The purpose of this course is to learn some basic spectral methods that are used to solve a wide range of problems in analytic number theory. One of the most significant results in this direction is the Kuznetsov trace formula, which establishes the relationship between Kloosterman sums and the sums of the Fourier coefficients of Hecke–Maass forms and Eisenstein series. We will study a detailed proof of the Kuznetsov formula and its various applications. The program includes the following topics:

1. Classical modular forms
2. Poincaré and Eisenstein series
3. Kloosterman sums
4. Petersson trace formula
5. Kuznetsov trace formula
6. L-functions

Financial support. The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, Agreement no. 075-15-2019-1614).

Lecturer
Balkanova Olga Germanovna

Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center