Abstract: An irreducible representation of an abstract group is called rigid, if it defines an isolated point in the moduli space of all representations.
Complex rigid representations are always defined over a number field. According to a conjecture by Simpson, for fundamental groups of smooth projective varieties one should expect furthermore that rigid representations can be defined over the ring of algebraic integers. I will report on joint work with H. Esnault where we prove this for so-called cohomologically rigid representations. Our argument is mostly arithmetic and passes through fields of positive characteristic and the p-adic numbers.