Spectral Methods of Automorphic Forms (Graduate Studies in Mathematics, V. 53)

Automorphic forms are one of the central topics of analytic number theory. In fact, they sit at the confluence of analysis, algebra, geometry, and number theory. In this book, Henryk Iwaniec once again displays his penetrating insight, powerful analytic techniques, and lucid writing style. The first edition of this volume was an underground classic, both as a textbook and as a respected source for results, ideas, and references. The book's reputation sparked a growing interest in the mathematical community to bring it back into print. The AMS has answered that call with the publication of this second edition. In the book, Iwaniec treats the spectral theory of automorphic forms as the study of the space $L^2 (H\Gamma)$, where $H$ is the upper half-plane and $\Gamma$ is a discrete subgroup of volume-preserving transformations of $H$. He combines various techniques from analytic number theory. Among the topics discussed are Eisenstein series, estimates for Fourier coefficients of automorphic forms, the theory of Kloosterman sums, the Selberg trace formula, and the theory of small eigenvalues. Henryk Iwaniec was awarded the 2002 AMS Cole Prize for his fundamental contributions to analytic number theory. Also available from the AMS by H. Iwaniec is Topics in Classical Automorphic Forms, Volume 17 in the Graduate Studies in Mathematics series. The book is designed for graduate students and researchers working in analytic number theory. Preface To The Ams Edition -- Ch. 0. Harmonic Analysis On The Euclidean Plane -- Ch. 1. Harmonic Analysis On The Hyperbolic Plane -- 1.1. The Upper Half-plane -- 1.2. H As Homogeneous Space -- 1.3. The Geodesic Polar Coordinates -- 1.4. Group Decompositions -- 1.5. The Classification Of Motions -- 1.6. The Laplace Operator -- 1.7. Eigenfunctions Of [delta] -- 1.8. The Invariant Integral Operators -- 1.9. The Green Function On H -- Ch. 2. Fuchsian Groups -- 2.1. Definitions -- 2.2. Fundamental Domains -- 2.3. Basic Examples -- 2.4. The Double Coset Decomposition -- 2.5. Kloosterman Sums -- 2.6. Basic Estimates -- Ch. 3. Automorphic Forms -- 3.1. Introduction -- 3.2. The Eisenstein Series -- 3.3. Cusp Forms -- 3.4. Fourier Expansion Of The Eisenstein Series -- Ch. 4. The Spectral Theorem. Discrete Part -- 4.1. The Automorphic Laplacian -- 4.2. Invariant Integral Operators Of C([gamma]) -- 4.3. Spectral Resolution Of [delta] In C([gamma]) -- Ch. 5. The Automorphic Green Function --^ 5.1. Introduction -- 5.2. The Fourier Expansion -- 5.3. An Estimate For The Automorphic Green Function -- 5.4. Evaluation Of Some Integrals -- Ch. 6. Analytic Continuation Of The Eisenstein Series -- 6.1. The Fredholm Equation For The Eisenstein Series -- 6.2. The Analytic Continuation Of E[subscript A](z,s) -- 6.3. The Functional Equations -- 6.4. Poles And Residues Of The Eisenstein Series -- Ch. 7. The Spectral Theorem. Continuous Part -- 7.1. The Eisenstein Transform -- 7.2. Bessel's Inequality -- 7.3. Spectral Decomposition Of [epsilon]([gamma]) -- 7.4. Spectral Expansion Of Automorphic Kernels -- Ch. 8. Estimates For The Fourier Coefficients Of Maass Forms -- 8.1. Introduction -- 8.2. The Rankin-selberg L-function -- 8.3. Bounds For Linear Forms -- 8.4. Spectral Mean-value Estimates -- 8.5. The Case Of Congruence Groups -- Ch. 9. Spectral Theory Of Kloosterman Sums -- 9.1. Introduction -- 9.2. Analytic Continuation Of Z[subscript S](m,n) -- 9.3. Bruggeman-kuznetsov Formula --^ 9.4. Kloosterman Sums Formula -- 9.5. Petersson's Formulas -- Ch. 10. The Trace Formula -- 10.1. Introduction -- 10.2. Computing The Spectral Trace -- 10.3. Computing The Trace For Parabolic Classes -- 10.4. Computing The Trace For The Identity Motion -- 10.5. Computing The Trace For Hyperbolic Classes -- 10.6. Computing The Trace For Elliptic Classes -- 10.7. Trace Formulas -- 10.8. The Selberg Zeta-function -- 10.9. Asymptotic Law For The Length Of Closed Geodesics -- Ch. 11. The Distribution Of Eigenvalues -- 11.1. Weyl's Law -- 11.2. The Residual Spectrum And The Scattering Matrix -- 11.3. Small Eigenvalues -- 11.4. Density Theorems -- Ch. 12. Hyperbolic Lattice-point Problems -- Ch. 13. Spectral Bounds For Cusp Forms -- 13.1. Introduction -- 13.2. Standard Bounds -- 13.3. Applying The Hecke Operator -- 13.4. Constructing An Amplifier -- 13.5. The Ergodicity Conjecture -- App. A. Classical Analysis -- App. B. Special Functions. Henryk Iwaniec. First Ed. Published In Revista Matemática Iberoamericana In 1995. Includes Bibliographical References And Index. Automorphic forms are one of the central topics of analytic number theory. In fact, they sit at the confluence of analysis, algebra, geometry, and number theory. In this book, Henryk Iwaniec once again displays his penetrating insight, powerful analytic techniques, and lucid writing style. The first edition of this book was an underground classic, both as a textbook and as a respected source for results, ideas, and references. Iwaniec treats the spectral theory of automorphic forms as the study of the space of $L^2$ functions on the upper half plane modulo a discrete subgroup. Key topics include Eisenstein series, estimates of Fourier coefficients, Kloosterman sums, the Selberg trace formula and the theory of small eigenvalues. Henryk Iwaniec was awarded the 2002 Cole Prize for his fundamental contributions to number theory.