Duke Mathematical Journal

A uniform open image theorem for -adic representations, II

Anna Cadoret and Akio Tamagawa

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Abstract

Let k be a field finitely generated over Q, and let X be a curve over k. Fix a prime . A representation ρ:π1(X)GLm(Z) is said to be geometrically Lie perfect if any open subgroup of ρ(π1(Xk)) has finite abelianization. Let G denote the image of ρ. Any closed point x on X induces a splitting x:Γκ(x):=π1(Spec(κ(x)))π1(Xκ(x)) of the restriction epimorphism π1(Xκ(x))Γκ(x) (here, κ(x) denotes the residue field of X at x) so one can define the closed subgroup Gx:=ρx(Γκ(x))G. The main result of this paper is the following uniform open image theorem. Under the above assumptions, for any geometrically Lie perfect representation ρ:π1(X)GLm(Z) and any integer d1, the set Xρ,d of all closed points xX such that Gx is not open in G and [κ(x):k]d is finite and there exists an integer Bρ,d1 such that [G:Gx]Bρ,d for any closed point xXXρ,d with [κ(x):k]d.

A key ingredient of our proof is that, for any integer γ1, there exists an integer ν=ν(γ)1 such that, given any projective system Yn+1YnY0 of curves (over an algebraically closed field of characteristic 0) with the same gonality γ and with Yn+1Yn a Galois cover of degree greater than 1, one can construct a projective system of genus 0 curves Bn+1BnBν and degree γ morphisms fn:YnBn, nν, such that Yn+1 is birational to Bn+1×Bn,fnYn, nν. This, together with the case for d=1 (which is the main result of part I of this paper), gives the proof for general d.

Our method also yields the following unconditional variant of our main result. With the above assumptions on k and X, for any -adic representation ρ:π1(X)GLm(Z) and integer d1, the set of all closed points xX such that Gx is of codimension at least 3 in G and [κ(x):k]d is finite.

Article information

Source
Duke Math. J., Volume 162, Number 12 (2013), 2301-2344.

Dates
First available in Project Euclid: 9 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1378729689

Digital Object Identifier
doi:10.1215/00127094-2323013

Mathematical Reviews number (MathSciNet)
MR3102481

Zentralblatt MATH identifier
1279.14056

Subjects
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

Citation

Cadoret, Anna; Tamagawa, Akio. A uniform open image theorem for $\ell$ -adic representations, II. Duke Math. J. 162 (2013), no. 12, 2301--2344. doi:10.1215/00127094-2323013. https://projecteuclid.org/euclid.dmj/1378729689


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