1 October 2012 A uniform open image theorem for -adic representations, I
Anna Cadoret, Akio Tamagawa
Duke Math. J. 161(13): 2605-2634 (1 October 2012). DOI: 10.1215/00127094-1812954


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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Anna Cadoret. Akio Tamagawa. "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


Published: 1 October 2012
First available in Project Euclid: 11 October 2012

zbMATH: 1305.14016
MathSciNet: MR2988904
Digital Object Identifier: 10.1215/00127094-1812954

Primary: 14K15
Secondary: 14H30 , 22E20

Rights: Copyright © 2012 Duke University Press


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Vol.161 • No. 13 • 1 October 2012
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