Maximality of Galois actions for abelian and hyper-Kähler varieties

Abstract

Let $\{{\rho }_{\ell }{\}}_{\ell }$ be the system of $\ell$-adic representations arising from the $i$th $\ell$-adic cohomology of a proper smooth variety $X$ defined over a number field $K$. Let ${\Gamma }_{\ell }$ and ${\mathbf{G}}_{\ell }$ be, respectively, the image and the algebraic monodromy group of ${\rho }_{\ell }$. We prove that the reductive quotient of ${\mathbf{G}}_{\ell }^{\circ }$ is unramified over every degree $12$ totally ramified extension of ${\mathbb{Q}}_{\ell }$ for all sufficiently large $\ell$. We give a necessary and sufficient condition $(*)$ on $\{{\rho }_{\ell }{\}}_{\ell }$ such that, for all sufficiently large $\ell$, the subgroup ${\Gamma }_{\ell }$ is in some sense maximal compact in ${\mathbf{G}}_{\ell }\left({\mathbb{Q}}_{\ell }\right)$. This is used to deduce Galois maximality results for $\ell$-adic representations arising from abelian varieties (for all $i$) and hyper-Kähler varieties ($i=2$) defined over finitely generated fields over $\mathbb{Q}$.