Eta products and theta series identities:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin ; Heidelberg [u.a.]
Springer
2011
|
| Ausgabe: | 1. ed. |
| Schriftenreihe: | Springer Monographs in Mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | XXI, 621 S. graph. Darst. 235 mm x 155 mm |
| ISBN: | 9783642161513 |
Internformat
MARC
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| 100 | 1 | |a Köhler, Günter |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Eta products and theta series identities |c Günter Köhler |
| 250 | |a 1. ed. | ||
| 264 | 1 | |a Berlin ; Heidelberg [u.a.] |b Springer |c 2011 | |
| 300 | |a XXI, 621 S. |b graph. Darst. |c 235 mm x 155 mm | ||
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| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Springer Monographs in Mathematics | |
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| 650 | 0 | 7 | |a Theta-Reihe |0 (DE-588)4206702-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Eisenstein-Reihe |0 (DE-588)4131762-2 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Dedekind-Etafunktion |0 (DE-588)4480070-8 |D s |
| 689 | 0 | 1 | |a Theta-Reihe |0 (DE-588)4206702-9 |D s |
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Datensatz im Suchindex
| _version_ | 1805095857252466688 |
|---|---|
| adam_text |
IMAGE 1
CONTENTS
PART I THEORETICAL BACKGROUND 1
1 DEDEKIND'S ETA FUNCTION AND MODULAR FORMS 3
1.1 IDENTITIES OF EULER, JACOBI AND GAUSS 3
1.2 THE SIGN TRANSFORM 10
1.3 THE MULTIPLIER SYSTEM OF T) 11
1.4 THE CONCEPT OF MODULAR FORMS 15
1.5 EISENSTEIN SERIES FOR THE FULL MODULAR GROUP 19
1.6 EISENSTEIN SERIES FOR TO (TV) AND FRICKE GROUPS 20
1.7 HECKE EIGENFORMS 24
1.8 IDENTIFICATION OF MODULAR FORMS 29
2 ETA PRODUCTS 31
2.1 LEVEL, WEIGHT, NOMINATOR AND DENOMINATOR OF AN ETA PRODUCT 31 2.2
ETA PRODUCTS ON THE FRICKE GROUP 33
2.3 EXPANSION AND ORDER AT CUSPS 34
2.4 CONDITIONS FOR HOLOMORPHIC ETA PRODUCTS 36
2.5 THE CONES AND SIMPLICES OF HOLOMORPHIC ETA PRODUCTS . . 37
3 ETA PRODUCTS AND LATTICE POINTS IN SIMPLICES 39
3.1 THE SIMPLICES S{N,K) OF ETA PRODUCTS 39
3.2 THE SETTING FOR PRIME POWER LEVELS 40
3.3 RESULTS FOR PRIME POWER LEVELS 41
3.4 KRONECKER PRODUCTS OF SIMPLICES 46
3.5 THE SIMPLICES FOR THE FRICKE GROUP 48
3.6 ETA PRODUCTS OF WEIGHT \ 50
4 AN ALGORITHM FOR LISTING LATTICE POINTS IN A SIMPLEX 55 4.1
DESCRIPTION OF THE ALGORITHM 55
4.2 IMPLEMENTATION 58
4.3 OUTPUT AND RUN TIMES 63
