Catálogo de la biblioteca Antonio Monteiro - INMABB

1 Introduction [1] --
1.1 Spectral Analysis on the Torus [1] --
1.2 The Hyperbolic Plane [4] --
1.3 Hyperbolic Surfaces [11] --
1.3.1 Fundamental Domains [12] --
1.3.2 Hyperbolic Laplacian [14] --
1.4 Description of the Main Results [16] --
1.4.1 Examples of Hyperbolic Surfaces [16] --
1.4.2 The Spectral Theorem [18] --
1.4.3 MaaB Forms [19] --
1.4.4 The Selberg Trace Formula [20] --
1.4.5 The Selberg Eigenvalue Conjecture [22] --
1.4.6 The Jacquet-Langlands Correspondence [23] --
1.4.7 Arithmetic Quantum Unique Ergodicity [24] --
1.5 Notation [24] --
1.6 Remarks and References [25] --
1.7 Exercises [25] --
2 Arithmetic Hyperbolic Surfaces [31] --
2.1 The Space of Lattices [31] --
2.2 Quaternion Algebras and Arithmetic Groups [35] --
2.2.1 An Exceptional Isomorphism [37] --
2.2.2 A Subset of the Space of Lattices [40] --
2.2.3 The Cocompactness Criterion [40] --
2.2.4 Commensurability Classes [42] --
2.3 Arithmetic Hyperbolic Surfaces [43] --
2.3.1 Eliminating Torsion [44] --
2.3.2 The Modular Surface and Its Covers [45] --
2.4 Commentary and References [49] --
2.5 Exercises [50] --
3 Spectral Decomposition [53] --
3.1 The Laplacian [53] --
3.2 Eigenfunctions of the Laplacian on H [56] --
3.2.1 Radial Functions [58] --
3.3 Invariant Integral Operators on H [60] --
3.4 The SelbergTransform [64] --
3.5 A Family of Examples: The Heat Kernel [67] --
3.6 The Laplacian on r\H [70] --
3.7 Integral Operators on r\H [73] --
3.7.1 The Heat Kernel [75] --
3.7.2 The Non-compact Case [76] --
3.8 Review of Functional Analysis [81] --
3.9 Proof of the Spectral Theorem [83] --
3.10 The Minimax Principle [89] --
3.10.1 Small Eigenvalues I: Geometric Existence Criterion [91] --
3.10.2 Small Eigenvalues II: The Selberg Conjecture [93] --
3.11 Commentary and References [94] --
3.12 Exercises [96] --
4 MaaB Forms [99] --
4.1 Eisenstein Series for SL(2, Z) [99] --
4.1.1 Eisenstein Series and the Riemann Zeta Function 1 [103] --
4.2 Eisenstein Series and the Spectrum of the Laplacian [106] --
4.2.1 Mellin Inversion Formula, Phragnruén-Lindelöf Principle [107] --
4.2.2 The Space of Incomplete Eisenstein Series [110] --
4.2.3 Regularity of E(z. 5) on the Vertical Line Re(s) = 1/2 [113] --
4.2.4 Eisenstein Series and the Riemann Zeta Function II [116] --
4.2.5 The Eisenstein Transform [118] --
4.2.6 The Spectral Theorem for the Modular Surface [121] --
4.3 Existence of Cusp Forms [123] --
4.3.1 Automorphic Wave Equation [123] --
4.3.2 Construction of Cusp Forms [125] --
4.4 Hyperbolic Periods of Eisenstein Series [127] --
4.4.1 Primitive Geodesics on the Modular Surface [127] --
4.4.2 Hyperbolic Fourier Series of E(z, s) [135] --
4.5 Explicit Construction of MaaB Forms [138] --
4.6 Commentary and References [145] --
4.7 Exercises [149] --
5 The Trace Formula [153] --
5.1 The Selberg Trace Formula I: General Framework [153] --
5.2 The Selberg Trace Formula II: The Case of Compact Surfaces [157] --
5.2.1 The Pretrace Formula [157] --
5.2.2 The Geometric Side of the Trace Formula [159] --
5.2.3 Contribution of Hyperbolic Elements [160] --
5.2.4 The Trace Formula [162] --
5.3 The Selberg Trace Formula III: The Case of SL(2, Z) [165] --
5.3.1 The Kernel KCont [167] --
5.3.2 The Kernel K — Kcont [167] --
5.3.3 The Spectral Side of the Trace Formula [171] --
5.3.4 The Geometric Side of the Trace Formula: --
Parabolic Term Contribution [175] --
5.3.5 The Trace Formula [179] --
5.4 Applications [181] --
5.4.1 The Weyl Law [181] --
5.4.2 The Prime Geodesic Theorem [183] --
5.5 Comments and References [188] --
5.6 Exercises [191] --
6 Multiplicity of -and the Selberg Conjecture [193] --
6.1 Point Counting in Arithmetic Lattices [193] --
6.2 Multiplicity of the First Eigenvalue [196] --
6.3 Representation Theory of PSL(2,Z/pZ) [205] --
