Let X be an algebraic variety over a field k. Which representations of π1(X) arise from geometry, e.g. as monodromy representations on the cohomology of a family of varieties over X? We study this question by analyzing the action of the Galois group of k on the fundamental group of X, where k is a finite or p-adic field.
As a sample application of our techniques, we show that if p is an odd prime and V is a (non-trivial) irreducible p-adic representation of the fundamental group of the projective line minus three points, which arises from geometry, then V must be non-trivial mod p.