Groups acting on hyperbolic space : harmonic analysis and number theory / J. Elstrodt, F. Grunewald, J. Mennicke.
Series Springer monographs in mathematicsEditor: Berlin ; New York : Springer, c1998Descripción: xv, 524 p. ; 25 cmISBN: 3540627456 (Berlin : alk. paper)Tema(s): Spectral theory (Mathematics) | Selberg trace formula | Automorphic forms | Functions, ZetaOtra clasificación: 11F72 (11F12 11M36 22E40 57M50 57S30) Recursos en línea: Publisher description | Table of contents onlyPreface VII -- 1. Three-Dimensional Hyperbolic Space [1] -- 1.1 The Upper Half-Space Model [1] -- 1.2 The Unit Ball Model [9] -- 1.3 The Exceptional Isomorphism [12] -- 1.4 The Hyperboloid Model [20] -- 1.5 The Kleinian Model [22] -- 1.6 Upper Half-Space as a Symmetric Space [28] -- 1.7 Notes and Remarks [30] -- 2. Groups Acting Discontinuously on Three-Dimensional Hyperbolic Space [33] -- 2.1 Discontinuity [33] -- 2.2 Fundamental Domains and Polyhedra [41] -- 2.3 Shimizu’s Lemma [48] -- 2.4 Jorgensen’s Inequality [53] -- 2.5 Covolumes [59] -- 2.6 Hyperbolic Lattice Point Problems [67] -- 2.7 Generators and Relations [71] -- 2.8 Conjugacy and Commensurability [75] -- 2.9 A Lemma of Selberg [78] -- 2.10 Notes and Remarks [80] -- 3. Automorphic Functions [83] -- 3.1 Definition and Elementary Properties of some Poincaré Series [84] -- 3.2 Definition and Elementary Properties of Eisenstein Series [98] -- 3.3 Fourier Expansion in Cusps and the MaaB-Selberg Relations [105] -- 3.4 Fourier Expansion of Eisenstein Series [110] -- 3.5 Expansion of Eigenfunctions and the Selberg Transform [114] -- 3.6 Behaviour of the Poincar4 Series at the Abscissa of Convergence [122] -- 3.7 Notes and Remarks [129] -- 4. Spectral Theory of the Laplace Operator [131] -- 4.1 Essential Self-Adjointness of the Laplace-Beltrami Operator [132] -- 4.2 The Resolvent Kernel [143] -- 4.3 Hilbert-Schmidt Type Resolvents [156] -- 4.4 Analytic Continuation of the Resolvent Kernel [162] -- 4.5 Approximation by Kernels of Hilbert-Schmidt Type [169] -- 5. Spectral Theory of the Laplace Operator for Cocompact Groups [185] -- 5.1 The Hyperbolic Lattice-Point Problem [187] -- 5.2 Computation of the Trace [190] -- 5.3 Huber’s Theorem [201] -- 5.4 The Selberg Zeta Function [205] -- 5.5 Weyl’s Asymptotic Law and the Hadamard Factorisation of the Zeta Function [210] -- 5.6 Analogue of the Lindelöf Hypothesis for the Selberg Zeta Function [218] -- 5.7 The Prime Geodesic Theorem [222] -- 5.8 Notes and Remarks [228] -- 6. Spectral Theory of the Laplace Operator for Cofinite Groups [231] -- 6.1 Meromorphic Continuation of the Eisenstein Series [231] -- 6.2 Generalities on Eigenfunctions and Eigenpackets [244] -- 6.3 Spectral Decomposition Theory [265] -- 6.4 Spectral Expansions of Integral Kernels and Poincaré Series [276] -- 6.5 The Trace Formula and some Applications [296] -- 6.6 Notes and Remarks [310] -- 7. PSL(2) over Rings of Imaginary Quadratic Integers [311] -- 7.1 Introduction of the Groups [311] -- 7.2 The Cusps [313] -- 7.3 Description of a Fundamental Domain [318] -- 7.4 Groups Commensurable with PSL(2,Q) [327] -- 7.5 The Group Theoretic Structure of PSL(2.Q) [334] -- 7.6 Spectral Theory of the Laplace Operator [346] -- 7.7 Notes and Remarks [356] -- 8. Eisenstein Series for PSL(2) over Imaginary Quadratic Integers [359] -- 8.1 Functions Closely Related to Eisenstein Series [359] -- 8.2 Fburier Expansion of Eisenstein Series for PSL(2,Q) [363] -- 8.3 Meromorphic Continuation by Fourier Expansion and the Kronecker Limit Formula [370] -- 8.4 Special Values of Eisenstein Series [380] -- 8.5 Applications to Zeta Functions and Asymptotics of Divisor Sums [393] -- 8.6 Non-Vanishing of L-Functions [397] -- 8.7 Meromorphic Continuation by Integral Representation [400] -- 8.8 Computation of the Volume [402] -- 8.9 Weyl’s Asymptotic Law [404] -- 9. Integral Binary Hermitian Forms [407] -- 9.1 Upper Half-Space and Binary Hermitian Forms [407] -- 9.2 Reduction Theory [410] -- 9.3 Representation Numbers of Binary Hermitian Forms [414] -- 9.4 Zeta Functions for Binary Hermitian Forms [425] -- 9.5 The Mass-Formula [430] -- 9.6 Computation of the Covolume of PSL(2, Q) à la Humbert [436] -- 9.7 Notes and Remarks [441] -- 10. Examples of Discontinuous Groups [443] -- 10.1 Groups of Quaternions [444] -- 10.2 Unit Groups of Quadratic Forms [456] -- 10.3 Arithmetic and Non-Arithmetic Discrete Groups [477] -- 10.4 The Tetrahedral Groups [480] -- 10.5 Notes and Remarks [494] -- References [497] -- Subject Index [521] --
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Includes bibliographical references (p. [497]-520) and index.
