Eta Products and Theta Series Identities (eBook)

Preface 5
Contents 7
Introduction 13
Theoretical Background 22
Dedekind's Eta Function and Modular Forms 23
Identities of Euler, Jacobi and Gauss 23
The Sign Transform 30
The Multiplier System of eta 31
The Concept of Modular Forms 35
Eisenstein Series for the Full Modular Group 39
Eisenstein Series for Gamma0(N) and Fricke Groups 40
Hecke Eigenforms 44
Identification of Modular Forms 49
Eta Products 51
Level, Weight, Nominator and Denominator of an Eta Product 51
Eta Products on the Fricke Group 53
Expansion and Order at Cusps 54
Conditions for Holomorphic Eta Products 56
The Cones and Simplices of Holomorphic Eta Products 57
Eta Products and Lattice Points in Simplices 58
The Simplices S(N, k) of Eta Products 58
The Setting for Prime Power Levels 59
Results for Prime Power Levels 60
Kronecker Products of Simplices 65
The Simplices for the Fricke Group 67
Eta Products of Weight 12 69
An Algorithm for Listing Lattice Points in a Simplex 74
Description of the Algorithm 74
Implementation 77
Output and Run Times 82
Theta Series with Hecke Character 85
Definition of Hecke Characters and Hecke L-functions 85
Hecke Theta Series for Quadratic Fields 87
Fourier Coefficients of Theta Series 88
More on Theta Series for Quadratic Fields 90
Description of Theta Series by Ideal Numbers 92
Coincidence of Theta Series of Weight 1 96
Groups of Coprime Residues in Quadratic Fields 98
Reduction to Prime Powers and One-units 98
One-units in Arbitrary Number Fields 100
Ramified Primes p > =3 in Quadratic Number Fields
The Ramified Prime 2 in Quadratic Number Fields 108
Examples 113
Ideal Numbers for Quadratic Fields 114
Class Numbers 1 and 2 114
Class Number 4 117
Class Number 8 120
Class Numbers 3, 6 and 12 124
Ideal Numbers for Some Real Quadratic Fields 126
Eta Products of Weight 12 and 32 128
Levels 1, 2 and 4 128
Levels 6 and 12 130
Eta Products of Weight 32 and the Concept of Superlacunarity 132
Level 1: The Full Modular Group 134
Weights k = 1, k 1 mod4 and k 1 mod6 134
Weights k = 2 and k 2 mod 6 137
Weights k = 3 and k 3 mod 4 140
Weights k = 4 and k 1 mod 3 143
Weights k 0 mod 6 146
The Prime Level N = 2 147
Weight 1 and Other Odd Weights for the Fricke Group Gamma*(2) 147
Weight 1 for Gamma0(2) 153
Even Weights for the Fricke Group Gamma*(2) 155
Weight k = 2 for Gamma0(2) 158
Lacunary Eta Products with Weight 3 for Gamma0(2) 160
Lacunary Eta Products with Weight 5 for Gamma0(2) 164
The Prime Level N = 3 168
Weight 1 and Other Weights k 1 mod 6 for Gamma*(3) and Gamma0(3) 168
Even Weights for the Fricke Group Gamma*(3) 171
Weights k 3, 5 mod 6 for the Fricke Group Gamma*(3) 176
Weight k = 2 for Gamma0(3) 180
Lacunary Eta Products with Weights k > 2 for Gamma0(3)
Prime Levels N = p > =5
Odd Weights for the Fricke Groups Gamma*(p), p = 5, 7,11, 23 185
Weight 1 for the Fricke Groups Gamma*(p), p = 13, 17, 19 192
Weight 2 for Gamma0(p) 194
Weights 3 and 5 for Gamma0(5) 197
Level N = 4 199
Odd Weights for the Fricke Group Gamma*(4) 199
Even Weights for the Fricke Group Gamma*(4) 203
