  Fix an abelian variety $A$ of dimension $g\geq 1$ defined over a number field
$K$. For each prime $\ell$, the Galois action on the $\ell$-power torsion
points of $A$ induces a representation $\rho_{A,\ell}\colon Gal_K \to
GL_{2g}(\mathbb{Z}_\ell)$. The $\ell$-adic monodromy group of $A$ is the
Zariski closure $G_{A,\ell}$ of the image of $\rho_{A,\ell}$ in
$GL_{2g,\mathbb{Q}_\ell}$. The image of $\rho_{A,\ell}$ is open in
$G_{A,\ell}(\mathbb{Q}_\ell)$ with respect to the $\ell$-adic topology and
hence the index $[G_{A,\ell}(\mathbb{Q}_\ell)\cap GL_{2g}(\mathbb{Z}_\ell):
\rho_{A,\ell}(Gal_K)]$ is finite. We prove that this index can be bounded in
terms of $g$ for all $\ell$ larger then some constant depending on certain
invariants of $A$.
