Families of abelian varieties and large Galois images

Abstract

Associated to an abelian variety A of dimension g over a number field K is a Galois representation ρ A : Gal(K/K) → GL 2g ( Z). The representation ρ 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 nonempty open subvariety U ⊆ P n K . We show that away from a set of density 0, the image of ρ 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 which assumed that the family of abelian varieties had "big monodromy". We also give a version for a family of abelian varieties with a more general base.

Abelian varieties not isogenous to Jacobians over global fields

“…Let π : C → P 1 denote a finite map of degree d, and consider B = Res C/P 1 (A), which is an abelian scheme over an open subset U ⊂ P 1 of dimension gd. Work of Zywina [16,Theorem 1.1] implies that for any number field L, there exists a constant c L such that for a density 1 set of points y ∈ U (L), the monodromy of B y has index bounded by c L in the monodromy of B U . In Section 1.2 of loc.…”