Spectral Theory and the Trace Formula

Time: the first 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, GSM Volume 53, Amer. Math. Soc., Providence, 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.