Traces of Hecke operators:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
[2006]
|
| Schriftenreihe: | Mathematical surveys and monographs
Volume 133 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Beschreibung: | ix, 378 Seiten Illustrationen, Diagramme |
| ISBN: | 0821837397 9780821837399 |
Internformat
MARC
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| 020 | |a 9780821837399 |9 978-0-8218-3739-9 | ||
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| 100 | 1 | |a Knightly, Andrew |d 1972- |e Verfasser |0 (DE-588)137749643 |4 aut | |
| 245 | 1 | 0 | |a Traces of Hecke operators |c Andrew Knightly, Charles Li |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c [2006] | |
| 264 | 4 | |c © 2006 | |
| 300 | |a ix, 378 Seiten |b Illustrationen, Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematical surveys and monographs |v Volume 133 | |
| 650 | 4 | |a Formules de trace | |
| 650 | 4 | |a Hecke, Opérateurs de | |
| 650 | 4 | |a Nombres, Théorie des | |
| 650 | 7 | |a Teoria dos números |2 larpcal | |
| 650 | 4 | |a Hecke operators | |
| 650 | 4 | |a Trace formulas | |
| 650 | 4 | |a Number theory | |
| 650 | 0 | 7 | |a Spurformel |0 (DE-588)4182612-7 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Hecke-Operator |0 (DE-588)4135665-2 |2 gnd |9 rswk-swf |
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| 689 | 0 | 1 | |a Spurformel |0 (DE-588)4182612-7 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Li, Charles |d 1965- |e Verfasser |0 (DE-588)118039512 |4 aut | |
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| 830 | 0 | |a Mathematical surveys and monographs |v Volume 133 |w (DE-604)BV000018014 |9 133 | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-016090601 | ||
Datensatz im Suchindex
| _version_ | 1804137151556747264 |
|---|---|
| adam_text | Contents
Traces of Hecke Operators
1. Introduction 1
2. The Arthur-Selberg trace formula for GL(2) 3
3. Cusp forms and Hecke operators 7
3.1. Congruence subgroups of SL2(Z) 7
3.2. Weak modular forms 10
3.3. Cusps and Fourier expansions of modular forms 12
3.4. Hecke rings 20
3.5. The level N Hecke ring 24
3.6. The elements T(n) 29
3.7. Hecke operators 34
3.8. The Petersson inner product 37
3.9. Adjoints of Hecke operators 42
3.10. Traces of the Hecke operators 45
Odds and Ends
4. Topological groups 49
5. Adeles and ideles 52
5.1. p-adic Numbers 52
5.2. Adeles and ideles 54
6. Structure theorems and strong approximation for GL2(A) 59
6.1. Topology of GL2(A) 59
6.2. The Iwasawa decomposition 61
6.3. Strong approximation for GL2(A) 63
6.4. The Cartan decomposition 66
6.5. The Bruhat decomposition 69
7. Haar measure 69
7.1. Basic properties of Haar measure 69
7.2. Invariant measure on a quotient space 74
7.3. Haar measure on a restricted direct product 82
7.4. Haar measure on the adeles and ideles 85
7.5. Haar measure on B 87
7.6. Haar measure on GL(2) 90
7.7. Haar measure on SL2(R) 92
7.8. Haar measure on GL(2) 94
7.9. Discrete subgroups and fundamental domains 95
7.10. Haar measure on Q A and Q* A* 102
7.11. Quotient measure on GL2(Q) GL2(A) 103
7.12. Quotient measure on J?(Q) G(A) 105
8. The Poisson summation formula 108
9. Tate zeta functions 119
9.1. Definition and meromorphic continuation 119
9.2. Functional equation and behavior at s = 1 124
10. Intertwining operators and matrix coefficients 128
10.1. Linear algebra 129
10.2. Representation theory 134
10.3. Orthogonality of matrix coefficients 140
11. The discrete series of GL2(R) 151
11.1. K-type decompositions 151
11.2. Representations of 0(2) 155
11.3. K-type decomposition of an induced representation 156
11.4. The (s, ^-modules (VX)K 162
11.5. Classification of irreducible (fl, if )-modules for GL2(R) 165
11.6. A detailed look at the Lie algebra action 181
11.7. Characterization of the weight k discrete series of GL2(R) 186
Groundwork
12. Cusp forms as elements of Lq(w) 195
12.1. Prom Dirichlet characters to Hecke characters 195
12.2. From cusp forms to functions on G(A) 197
12.3. Comparison of classical and adelic Fourier coefficients 198
12.4. Characterizing the image of S^(N,uj ) in Lq(co) 201
13. Construction of the test function / 206
13.1. The non-archimedean component of / 206
13.2. Spectral properties of R(f) 213
14. Explicit computations for /k and /oo 221
The Trace Formula
15. Introduction to the trace formula for R(f) 227
16. Terms that contribute to K(x, y) 230
17. Truncation of the kernel 231
18. Bounds for E7 I/( ?! S^) | 240
19. Integrability of kj(g) 248
20. The hyperbolic terms as weighted orbital integrals 259
21. Simplifying the unipotent term 270
