Bogomolov's paper "Sur l'algébricité des représentations l-adiques" proves that the image of the $\ell$-adic Galois representation associated to an abelian variety over a number field $K$ is open in the $\mathbb{Q}_ {\ell}$ points of its $\mathbb{Q}_{\ell}$-Zariski closure.

The Comptes-Rendus announcement has only sketched the proof. My French is limited, but I have tried to capture the key step below.

In the abelian case, it proceeds as follows. Let $H_{\ell}$ be the $\mathbb{Q}_ {\ell}$ Zariski-clsure of the image of the absolute Galois group. Replacing $K$ by a finite extension, we assume that the is connected. If $H_{\ell}$ is commutative, then $\rho$ is abelian. We apply properties of Hodge-Tate representations to deduce that $H_{\ell}$ is a torus. For otherwise, $H_{\ell}$ contains a factor of $\mathbb{G}_a$. Composing with $\rho$ would give an infinite unramified abelian extension of $K$. [** Why? **]

I do not understand the last step used to obtain an infinite unramified abelian extension. Perhaps somebody else can see it?

The Russian version is at http://www.mathnet.ru/links/c33c60182f2bcfa61bc2cb3a0b684c62/im1843.pdf, but I cannot read a word of Russian.