Spectral methods of automorphic forms / Henryk Iwaniec.

Format:

Book

Author/Creator:

Iwaniec, Henryk.

Series:

Graduate studies in mathematics 1065-7339 ; v. 53.

Graduate studies in mathematics, 1065-7339 ; v. 53

Language:

English

Subjects (All):

Automorphic functions.

Automorphic forms.

Spectral theory (Mathematics).

Physical Description:

xi, 220 pages : illustrations ; 26 cm.

Edition:

Second edition.

Place of Publication:

Providence, R.I. : American Mathematical Society, [2002]

Contents:

Chapter 0. Harmonic Analysis on the Euclidean Plane 3

Chapter 1. Harmonic Analysis on the Hyperbolic Plane 7

1.1. The upper half-plane 7

1.2. H as a homogeneous space 12

1.3. The geodesic polar coordinates 15

1.4. Group decompositions 16

1.5. The classification of motions 17

1.6. The Laplace operator 19

1.7. Eigenfunctions of [Delta] 19

1.8. The invariant integral operators 26

1.9. The Green function on H 32

Chapter 2. Fuchsian Groups 37

2.2. Fundamental domains 38

2.3. Basic examples 41

2.4. The double coset decomposition 45

2.5. Kloosterman sums 47

2.6. Basic estimates 49

Chapter 3. Automorphic Forms 53

3.2. The Eisenstein series 56

3.3. Cusp forms 57

3.4. Fourier expansion of the Eisenstein series 59

Chapter 4. The Spectral Theorem. Discrete Part 63

4.1. The automorphic Laplacian 63

4.2. Invariant integral operators on C([Gamma]\H) 64

4.3. Spectral resolution of [Delta] in C([Gamma]\H) 68

Chapter 5. The Automorphic Green Function 71

5.2. The Fourier expansion 72

5.3. An estimate for the automorphic Green function 75

5.4. Evaluation of some integrals 76

Chapter 6. Analytic Continuation of the Eisenstein Series 81

6.1. The Fredholm equation for the Eisenstein series 81

6.2. The analytic continuation of E[subscript a](z, s) 84

6.3. The functional equations 86

6.4. Poles and residues of the Eisenstein series 88

Chapter 7. The Spectral Theorem. Continuous Part 95

7.1. The Eisenstein transform 96

7.2. Bessel's inequality 98

7.3. Spectral decomposition of [varepsilon]([Gamma]\H] 101

7.4. Spectral expansion of automorphic kernels 104

Chapter 8. Estimates for the Fourier Coefficients of Maass Forms 107

8.2. The Rankin-Selberg L-function 109

8.3. Bounds for linear forms 110

8.4. Spectral mean-value estimates 113

8.5. The case of congruence groups 116

Chapter 9. Spectral Theory of Kloosterman Sums 121

9.2. Analytic continuation of Z[subscript s] (m, n) 122

9.3. Bruggeman-Kuznetsov formula 125

9.4. Kloosterman sums formula 128

9.5. Petersson's formulas 131

Chapter 10. The Trace Formula 135

10.2. Computing the spectral trace 139

10.3. Computing the trace for parabolic classes 142

10.4. Computing the trace for the identity motion 145

10.5. Computing the trace for hyperbolic classes 146

10.6. Computing the trace for elliptic classes 147

10.7. Trace formulas 150

10.8. The Selberg zeta-function 152

10.9. Asymptotic law for the length of closed geodesics 154

Chapter 11. The Distribution of Eigenvalues 157

11.1. Weyl's law 157

11.2. The residual spectrum and the scattering matrix 162

11.3. Small eigenvalues 164

11.4. Density theorems 168

Chapter 12. Hyperbolic Lattice-Point Problems 171

Chapter 13. Spectral Bounds for Cusp Forms 177

13.2. Standard bounds 178

13.3. Applying the Hecke operator 180

13.4. Constructing an amplifier 181

13.5. The ergodicity conjecture 183

Appendix A. Classical Analysis 185

A.1. Self-adjoint operators 185

A.2. Matrix analysis 187

A.3. The Hilbert-Schmidt integral operators 189

A.4. The Fredholm integral equations 190

A.5. Green function of a differential equation 194

Appendix B. Special Functions 197

B.1. The gamma function 197

B.2. The hypergeometric functions 199

B.3. The Legendre functions 200

B.4. The Bessel functions 202

B.5. Inversion formulas 205.

Notes:

"Revista matemática iberoamericana."

First ed. published in Revista matemática iberoamericana in 1995.

Includes bibliographical references (pages 209-213) and indexes.

ISBN:

0821831607

OCLC:

50280013