On the rationality of certain type A Galois representations

Abstract:

Let $X$ be a complete smooth variety defined over a number field $K$ and let $i$ be an integer. The absolute Galois group $\mathrm {Gal}_K$ of $K$ acts on the $i$th étale cohomology group $H^i_{\mathrm {\acute {e}t}}(X_{\bar K},\mathbb {Q}_\ell )$ for all primes $\ell$, producing a system of $\ell$-adic representations $\{\Phi _\ell \}_\ell$. The conjectures of Grothendieck, Tate, and Mumford-Tate predict that the identity component of the algebraic monodromy group of $\Phi _\ell$ admits a reductive $\mathbb {Q}$-form that is independent of $\ell$ if $X$ is projective. Denote by $\Gamma _\ell$ and $\mathbf {G}_\ell$ respectively the monodromy group and the algebraic monodromy group of $\Phi _\ell ^{\mathrm {ss}}$, the semisimplification of $\Phi _\ell$. Assuming that $\mathbf {G}_{\ell _0}$ satisfies some group theoretic conditions for some prime $\ell _0$, we construct a connected quasi-split $\mathbb {Q}$-reductive group $\mathbf {G}_{\mathbb {Q}}$ which is a common $\mathbb {Q}$-form of $\mathbf {G}_\ell ^\circ$ for all sufficiently large $\ell$. Let $\mathbf {G}_{\mathbb {Q}}^{\mathrm {sc}}$ be the universal cover of the derived group of $\mathbf {G}_{\mathbb {Q}}$. As an application, we prove that the monodromy group $\Gamma _\ell$ is big in the sense that $\Gamma _\ell ^{\mathrm {sc}}\cong \mathbf {G}_{\mathbb {Q}}^{\mathrm {sc}}(\mathbb {Z}_\ell )$ for all sufficiently large $\ell$.