A uniform open image theorem for ℓ-adic representations, II

Abstract

Let k be a field finitely generated over Q, and let X be a curve over k. Fix a prime ℓ. A representation ρ:π1(X)→GLm(Zℓ) 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:=ρ∘x(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(Zℓ) and any integer d≥1, the set Xρ,d of all closed points x∈X such that Gx is not open in G and [κ(x):k]≤d is finite and there exists an integer Bρ,d≥1 such that [G:Gx]≤Bρ,d for any closed point x∈X∖Xρ,d with [κ(x):k]≤d. A key ingredient of our proof is that, for any integer γ≥1, there exists an integer ν=ν(γ)≥1 such that, given any projective system ⋯→Yn+1→Yn→⋯→Y0 of curves (over an algebraically closed field of characteristic 0) with the same gonality γ and with Yn+1→Yn a Galois cover of degree greater than 1, one can construct a projective system of genus 0 curves ⋯→Bn+1→Bn→⋯→Bν and degree γ morphisms fn:Yn→Bn, n≥ν, such that Yn+1 is birational to Bn+1×Bn,fnYn, n≥ν. 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 ℓ-adic representation ρ:π1(X)→GLm(Zℓ) and integer d≥1, the set of all closed points x∈X such that Gx is of codimension at least 3 in G and [κ(x):k]≤d is finite.