An effective open image theorem for abelian varieties

Abstract

Fix an abelian variety $A$ of dimension $g\ge 1$ defined over a number field $K$. For each prime $\ell$, the Galois action on the $\ell$-power torsion points of $A$ induces a representation ${\rho }_{A,\ell }:Ga{l}_K\to G{L}_{2g}({\mathbb{Z}}_{\ell })$. The $\ell$-adic monodromy group of $A$ is the Zariski closure $G_{A,\ell }$ of the image of ${\rho }_{A,\ell }$ in $G{L}_{2g,{\mathbb{Q}}_{\ell }}$. 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 $[G_{A,\ell }({\mathbb{Q}}_{\ell })\cap G{L}_{2g}({\mathbb{Z}}_{\ell }):{\rho }_{A,\ell }(Ga{l}_K)]$ 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 $A$.

References in corpus (1)

• Expander graphs, gonality and variation of Galois representations