22. The trace formula 276
Computation of the Trace
23. The identity term 279
24. The hyperbolic terms 280
24.1. A lemma about orbital integrals 280
24.2. The archimedean orbital integral and weighted orbital integral 281
24.3. Simplification of the hyperbolic term 283
24.4. Calculation of the local orbital integrals 283
24.5. The global hyperbolic result 286
25. The unipotent term 288
25.1. Explicit evaluation of the zeta integral at oo 288
25.2. Computation of the non-archimedean local zeta functions 291
25.3. The global unipotent result 293
26. The elliptic terms 294
26.1. Properties of the elliptic orbital integrals 295
26.2. The archimedean elliptic orbital integral 300
26.3. Orders and lattices in an imaginary quadratic field 302
26.4. Local-global theory for lattices 307
26.5. Prom G(Afin) to lattices in E 310
26.6. The non-archimedean orbital integral for N = 1 314
26.7. The case of level TV 318
Applications
27. Dimension formulas 333
27.1. The elliptic terms 333
27.2. The unipotent term 337
27.3. The dimension of S^(N,tu ) and some examples 338
28. Computing Hecke eigenvalues 340
28.1. Obtaining eigenvalues from knowledge of the traces 340
28.2. Integrality of Hecke eigenvalues 343
28.3. The r-function 344
28.4. An example with nontrivial character 347
29. The distribution of Hecke eigenvalues 351
29.1. Bounds for the Eichler-Selberg trace formula 352
29.2. Chebyshev polynomials 355
29.3. Distribution of eigenvalues 356
29.4. Further applications and generalizations 359
30. A recursion relation for traces of Hecke operators 360
Bibliography 363
Tables of notation 368
Statement of the final result 370
Index 373
|
| adam_txt |
Contents
Traces of Hecke Operators
1. Introduction 1
2. The Arthur-Selberg trace formula for GL(2) 3
3. Cusp forms and Hecke operators 7
3.1. Congruence subgroups of SL2(Z) 7
3.2. Weak modular forms 10
3.3. Cusps and Fourier expansions of modular forms 12
3.4. Hecke rings 20
3.5. The level N Hecke ring 24
3.6. The elements T(n) 29
3.7. Hecke operators 34
3.8. The Petersson inner product 37
3.9. Adjoints of Hecke operators 42
3.10. Traces of the Hecke operators 45
Odds and Ends
4. Topological groups 49
5. Adeles and ideles 52
5.1. p-adic Numbers 52
5.2. Adeles and ideles 54
6. Structure theorems and strong approximation for GL2(A) 59
6.1. Topology of GL2(A) 59
6.2. The Iwasawa decomposition 61
6.3. Strong approximation for GL2(A) 63
6.4. The Cartan decomposition 66
6.5. The Bruhat decomposition 69
7. Haar measure 69
7.1. Basic properties of Haar measure 69
7.2. Invariant measure on a quotient space 74
7.3. Haar measure on a restricted direct product 82
7.4. Haar measure on the adeles and ideles 85
7.5. Haar measure on B 87
7.6. Haar measure on GL(2) 90
7.7. Haar measure on SL2(R) 92
7.8. Haar measure on GL(2) 94
7.9. Discrete subgroups and fundamental domains 95
7.10. Haar measure on Q\A and Q*\A* 102
7.11. Quotient measure on GL2(Q)\GL2(A) 103
7.12. Quotient measure on J?(Q)\G(A) 105
8. The Poisson summation formula 108
9. Tate zeta functions 119
9.1. Definition and meromorphic continuation 119
9.2. Functional equation and behavior at s = 1 124
10. Intertwining operators and matrix coefficients 128
10.1. Linear algebra 129
10.2. Representation theory 134
10.3. Orthogonality of matrix coefficients 140
11. The discrete series of GL2(R) 151
11.1. K-type decompositions 151
11.2. Representations of 0(2) 155
11.3. K-type decomposition of an induced representation 156
11.4. The (s, ^-modules (VX)K 162
11.5. Classification of irreducible (fl, if )-modules for GL2(R) 165
11.6. A detailed look at the Lie algebra action 181
11.7. Characterization of the weight k discrete series of GL2(R) 186
Groundwork
12. Cusp forms as elements of Lq(w) 195
12.1. Prom Dirichlet characters to Hecke characters 195
12.2. From cusp forms to functions on G(A) 197
12.3. Comparison of classical and adelic Fourier coefficients 198
12.4. Characterizing the image of S^(N,uj') in Lq(co) 201
13. Construction of the test function / 206
13.1. The non-archimedean component of / 206
13.2. Spectral properties of R(f) 213
14. Explicit computations for /k and /oo 221
The Trace Formula
15. Introduction to the trace formula for R(f) 227
16. Terms that contribute to K(x, y) 230
17. Truncation of the kernel 231
18. Bounds for E7 I/( ?!"S^) | 240
19. Integrability of kj(g) 248
20. The hyperbolic terms as weighted orbital integrals 259
21. Simplifying the unipotent term 270
22. The trace formula 276
Computation of the Trace
23. The identity term 279
24. The hyperbolic terms 280
24.1. A lemma about orbital integrals 280
