Maximality of galois actions for abelian and hyper-kähler varieties

Abstract

© 2020 Duke University Press. All rights reserved. Let {ρℓ} be the system of ℓ-adic representations arising from the i th ℓ-adic cohomology of a proper smooth variety X defined over a number field K. Let Γℓand Gℓbe, respectively, the image and the algebraic monodromy group of ρℓ. We prove that the reductive quotient of Gοℓis unramified over every degree 12 totally ramified extension of ℚℓfor all sufficiently large ℓ. We give a necessary and sufficient condition (∗) on {ρℓ}ℓsuch that, for all sufficiently large ℓ, the subgroup Γℓis in some sense maximal compact in Gℓ(ℚℓ). This is used to deduce Galois maximality results for ℓ-adic representations arising from abelian varieties (for all i) and hyper-Kähler varieties (i=2) defined over finitely generated fields over ℚ.