In "Mixed Hodge polynomials of character varieties," Tamas Hausel and Fernando Rodriguez-Villegas examine the representation and (twisted) character varieties of fundamental groups of compact Riemann surfaces. I will briefly sketch their motivation and the combinatorial techniques they use, before turning to the question: What can we say if instead of a compact Riemann surface, we consider a complete curve over a field of positive characteristic? I will share some preliminary results which give combinatorial answers to this question in various special cases.