On the Theory of Maass Wave Forms

Preface......Page 6
 Acknowledgements......Page 7
Contents......Page 9
List of Symbols......Page 14
 Chapter 1 – A Short Introduction to Modular Forms......Page 16
 Chapter 2 – Period Polynomials......Page 17
 Chapter 3 – Maass Wave Forms of Real Weight......Page 18
 Chapter 4 – Families of Maass Cusp Forms, L-Series, and Eichler Integrals......Page 19
 Chapter 6 – Continued Fractions and the Transfer Operator Approach......Page 21
 Chapter 7 – Weak Harmonic Maass Wave Forms......Page 23
 1.1 Notations and Some Simple Results......Page 25
  1.2.1 The Full Modular Group......Page 30
  1.2.2 Fundamental Domains for the Full Modular Group......Page 32
  1.2.3 Fundamental Domains for Subgroups of (1)......Page 33
  1.2.4 Cuspidal Points......Page 35
  1.3.1 Introduction – Modular Forms......Page 39
  1.3.2 Multiplier Systems......Page 41
  1.3.3 Modular Forms......Page 45
  1.3.4 Eisenstein Series......Page 50
  1.3.5 The Discriminant Function (z)......Page 52
  1.3.6 Petersson Scalar Product and the Hilbert Space of Cusp Forms......Page 54
 1.4 Holomorphic Poincaré Series......Page 56
  1.4.1 Further Remarks on Modular Forms......Page 61
  1.5.1 Jacobi Theta Function......Page 62
  1.5.2 Theta Series of Index m......Page 76
  1.5.3 Theta Series and Sums of Squares......Page 77
  1.6.1 The Slash Operator......Page 91
  1.6.2 Hecke Operators of Index n N......Page 93
  1.6.3 Eigenforms......Page 96
  1.7.1 The Mellin Transform......Page 98
  1.7.2 L-Series Associated with Cusp Forms on (1)......Page 102
  1.7.3 L-Functions of Hecke Eigenforms......Page 110
 1.8 Concluding Remarks......Page 116
 2.1 Various Ways to Period Polynomials......Page 117
  2.1.1 Period Polynomials via Integral Transformations......Page 118
  2.1.2 Eichler Integrals and Period Polynomials......Page 122
  2.1.3 The (k-1)-Fold Antiderivative......Page 126
  2.1.4 An Integral Representation of the Eichler Integral for Cusp Forms......Page 128
  2.1.5 An Integral Representation of the Eichler Integral for Entire Forms......Page 131
  2.1.6 The Niebur Integral Representation for Entire Forms......Page 134
 2.2 The Eichler Cohomology Group......Page 136
 2.3 The Eichler Cohomology Theorem......Page 142
  2.3.1 Injectivity of the Map μ......Page 144
  2.3.2 Surjectivity of the Map μ......Page 148
 2.4 Hecke Operators and Period Polynomials......Page 157
 2.5 Concluding Remarks......Page 171
 3.1 Multiplier Systems and Unitary Automorphic Factors......Page 172
 3.2 The Differential Operators k and Ek......Page 175
  3.2.1 Whittaker Functions......Page 182
  3.2.2 Whittaker-Fourier Expansions on (1)......Page 185
  3.2.3 Whittaker-Fourier Expansions for (1)......Page 187
  3.2.4 Maass Operators Ek and the Fourier-WhittakerExpansion......Page 192
 3.3 Maass Wave Forms of Real Weight......Page 195
  3.4.1 The Embedding of Classical Modular Forms......Page 203
  3.4.2 Non-holomorphic Eisenstein Series for the Full Modular Group......Page 209
  3.4.3 Maass Cusp Forms Derived from the Dedekindη-Function......Page 212
  3.5.1 Definition of the Petersson Scalar Product......Page 214
  3.5.2 Properties of the Petersson Scalar Product......Page 215
  3.5.3 The Petersson Scalar Product and Maass Cusp Forms......Page 219
 3.6 Hecke Operators on Maass Cusp Forms......Page 222
 3.7 The Laplace Operator and Its Self-Adjoint FriedrichsExtension......Page 226
  3.7.1 The Friedrichs Extension of 0 for the Full Modular Group......Page 227
  3.7.2 The Friedrichs Extension of k......Page 230
 3.8 Selberg\'s Conjecture......Page 235
 3.9 Concluding Remarks......Page 236
4 Families of Maass Cusp Forms, L-Series, and Eichler Integrals......Page 238
 4.1 Families of Maass Cusp Forms on the Full Modular Group......Page 239
 4.2 L-Functions Associated with Maass Cusp Forms......Page 250
  4.2.1 L-Functions for Sk,v(to.(1),ν)to.......Page 251
  4.2.2 A Simpler Representation of Families of Cusp Forms......Page 278
  4.2.3 D-Functions for Sk,v(to.(1),ν)to.......Page 296
 4.3 Nearly Periodic Functions and Maass Cusp Forms......Page 305
