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-Wüstholz theorem and Kleidman and Liebecks classification of the maximal subgroups of GL_2 F_l) and GSp_4 (F_l).
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Takashi Kawamura, The effective surjectivity of mod l Galois representations of 1- and 2-dimensional abelian varieties with trivial endomorphism ring. Comment. Math. Helv. 78 (2003), no. 3, pp. 486–493DOI 10.1007/S00014-003-0768-7