On the rationality of certain type A Galois representations

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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 ith l-adic etale cohomology of X for all l, producing a system of l-adic representations {Φₗ}. The conjectures of Grothendieck, Tate, and Mumford-Tate predict that the identity component of the algebraic monodromy group of Φℓ admits a common reductive Q-form for all l if X is projective. Denote by Γₗ and Gₗ respectively the monodromy group and the algebraic monodromy group of Φₗss, the semisimplification of Φℓ. Assuming that Gₗ₀ satisfies a group theoretic condition for some prime l₀ (Hypothesis A), we construct a connected quasi-split Q-reductive group GQ which is a common Q-form of Gₗ∘ for all sufficiently large l. Let GQsc be the universal cover of the derived group of GQ. As an application, we prove that the monodromy group Γℓ is big in the sense that Γℓsc GQsc(Zₗ) for all sufficiently large l.