On the rationality of certain type A Galois representations

Abstract

Let X X be a complete smooth variety defined over a number field K K and let i i be an integer. The absolute Galois group G a l K \mathrm {Gal}_K of K K acts on the i i th étale cohomology group H e ´ t i ( X K ¯ , Q ℓ ) H^i_{\mathrm {\acute {e}t}}(X_{\bar K},\mathbb {Q}_\ell ) for all primes ℓ \ell , producing a system of ℓ \ell -adic representations { Φ ℓ } ℓ \{\Phi _\ell \}_\ell . The conjectures of Grothendieck, Tate, and Mumford-Tate predict that the identity component of the algebraic monodromy group of Φ ℓ \Phi _\ell admits a reductive Q \mathbb {Q} -form that is independent of ℓ \ell if X X is projective. Denote by Γ ℓ \Gamma _\ell and G ℓ \mathbf {G}_\ell respectively the monodromy group and the algebraic monodromy group of Φ ℓ s s \Phi _\ell ^{\mathrm {ss}} , the semisimplification of Φ ℓ \Phi _\ell . Assuming that G ℓ 0 \mathbf {G}_{\ell _0} satisfies some group theoretic conditions for some prime ℓ 0 \ell _0 , we construct a connected quasi-split Q \mathbb {Q} -reductive group G Q \mathbf {G}_{\mathbb {Q}} which is a common Q \mathbb {Q} -form of G ℓ ∘ \mathbf {G}_\ell ^\circ for all sufficiently large ℓ \ell . Let G Q s c \mathbf {G}_{\mathbb {Q}}^{\mathrm {sc}} be the universal cover of the derived group of G Q \mathbf {G}_{\mathbb {Q}} . As an application, we prove that the monodromy group Γ ℓ \Gamma _\ell is big in the sense that Γ ℓ s c ≅ G Q s c ( Z ℓ ) \Gamma _\ell ^{\mathrm {sc}}\cong \mathbf {G}_{\mathbb {Q}}^{\mathrm {sc}}(\mathbb {Z}_\ell ) for all sufficiently large ℓ \ell .