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1006508015
DIGITALISIERT DURCH
IMAGE 2
VIII CONTENTS
5 THETA SERIES WITH HECKE CHARACTER 67
5.1 DEFINITION OF HECKE CHARACTERS AND HECKE L-FUNCTIONS . . . 67 5.2
HECKE THETA SERIES FOR QUADRATIC FIELDS 69
5.3 FOURIER COEFFICIENTS OF THETA SERIES 70
5.4 MORE ON THETA SERIES FOR QUADRATIC FIELDS 72
5.5 DESCRIPTION OF THETA SERIES BY IDEAL NUMBERS 74
5.6 COINCIDENCE OF THETA SERIES OF WEIGHT 1 78
6 GROUPS OF COPRIME RESIDUES IN QUADRATIC FIELDS 81
6.1 REDUCTION TO PRIME POWERS AND ONE-UNITS 81
6.2 ONE-UNITS IN ARBITRARY NUMBER FIELDS 83
6.3 RAMIFIED PRIMES P 3 IN QUADRATIC NUMBER FIELDS 87
6.4 THE RAMIFIED PRIME 2 IN QUADRATIC NUMBER FIELDS 91
PART II EXAMPLES 97
7 IDEAL NUMBERS FOR QUADRATIC FIELDS 99
7.1 CLASS NUMBERS 1 AND 2 99
7.2 CLASS NUMBER 4 102
7.3 CLASS NUMBER 8 105
7.4 CLASS NUMBERS 3, 6 AND 12 109
7.5 IDEAL NUMBERS FOR SOME REAL QUADRATIC FIELDS I LL
8 ETA PRODUCTS OF WEIGHT \ AND § 113
8.1 LEVELS 1, 2 AND 4 113
8.2 LEVELS 6 AND 12 115
8.3 ETA PRODUCTS OF WEIGHT § AND THE CONCEPT OF SUPERLACUNARITY 117
9 LEVEL 1: THE FULL MODULAR GROUP 119
9.1 WEIGHTS K = 1, K = 1 MOD 4 AND K = IMOD6 119
9.2 WEIGHTS K = 2 AND K = 2MOD6 122
9.3 WEIGHTS K = 3 AND A; = 3MOD4 125
9.4 WEIGHTS K = 4 AND IT = 1 MOD 3 128
9.5 WEIGHTS K = 0MOD6 131
10 THE PRIME LEVEL N = 2 133
10.1 WEIGHT 1 AND OTHER ODD WEIGHTS FOR THE FRICKE GROUP T*(2) 133 10.2
WEIGHT 1 FOR T O (2) 139
10.3 EVEN WEIGHTS FOR THE FRICKE GROUP T* (2) 141
10.4 WEIGHT K = 2 FOR R O (2) 144
10.5 LACUNARY ETA PRODUCTS WITH WEIGHT 3 FOR T O (2) 146 10.6 LACUNARY
ETA PRODUCTS WITH WEIGHT 5 FOR T O (2) 150
11 THE PRIME LEVEL N = 3 155
11.1 WEIGHT 1 AND OTHER WEIGHTS K =E 1 MOD 6 FOR R*(3) AND R O (3) 155
IMAGE 3
CONTENTS IX
11.2 EVEN WEIGHTS FOR THE FRICKE GROUP T*(3) 158
11.3 WEIGHTS K = 3, 5MOD6 FOR THE FRICKE GROUP T*(3) 163
11.4 WEIGHT K = 2 FOR R O (3) 167
11.5 LACUNARY ETA PRODUCTS WITH WEIGHTS K 2 FOR R O (3) 169
12 PRIME LEVELS N = P 5 173
12.1 ODD WEIGHTS FOR THE FRICKE GROUPS T*(P), P = 5,7,11,23 . . . 173
12.2 WEIGHT 1 FOR THE FRICKE GROUPS T*{P), P = 13,17,19 180
12.3 WEIGHT 2 FOR T 0 (P) 182
12.4 WEIGHTS 3 AND 5 FOR R O (5) 185
13 LEVEL N = 4 187
13.1 ODD WEIGHTS FOR THE FRICKE GROUP T* (4) 187
13.2 EVEN WEIGHTS FOR THE FRICKE GROUP T*(4) 191
13.3 WEIGHT 1 FOR T O (4) 193
13.4 WEIGHT 2 FOR T O (4), CUSP FORMS WITH DENOMINATORS T G . . 197
13.5 WEIGHT 2 FOR R O (4), CUSP FORMS WITH DENOMINATORS T = 8,12 201
