1. A survey of adelic theory for two-dimensional schemes. Algebraic surfaces over finite fields and arithmetical surfaces.

2. Discrete nilpotent groups arising from the adelic groups.

3. Representations of discrete Heisenberg groups. Moduli of induced representations. Characters as Jacobi forms on the moduli spaces.

4. Harmonic analysis on discrete groups and complex tori. Poisson formula as a sum of residues. Relation with explicit formulas.

5. Applications to unramified $L$-functions of two-dimensional schemes.

Institutions
St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences