Eisenstein Series, Trace Formulas and Applications（I）

Time/Location: 7-8 THU, B819

Syllabus: 

Classical Language

1. Classical Automorphic Forms

   1. Holomorphic automorphic forms

   2. Real analytic Maass forms

   3. Bounds of the Fourier coefficients of cusp forms

   4. Automorphic L-functions

   5. Hecke operator and Euler product

2. Classical Eisenstein Series and Applications

   1. Holomorphic Eisenstein series： Fourier expansion， linear space of modular forms

   2. Real analytic Eisenstein series： Definition, Fourier expansion, functional equation, analytic continuation

   3. Selberg Spectral decomposition: Maass-Selberg relation, spectral decomposition

   4. Nonvanishing of the Riemann zeta function: Nonvanishing, Arthur's truncated Eisenstein series, Sarnak's result

   5. Rankin-Selberg methods: Rankin-Selberg L-functions, bounds toward Ramanujan conjecture

   6. Volume of the fundamental domain: two methods: sharp cut-off, smooth cut-off

3. The Selberg Trace Formula and Applications

   1. The trace formula for cocompact groups

   2. The trace formula for Non-cocompact groups

   3. The Geometric side of the trace Formula

   4. The Selberg zeta function

   5. The Weyl law and the dimension of eigenspaces

   6. The prime geodesic theorem

Eisenstein Series, Trace Formulas and Applications(II)

References：

1. Ye, Modular forms and trace fromula.

2. Bump, Automorohic forms and Representations.

3. Bergeron, The Spectrum of Hyperbolic Surfaces.

4. Iwaniec, Spectral Methods of Automorphic Forms.

5. Sarnak, Nonvanishing of L-functions on Re(s)=1.

6. Zagier, Eisenstein series and Selberg trace formula.

7. Iwaniec, Prime Geodesic Theorem.