Let be a field finitely generated over , and let be a smooth, separated, and geometrically connected curve over . Fix a prime . A representation is said to be geometrically Lie perfect if the Lie algebra of is perfect. Typical examples of such representations are those arising from the action of on the generic -adic Tate module of an abelian scheme over or, more generally, from the action of on the -adic étale cohomology groups , , of the geometric generic fiber of a smooth proper scheme over . Let denote the image of . Any -rational point on induces a splitting of the canonical restriction epimorphism so one can define the closed subgroup . The main result of this paper is the following uniform open image theorem. Under the above assumptions, for every geometrically Lie perfect representation , the set of all such that is not open in is finite and there exists an integer such that for every .
"A uniform open image theorem for -adic representations, I." Duke Math. J. 161 (13) 2605 - 2634, 1 October 2012. https://doi.org/10.1215/00127094-1812954