\magnification=\magstep1
\outer\def\beginsection#1\par{\vskip0pt plus0.1\vsize\penalty-250
	\vskip0pt plus-.1\vsize\bigskip\vskip\parskip
	\message{#1}\leftline{\bf#1}\nobreak\smallskip\noindent}


\baselineskip=18pt
\lineskiplimit=3pt
\lineskip=3pt minus 1pt

\let\tempcirc=\circ
\def\circ{\mathord{\raise0.25ex\hbox{$\scriptscriptstyle\tempcirc$}}}


% establish font family for gothic letters

\font\tgoth=eufm10\font\sgoth=eufm7\font\ssgoth=eufm5
\newfam\gothfam \textfont\gothfam=\tgoth
\scriptfont\gothfam=\sgoth\scriptscriptfont\gothfam=\ssgoth 
\def\goth{\fam\gothfam}

  % maximal ideals represented by gothic letters:

        \def\m{{\goth m}}
        \def\wp{{\goth p}}
        \def\wP{{\goth P}}

% make a seven pt italic font for the author's address:
\font\sevenit=cmti7

% mathops 

\def\pico{\mathop{{\rm Pic}^o}}
\def\Aut{\mathop{\rm Aut}}
\def\End{\mathop{\rm End}}
\def\Gal{\mathop{\rm Gal}\nolimits}
\def\Hom{\mathop{\rm Hom}\nolimits}
\def\tr{\mathop{\rm tr}}


\let\Frob=\varphi  %Frobenius elements

\def\xo#1{X_o(#1)}  %modular curves
\def\jo#1{J_o(#1)}  %Jacobian of above

% fields and rings
\def\F{{\bf F}}
\def\Fp{\F_p}
\def\T{{\bf T}}
\def\Q{{\bf Q}}
\def\Qbar{\overline{\bf Q}}
\def\Qp{{\bf Q}_p}
\def\Qpbar{{\overline\Q}_p}
\def\Zp{{\bf Z}_p}
\def\Z{{\bf Z}}

\def\GalQ{\Gal(\Qbar/\Q)}
\def\GalQp{\Gal(\Qpbar/\Qp)}

% caligraphic letters

\def\L{{\cal L}}
\def\M{{\cal M}}
\def\Oh{{\cal O}}
\def\V{{\cal V}}


\def\seteq{\mathrel{:=}}
\def\ac{\nobreak\hskip.1111em\mathpunct{}\nonscript\mkern-\thinmuskip{:}\hskip
 .3333emplus.0555em\relax}  

\def\image{\pi({\cal R})}
\def\foom{\jo N^2}

\def\today{\ifcase\month\or
 January\or February\or March\or April\or May\or June\or
 July\or August\or September\or October\or November\or December\fi
 \space\number\day, \number\year}

% references

\def\BLR{1}     % Boston-Lenstra-Ribet
\def\DOI{2}
\def\KATZ{3}    % Katz at Bonn
\def\MAZE{4}    % Mazur's Eisenstein ideal paper
\def\MAZR{5}    % Mazur-Ribet on ``dimension two''
\def\RAYN{6}    % Raynaud's ppp article
\def\HECKE{7}   % Ribet on mod p Hecke operators
\def\ICM{8}     % ICM article
\def\BX{9}      % Bordeaux article
\def\FERMAT{10} % Ribet's ``Fermat'' article in Invent Math
\def\JPS{11}    % Serre Bourbaki
\def\DUKE{12}   % Serre's Duke article
\def\SHIM{13}   % Shimura's book
\def\HPF{14}    % Swinnerton-Dyer


