3-4 hours of lectures per week.
Lecturer
Alexei Venkov
Content
The purpose of the course is to study the important arithmetical functions by analytical methods. Modular forms are the most important generating functions for arithmetical functions. The content of the course is the following.
- Eisenstein series
- The discriminant function
- Theta series
- Modular forms of iteger and half-integer weight
- New forms and old forms
- Hecke L-series
- Hecke theory
- Non-holomorphic Eisenstein series
- Kronecker limit formulas
- Gauss and Kloostermann sums
- Rankin-Selberg convolution method of analytic continuation of L-series
- Modified Kummer conjecture for arguments of the Gauss sums
- Jacobi modular forms
- Jacobi forms and infinite dimensional Lie algebras
- Siegel modular forms
- Selberg trace formula
Prerequisites
A knowledge of elementary properties of the Riemann zeta function.
Literature
C. L. Siegel: Lectures on advanced analytic number theory.
J.-P. Serre: Cours d'arithmetique.
D. Zagier: Introduction to modular forms.
S. Lang: Elliptic curves.
E. Freitag: Siegelsche Modulfunktionen.
V. Kac: In nite dimensional Lie algebras.
A.Venkov: Spectral theory of automorphic functions and its applications.