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