A uniform open image theorem for -adic representations i
Résumé
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 . A representation ρ : π 1 (X) → GLm(Z ) is said to be geometrically Lie perfect if the Lie algebra of ρ(π 1 (X k )) is perfect. Typical examples of such representations are those arising from the action of π 1 (X) on the generic -adic Tate module T (Aη) of an abelian scheme A over X or, more generally, from the action of π 1 (X) on the -adic etale cohomology groups H i et (Y η , Q ), 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 :
The main result of this paper is the following uniform open image theorem. Under the above assumptions, for every geometrically Lie perfect representation ρ : π 1 (X) → GLm(Z ), 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.
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