Weight 1 for Gamma0(4) 205
Weight 2 for Gamma0(4), Cusp Forms with Denominators t < =6
Weight 2 for Gamma0(4), Cusp Forms with Denominators t =8, 12 213
Weight 2 for Gamma0(4), Cusp Forms with Denominator t =24 218
Weight 2 for Gamma0(4), Non-cuspidal Eta Products 222
A Remark on Weber Functions 225
Levels N = p2 with Primes p > =3
Weight 1 for Level N = 9 227
Weight 2 for the Fricke Group Gamma*(9) 228
Weight 2 for Gamma0(9) 229
Weight 2 for Levels N = p2, p > =5
Levels N = p3 and p4 for Primes p 234
Weights 1 and 2 for Gamma*(8) 234
Weight 1 for Gamma0(8), Cuspidal Eta Products 236
Weight 1 for Gamma0(8), Non-cuspidal Eta Products 240
Weight 1 for Gamma*(16) 242
Weight 2 for Gamma*(16) 244
Weight 1 for Gamma0(16), Cusp Forms with Denominators t = 3, 6, 8 251
Weight 1 for Gamma0(16), Cusp Forms with Denominator t = 24 253
Weight 1 for Gamma0(16), Non-cuspidal Eta Products 257
Levels N = pq with Primes 3 < =p <
Weight 1 for Fricke Groups Gamma*(3 q) 262
Weight 1 in the Case 5 < =p <
Weight 2 for Fricke Groups 268
Cuspidal Eta Products of Weight 2 for Gamma0(15) 270
Some Eta Products of Weight 2 for Gamma0(21) 274
Weight 1 for Levels N = 2p with Primes p > =5
Eta Products for Fricke Groups 277
Cuspidal Eta Products for Gamma0(10) 284
Non-cuspidal Eta Products for Gamma0(10) 288
Eta Products for Gamma0(14) 290
Eta Products for Gamma0(22) 292
Weight 1 for Levels 26, 34 and 38 294
Level N = 6 301
Weights 1 and 2 for Gamma*(6) 301
Weight 1 for Gamma0(6), Cusp Forms with Denominators t = 4, 6, 8 304
Weight 1 for Gamma0(6), Cusp Forms with Denominators t = 12, 24 306
Non-cuspidal Eta Products with Denominators t > =4
Non-cuspidal Eta Products with Denominators t < =3
Weight 1 for Prime Power Levels p5 and p6 315
Weight 1 for Gamma*(32) 315
Cuspidal Eta Products of Weight 1 for Gamma0(32) 316
Non-cuspidal Eta Products of Weight 1 for Gamma0(32) 321
Weight 1 for Level 64 324
Levels p2 q for Distinct Primes p < >
The Case of Odd Primes p and q 329
Levels 2 p2 for Primes p > =7
Eta Products of Level 50 332
Eta Products for the Fricke Group Gamma*(18) 337
Cuspidal Eta Products of Level 18 with Denominators t < =8
Cuspidal Eta Products of Level 18 with Denominators t > =12
Non-cuspidal Eta Products of Level 18, Denominators t > =4
Non-cuspidal Eta Products, Level 18, Denominators 3 and 2 350
Non-cuspidal Eta Products of Level 18 with Denominator 1 353
Levels 4 p for the Primes p = 23 and 19 357
An Overview 357
Eta Products for the Fricke Groups Gamma*(92) and Gamma*(76) 358
Cuspidal Eta Products for Gamma0(92) with Denominators t < =12
Cuspidal Eta Products for Gamma0(92) with Denominator 24 364
Non-cuspidal Eta Products for Gamma0(92) and Gamma0(76) 369
Cuspidal Eta Products for Gamma0(76) 372
Levels 4 p for p = 17 and 13 379
Eta Products for the Fricke Groups Gamma*(68) and Gamma*(52) 379
Cuspidal Eta Products for Gamma0(68) with Denominators t < =12
Cuspidal Eta Products for Gamma0(68) with Denominator 24 388
Non-cuspidal Eta Products for Gamma0(68) 390
Cuspidal Eta Products for Gamma0(52) with Denominators t < =12
