Anna Cadoret, Akio Tamagawa

Duke Math. J. 162 (12), 2301-2344, (15 September 2013) DOI: 10.1215/00127094-2323013
KEYWORDS: 14K15, 14H30, 22E20

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.