\noindent


\hrule
\vskip0.8cm
\noindent
\bf
Multiplicities of $p$-finite mod~$p$ Galois representations in $\jo{Np}$
\medskip\noindent\rm
Kenneth A. Ribet 
\medskip\noindent
\baselineskip=10pt
\sevenit
Department of Mathematics\hfil\break
University of California\hfil\break
Berkeley, CA 94720 USA
\bigskip\noindent$$
%\shadowbox
{\vbox{\hsize=325pt\parfillskip0pt\noindent
   \vrule height 10pt depth 0pt width 0pt
%\sevenbf Acknowledgements.
\sevenrm This article was prepared while the author was visiting the
I.H.E.S., and is based partially on the author's lecture 
at I.M.P.A. during an algebraic geometry workshop in April, 1990.
It is a pleasure to thank both institutes for their hospitality,
as well as Professor B.~Mazur for helpful suggestions.
The  author was partially supported by   NSF contract DMS 88-06815.}}$$
\rm
\baselineskip=18pt
\beginsection 1. Introduction.\par
Let $M\ge1$ be an integer.  Let $\jo M$ be the Jacobian 
$\pico(\xo M)$ 
of the modular
curve $\xo M_\Q$.  For all $n\ge1$, let $T_n$ be the $n^{\rm th}$
Hecke correspondence on $\xo M$.  Write again $T_n$ for
the endomorphism $T_n^*$ of $\jo M$ induced by the correspondence $T_n$.
Let $\T_M=\Z[\ldots,T_n,\ldots]$ be the subring of $\End(\jo M)$
generated by the $T_n$ with $n\ge1$.  The ring $\T_M$ 
may be identified with
the  ring generated by the Hecke operators $T_n$ acting on the 
complex vector space
of weight-2 cusp forms on $\Gamma_o(M)$.

Suppose that $\wp$ is a maximal ideal of $\T_M$. The residue field $\T_M/\wp$
is a finite field $k$, whose characteristic will be denoted~$p$.
Attached to $\wp$ is a semisimple continuous representation
$$ \rho_\wp\ac\GalQ \to {\bf GL}(2,k),$$
unramified outside $Mp$, with the property that $\rho_\wp(\Frob_r)$
has characteristic polynomial 
     $X^2 - T_r X + r$ (mod~$\wp$)
for each prime
number $r$ prime to~$Mp$.  (Here, $\Frob_r$ denotes a Frobenius
element of $\GalQ$ for the prime number~$r$.)  The representation 
$\rho_\wp$ is
well defined up to isomorphism.

Consider the group of $p$-division points
%
$$ \jo M[\wp] \seteq  \bigl\{\, x \in \jo M(\Qbar) \bigm| 
   \lambda x=0 \hbox{ for all $\lambda\in\wp$}\,\bigr\},$$
i.e., the ``kernel of $\wp$ on $\jo M$."
{\parindent 80pt Assume:
\item{i.} The representation $\rho_\wp$ is irreducible over~$k$.
\item{ii.} The prime number $p$ is odd.\par}

Then $\rho_\wp$ is an absolutely irreducible representation of
$\GalQ$, and a theorem of Boston, Lenstra, and the author [\BLR]
shows that
$\jo M[\wp]$ is  a direct sum of representations isomorphic to
$\rho_\wp$.  The associated {\it multiplicity\/} $\mu_\wp$
is the number of summands in the direct sum:  
$$ \mu_\wp =  {1\over2}\dim_k\jo M[\wp].$$
Since $\jo M[\wp]$ is non-zero, $\mu_\wp$ is a positive integer.
Thus $\jo M[\wp]$ is ``multiplicity free" (in the sense that 
$\mu_\wp\le1$) if and only if we have $\mu_\wp=1$.

Using ideas of Mazur [\MAZE], the author proved that $\mu_\wp=1$
whenever $M$ is prime to~$p$ ([\FERMAT], Theorem~5.2).  In
[\MAZR], Mazur and the author considered the case where $p$
exactly divides~$M$, i.e., where $M=Np$ for some integer $N\ge1$
which is prime to~$p$.  The principal result of that article  states
that $\mu_\wp=1$ provided that $\rho_\wp$ is not finite at~$p$
(in the sense of Serre [\DUKE], p.~189), or equivalently ([\FERMAT],
Theorem~6.1) provided that $\rho_\wp$ is not modular of level~$N$.
These results were motivated by applications to Serre's conjectures
[\DUKE], and specifically are used in showing that certain mod~$p$
Galois representations which are known to be modular of some level
are in fact modular of the conjectured level (see [\FERMAT]).

