Independence of $\boldsymbol {\ell }$ of monodromy groups

Abstract:

Let $X$ be a smooth curve over a finite field of characteristic $p$, let $E$ be a number field, and let $\mathbf {L} = \{\mathcal {L}_\lambda \}$ be an $E$-compatible system of lisse sheaves on the curve $X$. For each place $\lambda$ of $E$ not lying over $p$, the $\lambda$-component of the system $\mathbf {L}$ is a lisse $E_\lambda$-sheaf $\mathcal {L}_\lambda$ on $X$, whose associated arithmetic monodromy group is an algebraic group over the local field $E_\lambda$. We use Serre’s theory of Frobenius tori and Lafforgue’s proof of Deligne’s conjecture to show that when the $E$-compatible system $\mathbf {L}$ is semisimple and pure of some integer weight, the isomorphism type of the identity component of these monodromy groups is “independent of $\lambda$”. More precisely, after replacing $E$ by a finite extension, there exists a connected split reductive algebraic group $G_0$ over the number field $E$ such that for every place $\lambda$ of $E$ not lying over $p$, the identity component of the arithmetic monodromy group of $\mathcal {L}_\lambda$ is isomorphic to the group $G_0$ with coefficients extended to the local field $E_\lambda$.