6.3.1 Review of the Representation Theory of Finite Groups [205] --
6.3.2 Proof of Theorem 6.8 [207] --
6.4 Lower Bound on the First Non-zero Eigenvalue [210] --
6.5 Comments and References [211] --
7 L-Functions and the Selberg Conjecture [213] --
7.1 The L-Function Attached to a MaaB Form [214] --
7.2 Hecke Operators and Applications [217] --
7.2.1 Hecke Operators [217] --
7.2.2 Atkin-Lehner Theory [221] --
7.2.3 Multiplicative Properties of Fourier Coefficients [227] --
7.3 Dirichlet Characters and Twisted MaaB Forms [230] --
7.3.1 Dirichlet Characters [230] --
7. 3.2 MaaB Forms Twisted by a Character [232] --
7.3.3 Twisted Eisenstein Series [234] --
7.4 Rankin-Selberg L-Functions [239] --
7.5 The Luo-Rudnick-Sarnak Theorem [253] --
7.6 Bounds on Fourier Coefficients [260] --
7.7 Comments and References [262] --
7.8 Exercises [265] --
8 Jacquet-Langlands Correspondence [267] --
8.1 Arithmetic of Quaternion Algebras [267] --
8.1.1 Orders in Quaternion Algebras [267] --
8.1.2 Orders in Quadratic Extensions of Q [270] --
8.1.3 Strong Approximation Theorems [272] --
8.2 Optimal Embeddings of Quadratic Fields [273] --
8.3 The Trace Formula [282] --
8.4 Jacquet-Langlands Correspondence and Applications [286] --
8.5 Commentary and References [292] --
9 Arithmetic Quantum Unique Ergodicty [295] --
9.1 Quantization of the Geodesic Flow [295] --
9.1.1 The Classical System: The Geodesic Flow [295] --
9.1.2 Quantum Systems [299] --
9.1.3 Quantum Mechanics on the Poincaré Upper Half-Plane [300] --
9.2 Microlocal Lift [302] --
9.2.1 The Microlocal Lift [304] --
9.2.2 Quantum Ergodicity [308] --
9.3 First Links with Ergodic Theory [310] --
9.4 Multiplication by 2 and 3 on the Circle [312] --
9.4.1 The Circle as a Foliated Space [312] --
9.4.2 Conditional Measures [313] --
9.4.3 Recurrence [315] --
9.4.4 Invariant Measures [317] --
9.4.5 Entropy [319] --
9.5 Hecke Operators and Lindenstrauss's Theorem [322] --
9.5.1 The Tree of PGL(2. Qp) [323] --
9.5.2 Hecke Operators [325] --
9.5.3 Lindenstrauss's Theorem [326] --
9.6 Use of Hecke Operators [327] --
9.6.1 Local Contributions [328] --
9.6.2 Intersections of Hecke Translates [333] --
9.6.3 Strongly Positive Entropy [337] --
9.6.4 Tp-Recurrence [339] --
9.7 Commentary and References [339] --
9.8 Exercises [342] --
Appendices --
A Three Coordinate Systems for H [343] --
B The Gamma Function and Bessel Functions [345] --
C Elementary Bounds on Hyper-Kloosterman Sums by Valentin Biome r and Farrell Brumley [349] --
References [355] --
Index of notation [363] --
Index [365] --
Index of names [369] --

MR, REVIEW #

This text is an introduction to the spectral theory of the Laplacian on compact or finite area hyperbolic surfaces. For some of these surfaces, called "arithmetic hyperbolic surfaces", the eigenfunctions are of arithmetic nature, and one may use analytic tools as well as powerful methods in number theory to study them. After an introduction to the hyperbolic geometry of surfaces, with a special emphasis on those of arithmetic type, and then an introduction to spectral analytic methods on the Laplace operator on these surfaces, the author develops the analogy between geometry (closed geodesics) and arithmetic (prime numbers) in proving the Selberg trace formula. Along with important number theoretic applications, the author exhibits applications of these tools to the spectral statistics of the Laplacian and the quantum unique ergodicity property. The latter refers to the arithmetic quantum unique ergodicity theorem, recently proved by Elon Lindenstrauss. The fruit of several graduate level courses at Orsay and Jussieu,The Spectrum of Hyperbolic Surfacesallows the reader to review an array of classical results and then to be led towards very active areas in modern mathematics.

Description based on publisher-supplied MARC data.