This book deals with a broad range of topics from the theory of automorphic functions on three-dimensional hyperbolic space and its arithmetic group theoretic and geometric ramifications. Starting off with several models of hyperbolic space and its group of motions the authors discuss the spectral theory of the Laplacian and Selberg's theory for cofinite groups. This culminates in explicit versions of the Selberg trace formula and the Selberg zeta-function. The interplay with arithmetic is demonstrated by means of the groups PSL(2) over rings and of quadratic integers, their Eisenstein series and their associated Hermitian forms. A rich chapter on concrete examples of arithmetic and non-arithmetic cofinite groups enhances the usefulness of this work for a wide circle of mathematicians.
Author(s): Juergen Elstrodt, Fritz Grunewald, Jens Mennicke
Series: Springer Monographs in Mathematics
Edition: 1
Publisher: Springer
Year: 1997
Language: English
Pages: 539
Table of Contents......Page 12
Preface......Page 6
1.1 The Upper Half-Space Model......Page 16
1.2 The Unit Ball Model......Page 24
1.3 The Exceptional Isomorphism......Page 27
1.4 The Hyperboloid Model......Page 35
1.5 The Kleinian Model......Page 37
1.6 Upper Half-Space as a Symmetric Space......Page 43
1.7 Notes and Remarks......Page 45
2.1 Discontinuity......Page 48
2.2 Fundamental Domains and Polyhedra......Page 56
2.3 Shimizu's Lemma......Page 63
2.4 J0rgensen's Inequality......Page 68
2.5 Covolumes......Page 74
2.6 Hyperbolic Lattice Point Problems......Page 82
2.7 Generators and Relations......Page 86
2.8 Conjugacy and Commensurability......Page 90
2.9 A Lemma of Selberg......Page 93
2.10 Notes and Remarks......Page 95
3. Automorphic Functions......Page 98
3.1 Definition and Elementary Properties of some Poincaré Series......Page 99
3.2 Definition and Elementary Properties of Eisenstein Series......Page 113
3.3 Fourier Expansion in Cusps and the Maafi-Selberg Relations......Page 120
3.4 Fourier Expansion of Eisenstein Series......Page 125
3.5 Expansion of Eigenfunctions and the Selberg Transform......Page 129
3.6 Behaviour of the Poincaré Series at the Abscissa of Convergence......Page 137
3.7 Notes and Remarks......Page 144
4. Spectral Theory of the Laplace Operator......Page 146
4.1 Essential Self-Adjointness of the Laplace-Beltrami Operator......Page 147
4.2 The Resolvent Kernel......Page 158
4.3 Hilbert-Schmidt Type Resolvents......Page 171
4.4 Analytic Continuation of the Resolvent Kernel......Page 177
4.5 Approximation by Kernels of Hilbert-Schmidt Type......Page 184
5. Spectral Theory of the Laplace Operator for Cocompact Groups......Page 200
5.1 The Hyperbolic Lattice-Point Problem......Page 202
5.2 Computation of the Trace......Page 205
5.3 Huber's Theorem......Page 216
5.4 The Selberg Zeta Function......Page 220
5.5 Weyl's Asymptotic Law and the Hadamard Factorisation of the Zeta Function......Page 225
5.6 Analogue of the Lindelof Hypothesis for the Selberg Zeta Function......Page 233
5.7 The Prime Geodesic Theorem......Page 237
5.8 Notes and Remarks......Page 243
6.1 Meromorphic Continuation of the Eisenstein Series......Page 246
6.2 Generalities on Eigenfunctions and Eigenpackets......Page 259
6.3 Spectral Decomposition Theory......Page 280
6.4 Spectral Expansions of Integral Kernels and Poincaré Series......Page 291
6.5 The Trace Formula and some Applications......Page 311
7.1 Introduction of the Groups......Page 326
7.2 The Cusps......Page 328
7.3 Description of a Fundamental Domain......Page 333
7.4 Groups Commensurable with PSL(2,O)......Page 342
7.5 The Group Theoretic Structure of PSL(2,O)......Page 349
7.6 Spectral Theory of the Laplace Operator......Page 361
7.7 Notes and Remarks......Page 371
8.1 Functions Closely Related to Eisenstein Series......Page 374
8.2 Fourier Expansion of Eisenstein Series for PSL(2,O)......Page 378
8.3 Meromorphic Continuation by Fourier Expansion and the Kronecker Limit Formula......Page 385
8.4 Special Values of Eisenstein Series......Page 395
8.5 Applications to Zeta Functions and Asymptotics of Divisor Sums......Page 408
8.6 Non-Vanishing of L-Functions......Page 412
8.7 Meromorphic Continuation by Integral Representation......Page 415
8.8 Computation of the Volume......Page 417
8.9 Weyl's Asymptotic Law......Page 419
9.1 Upper Half-Space and Binary Hermitian Forms......Page 422
9.2 Reduction Theory......Page 425
9.3 Representation Numbers of Binary Hermitian Forms......Page 429
9.4 Zeta Functions for Binary Hermitian Forms......Page 440
9.5 The Mass-Formula......Page 445
9.6 Computation of the Covolume of PSL(2,O) a la Humbert......Page 451
9.7 Notes and Remarks......Page 456
10. Examples of Discontinuous Groups......Page 458
10.1 Groups of Quaternions......Page 459
10.2 Unit Groups of Quadratic Forms......Page 471
10.3 Arithmetic and Non-Arithmetic Discrete Groups......Page 492
10.4 The Tetrahedral Groups......Page 495
10.5 Notes and Remarks......Page 509
References......Page 512
Subject Index......Page 536
