Quadratic forms and Hecke operators:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
1987
|
| Schriftenreihe: | Grundlehren der mathematischen Wissenschaften
286 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XII, 374 S. |
| ISBN: | 3540152946 0387152946 |
Internformat
MARC
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| 100 | 1 | |a Andrianov, Anatolij N. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Quadratic forms and Hecke operators |c Anatolij N. Andrianov |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1987 | |
| 300 | |a XII, 374 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Grundlehren der mathematischen Wissenschaften |v 286 | |
| 650 | 4 | |a Formes modulaires | |
| 650 | 4 | |a Hecke, Opérateurs de | |
| 650 | 7 | |a Hecke-operatoren |2 gtt | |
| 650 | 7 | |a Kwadratische vormen |2 gtt | |
| 650 | 4 | |a Séries thêta | |
| 650 | 4 | |a Forms, Modular | |
| 650 | 4 | |a Hecke operators | |
| 650 | 4 | |a Series, Theta | |
| 650 | 0 | 7 | |a Quadratische Form |0 (DE-588)4128297-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Hecke-Operator |0 (DE-588)4135665-2 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Hecke-Operator |0 (DE-588)4135665-2 |D s |
| 689 | 0 | 1 | |a Quadratische Form |0 (DE-588)4128297-8 |D s |
| 689 | 0 | |5 DE-604 | |
| 689 | 1 | 0 | |a Hecke-Operator |0 (DE-588)4135665-2 |D s |
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Datensatz im Suchindex
| _version_ | 1804115090471911424 |
|---|---|
| adam_text | Table of Contents
Preface VII
Chapter 1. Theta Series 1
§1.1. Definition of Theta Series 1
1.1.1. Representations of Quadratic Forms by Quadratic Forms . 1
1.1.2. Definition of Theta Series 3
§1.2. Symplectic Transformations 4
1.2.1. The Symplectic Group 4
1.2.2. The Siegel Upper Half Plane 7
§1.3. Symplectic Transformations of Theta Series 10
1.3.1. Transformations of Theta Functions 10
1.3.2. The Siegel Modular Group and the Theta Group 17
1.3.3. Symplectic Transformations of Theta Series 20
§1.4. Computation of the Multiplier 26
1.4.1. Factors of Automorphy 27
1.4.2. Quadratic Forms of Level 1 28
1.4.3. The Multiplier as a Gaussian Sum 30
1.4.4. Quadratic Forms in an Even Number of Variables .... 34
1.4.5. Quadratic Forms in an Odd Number of Variables 40
Chapter 2. Modular Forms 43
§2.1. Fundamental Domains for Subgroups of the Modular Group . . 43
2.1.1. The Modular Triangle 43
2.1.2. The Minkowski Domain 46
2.1.3. The Fundamental Domain for the Siegel Modular Group . 53
2.1.4. Subgroups of Finite Index 60
§ 2.2. Definition of Modular Forms 62
2.2.1. Congruence Subgroups of the Modular Group 62
2.2.2. Modular Forms of Integral Weight 62
2.2.3. Modular Forms of Half Integral Weight 63
2.2.4. Theta Series as Modular Forms 64
§ 2.3. Fourier Expansions 64
2.3.1. Modular Forms for Triangular Subgroups 64
X Table of Contents
2.3.2. Koecher s Effect 65
2.3.3. Fourier Expansions of Modular Forms 69
2.3.4. The Siegel Operator 74
2.3.5. Cusp Forms 78
2.3.6. Singular Forms 82
§ 2.4. Spaces of Modular Forms 87
2.4.1. Zeroes of Modular Forms for F1 88
2.4.2. Modular Forms Whose Initial Fourier Coefficients
Equal Zero 91
2.4.3. Dimension of Spaces of Modular Forms 95
§2.5. Scalar Product and Orthogonal Decomposition 96
2.5.1. Scalar Product 97
2.5.2. Orthogonal Decomposition 101
Chapter 3. Hecke Rings 105
§3.1. Abstract Hecke Rings 105
3.1.1. Averaging over Double Cosets 105
3.1.2. Hecke Rings 106
3.1.3. The Imbedding i 110
3.1.4. The Anti Isomorphism j Ill
3.1.5. Representations on Automorphic Functions 114
3.1.6. Hecke Rings over a Commutative Ring 115
§3.2. Hecke Rings of the General Linear Group 116
3.2.1. Global Rings 116
3.2.2. Local Rings 123
3.2.3. The Spherical Mapping 129
§3.3. Hecke Rings of the Symplectic Group 134
3.3.1. Global Rings 134
3.3.2. Local Rings 145
3.3.3. The Spherical Mapping 150
§ 3.4. Hecke Rings of the Triangular Subgroup of the
Symplectic Group 168
3.4.1. Global Rings 168
3.4.2. Local Rings 175
3.4.3. Decomposition of Elements T (a) for n = 1, 2 178
§3.5. Factorization of Symplectic Polynomials 180
3.5.1. Negative Powers of Frobenius Elements 180
3.5.2. Factorization of Symplectic Polynomials 186
3.5.3. A Symmetric Factorization of Polynomials Q£(u) for
m = 1, 2 190
3.5.4. Coefficients of Factors of Polynomials R£(i;) 192
3.5.5. A Symmetric Factorization of Polynomials R£(r) 208
Table of Contents XI
Chapter 4. Hecke Operators 212
§4.1. Hecke Operators for Congruence Subgroups of the
Modular Group 212
4.1.1. Hecke Operators 212
4.1.2. Invariant Subspaces and Eigenfunctions 216
