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Let $A$ be an absolutely simple abelian variety without (potential) complexmultiplication, defined over the number field $K$. Suppose that either $\dimA=2$ or $A$ is of $\operatorname{GL}_2$-type: we give an explicit bound$\ell_0(A,K)$ such that, for every prime $\ell>\ell_0(A,K)$, the image of theabsolute Galois group of $K$ in $\operatorname{Aut}(T_\ell(A))$ is as large asit is allowed to be by endomorphisms and polarizations.

Explicit surjectivity of Galois representations attached to abelian surfaces and $\operatorname{GL}_2$-varieties