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 the 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.Ministerio de Educación y CienciaDeutsche ForschungsgemeinschaftAlexander von Humboldt Research FellowshipMinistry of Science and Higher Education (Poland