13.6 WEIGHT 2 FOR R O (4), CUSP FORMS WITH DENOMINATOR T = 24 . . 206
13.7 WEIGHT 2 FOR R O (4), NON-CUSPIDAL ETA PRODUCTS 210
13.8 A REMARK ON WEBER FUNCTIONS 213
14 LEVELS N = P 2 WITH PRIMES P 3 215
14.1 WEIGHT 1 FOR LEVEL N = 9 215
14.2 WEIGHT 2 FOR THE FRICKE GROUP T*(9) 216
14.3 WEIGHT 2 FOR R O (9) 217
14.4 WEIGHT 2 FOR LEVELS N = P 2 , P 5 220
15 LEVELS JV = P 3 AND P 4 FOR PRIMES P 223
15.1 WEIGHTS 1 AND 2 FOR T*(8) 223
15.2 WEIGHT 1 FOR R O (8), CUSPIDAL ETA PRODUCTS 225
15.3 WEIGHT 1 FOR R O (8), NON-CUSPIDAL ETA PRODUCTS 229
15.4 WEIGHT 1 FOR T*(16) 231
15.5 WEIGHT 2 FOR T*(16) 233
15.6 WEIGHT 1 FOR RO(16), CUSP FORMS WITH DENOMINATORS = 3,6,8 240
15.7 WEIGHT 1 FOR R O (16), CUSP FORMS WITH DENOMINATOR T = 24 . 242
15.8 WEIGHT 1 FOR R O (16), NON-CUSPIDAL ETA PRODUCTS 246
16 LEVELS N = PQ WITH PRIMES 3 P Q 251
16.1 WEIGHT 1 FOR FRICKE GROUPS T*(3Q) 251
16.2 WEIGHT 1 IN THE CASE 5 P Q 254
16.3 WEIGHT 2 FOR FRICKE GROUPS 257
16.4 CUSPIDAL ETA PRODUCTS OF WEIGHT 2 FOR R O (15) 259
16.5 SOME ETA PRODUCTS OF WEIGHT 2 FOR R O (21) 263
IMAGE 4
X CONTENTS
17 WEIGHT 1 FOR LEVELS TV = 2P WITH PRIMES P 5 267
17.1 ETA PRODUCTS FOR FRICKE GROUPS 267
17.2 CUSPIDAL ETA PRODUCTS FOR R 0 (10) 274
17.3 NON-CUSPIDAL ETA PRODUCTS FOR R 0 (10) 278
17.4 ETA PRODUCTS FOR R O (14) 280
17.5 ETA PRODUCTS FOR R O (22) 282
17.6 WEIGHT 1 FOR LEVELS 26, 34 AND 38 284
18 LEVEL TV = 6 291
18.1 WEIGHTS 1 AND 2 FOR P*(6) 291
18.2 WEIGHT 1 FOR R O (6), CUSP FORMS WITH DENOMINATORS T = 4,6,8 294
18.3 WEIGHT 1 FOR R O (6), CUSP FORMS WITH DENOMINATORS T = 12,24 296
18.4 NON-CUSPIDAL ETA PRODUCTS WITH DENOMINATORS T 4 299
18.5 NON-CUSPIDAL ETA PRODUCTS WITH DENOMINATORS T 3 301
19 WEIGHT 1 FOR PRIME POWER LEVELS P 5 AND P 6 305
19.1 WEIGHT 1 FOR T*(32) 305
19.2 CUSPIDAL ETA PRODUCTS OF WEIGHT 1 FOR T O (32) 306
19.3 NON-CUSPIDAL ETA PRODUCTS OF WEIGHT 1 FOR R 0 (32) 311
19.4 WEIGHT 1 FOR LEVEL 64 314
20 LEVELS P 2 Q FOR DISTINCT PRIMES P ^ 2 AND Q 319
20.1 THE CASE OF ODD PRIMES P AND Q 319
20.2 LEVELS 2P 2 FOR PRIMES P 7 321
20.3 ETA PRODUCTS OF LEVEL 50 322
20.4 ETA PRODUCTS FOR THE FRICKE GROUP T*(18) 327
20.5 CUSPIDAL ETA PRODUCTS OF LEVEL 18 WITH DENOMINATORS T 8 . 329
20.6 CUSPIDAL ETA PRODUCTS OF LEVEL 18 WITH DENOMINATORS T 12 332 20.7
NON-CUSPIDAL ETA PRODUCTS OF LEVEL 18, DENOMINATORS T 4 . 337 20.8
NON-CUSPIDAL ETA PRODUCTS, LEVEL 18, DENOMINATORS 3 AND 2 . 340 20.9
NON-CUSPIDAL ETA PRODUCTS OF LEVEL 18 WITH DENOMINATOR 1 . 343
21 LEVELS 4P FOR THE PRIMES P = 23 AND 19 347
