Comparison of component groups of $\ell$-adic and mod $\ell$ monodromy groups

Abstract

Let $\{\rho_{\ell}:\mathrm{Gal}_K\to\mathrm{GL}_n(\mathbb{Q}_{\ell})\}_{\ell}$ be a semisimple compatible system of $\ell$-adic representations of a number field $K$ that is arising from geometry. Let $\textbf{G}_{\ell}\subset\mathrm{GL}_{n,\mathbb{Q}_{\ell}}$ and $\widehat{\underline{G_{\ell}}}\subset\mathrm{GL}_{n,\mathbb{F}_\ell}$ be respectively the algebraic monodromy group and full algebraic envelope of $\rho_{\ell}$. We prove that there is a natural isomorphism between the component groups $\pi_0(\textbf{G}_{\ell}) \simeq \pi_0(\widehat{\underline{G_\ell}})$ for all sufficiently large $\ell$.