Arithmetic Restrictions on Geometric Monodromy
  Let X be a normal complex algebraic variety, and p a prime. We show that there exists an integer N=N(X, p) such that: any non-trivial, irreducible representation of the fundamental group of X, which arises from geometry, must be non-trivial mod p^N.   The proof involves an analysis of the action of the Galois group of a finitely generated field on the etale fundamental group of X. We also prove many arithmetic statements about fundamental groups which are of independent interest, and give several applications. 
Daniel Litt
math.AG math.NT math.RT