Type A images of Galois representations and maximality

Abstract

© 2016, Springer-Verlag Berlin Heidelberg. Given a compatible subsystem {ρℓ}ℓ of n-dimensional ℓ-adic Galois representations arising from étale cohomology of any complete, non-singular variety over a number field K, we define Γ ℓ: = ρℓ(Gal (K¯ / K)) and let Gℓ denote the Zariski closure of Γ ℓ in GL n. If Gℓ is of Type A in the sense that all simple composition factors are of type A in the Cartan-Killing classification, then Γ ℓ is, in a suitable sense, maximal in Gℓ for all ℓ≫ 0. As a corollary, if ρℓ is semisimple and ℓ is sufficiently large, then Gℓ is unramified.