On the rationality of certain type a galois representations

Abstract

© 2018 American Mathematical Society. Let X be a complete smooth variety defined over a number field K and let i be an integer. The absolute Galois group GalK of K acts on the ith étale cohomology group (Forumala Presented). for all primes ℓ, producing a system of ℓ-adic representations {Φℓ }ℓ. The conjectures of Grothendieck, Tate, and Mumford-Tate predict that the identity component of the algebraic monodromy group of Φℓ admits a reductive Q-form that is independent of ℓ if X is projective. Denote by Γℓ and Gℓ respectively the monodromy group and the algebraic monodromy group of Φss ℓ,the semisimplification ofΦℓ. Assuming that Gℓ0 satisfies some group theoretic conditions for some prime ℓ0, we construct a connected quasi-split Q-reductive group GQ 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 GQ. As an application, we prove that the monodromy group Γℓ is big in the sense that (forumala Presented). for all sufficiently large ℓ.