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\title{Letter To Amod on Component Groups}
\author{Ken Ribet}
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\begin{verbatim}
Date: Mon, 30 Nov 1998 06:21:14 -0800 (PST)
From: "Kenneth A. Ribet" <ribet@math.berkeley.edu>
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\begin{quote}
 (From Amod)\\
 Did you get my email asking you about the number of connected
 components of the special fiber of the Neron model of $J_e(p)$ at $p$?  You
 had said that the only primes dividing it should be the Eisenstein
 primes and I was wondering if you know that as a fact or that you
 believe it follows from the stuff you gave to William?
\end{quote}

The more I think about this question, the more I realize that I don't
know the answer completely.  It's an interesting question.  Briefly,
if $A$ is an abelian subvariety of $J=J_0(N)$ ($N$ prime), you want to
know that the maximal ideals of the Hecke ring in the support of the
component group of $A$ are Eisenstein.  You probably won't complain if
we know this only for maximal ideals prime to $2$.  So let $\m$ be a maximal
ideal which is non-Eisenstein and of odd residue characteristic.
Then $J[\m]$ is irreducible and of of dimension 2, so $A[\m]$ is either 0 or
$J[\m]$.  In the former case, $\m$ has nothing to do with $A$, so we have
nothing to prove.  In the latter case, the idea would be to show that
if $\Psi[\m]$ is non-zero, where $\Psi$ is the component group, then $A[\m]$ 
is finite at $N$.  By my level-lowering result, this would be a contradiction.

There is an exact sequence
 $$0 \ra \Hom(X/\m X,\bmu_{\ell}) \ra A[\m] \ra (Y/\ell Y)[\m]$$
where $X$ is the character group of the torus which is the connected
component of $A$ in characteristic $N$, and $Y$ is the analogue of $X$ with
$A$ replaced by its dual.  Here $\ell$ is the residue characteristic of $\m$,
of course.  The group $\Hom(X/\m X,\bmu_\ell)$ is 1-dimensional, and it is
natural to expect that $(Y/\ell Y)[\m]$ is also 1-dimensional, i.e., that
the sequence is right-exact.  We further have a natural inclusion
$\Psi[\m] \hookrightarrow (Y/\ell Y)[\m]$, and the maximal finite part of $A[\m]$ is the
inverse image of $\Psi[\m]$ in $A[\m]$.  In other words, the intersection
of $\Psi[\m]$ and the image of $A[\m]$, taken in $(Y/\ell Y)[\m]$, is
zero by my
level-lowering result.  Thus we know that $\Psi[\m]=0$ at least whenever
$(Y/\ell Y)[\m]$ is 1-dimensional.

In the simplest case you can imagine, $A$ is an elliptic curve and $Y$
is isomorphic to $\Z$.  Then $Y/\ell Y$ is $\Z/\ell\Z$, 
and $(Y/\ell Y)[\m]$ is also $\Z/\ell\Z$;
i.e., it's 1-dimensional.  It should be easy to prove that                
$A[\m] \ra (Y/\ell Y)[\m]$ is surjective when $\ell$ 
doesn't divide $N-1$, and also
when the image of the Hecke ring in $\End(A)$ has discriminant prime to $\ell$.
But, in general, I don't see a quick argument that proves the
desired surjectivity.

-ken                   

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