An effective open image theorem for abelian varieties

Abstract

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 $A$.