In the present article, we explore the situation
when $p$ exactly divides~$M$ but $\rho_\wp$ is 
allowed to be finite.  Although this situation does not appear in applications
to Serre's conjectures, the author hopes that the present paper will
shed light on the general problem of computing $\mu_\wp$.

In order to state our results, we  introduce
the $p$-old and $p$-new subvarieties $A$ and~$B$
of $\jo{M}$.  We assume that $M=Np$, and that $p$ is prime to~$N$.
Recall [\ICM] that the abelian varieties $\jo N$ and~$\jo{Np}$ are
connected by a natural ``degeneracy map"
$$ d\ac \jo N\times\jo N \longrightarrow \jo{Np},$$
whose kernel $\Sigma$ is the 
finite group consisting
of all pairs $(\sigma,-\sigma)$,
where $\sigma$ runs over the Shimura subgroup of $\jo N$, i.e., the
kernel of the map $\jo N\to J_1(N)$
induced by the standard covering $X_1(N)\to X_o(N)$.  The $p$-old
subvariety of $\jo{Np}$ is the image $A$ of~$d$.  The $p$-new
subvariety of $\jo{Np}$ is the ``orthogonal complement" $B$ to $A$
under the natural autoduality $\jo{Np} \mathrel{\widetilde\to} \jo{Np}^\vee$.
In other words, the image of $B$ in $\jo{Np}^\vee$ under the
isomorphism 
$\jo{Np}   \mathrel{\widetilde\to}  \jo{Np}^\vee$ 
is the
kernel of the map $\jo{Np}^\vee \to A^\vee$ 
which is dual to the inclusion of $A$
in~$\jo{Np}$.  One has $\jo{Np} = A + B$, and the group $\Delta\seteq
A\cap B$ is finite; this group was calculated in [\ICM].

In the following discussion, we will write simply $J$ for~$\jo{Np}$.
Also, we will let $A[\wp]$ and~$B[\wp]$ be the analogues of
$\jo{Np}[\wp]$ for $A$ and~$B$, respectively. 
When (i) and~(ii) are satisfied,  $A[\wp]$ and~$B[\wp]$ are each direct
sums of representations isomorphic to  $\rho_\wp$. We say that
$A[\wp]$ or~$B[\wp]$ is {\it multiplicity free\/} if the corresponding
direct sum has at most one term.  Thus $B[\wp]$, for example, is
multiplicity free if it is either zero, or else of dimension two over~$k$.
 Finally, we write
$\foom$ for the product $\jo N\times\jo N$.
\medbreak 
{\it Assume that the above
hypotheses (i) and (ii) are satisfied.}
%
\proclaim Proposition 1.  
The representation $\rho_\wp$ is finite at $p$ if and only if
$A[\wp]$ is non-zero.

\proclaim Theorem 1.
If $A[\wp]=0$, then $\mu_\wp=1$.

\noindent
Proposition 1 results easily from known facts; its proof will be given
in \S2.  Using the Proposition, we see immediately that
Theorem~1 is a restatement of the  result of [\MAZR] which was
introduced above.  

In this article, we shall prove the following
complement:
%
\proclaim Theorem 2.  Suppose that (i) and (ii) are satisfied.  
Then the group $J[\wp]$ is multiplicity free
if and only if the group $B[\wp]$ is multiplicity free.
\par\noindent
As mentioned above, the condition that $J[\wp]$ be
multiplicity free is equivalent to the equality $\mu_\wp=1$.
Hence Theorem~2 provides a means of verifying this equality
even when the condition
$A[\wp]=0$ of
Theorem~1 is not necessarily satisfied.

\proclaim Corollary.  Suppose that  (i) and (ii) are satisfied and that
$B$ is an elliptic curve.  Then $J[\wp]$ is multiplicity free.

\noindent The Corollary follows directly from the Theorem.
Indeed,
suppose that $B$ is an elliptic curve.  If $B[\wp]$ is non-zero,
then $\wp\subset\T_{pN}$ generates a non-trivial ideal in the image of
$\T_{pN}$ in $\End B$.  Since this image is isomorphic to $\Z$, we have
$B[\wp]=B[p]$, and the residue field of $\wp$ is the prime field $\Fp$.
Hence $B[\wp]$ is either 0, or else of dimension two over $\Fp$.
In particular, $B[\wp]$ is multiplicity free.

