A UNIFORM OPEN IMAGE THEOREM FOR ?-ADIC REPRESENTATIONS, I

摘要

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 p: π_1(X) → GL_m(Z_?) is said to be geometrically Lie perfect if the Lie algebra of p(π_1(X-j?j) is perfect. Typical examples of such representations are those arising from the action of π_1(X) on the generic ?-adic Tate module T?(A_η) of an abelian scheme A over X or, more generally, from the action of π_1(X) on the ?-adic étale cohomology groups H_et~i(?%> ??), i ≥ 0, of the geometric generic fiber of a smooth proper scheme Y over X. Let G denote the image of p. Any k-rational point x on X induces a splitting x: Γ_k := π_1 (Spec(k)) →-π_1 (X) of the canonical restriction epimorphism π_1 (X) →?Γ_k so one can define the closed subgroup Gx :— p ¤ x(Γfc) c G. The main result of this paper is the following uniform open image theorem. Under the above assumptions, for every geometrically Lie perfect representation p : π_1 (X) → GL_m (Z?), the set X_p of all x ? X(k) such that G_x is not open in G is finite and there exists an integer B_p ≥ 1 such that [G : Gx]≤B_pfor every x e X(k) X_p.