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 $\ell$.