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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

$G_K \rightarrow GSp(\mathbf{F}_l)$

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.

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.")

In 2010, is there a good reference for this generalization?

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I always wonder how these private "letters" circulate on to the possession of a large number of people. This time I must try to find an answer. How did it reach your hands? :D –  Anweshi Jul 13 '10 at 20:04
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Anweshi, it's item 137 in volume IV of Serre's Oeuvres, near which are other letters too. JSE, is it just a matter of look at the proofs and applying more general "spreading out" and Chebotarev arguments with Faltings' generalization of Mordell-stuff to finitely generated ground fields (in char. 0)? If so, probably there's no reference, so you should write one as an appendix if you need it for a paper. Or should I write one for you (maybe an exercise as part of the Mordell seminar with AV next year...)? –  BCnrd Jul 13 '10 at 21:23
    
I find it likely that a proof can be given by the sort of arguments BCnrd mentions above. However, I would also be interested in seeing the details. –  Pete L. Clark Jul 13 '10 at 21:54
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BCnrd, that is what we think, but wanted to save ourselves the peu plus attention if someone has already written the appendix for us. Will report back. –  JSE Jul 13 '10 at 22:01
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1 Answer 1

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. http://arxiv.org/abs/1008.3675

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I think the E in JSE stands for Ellenberg! –  Hailong Dao Oct 22 '10 at 10:03
    
Oh, that's funny. So maybe he has written up what he needs in the meantime, and I presented to him only a link to his own paper :-) But how could I know ... –  Sebastian Petersen Oct 22 '10 at 10:26
    
He has a link on his profile page (but we don't expect everyone to check). –  S. Carnahan Oct 22 '10 at 14:55
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And this is why Scott pesters me to use my real name on Overflow! –  JSE Oct 23 '10 at 23:00
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