Explicit surjectivity for Galois representations attached to abelian surfaces

Davide Lombardo

Published 2014-11-06Version 1

Let $A$ be an absolutely simple abelian surface without (potential) complex multiplication, defined over the number field $K$. We explicitly give a bound $\ell_0(A,K)$ such that, for every prime $\ell>\ell_0(A,K)$, the image of $\operatorname{Gal}\left(\overline{K}/K\right)$ in $\operatorname{Aut}(T_\ell(A))$ is as large as it is allowed to be by endomorphisms and polarizations.