Preface VII --
1. Three-Dimensional Hyperbolic Space [1] --
1.1 The Upper Half-Space Model [1] --
1.2 The Unit Ball Model [9] --
1.3 The Exceptional Isomorphism [12] --
1.4 The Hyperboloid Model [20] --
1.5 The Kleinian Model [22] --
1.6 Upper Half-Space as a Symmetric Space [28] --
1.7 Notes and Remarks [30] --
2. Groups Acting Discontinuously on Three-Dimensional Hyperbolic Space [33] --
2.1 Discontinuity [33] --
2.2 Fundamental Domains and Polyhedra [41] --
2.3 Shimizu’s Lemma [48] --
2.4 Jorgensen’s Inequality [53] --
2.5 Covolumes [59] --
2.6 Hyperbolic Lattice Point Problems [67] --
2.7 Generators and Relations [71] --
2.8 Conjugacy and Commensurability [75] --
2.9 A Lemma of Selberg [78] --
2.10 Notes and Remarks [80] --
3. Automorphic Functions [83] --
3.1 Definition and Elementary Properties of some Poincaré Series [84] --
3.2 Definition and Elementary Properties of Eisenstein Series [98] --
3.3 Fourier Expansion in Cusps and the MaaB-Selberg Relations [105] --
3.4 Fourier Expansion of Eisenstein Series [110] --
3.5 Expansion of Eigenfunctions and the Selberg Transform [114] --
3.6 Behaviour of the Poincar4 Series at the Abscissa of Convergence [122] --
3.7 Notes and Remarks [129] --
4. Spectral Theory of the Laplace Operator [131] --
4.1 Essential Self-Adjointness of the Laplace-Beltrami Operator [132] --
4.2 The Resolvent Kernel [143] --
4.3 Hilbert-Schmidt Type Resolvents [156] --
4.4 Analytic Continuation of the Resolvent Kernel [162] --
4.5 Approximation by Kernels of Hilbert-Schmidt Type [169] --
5. Spectral Theory of the Laplace Operator for Cocompact Groups [185] --
5.1 The Hyperbolic Lattice-Point Problem [187] --
5.2 Computation of the Trace [190] --
5.3 Huber’s Theorem [201] --
5.4 The Selberg Zeta Function [205] --
5.5 Weyl’s Asymptotic Law and the Hadamard Factorisation of the Zeta Function [210] --
5.6 Analogue of the Lindelöf Hypothesis for the Selberg Zeta Function [218] --
5.7 The Prime Geodesic Theorem [222] --
5.8 Notes and Remarks [228] --
6. Spectral Theory of the Laplace Operator for Cofinite Groups [231] --
6.1 Meromorphic Continuation of the Eisenstein Series [231] --
6.2 Generalities on Eigenfunctions and Eigenpackets [244] --
6.3 Spectral Decomposition Theory [265] --
6.4 Spectral Expansions of Integral Kernels and Poincaré Series [276] --
6.5 The Trace Formula and some Applications [296] --
6.6 Notes and Remarks [310] --
7. PSL(2) over Rings of Imaginary Quadratic Integers [311] --
7.1 Introduction of the Groups [311] --
7.2 The Cusps [313] --
7.3 Description of a Fundamental Domain [318] --
7.4 Groups Commensurable with PSL(2,Q) [327] --
7.5 The Group Theoretic Structure of PSL(2.Q) [334] --
7.6 Spectral Theory of the Laplace Operator [346] --
7.7 Notes and Remarks [356] --
8. Eisenstein Series for PSL(2) over Imaginary Quadratic Integers [359] --
8.1 Functions Closely Related to Eisenstein Series [359] --
8.2 Fburier Expansion of Eisenstein Series for PSL(2,Q) [363] --
8.3 Meromorphic Continuation by Fourier Expansion and the Kronecker Limit Formula [370] --
8.4 Special Values of Eisenstein Series [380] --
8.5 Applications to Zeta Functions and Asymptotics of Divisor Sums [393] --
8.6 Non-Vanishing of L-Functions [397] --
8.7 Meromorphic Continuation by Integral Representation [400] --
8.8 Computation of the Volume [402] --
8.9 Weyl’s Asymptotic Law [404] --
9. Integral Binary Hermitian Forms [407] --
9.1 Upper Half-Space and Binary Hermitian Forms [407] --
9.2 Reduction Theory [410] --
9.3 Representation Numbers of Binary Hermitian Forms [414] --
9.4 Zeta Functions for Binary Hermitian Forms [425] --
9.5 The Mass-Formula [430] --
9.6 Computation of the Covolume of PSL(2, Q) à la Humbert [436] --
9.7 Notes and Remarks [441] --
10. Examples of Discontinuous Groups [443] --
10.1 Groups of Quaternions [444] --
10.2 Unit Groups of Quadratic Forms [456] --
10.3 Arithmetic and Non-Arithmetic Discrete Groups [477] --
10.4 The Tetrahedral Groups [480] --
10.5 Notes and Remarks [494] --
References [497] --
Subject Index [521] --
MR, REVIEW #
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