§4.2. Action of Hecke Operators 221
4.2.1. Hecke Operators for rg(q) 221
4.2.2. Hecke Operators for To 222
4.2.3. Relations with Hecke Operators for G !_„ 229
4.2.4. Hecke Operators and the Siegel Operator 234
4.2.5. The Action of the Middle Factor of the Symmetric
Factorization of Polynomials Rp{v) 242
§ 4.3. Multiplicative Properties of Fourier Coefficients 252
4.3.1. Modular Forms of One Variable 253
4.3.2. Modular Forms of Genus 2, Gaussian Composition, and
Zeta Functions 264
4.3.3. Modular Forms of Arbitrary Genus and Even
Zeta Functions 283
Chapter 5. The Action of Hecke Operators on Theta Series 291
§5.1. The Action of Hecke Operators on Theta Series 292
5.1.1. Theta Series and e Series 292
5.1.2. The Basic Case 299
5.1.3. The General Case 306
§ 5.2. Theta Matrices of Hecke Operators and Eichler Matrices .... 310
5.2.1. The Comparison of Fourier Coefficients 311
5.2.2. Theta Matrices of Elements T (p) 314
5.2.3. Theta Matrices of Coefficients of Even Local Zeta Functions 319
5.2.4. Theta Matrices of Generators of Hecke Rings 322
5.2.5. Relations, Relations 325
Appendix 1. Symmetric Matrices over Fields 330
A.I.I. Arbitrary Fields 330
A.1.2. The Field R 331
Appendix 2. Quadratic Spaces 334
A.2.1. The Geometric Language 334
A.2.2. Non Degenerate Spaces 338
A.2.3. Gaussian Sums 339
A.2.4. Isotropic Subspaces of Non Degenerate Spaces over
the Fields Fp 342
A.2.5. Non Singular Spaces over Residue Class Rings 345
A.2.6. The Genus of Quadratic Spaces over Z 348
XII Table of Contents
Appendix 3. Modules in Quadratic Fields and Binary
Quadratic Forms 350
A.3.1. Modules of Algebraic Number Fields 350
A.3.2. Modules in Quadratic Fields and Prime Numbers .... 351
A.3.3. Modules in Imaginary Quadratic Fields and
Quadratic Forms 356
Notes 360
On Chapter 1 360
On Chapter 2 360
On Chapter 3 361
On Chapter 4 362
On Chapter 5 363
References 364
Index of Terminology 367
Index of Notation 370
|
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| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:17:05Z |
| institution | BVB |
| isbn | 3540152946 0387152946 |
| language | English |
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| physical | XII, 374 S. |
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| series | Grundlehren der mathematischen Wissenschaften |
| series2 | Grundlehren der mathematischen Wissenschaften |
| spelling | Andrianov, Anatolij N. Verfasser aut Quadratic forms and Hecke operators Anatolij N. Andrianov Berlin [u.a.] Springer 1987 XII, 374 S. txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 286 Formes modulaires Hecke, Opérateurs de Hecke-operatoren gtt Kwadratische vormen gtt Séries thêta Forms, Modular Hecke operators Series, Theta Quadratische Form (DE-588)4128297-8 gnd rswk-swf Hecke-Operator (DE-588)4135665-2 gnd rswk-swf Hecke-Operator (DE-588)4135665-2 s Quadratische Form (DE-588)4128297-8 s DE-604 Grundlehren der mathematischen Wissenschaften 286 (DE-604)BV000000395 286 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000400673&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Andrianov, Anatolij N. Quadratic forms and Hecke operators Grundlehren der mathematischen Wissenschaften Formes modulaires Hecke, Opérateurs de Hecke-operatoren gtt Kwadratische vormen gtt Séries thêta Forms, Modular Hecke operators Series, Theta Quadratische Form (DE-588)4128297-8 gnd Hecke-Operator (DE-588)4135665-2 gnd |
| subject_GND | (DE-588)4128297-8 (DE-588)4135665-2 |
| title | Quadratic forms and Hecke operators |
| title_auth | Quadratic forms and Hecke operators |
| title_exact_search | Quadratic forms and Hecke operators |
| title_full | Quadratic forms and Hecke operators Anatolij N. Andrianov |
| title_fullStr | Quadratic forms and Hecke operators Anatolij N. Andrianov |
| title_full_unstemmed | Quadratic forms and Hecke operators Anatolij N. Andrianov |
| title_short | Quadratic forms and Hecke operators |
| title_sort | quadratic forms and hecke operators |
| topic | Formes modulaires Hecke, Opérateurs de Hecke-operatoren gtt Kwadratische vormen gtt Séries thêta Forms, Modular Hecke operators Series, Theta Quadratische Form (DE-588)4128297-8 gnd Hecke-Operator (DE-588)4135665-2 gnd |
| topic_facet | Formes modulaires Hecke, Opérateurs de Hecke-operatoren Kwadratische vormen Séries thêta Forms, Modular Hecke operators Series, Theta Quadratische Form Hecke-Operator |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000400673&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
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