CoCalc Shared Fileswww / papers / pcmi / email-jul07-reply.txt
Author: William A. Stein
1Dear Ken,
2
3> p. 10 (+7) Perhaps explain why this is a concrete consequence,
4> saying a little about J_0(Nl^2); also, the end of the line
5> sticks out badly into the margin.
6
7That this is a concrete consequence is explained in Section 3.1 of chapter 3.
8Why does Brian want level Nl^2 instead of level Nl?
9
10RIBET: If you want weights beyond l+1, then you have to twist
11by powers of the cyclotomic character, which you view as a character
12of conductor l.  That twisting raises level from ...l to ...l^2.
13Whether Brian is right or not depends on the context -- were
14high weights contemplated at that juncture?
15
16WAS: -- YES -- It says:
17   A concrete consequence of the conjecture is that all odd
18   irreducible 2-dimensional~$\rho$ come from abelian varieties
19   over~$\Q$.  Given~$\rho$, one should be able to find a totally
20   real or CM number field~$E$, an abelian variety~$A$ over~$\Q$
21   of dimension [...]
22
23I now see why it is necessary to introduce ell^2.  However,
24I'm not sure we should explain further, at least at the late point
25in the writing of the paper.
26
27*However* maybe we should mention "Theorem F" (page 4) of Taylor's amazing new
28paper "Remarks on a Conjecture of Fontaine and Mazur", which seems to prove the
29above concrete consequence under a reasonable local hypothesis.
30
31William
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33