Transport of Zariski density in compatible collections of $G$-representations

Abstract

Let $X$ be a connected normal scheme of finite type over $\mathbf{Z}$, let $G$ be a connected reductive group over $\mathbf{Q}$, and let $\{{\rho }_{\ell }:{\pi }_1(X[1/\ell ])\to G({\mathbf{Q}}_{\ell }){\}}_{\ell }$ be a Frobenius-compatible collection of continuous homomorphisms indexed by the primes. Assume $\mathrm{Img}({\rho }_{\ell })$ is Zariski-dense in $G_{{\mathbf{Q}}_{\ell }}$ for all $\ell$ in a nonempty finite set $\mathcal{R}$. We prove that, under certain hypotheses on $\mathcal{R}$ (depending only on $G$), $\mathrm{Img}({\rho }_{\ell })$ is Zariski-dense in $G_{{\mathbf{Q}}_{\ell }}$ for all $\ell$ in a set of Dirichlet density $1$. As an application, we combine this result with a version of Hilbert's irreducibility theorem and recent work of Klevdal--Patrikis to obtain new information about the "canonical" local systems attached to Shimura varieties not of Abelian type.

18 pages. comments welcome!