On Mon, 03 Oct 2005 16:38:12 -0700, Amod Agashe <agashe@mail.math.fsu.edu> wrote:
> Your article is very interesting. However, I think the right Eisenstein
> ideal should be generated by T_\ell - 1 - \ell for \ell not dividing N,
> and by U_p - p for p dividing N. This is because that is the kernel
> of the action of T on (0)-(\infty).

I think there are several Eisenstein ideals for general level, since there are
other interesting subgroups of the cuspidal subgroup besides (0)-(infty).  

> Does it theoretically have anything to do with "self-fusion" primes at all
> (I have not read your paper fully)? Are there examples of level< yours
> where there is congruence mod an Eis prime with a form of lower level
> (but no self-fusion, necessarily)?

Yes, in fact 7^2 exactly divides the
discriminant of the quotient of Hecke algebra associated to A, so there
is a prime over 7 of self-fusion.   This is discussed in the middle
of page 3 of the paper. 

> Also, at the beginning of 2.1, you say that A[m] has a filtration
> by modules of the form Z/7Z and \mu_7, giving p.112 of Mazur's
> Eisenstein ideal paper as a reference. Mazur does discuss the
> filtration there, but he does not say they have to be constant
> or multiplicative group schemes (on p. 112, unless I missed something).
> How does one see that there are no other possibilities?

This is just a claim about *Galois* modules (which by Fontaine is
essentially equivalent to a claim about group schemes since 7=/=2, 
as explained in Mazur).    A one-dimensional Galois module is given
by a Dirichlet character, so it's either Z/7Z (the trivial character),
or some power of the mod-7 cyclotomic character chi_7.  But 
consideration of the determinant (since the weight is 2) implies 
that the character is chi_7, which corresponds to \mu_7.   Does
that help?  Thanks for your question -- I should add some more explanation. 

> P.S.: I was hoping to write the names Gorenstein, Eisenstein,
> and William Stein on a single page in a report/paper,
> and you beat me to it: they are there in the top two lines of your paper!

Maybe you can fit it all into one line in your paper by writing,
"...in view of Calegari and Stein's ``non-Gorenstein Eisenstein descent''... :-)

William