures 79
Contents vjj
8. References 80
Lecture 11. Functoriality 81
1. The Weil Deligne group 81
2. The dual group 82
3. The local Langlands conjecture 82
4. Local functoriality 83
5. Global functoriality 83
6. Functoriality and the Converse Theorem 84
7. References 85
Lecture 12. Functoriality for the Classical Groups 87
1. The results 87
2. Construction of a candidate lift 88
3. Analytic properties of L functions 90
4. Apply the Converse Theorem 90
5. References 90
Lecture 13. Functoriality for the Classical Groups, II 91
1. Functoriality 91
2. Descent 92
3. Bounds towards Ramanujan 94
4. The local converse theorem 94
5. Further applications 95
6. References 96
Automorphic L functions
Henry H. Kim
Introduction 99
Chapter 1. Chevalley Groups and their Properties 101
1. Algebraic groups 101
2. Roots and coroots 103
3. Classification of root systems 104
4. Construction of Chevalley groups: simply connected type 107
5. Structure of parabolic subgroups 108
Chapter 2. Cuspidal Representations 113
Chapter 3. L groups and Automorphic L functions 115
Chapter 4. Induced Representations 119
1. Harish Chandra homomorphisms 119
2. Induced representations: F local 121
3. Intertwining operators for I(s, it) 122
4. Digression on admissible representations 123
5. Induced representations: F global 126
viii Contents
6. Induced representations as holomorphic fiber bundles 126
Chapter 5. Eisenstein Series and Constant Terms 129
1. Definition of Eisenstein series 129
2. Constant terms 130
3. Psuedo Eisenstein series 132
Chapter 6. L functions in the Constant Terms 137
List of L functions via Langlands Shahidi method 143
Chapter 7. Meromorphic Continuation of L functions 145
Chapter 8. Generic Representations and their Whittaker Models 147
1. General case 147
2. Whittaker models for induced representations 149
Chapter 9. Local Coefficients and Non constant Terms 153
1. Non constant terms of Eisenstein series 153
2. Local coefficients and crude functional equation 158
Chapter 10. Local Langlands Correspondence 161
Chapter 11. Local L functions and Functional Equations 165
1. Definition of local L functions 169
2. Properties of local L functions; supercuspidal representations 170
Chapter 12. Normalization of Intertwining Operators 171
1. 7T is supercuspidal 171
2. it is tempered, generic 171
3. it is non tempered, generic 172
4. Application to reducibility criterion 175
Chapter 13. Holomorphy and Bounded in Vertical Strips 177
1. Holomorphy of L functions 177
2. Boundedness in vertical strips of L functions 177
Chapter 14. Langlands Functoriality Conjecture 181
Chapter 15. Converse Theorem of Cogdell and Piatetski Shapiro 183
Chapter 16. Functoriality of the Symmetric Cube 187
1. Weak Ramanujan property 187
2. Functoriality of the symmetric square 187
3. Functoriality of the tensor product of GL2 x GL3 188
4. Functoriality of the symmetric cube 190
Chapter 17. Functoriality of the Symmetric Fourth 193
1. Functoriality of the exterior square 193
Contents
IX
2. Functoriality of the symmetric fourth 194
Bibliography !99
Applications of Symmetric Power L functions
M. Ram Murty
Preface 205
Lecture 1. The Sato Tate Conjecture 207
1. Introduction 207
2. Uniform distribution 208
3. Wiener Ikehara Tauberian theorem 209
4. Weyl's theorem for compact groups 210
Lecture 2. Maass Wave Forms 213
1. Maass forms of weight zero 213
2. Maass forms with weight 214
3. Eisenstein series 214
4. Upper bound for Fourier coefficients and eigenvalue estimators 216
Lecture 3. The Rankin Selberg Method 219
1. Eisenstein series and non vanishing of £(s) on 9\(s) = 1 219
2. Explicit construction of Maass cusp forms 221
3. The Rankin Selberg L function 222
4. Rankin Selberg L functions for GLn 225
Lecture 4. Oscillations of Fourier Coefficients of Cusp Forms 227
1. Preliminaries 227
