On the rationality of algebraic monodromy groups of compatible systems

Abstract

Let $E$ be a number field and $X$ a smooth geometrically connected variety defined over a characteristic $p$ finite field. Given an $n$-dimensional pure $E$-compatible system of semisimple $\lambda$-adic representations of the étale fundamental group of $X$ with connected algebraic monodromy groups ${\mathbf{G}}_{\lambda }$, we construct a common $E$-form $\mathbf{G}$ of all the groups ${\mathbf{G}}_{\lambda }$ and in the absolutely irreducible case, a common $E$-form $\mathbf{G}\hookrightarrow {\text{GL}}_{n,E}$ of all the tautological representations ${\mathbf{G}}_{\lambda }\hookrightarrow {\mathrm{GL}}_{n,E_{\lambda }}$. Analogous rationality results in characteristic $p$ assuming the existence of crystalline companions in ${\text{F-Isoc}}^{†}(X)\otimes E_v$ for all $v|p$ and in characteristic zero assuming ordinariness are also obtained. Applications include the construction of a $\mathbf{G}$-compatible system from some ${\mathrm{GL}}_n$-compatible system and some results predicted by the Mumford–Tate conjecture.