On the rationality of certain type a galois representations

Let X be a complete smooth variety defined over a number field K and let i be an integer. The absolute Galois group Gal_K of K acts on the ith étale cohomology group (Forumala Presented). for all primes ℓ, producing a system of ℓ-adic representations {Φ_ℓ }_ℓ. The conjectures of Grothendieck, Tate, and Mumford-Tate predict that the identity component of the algebraic monodromy group of Φ_ℓ admits a reductive Q-form that is independent of ℓ if X is projective. Denote by Γ_ℓ and G_ℓ respectively the monodromy group and the algebraic monodromy group of Φ^ss ℓ^,the semisimplification of^Φℓ. Assuming that G_ℓ0 satisfies some group theoretic conditions for some prime ℓ₀, we construct a connected quasi-split Q-reductive group G_Q which is a common Q-form of G^◦ℓ for all sufficiently large ℓ. Let GscQ be the universal cover of the derived group of G_Q. As an application, we prove that the monodromy group Γ_ℓ is big in the sense that (forumala Presented). for all sufficiently large ℓ.