Big monodromy theorem for abelian varieties over finitely generated   fields
  An abelian variety over a field K is said to have big monodromy, if the image of the Galois representation on l-torsion points, for almost all primes l contains the full symplectic group. We prove that all abelian varieties over a finitely generated field K with endomorphism ring Z and semistable reduction of toric dimension one at a place of the base field K have big monodromy. We make no assumption on the transcendence degree or on the characteristic of K. This generalizes a recent result of Chris Hall. 
Sara Arias-de-Reyna, Wojciech Gajda, Sebastian Petersen
math.AG math.RT