24.2. The archimedean orbital integral and weighted orbital integral 281
24.3. Simplification of the hyperbolic term 283
24.4. Calculation of the local orbital integrals 283
24.5. The global hyperbolic result 286
25. The unipotent term 288
25.1. Explicit evaluation of the zeta integral at oo 288
25.2. Computation of the non-archimedean local zeta functions 291
25.3. The global unipotent result 293
26. The elliptic terms 294
26.1. Properties of the elliptic orbital integrals 295
26.2. The archimedean elliptic orbital integral 300
26.3. Orders and lattices in an imaginary quadratic field 302
26.4. Local-global theory for lattices 307
26.5. Prom G(Afin) to lattices in E 310
26.6. The non-archimedean orbital integral for N = 1 314
26.7. The case of level TV 318
Applications
27. Dimension formulas 333
27.1. The elliptic terms 333
27.2. The unipotent term 337
27.3. The dimension of S^(N,tu') and some examples 338
28. Computing Hecke eigenvalues 340
28.1. Obtaining eigenvalues from knowledge of the traces 340
28.2. Integrality of Hecke eigenvalues 343
28.3. The r-function 344
28.4. An example with nontrivial character 347
29. The distribution of Hecke eigenvalues 351
29.1. Bounds for the Eichler-Selberg trace formula 352
29.2. Chebyshev polynomials 355
29.3. Distribution of eigenvalues 356
29.4. Further applications and generalizations 359
30. A recursion relation for traces of Hecke operators 360
Bibliography 363
Tables of notation 368
Statement of the final result 370
Index 373 |
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| id | DE-604.BV022885691 |
| illustrated | Illustrated |
| index_date | 2024-07-02T18:51:53Z |
| indexdate | 2024-07-09T21:07:44Z |
| institution | BVB |
| isbn | 0821837397 9780821837399 |
| language | English |
| lccn | 2006047814 |
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| physical | ix, 378 Seiten Illustrationen, Diagramme |
| publishDate | 2006 |
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| publisher | American Mathematical Society |
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| spelling | Knightly, Andrew 1972- Verfasser (DE-588)137749643 aut Traces of Hecke operators Andrew Knightly, Charles Li Providence, Rhode Island American Mathematical Society [2006] © 2006 ix, 378 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Mathematical surveys and monographs Volume 133 Formules de trace Hecke, Opérateurs de Nombres, Théorie des Teoria dos números larpcal Hecke operators Trace formulas Number theory Spurformel (DE-588)4182612-7 gnd rswk-swf Hecke-Operator (DE-588)4135665-2 gnd rswk-swf Hecke-Operator (DE-588)4135665-2 s Spurformel (DE-588)4182612-7 s DE-604 Li, Charles 1965- Verfasser (DE-588)118039512 aut Erscheint auch als Online-Ausgabe 978-1-4704-1360-6 Mathematical surveys and monographs Volume 133 (DE-604)BV000018014 133 http://www.loc.gov/catdir/toc/fy0707/2006047814.html Inhaltsverzeichnis HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016090601&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Knightly, Andrew 1972- Li, Charles 1965- Traces of Hecke operators Mathematical surveys and monographs Formules de trace Hecke, Opérateurs de Nombres, Théorie des Teoria dos números larpcal Hecke operators Trace formulas Number theory Spurformel (DE-588)4182612-7 gnd Hecke-Operator (DE-588)4135665-2 gnd |
| subject_GND | (DE-588)4182612-7 (DE-588)4135665-2 |
| title | Traces of Hecke operators |
| title_auth | Traces of Hecke operators |
| title_exact_search | Traces of Hecke operators |
| title_exact_search_txtP | Traces of Hecke operators |
| title_full | Traces of Hecke operators Andrew Knightly, Charles Li |
| title_fullStr | Traces of Hecke operators Andrew Knightly, Charles Li |
| title_full_unstemmed | Traces of Hecke operators Andrew Knightly, Charles Li |
| title_short | Traces of Hecke operators |
| title_sort | traces of hecke operators |
| topic | Formules de trace Hecke, Opérateurs de Nombres, Théorie des Teoria dos números larpcal Hecke operators Trace formulas Number theory Spurformel (DE-588)4182612-7 gnd Hecke-Operator (DE-588)4135665-2 gnd |
| topic_facet | Formules de trace Hecke, Opérateurs de Nombres, Théorie des Teoria dos números Hecke operators Trace formulas Number theory Spurformel Hecke-Operator |
| url | http://www.loc.gov/catdir/toc/fy0707/2006047814.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016090601&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000018014 |
| work_keys_str_mv | AT knightlyandrew tracesofheckeoperators AT licharles tracesofheckeoperators |
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