Specialization of monodromy group and ℓ-independence

Abstract

Let E be an abelian scheme over a geometrically connected, smooth variety X defined over k, a finitely generated field over Q. Let η be the generic point of X and x∈X a closed point. If gℓ and (gℓ)x are the Lie algebras of the ℓ-adic Galois representations for abelian varieties E η and E x, then (gℓ)x is embedded in gℓ by specialization. We prove that the set {x∈X closed point|(gℓ)x{subset of with not equal to}gℓ} is independent of ℓ and confirm Conjecture 5.5 in Cadoret and Tamagawa [3]. © 2011 Académie des sciences.