Title: Maximality of Galois actions for abelian varieties,
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 $K$. Denote the image of $\rho_\ell$ by $\Gamma_\ell$ and
its Zariski closure, which is a linear algebraic group over $\mathbb{Q}_\ell$,
by $\mathbf{G}_\ell$. We prove that $\mathbf{G}_\ell^{red}$, the quotient of
$\mathbf{G}_\ell^\circ$ by its unipotent radical, is unramified over a totally
ramified extension of $\mathbb{Q}_\ell$ for all sufficiently large $\ell$. We
give a sufficient condition on $\{\rho_\ell\}_\ell$ such that for all
sufficiently large $\ell$, $\Gamma_\ell$ is in some sense maximal compact in
$\mathbf{G}_\ell(\mathbb{Q}_\ell)$. Since the condition is satisfied when $X$
is an abelian variety by the Tate conjecture, we obtain maximality of Galois
actions for abelian varieties.