Duke Mathematical Journal
- Duke Math. J.
- Volume 161, Number 13 (2012), 2605-2634.
A uniform open image theorem for -adic representations, I
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 .
Duke Math. J., Volume 161, Number 13 (2012), 2605-2634.
First available in Project Euclid: 11 October 2012
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Primary: 14K15: Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx]
Secondary: 14H30: Coverings, fundamental group [See also 14E20, 14F35] 22E20: General properties and structure of other Lie groups
Cadoret, Anna; Tamagawa, Akio. A uniform open image theorem for $\ell$ -adic representations, I. Duke Math. J. 161 (2012), no. 13, 2605--2634. doi:10.1215/00127094-1812954. https://projecteuclid.org/euclid.dmj/1349960278