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

Abstract

Let $k$ be a field finitely generated over $\mathbb{Q}$, and let $X$ be a curve over $k$. Fix a prime $\ell$. A representation $\rho :{\pi }_1\left(X\right)\to {GL}_m\left({\mathbb{Z}}_{\ell }\right)$ is said to be geometrically Lie perfect if any open subgroup of $\rho \left({\pi }_1\right(X_k\left)\right)$ has finite abelianization. Let $G$ denote the image of $\rho$. Any closed point $x$ on $X$ induces a splitting $x:{\Gamma }_{\kappa \left(x\right)}:={\pi }_1(Spec(\kappa \left(x\right)\left)\right)\to {\pi }_1\left(X_{\kappa \left(x\right)}\right)$ of the restriction epimorphism ${\pi }_1\left(X_{\kappa \left(x\right)}\right)\to {\Gamma }_{\kappa \left(x\right)}$ (here, $\kappa \left(x\right)$ denotes the residue field of $X$ at $x$) so one can define the closed subgroup $G_x:=\rho \circ x\left({\Gamma }_{\kappa \left(x\right)}\right)\subset G$. The main result of this paper is the following uniform open image theorem. Under the above assumptions, for any geometrically Lie perfect representation $\rho :{\pi }_1\left(X\right)\to {GL}_m\left({\mathbb{Z}}_{\ell }\right)$ and any integer $d\ge 1$, the set $X_{\rho ,d}$ of all closed points $x\in X$ such that $G_x$ is not open in $G$ and $\left[\kappa \right(x):k]\le d$ is finite and there exists an integer $B_{\rho ,d}\ge 1$ such that $[G:G_x]\le B_{\rho ,d}$ for any closed point $x\in X\setminus X_{\rho ,d}$ with $\left[\kappa \right(x):k]\le d$.

A key ingredient of our proof is that, for any integer $\gamma \ge 1$, there exists an integer $\nu =\nu \left(\gamma \right)\ge 1$ such that, given any projective system $\cdots \to Y_{n+1}\to Y_n\to \cdots \to Y_0$ of curves (over an algebraically closed field of characteristic $0$) with the same gonality $\gamma$ and with $Y_{n+1}\to Y_n$ a Galois cover of degree greater than $1$, one can construct a projective system of genus $0$ curves $\cdots \to B_{n+1}\to B_n\to \cdots \to B_{\nu }$ and degree $\gamma$ morphisms $f_n:Y_n\to B_n$, $n\ge \nu$, such that $Y_{n+1}$ is birational to $B_{n+1}{\times }_{B_n,f_n}{Y}_n$, $n\ge \nu$. 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 $\ell$-adic representation $\rho :{\pi }_1\left(X\right)\to {GL}_m\left({\mathbb{Z}}_{\ell }\right)$ and integer $d\ge 1$, the set of all closed points $x\in X$ such that $G_x$ is of codimension at least $3$ in $G$ and $\left[\kappa \right(x):k]\le d$ is finite.