Theorem~2 is obtained as a consequence of two results proved below.
Namely, in \S2, we prove, under the hypotheses of Theorem~2,
that $A[\wp]$ is  multiplicity free (Theorem~3).  In \S3, we show
that $B[\wp]=J[\wp]$ whenever $B[\wp]$ is non-zero (Theorem~4).  
These together imply Theorem~2  by an elementary argument,
in which we consider separately the case  where $B[\wp]$
is~0 and the case where it is non-zero.
Indeed, suppose first that $B[\wp]=0$.
Then $B[\wp]$ is
certainly multiplicity free, and Theorem~2 states that
$J[\wp]$ is multiplicity free.  However, $\Delta[\wp]=0$, from which it
follows that the $p$-divisible group
$ \cup_i J[\wp^i]$ is the direct sum of the analogous
$p$-divisible groups for $A$ and~$B$.  This gives, in particular,
$J[\wp] = A[\wp]\oplus B[\wp]$, so that $J[\wp]=A[\wp]$, because
$B[\wp]$ is~0.  Theorem~3 then implies that $J[\wp]$ is
multiplicity free, as desired.  
On the other hand, if $B[\wp]$ is non-zero,
then Theorem~4 states that $J[\wp]$ and~$B[\wp]$ are equal, so that
one is multiplicity free if and only if the other is.

It is perhaps worth stressing that our results prove that
$\mu_\wp=1$ whenever one of $A[\wp]$ and~$B[\wp]$ vanishes
and the hypotheses (i) and~(ii) are satisfied.
Indeed, suppose this to be the case, and assume first
that $A[\wp]$ vanishes.    Then an
argument analogous to that just given shows that
 $J[\wp]=B[\wp]$, so that
Theorem~2 is trivially true (as is Theorem~4).
By Theorem~1, we   have $\mu_\wp=1$; i.e., $J[\wp]=B[\wp]$
is multiplicity free. 
Next, assume that $B[\wp]$ vanishes.
Then, as noted above, we have $A[\wp]=J[\wp]$.  Also,
$A[\wp]$ is multiplicity free by Theorem~3.  Hence $\mu_\wp=1$
in this case as well.


Assume now that  both
 $A[\wp]$ and~$B[\wp]$ are
non-zero, and that (i) and~(ii) are satisfied.  
Then our results  give the relations
$$ 0 \subset A[\wp] \subseteq B[\wp]=J[\wp],$$
and show that $A[\wp]$ is multiplicity free.
We have $\mu_\wp=1$ in this situation if and only if all three
groups $A[\wp]$, $B[\wp]$ and~$J[\wp]$ coincide.
By the Corollary above, these
  groups do in fact coincide if $B$ is an elliptic curve, although
perhaps not in general.
(The author knows of no example where they fail to coincide.)



We close this Introduction with a numerical example.  Take $N=11$
and~$p=3$.
The dimensions of $\jo{11}$ and $\jo{33}$
are respectively 1 and~3.  Thus, $A$ is isogenous to a product of
two copies of
the elliptic curve $\jo{11}$, while $B$ is an elliptic curve of
conductor~33.  By the Corollary to Theorem~2, $J[\wp]$ is 
multiplicity free for each prime $\wp$ of $\T_{33}$ which divides~3.