Cuspidal Eta Products for Gamma0(52) with Denominator 24 400
Non-cuspidal Eta Products for Gamma0(52) 405
Levels 4 p for p = 11 and 7 407
Eta Products for the Fricke Groups Gamma*(44) and Gamma*(28) 407
Cuspidal Eta Products for Gamma0(44) with Denominators t < =12
Cuspidal Eta Products for Gamma0(44) with Denominator 24 414
Non-cuspidal Eta Products for Gamma0(44) 418
Cuspidal Eta Products for Gamma0(28) with Denominators t < =12
Cuspidal Eta Products for Gamma0(28) with Denominator 24 427
Non-cuspidal Eta Products for Gamma0(28) 432
Weight 1 for Level N = 20 436
Eta Products for the Fricke Group Gamma*(20) 436
Cuspidal Eta Products for Gamma0(20) with Denominators t < =6
Cuspidal Eta Products with Denominators 8 and 12 443
Cuspidal Eta Products with Denominator 24, First Part 447
Cuspidal Eta Products with Denominator 24, Second Part 452
Non-cuspidal Eta Products with Denominators t > 1
Non-cuspidal Eta Products with Denominator 1 460
Cuspidal Eta Products of Weight 1 for Level 12 464
Eta Products for the Fricke Group Gamma*(12) 464
Cuspidal Eta Products for Gamma0(12) with Denominators t = 2, 3 468
Cuspidal Eta Products with Denominator 4 471
Cuspidal Eta Products with Denominator 6 473
Cuspidal Eta Products with Denominator 8 475
Cuspidal Eta Products with Denominator 12 481
Cuspidal Eta Products with Denominator 24, First Part 484
Cuspidal Eta Products with Denominator 24, Second Part 489
Non-cuspidal Eta Products of Weight 1 for Level 12 494
Non-cuspidal Eta Products with Denominator 24 494
Non-cuspidal Eta Products with Denominators 6 and 12 496
Non-cuspidal Eta Products with Denominator 8 501
Non-cuspidal Eta Products with Denominator 4 505
Non-cuspidal Eta Products with Denominator 3 509
Non-cuspidal Eta Products with Denominator 2 513
Denominator 1, First Part 514
Denominator 1, Second Part 518
Weight 1 for Fricke Groups Gamma*(q3 p) 522
An Overview, and the Case p = 2 522
Levels N = 8 p for Primes p > =7
Eta Products for Gamma*(40) 528
Cuspidal Eta Products of Weight 1 for Gamma*(24) 531
Non-cuspidal Eta Products of Weight 1 for Gamma*(24) 533
Weight 1 for Fricke Groups Gamma*(2 p q) 535
Levels N = 2 p q for Primes p > q >
Levels 30 and 42 542
Levels 6 p for Primes p = 11, 13 545
Levels 6 p for Primes p = 17, 19, 23 548
Weight 1 for Fricke Groups Gamma*(p2 q2) 554
An Overview, and an Example for Level 196 554
Some Examples for Level 100 555
Cuspidal Eta Products for Gamma*(36) 559
Non-cuspidal Eta Products for Gamma*(36) 563
Weight 1 for the Fricke Groups Gamma*(60) and Gamma*(84) 566
An Overview 566
Cuspidal Eta Products for Gamma*(60) 566
Non-cuspidal Eta Products for Gamma*(60) 570
Cuspidal Eta Products for Gamma*(84) 573
Non-cuspidal Eta Products for Gamma*(84) 576
Some More Levels 4pq with Odd Primes p < >
Weight 1 for Gamma*(132) 578
Weight 1 for Gamma*(156) 582
Weight 1 for Gamma*(228) 587
Weight 1 for Gamma*(276) 592
Weight 1 for Gamma*(140) 594
Weight 1 for Gamma*(220) 597
Appendix 599
A Directory of Characters 599
B Index of Notations 614
References 616
Index 624