2. Rankin's theorem 228
3. A review of symmetric power L functions 230
4. Proof of Theorem 4.1 232
Lecture 5. Poincare Series 237
1. Poincare series for SL2(Z) 237
2. Fourier coefficients and Kloosterman sums 239
3. The Kloosterman Selberg zeta function 242
Lecture 6. Kloosterman Sums and Selberg's Conjecture 243
1. Petersson's formula 243
2. Selberg's theorem 244
3. The Selberg Linnik conjecture 245
Lecture 7. Refined Estimates for Fourier Coefficients of Cusp Forms 247
1. Sieve theory and Kloosterman sums 247
2. Gauss sums and hyper Kloosterman sum 248
3. The Duke Iwaniec method 248
x Contents
Lecture 8. Twisting and Averaging of L series 253
1. Selberg conjectures for GLn 253
2. Ramanujan conjecture for Gln 254
3. The method of averaging L functions 255
Lecture 9. The Kim Sarnak Theorem 257
1. Preliminaries 257
2. Rankin Selberg theory 258
3. An application of the Duke Iwaniec method 259
Lecture 10. Introduction to Artin L functions 265
1. Hecke L functions 265
2. Artin L functions 266
3. Automorphic induction and Artin's conjecture 268
Lecture 11. Zeros and Poles of Artin L functions 271
1. The Heilbronn character 271
2. The fundamental inequality 272
3. Rankin Selberg property for Galois representations 273
Lecture 12. The Langlands Tunnell Theorem 275
1. Review of some group theory 275
2. Some representation theory 276
3. An application of the Deligne Serre theory 277
4. The general case 277
5. Sarnak's theorem 278
Bibliography 281 |
| any_adam_object | 1 |
| author | Cogdell, James W. 1953- Kim, Henry H. 1964- Murty, Maruti Ram 1953- |
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| spelling | Cogdell, James W. 1953- Verfasser (DE-588)130465259 aut Lectures on automorphic L-functions James M. Cogdell, Henry H. Kim, M. Ram Murty Providence, R.I. American Mathematical Society 2004 XII, 283 Seiten txt rdacontent n rdamedia nc rdacarrier Fields Institute monographs 20 Includes bibliographical references Fonctions L Fonctions automorphes Automorphic functions L-functions Automorphe Funktion (DE-588)4143706-8 gnd rswk-swf L-Funktion (DE-588)4137026-0 gnd rswk-swf L-Funktion (DE-588)4137026-0 s Automorphe Funktion (DE-588)4143706-8 s b DE-604 Kim, Henry H. 1964- Verfasser (DE-588)17371983X aut Murty, Maruti Ram 1953- Verfasser (DE-588)120837307 aut Fields Institute monographs 20 (DE-604)BV009737926 20 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012817885&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Cogdell, James W. 1953- Kim, Henry H. 1964- Murty, Maruti Ram 1953- Lectures on automorphic L-functions Fields Institute monographs Fonctions L Fonctions automorphes Automorphic functions L-functions Automorphe Funktion (DE-588)4143706-8 gnd L-Funktion (DE-588)4137026-0 gnd |
| subject_GND | (DE-588)4143706-8 (DE-588)4137026-0 |
| title | Lectures on automorphic L-functions |
| title_auth | Lectures on automorphic L-functions |
| title_exact_search | Lectures on automorphic L-functions |
| title_full | Lectures on automorphic L-functions James M. Cogdell, Henry H. Kim, M. Ram Murty |
| title_fullStr | Lectures on automorphic L-functions James M. Cogdell, Henry H. Kim, M. Ram Murty |
| title_full_unstemmed | Lectures on automorphic L-functions James M. Cogdell, Henry H. Kim, M. Ram Murty |
| title_short | Lectures on automorphic L-functions |
| title_sort | lectures on automorphic l functions |
| topic | Fonctions L Fonctions automorphes Automorphic functions L-functions Automorphe Funktion (DE-588)4143706-8 gnd L-Funktion (DE-588)4137026-0 gnd |
| topic_facet | Fonctions L Fonctions automorphes Automorphic functions L-functions Automorphe Funktion L-Funktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012817885&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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