Spectral Theory and the Trace Formula

Time: the third semester

Course Classification: elective course

Credit: 3 hours

Period: 54 hours

Previous courses: Basic Analytic Number Theory, Classical Automorphic Forms

Syllabus:

(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) N. Bergeron, The spectrum of hyperbolic surfaces, Springer.

(2) H. Iwaniec, Spectral methods of automorphic forms, Second Edtion, AMS, 2002.

(3) Y. Motohashi, Spectral theory of the Riemann zeta-function, Cambridge University Press, 1997.

(4) Yangbo Ye, Modular forms and trace formula (in Chinese), Peking University Press, 2001.