Families of Automorphic Forms. Modern Birkhäuser Classics (eBook)

Contents 6
Preface 9
Chapter 1 Modular introduction 11
1.1 The modular group 11
1.2 Maass forms 13
1.3 Holomorphic modular forms 15
1.4 Fourier expansion of modular forms 17
1.5 More modular forms 19
1.6 Truncation and perturbation 24
1.7 Further remarks 28
Part I General theory 32
Chapter 2 Automorphic forms on the universal covering group 33
2.1 Automorphic forms on the upper half plane 33
2.2 The universal covering group 35
2.3 Automorphic forms on G 39
Chapter 3 Discrete subgroups 41
3.1 Cofinite groups 41
3.2 The quotient 43
3.3 Canonical generators 45
3.4 Characters 50
3.5 Notations 52
Chapter 4 Automorphic forms 55
4.1 Fourier expansion 56
4.2 Spaces of Fourier terms 59
4.3 Automorphic forms with growth condition 66
4.4 Differentiation of Fourier terms 69
4.5 Differentiation of automorphic forms 73
4.6 Maass-Selberg relation 74
Chapter 5 Poincar´e series 79
5.1 Construction of Poincar´e series 79
5.2 Fourier coefficients 86
Chapter 6 Selfadjoint extension of the Casimir operator 93
6.1 The Hilbert space of equivariant functions 94
6.2 The subspace for the energy norm 96
6.3 Fourier coefficients 99
6.4 Compactness 102
6.5 Extension of the Casimir operator 105
6.6 Relation to automorphic forms 107
6.7 The discrete spectrum 109
Chapter 7 Families of automorphic forms 114
7.1 Parameter spaces 114
7.2 Holomorphic families of automorphic forms and Fourier terms 116
7.3 Families of eigenfunctions 120
7.4 Families of eigenfunctions with automorphic transformation behavior 124
7.5 Families of automorphic forms 130
7.6 Families of Fourier terms 131
7.7 Differentiation 137
Chapter 8 Transformation and truncation 139
8.1 Parameter space 139
8.2 Transformation 141
8.3 Truncation 144
8.4 The subspace for the energy norm 147
8.5 Families of automorphic forms 153
Chapter 9 Pseudo Casimir operator 156
9.1 Sesquilinear form 156
9.2 Pseudo Casimir operator 161
9.3 Meromorphy of the resolvent 165
9.4 Meromorphic families of automorphic forms 172
9.5 Dimension results for spaces of automorphic forms 179
Chapter 10 Meromorphic continuation of Poincar´e series 182
10.1 Cells of continuation 182
10.2 Meromorphic continuation 185
10.3 Relations and functional equations 190
Chapter 11 Poincar´e families along vertical lines 195
11.1 General results 195
11.2 Eisenstein families 202
11.3 Other Poincar´e families 207
Chapter 12 Singularities of Poincar´e families 216
12.1 Local curves 216
12.2 Value sets 220
12.3 General results on singularities 224
12.4 General parameter spaces 230
12.5 Restricted parameter spaces 233
Part II Examples 240
Chapter 13 Automorphic forms for the modular group 241
13.1 The covering group and its characters 241
13.2 Fourier expansions of modular forms 246
13.3 The modular spectrum 249
13.4 Families of modular forms 251
13.5 Derivatives of Eisenstein families 254
13.6 Distribution results 260
Chapter 14 Automorphic forms for the theta group 267
14.1 Theta function and theta group 267
14.2 The covering group 268
14.3 Fourier expansions 270
14.4 Eisenstein series 273
14.5 More than one parameter 276
Chapter 15 Automorphic forms for the commutator subgroup 277
15.1 Commutator subgroup 277
15.2 Automorphic forms for 280
15.3 The period map 282
15.4 Poincar´e series 286
15.5 Eisenstein family of weight 0 288
15.6 Harmonic automorphic forms 292
15.7 Maass forms and singularities of the Eisenstein family 304
Bibliography 309
Index 313