On the semisimplicity of reductions and adelic openness for $E$-rational compatible systems over global function fields

Abstract

Let $X$ be a normal geometrically connected variety over a finite field $\kappa$ of characteristic~$p$. Let $E$ be a number field. Using automorphic methods over global function fields, we derive properties of the geometric monodromy groups of arbitrary connected $E$-rational semisimple compatible systems $(\rho_\lambda)$ of $n$-dimensional representations of the arithmetic fundamental group $\pi_1(X)$, where $\lambda$ ranges over the finite places of $E$ not above $p$: Let $\Lambda_\lambda$ be any $\pi_1(X)$-stable lattice in $E_\lambda^n$ under $\rho_\lambda$. Then for almost all $\lambda$, the schematic closure of the geometric monodromy $\rho_\lambda(\pi_1(X_{\overline{\kappa}}))$ in $\mathrm{Aut}_{\mathcal{O}_\lambda}(\Lambda_\lambda)$ is a semisimple $\mathcal{O}_\lambda$-group scheme, and its special fiber agrees with the Nori envelope of the geometric monodromy of the mod-$\lambda$ reduction of $\rho_\lambda$. A comparable result under different hypotheses was recently proved by Cadoret, Hui and Tamagawa by other methods. We also provide natural criteria for the image of $\pi_1(X_{\overline{\kappa}})$ under $\prod_\lambda\rho_\lambda$ to have adelic open image in an appropriate sense.