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 of l(0) by an "elementary" proof of the surjectivity. The proof uses the Masser Wilstholz theorem and Kleidman and Liebeck's classification of the maximal subgroups of GL(2)(F-1) and GSp(4)(F-l).