Monodromy groups of families of abelian varieties: a reference request - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T07:31:17Z http://mathoverflow.net/feeds/question/31756 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/31756/monodromy-groups-of-families-of-abelian-varieties-a-reference-request Monodromy groups of families of abelian varieties: a reference request JSE 2010-07-13T19:40:31Z 2010-11-19T10:22:14Z <p>In Serre's letter to Vigneras of 2 Oct 1986, he summarizes a course he's giving in Paris, explaining how to control the image of the mod-l Galois representations attached to abelian varieties. In particular, given an abelian variety A over a number field K, and a (rational) prime l, he constructs an algebraic group H_l such that the image of the map</p> <p>$G_K \rightarrow GSp(\mathbf{F}_l)$</p> <p>has image contained in H_l(F_l) with bounded index, for all but finitely many l. The group H_l is constructed as the product of a semisimple group S_l and a torus C_l.</p> <p>When K is instead a field of finite type over Q, Serre remarks in section 8.1 that all the theorems in the letter should still be true, but one has to be a little more careful ("il faut faire un peu plus attention.") </p> <p>In 2010, is there a good reference for this generalization? </p> http://mathoverflow.net/questions/31756/monodromy-groups-of-families-of-abelian-varieties-a-reference-request/43156#43156 Answer by Sebastian Petersen for Monodromy groups of families of abelian varieties: a reference request Sebastian Petersen 2010-10-22T10:00:37Z 2010-10-22T10:00:37Z <p>Part of what you are looking for seems to have been done recently. See Appendix B and Section 4 (especially Thm. 14) of the recent preprint "Expander graphs, gonality and variation of Galois representations" of Ellenberg, Hall and Kowalski. <a href="http://arxiv.org/abs/1008.3675" rel="nofollow">http://arxiv.org/abs/1008.3675</a></p>