Welcome to MathOverflow

Question

6

2

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?

edited Jul 13 2010 at 21:05

BCnrd
4,156●1●38●58

asked Jul 13 2010 at 19:40

JSE
13.6k●33●85

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 2010 at 20:04

6

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 2010 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 2010 at 21:54

3

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 2010 at 22:01

Sebastian Petersen · Answer 1 · 2010-10-22 10:00:37Z UTC

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

answered Oct 22 2010 at 10:00

Sebastian Petersen
1,271●2●9

4

I think the E in JSE stands for Ellenberg! – Hailong Dao Oct 22 2010 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 2010 at 10:26

He has a link on his profile page (but we don't expect everyone to check). – S. Carnahan♦ Oct 22 2010 at 14:55

1

And this is why Scott pesters me to use my real name on Overflow! – JSE Oct 23 2010 at 23:00

Monodromy groups of families of abelian varieties: a reference request

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.