One can verify that there are in fact two $\wp$'s,
and that $\rho_\wp$ is finite for each~$\wp$.  
(In other words, Theorem~1 applies for neither of them.)
To describe the two $\wp$,
we let $E=\jo{11}$ and use the degeneracy map $d$ to identify
$A[3]$ with the product $E[3]\times E[3]$. The Hecke operator $T_3$
of $J$ %(which is often denoted $U_3$)
induces the operator
$(x,y)\mapsto (-x,-x)$ on $E[3]\times E[3]$.  Accordingly, the
diagonal image $V_1$ of $E[3]$ in $E[3]\times E[3]$ is $\T_{33}$-stable,
and $\T_{33}$ acts on $V_1$ via a homomorphism 
$\phi_1\ac\T_{33}\to\F_3$ which maps
$T_3$ to $-1$.  If $\wp_1=\ker\phi_1$, then 
Theorem~4 implies that
$B[\wp_1]=J[\wp_1]=A[\wp_1]=V.$  Similarly, if $V_2$ is the subspace
$0\oplus E[3]$ of $A[3]$, then $\T_{33}$ acts on $V_2$ via a homomorphism
$\phi_2\ac\T_{33}\to\F_3$ which maps $T_3$ to~0.  If $\wp_2=\ker\phi_2$,
then $B[\wp_2]=0$, and we have $J[\wp_2]=A[\wp_2]=V'$.
%
%
\beginsection 2.  Study of $A[\wp]$.\par
Let $p$ be an odd prime, and suppose that $N$ is a positive integer
prime to $p$. 
Write $\T$ for the ring $\T_{Np}$.  The $p$-old
subvariety $A$ of $\jo{pN}$ is $\T$-stable.
We define the
{\it $p$-old quotient of $\T$} to be the
image $\T_A$ of $\T$ in $\End(A)$.
Similarly, we define
a maximal ideal $\wp$ of $\T$ to be {\it $p$-old\/} if and only if it
arises by pullback from a maximal ideal of $\T_A$.  
It is clear that $\wp$ is $p$-old
if and only if $A[\wp]$ is non-zero.  Thus, Proposition 1 states,
for $\wp|p$ such that $\rho_\wp$ is irreducible,
that
$\rho_\wp$ is finite at $p$ if and only $\wp$ is $p$-old.

We now prove this Proposition.
Assume, first, that $\rho_\wp$ is finite at $p$.
Theorem~6.1 of [\FERMAT] 
then states that $\rho_\wp$ is ``modular of level~$N$."
(The hypotheses of that theorem are satisfied because the residue
characteristic of $\wp$, which is~$p$, is not congruent to~1  mod~$p$.)
In fact, the
argument given in [\FERMAT] proves the apparently stronger
fact that $\wp$ is
$p$-old.  Namely, under the finiteness assumption, the author constructs
a certain subgroup $\cal V$ of the fiber at~$p$ of the N\'eron model
for $\jo{Np}$. Write ${\cal J}_{\Fp}$ for this fiber.
The discussion on pp.~470--471 of [\FERMAT] shows
that $\cal V$ cannot be a subgroup of the 
maximal torus $T$ of ${\cal J}_{\Fp}$.  Furthermore, the argument given
to prove Lemma~6.3 of [\FERMAT] shows then that $\cal V$ must map
non-trivially to the product $\jo N_{\Fp}\times \jo N_{\Fp}$, which is
the maximal abelian variety quotient of the connected component of the
identity in ${\cal J}_{\Fp}$.  
Since  $\cal V$ is annihilated by $\wp$, it follows from
[\FERMAT], Theorem~3.11 that $\wp$ is $p$-old.

Suppose, conversely, that $\wp$ is $p$-old.  Then $A[\wp]$ is non-zero.
Since $A[\wp]$ isomorphic
to a direct sum of copies of the representation $\rho_\wp$, we may
choose a two-dimensional $\GalQ$-stable submodule $V$ of $A[\wp]$ which
is a model for the representation $\rho_\wp$.  To show that $\rho_\wp$ is
finite, we must produce a finite flat $\T/\wp$-vector space scheme
$\cal V$ over $\Zp$ such that ${\cal V}(\Qpbar)$ and $V$ are isomorphic
representations of $\GalQp$.  (We identify this latter Galois group with
a decomposition group for $p$ in $\GalQ$.)
One has inclusions of $\GalQp$-modules
$V\subseteq A[\wp]\subseteq A[p]$, and the latter group extends to a finite
flat group scheme $\cal S$
of type $(p,\ldots,p)$ over $\Zp$ because $A$ has
good reduction at~$p$.  Hence $V$ extends to a finite flat group scheme
$\cal V$ of this type: the Zariski closure of $V$ in $\cal S$.  Because
$p>2$, the action of $\T/\wp$ on $V$ extends uniquely to $\cal V$ in
view of Raynaud [\RAYN], 3.3.6.  This completes the
proof of Proposition~1.\medskip\noindent
%
%
Consider the subring $R$
of $\T_N$ which is generated by all $T_n$ with $n$
prime to~$p$. For each maximal ideal $\m$ of $R$, we denote by $\rho_\m$ the
semisimple two-dimensional representation of $\GalQ$ over $R/\m$
which is characterized by the property analogous to that given above
for the $\rho_\wp$.

