Maximality of Galois actions for abelian and hyperkahler varieties

Understand

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 $\mathbf{G}_\ell(\mathbb{Q}_\ell)$.