21.1 AN OVERVIEW 347
21.2 ETA PRODUCTS FOR THE FRICKE GROUPS T*(92) AND T*(76) . . . 348
21.3 CUSPIDAL ETA PRODUCTS FOR T O (92) WITH DENOMINATORS T 12 . 350
21.4 CUSPIDAL ETA PRODUCTS FOR T O (92) WITH DENOMINATOR 24 . . 354
21.5 NON-CUSPIDAL ETA PRODUCTS FOR T 0 (92) AND R O (76) 359
21.6 CUSPIDAL ETA PRODUCTS FOR T O (76) 362
22 LEVELS AP FOR P = 17 AND 13 369
22.1 ETA PRODUCTS FOR THE FRICKE GROUPS T*(68) AND T*(52) . . . 369
22.2 CUSPIDAL ETA PRODUCTS FOR T O (68) WITH DENOMINATORS T 12 . 374
22.3 CUSPIDAL ETA PRODUCTS FOR R O (68) WITH DENOMINATOR 24 . . 378
22.4 NON-CUSPIDAL ETA PRODUCTS FOR R O (68) 380
22.5 CUSPIDAL ETA PRODUCTS FOR T O (52) WITH DENOMINATORS T 12 . 384
IMAGE 5
CONTENTS XI
22.6 CUSPIDAL ETA PRODUCTS FOR T O (52) WITH DENOMINATOR 24 . . 390
22.7 NON-CUSPIDAL ETA PRODUCTS FOR T 0 (52) 395
23 LEVELS 4P FOR P = 11 AND 7 397
23.1 ETA PRODUCTS FOR THE FRICKE GROUPS F*(44) AND T*(28) . . . 397
23.2 CUSPIDAL ETA PRODUCTS FOR T O (44) WITH DENOMINATORS T 12 . 401
23.3 CUSPIDAL ETA PRODUCTS FOR TO(44) WITH DENOMINATOR 24 . . 404 23.4
NON-CUSPIDAL ETA PRODUCTS FOR T O (44) 408
23.5 CUSPIDAL ETA PRODUCTS FOR TO(28) WITH DENOMINATORS T 12 . 410
23.6 CUSPIDAL ETA PRODUCTS FOR T O (28) WITH DENOMINATOR 24 . . 417
23.7 NON-CUSPIDAL ETA PRODUCTS FOR T 0 (28) 422
24 WEIGHT 1 FOR LEVEL N = 20 427
24.1 ETA PRODUCTS FOR THE FRICKE GROUP T*(20) 427
24.2 CUSPIDAL ETA PRODUCTS FOR R O (2O) WITH DENOMINATORS T 6 . 430
24.3 CUSPIDAL ETA PRODUCTS WITH DENOMINATORS 8 AND 12 434 24.4 CUSPIDAL
ETA PRODUCTS WITH DENOMINATOR 24, FIRST PART . . . 438 24.5 CUSPIDAL ETA
PRODUCTS WITH DENOMINATOR 24, SECOND PART . . 443
24.6 NON-CUSPIDAL ETA PRODUCTS WITH DENOMINATORS I 1 447 24.7
NON-CUSPIDAL ETA PRODUCTS WITH DENOMINATOR 1 451
25 CUSPIDAL ETA PRODUCTS OF WEIGHT 1 FOR LEVEL 12 455
25.1 ETA PRODUCTS FOR THE FRICKE GROUP T* (12) 455
25.2 CUSPIDAL ETA PRODUCTS FOR T O (12) WITH DENOMINATORS T = 2,3 459
25.3 CUSPIDAL ETA PRODUCTS WITH DENOMINATOR 4 462
25.4 CUSPIDAL ETA PRODUCTS WITH DENOMINATOR 6 464
25.5 CUSPIDAL ETA PRODUCTS WITH DENOMINATOR 8 466
25.6 CUSPIDAL ETA PRODUCTS WITH DENOMINATOR 12 472
25.7 CUSPIDAL ETA PRODUCTS WITH DENOMINATOR 24, FIRST PART . . 475 25.8
CUSPIDAL ETA PRODUCTS WITH DENOMINATOR 24, SECOND PART . 480
26 NON-CUSPIDAL ETA PRODUCTS OF WEIGHT 1 FOR LEVEL 12 485
26.1 NON-CUSPIDAL ETA PRODUCTS WITH DENOMINATOR 24 485
26.2 NON-CUSPIDAL ETA PRODUCTS WITH DENOMINATORS 6 AND 12 . . 487 26.3
