A note on images of Galois representations (with an application to a result of Litt)

Abstract

Let $X$ be a variety (possibly non-complete or singular) over a finitely generated field $k$ of characteristic $0$. For a prime number $\ell$, let $\rho_\ell$ be the Galois representation on the first $\ell$-adic cohomology of $X$. We show that if $\ell$ varies the image of $\rho_\ell$ is of bounded index in the group of $\mathbb{Z}_\ell$-points of its Zariski closure. We use this to improve a recent result of Litt about arithmetic representations of geometric fundamental groups. Litt's result says that there exist constants $N = N(X,\ell)$ such that every arithmetic representation $\pi_1(X_{\bar{k}}) \to \mathrm{GL}_n(\mathbb{Z}_\ell)$ that is trivial modulo $\ell^N$ is unipotent. We show that these constants can in fact be chosen independently of $\ell$.