Some counting techniques for character varieties

MARTIN MEREB

París

Congreso; Quiver varieties, character varieties and DT-invariants; 2020

Institut Henri Poincaré

We will go over some techniques to count points of character varieties over finite fields. The goal is to find polynomials that give, at least for sufficiently many primes, the exact number of points. This is usually done through arithmetic harmonic analysis, a sort of Fourier transform involving the character tables of finite groups, or some Hecke algebras, depending on the problem. These counting functions relate to certain geometric invariants of the complex variety and permit to conjecture their Hodge structure.