NON-CUSPIDAL ETA PRODUCTS WITH DENOMINATOR 8 492
26.4 NON-CUSPIDAL ETA PRODUCTS WITH DENOMINATOR 4 496
26.5 NON-CUSPIDAL ETA PRODUCTS WITH DENOMINATOR 3 500
26.6 NON-CUSPIDAL ETA PRODUCTS WITH DENOMINATOR 2 504
26.7 DENOMINATOR 1, FIRST PART 505
26.8 DENOMINATOR 1, SECOND PART 509
27 WEIGHT 1 FOR FRICKE GROUPS T*(Q 3 P) 513
27.1 AN OVERVIEW, AND THE CASE P = 2 513
27.2 LEVELS N = 8P FOR PRIMES P 7 515
27.3 ETA PRODUCTS FOR T*(40) 519
IMAGE 6
XII CONTENTS
27.4 CUSPIDAL ETA PRODUCTS OF WEIGHT 1 FOR T*(24) 522
27.5 NON-CUSPIDAL ETA PRODUCTS OF WEIGHT 1 FOR T* (24) 524
28 WEIGHT 1 FOR FRICKE GROUPS T*(2PQ) 527
28.1 LEVELS N = 2PQ FOR PRIMES P Q 5 527
28.2 LEVELS 30 AND 42 534
28.3 LEVELS 6P FOR PRIMES P= 11,13 537
28.4 LEVELS 6P FOR PRIMES P= 17,19,23 540
29 WEIGHT 1 FOR FRICKE GROUPS T*(P 2 Q 2 ) 547
29.1 AN OVERVIEW, AND AN EXAMPLE FOR LEVEL 196 547
29.2 SOME EXAMPLES FOR LEVEL 100 548
29.3 CUSPIDAL ETA PRODUCTS FOR T*(36) 552
29.4 NON-CUSPIDAL ETA PRODUCTS FOR T*(36) 556
30 WEIGHT 1 FOR THE FRICKE GROUPS T*(60) AND T*(84) 559 30.1 AN OVERVIEW
559
30.2 CUSPIDAL ETA PRODUCTS FOR T*(60) 559
30.3 NON-CUSPIDAL ETA PRODUCTS FOR T*(60) 563
30.4 CUSPIDAL ETA PRODUCTS FOR T*(84) 566
30.5 NON-CUSPIDAL ETA PRODUCTS FOR T*(84) 569
31 SOME MORE LEVELS 4PQ WITH ODD PRIMES P ^ Q 571
31.1 WEIGHT 1 FOR T*(132) 571
31.2 WEIGHT 1 FOR T*(156) 575
31.3 WEIGHT 1 FOR T*(228) 580
31.4 WEIGHT 1 FOR T*(276) 585
31.5 WEIGHT 1 FOR T*(140) 587
31.6 WEIGHT 1 FOR T*(220) 590
APPENDIX 593
A DIRECTORY OF CHARACTERS 593
B INDEX OF NOTATIONS 608
REFERENCES 611
INDEX 619 |
| any_adam_object | 1 |
| author | Köhler, Günter |
| author_facet | Köhler, Günter |
| author_role | aut |
| author_sort | Köhler, Günter |
| author_variant | g k gk |
| building | Verbundindex |
| bvnumber | BV037349438 |
| classification_rvk | SK 180 |
| ctrlnum | (OCoLC)697829888 (DE-599)DNB1006508015 |
| dewey-full | 512.7 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.7 |
| dewey-search | 512.7 |
| dewey-sort | 3512.7 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. ed. |
| format | Book |
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| id | DE-604.BV037349438 |
| illustrated | Illustrated |
| indexdate | 2024-07-20T11:05:57Z |
| institution | BVB |
| isbn | 9783642161513 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-022502955 |
| oclc_num | 697829888 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-20 |
| owner_facet | DE-19 DE-BY-UBM DE-20 |
