We report on arithmetic techniques introduced by Hausel and Rodriguez-Villegas for computing the E-polynomial of certain families of algebraic varieties, called character varieties of surface groups. The construction of such varieties follows from Geometric Invariant Theory. We will discuss these arithmetic techniques based on a fundamental theorem of Katz, that reduces the computation of E-polynomials to point counting over finite fields. Then we give an idea on how to apply them tomore general character varieties.