Maximality of Galois actions for abelian and hyperkahler varieties

TL;DR

This paper establishes conditions under which the Galois representations from abelian and hyperk"ahler varieties are maximal, showing their algebraic monodromy groups have unramified reductive quotients over certain extensions.

Contribution

It introduces a necessary and sufficient condition for the maximality of Galois images in $ ext{GL}_n(Q_ell)$ for abelian and hyperk"ahler varieties, extending previous maximality results.

Findings

Reductive quotients are unramified over specific extensions for large $ell$

Condition $(igstar)$ characterizes when Galois images are maximal

Galois maximality results for abelian and hyperk"ahler varieties over finitely generated fields

Abstract

Let $\{\rho_\ell\}_\ell$ be the system of $\ell$-adic representations arising from the $i$th $\ell$-adic cohomology of a complete 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 $(\ast)$ on $\{\rho_\ell\}_\ell$ such that for all sufficiently large $\ell$, the subgroup $\Gamma_\ell$ is in some sense maximal compact in $\mathbf{G}_\ell(\mathbb{Q}_\ell)$. This is used to deduce Galois maximality results for $\ell$-adic representations arising from abelian varieties (for all $i$) and hyperk\"ahler varieties ($i=2$) defined over finitely generated fields over $\mathbb{Q}$.