On the rationality of certain type A Galois representations

Abstract: Let $X$ be a complete smooth variety defined over number field $K$ and $i$ an integer. The absolute Galois group of $K$ acts on the $i$th $l$-adic etale cohomology of $X$ for all $l$, producing a system of $l$-adic representations ${\Phi_l}$. The conjectures of Grothendieck, Tate, and Mumford-Tate predict that the identity component of the algebraic monodromy group of $\Phi_\ell$ admits a common reductive $Q$-form for all $l$ if $X$ is projective. Denote by $\Gamma_l$ and $G_l$ respectively the monodromy group and the algebraic monodromy group of $\Phi_l^{ss}$, the semisimplification of $\Phi_\ell$. Assuming that $G_{l_0}$ satisfies a group theoretic condition for some prime $l_0$ (Hypothesis A), we construct a connected quasi-split $Q$-reductive group $G_Q$ which is a common $Q$-form of $G_l^\circ$ for all sufficiently large $l$. Let $G_Q^{sc}$ be the universal cover of the derived group of $G_Q$. As an application, we prove that the monodromy group $\Gamma_\ell$ is big in the sense that $\Gamma_\ell^{sc}\cong G_Q^{sc}(Z_l)$ for all sufficiently large $l$.