Duke Mathematical Journal

A uniform open image theorem for -adic representations, I

Anna Cadoret and Akio Tamagawa

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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:π1(X)GLm(Z) 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 -adic Tate module T(Aη) of an abelian scheme A over X or, more generally, from the action of π1(X) on the -adic étale cohomology groups Héti(Yη¯,Q), i0, 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:=ρx(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(Z), the set Xρ of all xX(k) such that Gx is not open in G is finite and there exists an integer Bρ1 such that [G:Gx]Bρ for every xX(k)Xρ.

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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

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