| physical | XXI, 621 S. graph. Darst. 235 mm x 155 mm |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer Monographs in Mathematics |
| spelling | Köhler, Günter Verfasser aut Eta products and theta series identities Günter Köhler 1. ed. Berlin ; Heidelberg [u.a.] Springer 2011 XXI, 621 S. graph. Darst. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Springer Monographs in Mathematics Quadratischer Zahlkörper (DE-588)4385849-1 gnd rswk-swf Modulform (DE-588)4128299-1 gnd rswk-swf Dedekind-Etafunktion (DE-588)4480070-8 gnd rswk-swf Theta-Reihe (DE-588)4206702-9 gnd rswk-swf Eisenstein-Reihe (DE-588)4131762-2 gnd rswk-swf Dedekind-Etafunktion (DE-588)4480070-8 s Theta-Reihe (DE-588)4206702-9 s Quadratischer Zahlkörper (DE-588)4385849-1 s Eisenstein-Reihe (DE-588)4131762-2 s Modulform (DE-588)4128299-1 s DE-604 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3532043&prov=M&dok%5Fvar=1&dok%5Fext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022502955&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Köhler, Günter Eta products and theta series identities Quadratischer Zahlkörper (DE-588)4385849-1 gnd Modulform (DE-588)4128299-1 gnd Dedekind-Etafunktion (DE-588)4480070-8 gnd Theta-Reihe (DE-588)4206702-9 gnd Eisenstein-Reihe (DE-588)4131762-2 gnd |
| subject_GND | (DE-588)4385849-1 (DE-588)4128299-1 (DE-588)4480070-8 (DE-588)4206702-9 (DE-588)4131762-2 |
| title | Eta products and theta series identities |
| title_auth | Eta products and theta series identities |
| title_exact_search | Eta products and theta series identities |
| title_full | Eta products and theta series identities Günter Köhler |
| title_fullStr | Eta products and theta series identities Günter Köhler |
| title_full_unstemmed | Eta products and theta series identities Günter Köhler |
| title_short | Eta products and theta series identities |
| title_sort | eta products and theta series identities |
| topic | Quadratischer Zahlkörper (DE-588)4385849-1 gnd Modulform (DE-588)4128299-1 gnd Dedekind-Etafunktion (DE-588)4480070-8 gnd Theta-Reihe (DE-588)4206702-9 gnd Eisenstein-Reihe (DE-588)4131762-2 gnd |
| topic_facet | Quadratischer Zahlkörper Modulform Dedekind-Etafunktion Theta-Reihe Eisenstein-Reihe |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3532043&prov=M&dok%5Fvar=1&dok%5Fext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022502955&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT kohlergunter etaproductsandthetaseriesidentities |