Spectral Methods of Automorphic FormsAutomorphic 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 inthe 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 $L2 (H\Gamma)$, where $H$ is the upper half-plane and $\Gamma$ is a discretesubgroup 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 ClassicalAutomorphic Forms, Volume 17 in the Graduate Studies in Mathematics series. The book is designed for graduate students and researchers working in analytic number theory. |
Contents
| 3 | |
Fuchsian Groups | 37 |
Automorphic Forms | 53 |
The Spectral Theorem Discrete Part | 63 |
The Automorphic Green Function | 71 |
Analytic Continuation of the Eisenstein Series | 81 |
The Spectral Theorem Continuous Part | 95 |
Estimates for the Fourier Coefficients of Maass Forms | 107 |
The Trace Formula | 135 |
The Distribution of Eigenvalues | 157 |
Hyperbolic LatticePoint Problems | 171 |
Spectral Bounds for Cusp Forms | 177 |
Appendix A Classical Analysis | 185 |
Appendix B Special Functions | 197 |
| 209 | |
| 215 | |
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Common terms and phrases
analytic continuation asymptotic automorphic forms automorphic functions automorphic Green function automorphic kernel coefficients computations congruence groups conjugacy class constant is absolute converges absolutely cusp forms cuspidal zones defined denote derived discrete spectrum Ea(z eigenfunction eigenvalue Eisenstein series elliptic estimate Fa(Y finite finite volume group first fixed fixed points Fourier coefficients Fourier expansion Fuchsian group functional equation fundamental domain given Green function Gs(u half-plane Hecke Hecke operator Hence holomorphic hyperbolic plane implied constant depending inequality infer inner product invariant integral operators inversion Kloosterman sums L-functions Laplace operator Lemma line Res lower bound Maass forms modular group obtain orthogonal parabolic point-pair invariant poles proof Proposition residue Riemann Sarnak satisfies scattering matrix segment Selberg space spectral decomposition spectral theorem stability group subgroup subspace symmetric test function Theorem trace formula transform uj(z zero zero-th term zeta-function