Jacobi-like forms, pseudodifferential operators, and quasimodular forms

Preface......Page 6
Contents......Page 8
Introduction......Page 11
 1.1 Correspondences of Power Series......Page 16
 1.2 Actions of SL(2,R) on Power Series......Page 21
 1.3 Jacobi-like Forms and Modular Series......Page 27
 1.4 Pseudodifferential Operators......Page 33
 2.1 Power Series Bundles......Page 40
 2.2 Automorphic Pseudodifferential Operators of Mixed Weight......Page 45
 2.3 Bundles of Pseudodifferential Operators......Page 49
 2.4 Poincaré Series......Page 52
 2.5 Linear Maps and Rankin–Cohen Brackets......Page 55
 3.1 Jacobi-like Forms, Modular Series, and Hecke Operators......Page 62
 3.2 Hecke Operators on Pseudodifferential Operators......Page 69
 3.3 Differential Equations and Modular Forms......Page 76
 3.4 Hecke Operators and Differential Equations......Page 81
 4.1 Lie Algebras of Power Series......Page 86
 4.2 Lie Algebra Homomorphisms......Page 89
 4.3 Equivariant Splittings......Page 92
 4.4 Lie Algebras of Jacobi-like Forms......Page 96
 5.1 Radial Heat Operators......Page 99
 5.2 Modular Series......Page 108
 5.3 Pseudodifferential Operators......Page 110
 6.1 Cohomology of Groups......Page 115
 6.2 Hecke Operators......Page 119
 6.3 Jacobi-like Forms and Group Cohomology......Page 123
 7.1 Quasimodular and Modular Forms......Page 129
 7.2 Polynomials and Formal Power Series......Page 135
 7.3 Quasimodular Polynomials......Page 138
 7.4 Hecke Operators......Page 145
 7.5 Vector Bundles......Page 149
 8.1 Correspondences of Polynomials......Page 156
 8.2 Modular and Quasimodular Polynomials......Page 160
 8.3 Poincare ´Series......Page 164
 8.4 Heat Operators......Page 168
 9.1 Liftings of Modular Forms to Quasimodular Forms......Page 172
 9.2 Liftings of Quasimodular Forms to Jacobi-like Forms......Page 175
 9.3 Lie Brackets......Page 183
 9.4 Rankin–Cohen Brackets on Quasimodular Forms......Page 188
 10.1 Vector-valued Forms......Page 192
 10.2 Symmetric Tensor Representations......Page 196
 10.3 Scalar- and Vector-valued Modular Forms......Page 201
 10.4 Quasimodular Polynomials and Vector-valued Modular Forms......Page 209
 11.1 Jacobi-like Forms and Quasimodular Forms with Character......Page 214
 11.2 Differential Operators on Modular Forms......Page 218
 11.3 Theta Functions......Page 222
 11.4 Differential Operators Associated to Theta Functions......Page 225
 12.1 Quasimodular Forms of Half-integral Weight......Page 230
 12.2 Shimura Correspondences......Page 234
 12.3 Shintani Liftings......Page 238
 13.1 Projective Structures on Riemann Surfaces......Page 242
 13.2 Pseudodifferential Operators......Page 245
 13.3 The Degree Four Case......Page 251
 13.4 The Degree Five Case......Page 253
 14.1 Quasimodular Forms for the Full Modular Group......Page 257
 14.2 The Work of Kaneko–Zagier......Page 262
 14.3 Covers of Elliptic Curves......Page 269
 14.4 Partitions......Page 274
 14.5 Modular Linear Differential Equations......Page 282
 14.6 Holomorphic Anomaly Equations......Page 287
 14.7 Other Applications......Page 290
References......Page 297
 List of Notation......Page 302
 Index......Page 306