\proclaim Proposition 2.  The index of $R$ in $\T_N$ is finite and
prime to $p$.\par\noindent
To prove the Proposition, we can (and shall) assume that $N\ge11$,
since otherwise $\T_N=0$.  This enables us to apply a result of
N.~Katz 
[\KATZ] concerning the operator $\theta=q{d\over dq}$ on weight-$k$
mod~$p$
modular forms, for which the assumption $N\ge3$ is made.  The result
proved by Katz had been proved by Serre and Swinnerton-Dyer (see 
[\JPS], [\HPF])
in the case $N=1$.

Let $S$ be the space of
weight-2 cusp forms on $\Gamma_o(N)$ with coefficients in $\F_p$.
For each $f\in S$, let $\sum_{n\ge1} a_n(f)q^n \in \F_p[[q]]$ be the
$q$-expansion of $f$.
As is well known (cf.~[\HECKE], \S2), 
an argument due to Shimura ([\SHIM], Chapter 3) shows
that the pairing
$$ \T_N/p\T_N \times S \longrightarrow \F_p,\qquad (T,f)\mapsto a_1(f|T)$$
is perfect.  Let $\cal A$ be the subring of $\T_N/p\T_N$ generated by
the $T_n$ with $n$ prime to~$p$.  Suppose that $f$ is orthogonal to
$\cal A$ in the sense that
$a_1(f|T_n)=0$ for all $n$ prime to
$p$. Since $a_1(f|T_n)=a_n(f)$, the power series $\sum_{n\ge1} a_n(f)q^n$
is then annihilated by $\theta$. By the theorem
of Katz,
the sum $\sum_{n\ge1} a_n(f)q^n$ vanishes.  (The point
is that $\sum_{n\ge1} a_n(f)q^n$, viewed as the mod~$p$ modular form $f$,
otherwise has filtration $w(f)=2$.  Since 2 is prime to~$p$, the filtration
of $\theta f$ is then $2+p+1$.  This contradicts the vanishing of
$\theta f$.)  The subspace of $S$ orthogonal to $\cal A$ is then~0, which
proves that ${\cal A}=\T_N/p\T_N$.  The Proposition now follows immediately,
since $\T_N$ is free of finite rank over $\Z$, and since $\cal A$ is the
image of $R$ in $\T_N/p\T_N$. 
%
\proclaim Corollary.  Let $\wp$ be a maximal ideal of $\T_N$
of residue characteristic $p$.    Let $\m=R\cap\wp$.  Then
the natural inclusion $R/\m\hookrightarrow \T_N/\wp$ is an isomorphism,
and we have
$\jo N[\wp]=\jo N[\m]$.  
\par\noindent
%
The first statement is an immediate consequence of the Proposition,
since the index of $R/\m$ in $\T_N/\wp$ is simultaneously a power
of~$p$ and a divisor of the index $i=(\T_N\colon R)$
of $R$ in $\T_N$.
For the second, we must show that a point $x$ of $\jo N$ which is annihilated
by $\m$ is automatically annihilated by $\wp$.  Let $\lambda$ be an
element of $\wp$.  Then $i\lambda\in\m$.
Hence $x$ in annihilated by $i\lambda$, and it is also annihilated by
$p\lambda$, since   $p\in\m$.  Thus $x$ is annihilated by $\lambda$,
since $(i,p)=1$.
\proclaim Proposition 3.  Let $\m$ be a maximal ideal of $R$
of residue characteristic $p$. 
Suppose that the  associated
representation $\rho_\m$ of $\GalQ$ is irreducible.  Then 
$\jo N[\m]$ has dimension two over $R/\m$.\par\noindent
By a theorem of Cohen-Seidenberg, we may find a maximal ideal
$\wp$ of~$\T_N$ such that $\m=\wp\cap R$.
The Proposition follows from the above Corollary, together with the
fact that $\jo N[\wp]$ has dimension two over $\T_N/\wp$
([\FERMAT], Theorem~5.2b).
\medskip\noindent
%
{\bf Remark:} Proposition~2 becomes false if we allow the case $p=2$.
Indeed, if $N=23$, then $\T_N$ is isomorphic to the ring of integers
of the quadratic field $\Q(\sqrt{5})$. 
Let $R$ be the subring of $\T_N$ which is
generated by the $T_n$ with $n$ odd. Then the index of $R$ 
in $\T_N$
is divisible by~2.  For more details concerning this example, 
see [\DOI], \S4.  (The author wishes
to thank G.~Shimura for pointing out this example to him some years
ago.)
%
\medskip\noindent
%
Recall now that
the subring of $\T_A$ generated by the Hecke operators $T_n$
with $n$ prime to~$p$ may be identified with the ring $R\subseteq \T_N$
which appears above.  Indeed, $\T_N$ is a subring of the ring
of endomorphisms of $\jo N$, which acts diagonally on the product
$\foom=\jo N\times\jo N$.  This gives a natural action of $\T_N$ on
$\foom$.  This action preserves the finite subgroup $\Sigma$ of
$\foom$, and so 
descends to a faithful action of $\T_N$ on
the quotient $A$.  If $n$ is prime to~$p$, the operator
$T_n$ of $\T_N$ induces the operator labeled $T_n$ in $\T_A$.  Hence
the subring $R$ of $\T_N$ generated by such $T_n$ maps isomorphically
onto the indicated subring of $\T_A$.  Note, however, 
that the $p^{\rm th}$
Hecke
operator in $\T_N$ maps to an endomorphism  of $A$ which is not
necessarily an element of $\T_A$. Let $\tau$ denote this element
of $\T_N$.  Then, with our conventions,
the operator $T_p\in\T_A$ is induced by the endomorphism
$(x,y)\mapsto (\tau x + py, -x)$ of $\foom$, which we regard
as the matrix $\pmatrix{\hfill\tau&p\cr-1&0\cr}$ of endomorphisms
of $\jo N$.  

