On the rationality of algebraic monodromy groups of compatible systems

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 'etale fundamental group of $X$ with connected algebraic monodromy groups $G_{\lambda }$, we construct a common $E$-form $G$ of all the groups $G_{\lambda }$ and in the absolutely irreducible case, a common $E$-form $G\hookrightarrow {\text{GL}}_{n,E}$ of all the tautological representations $G_{\lambda }\hookrightarrow {\text{GL}}_{n,E_{\lambda }}$ (Theorem 1.1). Analogous rationality results in characteristic $p$ assuming the existence of crystalline companions in ${\text{F-Isoc}}^{†}(X)\otimes E_v$ for all $v\mathrm{\mid }p$ (Theorem 1.5) and in characteristic zero assuming ordinariness (Theorem 1.6) are also obtained. Applications include a construction of $G$-compatible system from some ${\text{GL}}_n$-compatible system and some results predicted by the Mumford-Tate conjecture.