Time: the third Semester.
Credit: 3 hours.
Period: 54 hours.
Previous courses: Basic Analytic Number Theory,
                                 Classical Automorphic Forms

Course contents:

(1) Harmonic Analysis on the Hyperbolic Plane;
(2) Fuchsian Groups;
(3) Automorphic Forms;
(4) The Spectra Theorem: Discrete Part;
(5) The automorphic Green Function;
(6) Analytic Continuation of Eisenstein Series;
(7) The Spectral Theorem: Continuous Part;
(8) Estimates for the Fourier Coefficients of Maass Forms;
(9) Spectral Theory of Kloosterman Sums;
(10) The Trace Formula;
(11) The Distribution of Eigenvalues;
(12) Hyperbolic Lattice-Point Problems;
(13) Spectral Bounds for Cusp Forms.

References:

(1) Roger C. Baker, Kloosterman sums and maass forms, Volume 1, Kendrick       Press, 2003.
(2) Daniel Bump, Spectral theory and the trace formula, available at       http://match.stanford.edu/bump/trace.pdf.
(3) D. A. Hejhal, The Selberg trace formula for PSL (2, R), Volume 1 & 2,       Lectures Notes in Math., No. 548, & No. 1001, Springer, 1976 and 1983.
(4) H. Iwaniec, Spectral methods of automorphic forms, Second Edtion, GSM      Volume 53, Amer. Math. Soc., Providence, 2002.
(5) Y. Motohashi, Spectral theory of the Riemann zeta-function, Cambridge       University Press, 1997.
(6) A. Terras, Harmonic analysis on symmetric spaces and applications I,       Springer-Verlag, New York, 1985.
(7) A. B. Venkov, Spectral theory of automorphic functions and its applications,       Kluwer, Dordrecht, The Netherlands, 1990.
(8) Yangbo Ye, Modular forms and trace formula (in Chinese), Peking       University Press, 2001.