The effective surjectivity of mod l Galois representations of 1- and 2-dimensional abelian varieties with trivial endomorphism ring

Mod l Galois representations of 1- and 2-dimensional abelian varieties with trivial endomorphism ring are surjective for sufficiently large prime l as Serre proved. But he did not give an effective lower bound of l ₀ such that they are surjective for l > l ₀. We supply an effective evaluation ofl ₀ by anelementary proof of the surjectivity. The proofuses the Masser-Wüstholz theorem and Kleidman and Liebecksclassification of the maximal subgroups of GL ₂ F _l )and GSp ₄(F _l ).