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 ${\text{G}}_{\ell }\subset {\mathrm{GL}}_{n,{\mathbb{Q}}_{\ell }}$ 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 ${\pi }_0({\text{G}}_{\ell })\simeq {\pi }_0(\widehat{\underline{G_{\ell }}})$ for all sufficiently large $\ell$.

arXiv comment10 pages. To appear in Journal of Number Theory