A uniform open image theorem for ℓ-adic representations, I

Abstract

Let k be a field finitely generated over Q, and let X be a smooth, separated, and geometrically connected curve over k. Fix a prime ℓ. A representation ρ:π1(X)→GLm(Zℓ) is said to be geometrically Lie perfect if the Lie algebra of ρ(π1(X k¯)) is perfect. Typical examples of such representations are those arising from the action of π1(X) on the generic ℓ-adic Tate module Tℓ(Aη) of an abelian scheme A over X or, more generally, from the action of π1(X) on the ℓ-adic étale cohomology groups Héti(Yη¯,Qℓ), i≥0, of the geometric generic fiber of a smooth proper scheme Y over X. Let G denote the image of ρ. Any k-rational point x on X induces a splitting x:Γk:=π1(Spec(k))→π1(X) of the canonical restriction epimorphism π1(X)→Γk so one can define the closed subgroup Gx:=ρ∘x(Γk)⊂G. The main result of this paper is the following uniform open image theorem. Under the above assumptions, for every geometrically Lie perfect representation ρ:π1(X)→GLm(Zℓ), the set Xρ of all x∈X(k) such that Gx is not open in G is finite and there exists an integer Bρ≥1 such that [G:Gx]≤Bρ for every x∈X(k)∖Xρ.