Commentarii Mathematici Helvetici
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Volume 78, Issue 3, 2003, pp. 486–493
DOI: 10.1007/s00014-003-0768-7
The effective surjectivity of mod l Galois representations of 1- and 2-dimensional abelian varieties with trivial endomorphism ring
TAKASHI KAWAMURA (1)
(1) GRADUATE SCHOOL OF MATHEMATICAL SCIENCES, UNIVERSITY OF TOKYO, 3-8-1 KOMABA MEGURO-KU, 153-8914 MZ, TOKYO, JAPANMod 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).
Keywords: Abelian variety - mod l Galois representation - Effective surjectivity