Project A4: Combinatorial Euler products
Project A4 takes a broader look at counting problems arising from various algebraic and geometric sources. For instance, if an arithmetic group satisfies the congruence subgroup property, the number of its irreducible complex representations grows at most polynomially as a function of the degree and the associated Dirichlet series converges in a half plane. Other constructions involve zeta functions for local and global fields, dynamical systems, or graphs. Conjecturally, there is a homological reason for the various symmetries and functional equations, which is responsible for the uniform behaviour of these counting problems of completely different origin. We aim to identify and understand which mechanisms under the surface are responsible for the uniformity.