  4.3.1 D-Functions Giving Rise to Nearly Periodic Functions......Page 306
  4.3.2 Nearly Periodic Functions Giving Rise to D-Functions......Page 327
   4.3.3.1 The R-Function......Page 335
   4.3.3.2 The Maass-Selberg Differential Form......Page 341
   4.3.3.3 Everything Combined......Page 344
   4.3.3.4 Nearly Periodic Functions via Integral Transformations......Page 345
  4.3.4 Nearly Periodic Functions and Eichler Integrals......Page 349
 4.4 Concluding Remarks......Page 351
5 Period Functions......Page 353
 5.1 Period Functions Associated with Maass Cusp Forms......Page 354
  5.1.1 Period Functions and Nearly Periodic Functions......Page 356
  5.1.2 Period Functions via Integral Transformations......Page 362
  5.1.3 Period Functions via Nearly Periodic Functions Given by an Integral Transform......Page 366
  5.1.4 Period Functions and Families of Maass Cusp Forms......Page 368
  5.1.5 Period Functions and Period Polynomials......Page 373
  5.1.6 Some Remarks on Period Functions......Page 375
  5.2.1 The Spaces Vν∞, Vνω, and Vνω*,∞......Page 376
  5.2.2 Mixed Parabolic Cohomology Group H1par(to.(1), Vνω, Vνω*,∞)to.......Page 377
  5.2.3 Zagier\'s Cohomology Theorem......Page 379
 5.3 Hecke Operators on Period Functions......Page 380
  5.3.1 On Farey Sequences......Page 381
  5.3.2 Left Neighbor Sequences......Page 383
  5.3.3 Paths in the Upper Half-Plane......Page 387
  5.3.4 Hecke Operators for Period Functions for (1)......Page 389
 5.4 Concluding Remarks......Page 393
 6.1 The Artin Billiard and Continued Fractions......Page 394
  6.1.1 Continued Fractions and the Gauss Map......Page 395
  6.1.2 GL2( Z ) and the Artin Billiard......Page 398
   6.1.2.1 The Group GL2( Z )......Page 399
   6.1.2.2 GL2( Z ) and Möbius Transformations......Page 401
   6.1.2.3 The Artin Billiard......Page 402
  6.1.3 Geodesics on the Artin Billiard......Page 405
   6.1.3.1 GL2( Z )-Equivalent Geodesics......Page 406
   6.1.3.2 GL2( Z )-Equivalent Geodesics and Continued Fractions......Page 409
  6.1.4 Continued Fractions, Geodesics, and the Full Modular Group......Page 415
  6.1.5 Closed Geodesics Under SL2( Z ) and Periodic Continued Fraction Expansions......Page 416
  6.2.1 Motivation of Dynamical Zeta Functions and the Associated Transfer Operator......Page 419
  6.2.2 The Transfer Operator for the Gauss Map......Page 420
  6.2.3 Mayer\'s Transfer Operator......Page 430
 6.3 The Transfer Operator and Period Functions on (1)......Page 432
  6.3.1 Eigenfunctions of the Transfer Operator and Solutions of a Three-Term Equation......Page 435
  6.3.2 Solutions of a Three-Term Equation and PeriodFunctions......Page 442
  6.4.1 Geodesics on H, SL2( Z ) Periodic Orbits, and Continued Fractions......Page 446
  6.4.2 The Dynamical Zeta Function by Ruelle and the Selberg Zeta Function......Page 449
 6.5 Concluding Remarks......Page 451
7 Weak Harmonic Maass Wave Forms......Page 453
 7.1 Weak Harmonic Maass Wave Forms......Page 454
  7.2.1 Pure Mock Modular Forms......Page 463
  7.2.2 Mock Jacobi Forms......Page 466
  7.2.3 General Mock Modular Forms......Page 467
  7.3.1 Regularized L-Series for S!k,1(to.(1))to.......Page 471
  7.3.2 Regularized L-Series for Hk(to.(1))to.......Page 475
  7.4.1 The Mock Modular Period Function......Page 476
  7.4.2 Period Functions via Formal Eichler Integrals......Page 477
 7.5 Concluding Remarks......Page 485
 A.2 Phragmén-Lindelöf principle......Page 486
 A.4 Cauchy-Riemann Equations......Page 487
 A.6 Rouché\'s Theorem......Page 488
 A.8 Unique Continuation Principle For Holomorphic Functions......Page 489
 A.9 Whittaker Functions......Page 490
 A.10 Generalized Hypergeometric Functions......Page 491
 A.12 Liouville\'s Theorem......Page 492
 A.14 Geometric Series......Page 493
 A.15 Continued Fractions......Page 495
 A.16 Hurwitz Zeta Function......Page 499
 A.18 Properties of Linear Fractional Transformations......Page 500
 A.19 Nuclear Operator......Page 502
 A.20 Geodesics on the Upper Half-Plane (Poincaré Model of a Hyperbolic Plane)......Page 503
References......Page 509
Symbol Index......Page 519
Index......Page 521