In this course I will talk about representation zeta functions and related topics. The representation zeta function of a group is the formal Dirichlet generating series whose coefficients enumerate isomorphism classes of finite dimensional irreducible representations of the group according to their dimension. Special values of these zeta functions are related the character varieties. In the course we will focus on properties of representation zeta functions associated with arithmetic and p-adic analytic groups. Among the topics that will be covered are p-adic analytic groups and Lazard’s correspondence, the orbit method, arithmetic groups and the congruence subgroup property, and applications of Model theory to rationality.
I will assume familiarity with basic notions in: