Families of Abelian Varieties and Large Galois Images

Associated with an abelian variety A of dimension g over a number field K is a Galois representation rho(A): Gal((K) over bar /K) -> GL(2g)((Z) over cap). The representation rho(A) encodes the Galois action on the torsion points of A and its image is an interesting invariant of A that contains a lot of arithmetic information. We consider abelian varieties over K parametrized by the K-points of a non-empty open subvariety U subset of P-K(n). We show that away from a set of density 0, the image of rho(A) will be very large; more precisely, it will have uniformly bounded index in a group obtained from the family of abelian varieties. This generalizes earlier results that assumed that the family of abelian varieties has "big monodromy". We also give a version for families of abelian varieties with a more general base.