A uniform open image theorem for l-adic representations in positive characteristic

TL;DR

This paper proves a uniform open image theorem for l-adic representations over positive characteristic fields, showing finiteness and boundedness properties of Galois images associated with rational points on algebraic curves.

Contribution

It extends previous characteristic zero results to positive characteristic, establishing finiteness of exceptional points and uniform bounds on Galois image indices for l-adic representations.

Findings

Finiteness of the set of rational points with non-open Galois image.

Existence of a uniform bound on the index of Galois images.

Application to uniform bounds on torsion in algebraic families.

Abstract

Let $k$ be a finitely generated field of characteristic $p > 0$ and $\ell$ a prime. Let $X$ be a smooth, separated, geometrically connected curve of finite type over $k$ and $\rho: \pi_1(X)\rightarrow GL_r(\mathbb Z_{\ell})$ a continuous representation of the \etale fundamental group of $X$ with image $G$. Any $k$-rational point $x:Spec(k)\rightarrow X$ induces a local representation $\rho_x: \pi_1(Spec(k)) \rightarrow \pi_1(X) \rightarrow GL_r(\mathbb Z_{\ell})$ with image $G_x$. The goal of this paper is to study how $G_x$ varies with $x\in X(k)$. In particular we prove that if $\ell\neq p$ and every open subgroup of $\rho(\pi_1(X_{\overline k}))$ has finite abelianization, then the set $X_{\rho}^{ex}(k)$ of $k$-rational points such that $G_x$ is not open in $G$ is finite and there exists a constant $C\geq 0$ such that $[G:G_x]\leq C$ for all $x\in X(k)-X_{\rho}^{ex}(k)$. This result can be applied to obtain uniform bounds for the $\ell$-primary torsion of groups theoretic invariants in one dimensional families of varieties. For example, torsion of abelian varieties and the Galois invariants of the geometric Brauer group. This extends to positive characteristic previous results of Anna Cadoret and Akio Tamagawa in characteristic 0.