\proclaim Theorem 3.  Let $\wp$ be a maximal ideal of $\T$ with
residue characteristic~$p$.  Assume that $\rho_\wp$ is irreducible,
and that $A[\wp]$ is non-zero.
Then $A[\wp]$ is of dimension two over $\T/\m$.\par\noindent
%
Since $A[\wp]$ is non-zero, the maximal ideal $\wp$ is  $p$-old.
By abuse of notation, we consider that $\wp$ is a maximal ideal of
$\T_A$.  Set $\m=R\cap\wp$.  Considering first the action of $R$
on $\jo N$, we find that $\jo N[\m]$ has dimension two over $R/\m$
(Proposition~3).  The kernel $\Sigma$ of the quotient map
$\foom\to A$ is Eisenstein [\BX], and therefore
prime to $\m$ (cf.~[\FERMAT], Theorem~5.2c).  Therefore, $A[\m]$ may
be identified with $\jo N[\m]^2$, and in particular has dimension four
over $R/\m$.

We have $\T_A/\wp \supseteq R/\m$ and $A[\m]\supseteq A[\wp]$.  Also, the
dimension of $A[\wp]$ over $\T_A/\wp$ is a multiple of~2.  Hence, this
dimension is either 2 or 4, with the latter case occurring if and only
if we have both $A[\wp]=A[\m]$ and $\T_A/\wp=R/\m$.  However, if these
equalities are satisfied, then $T_p\in\T_A$ operates on $\jo N[\m]^2$
as a scalar, i.e., as an element of $R/\m$.  This is impossible, since
the operation of $T_p$ is given by the matrix
$\pmatrix{\hfill\tau&0\cr -1&0\cr}$, which is clearly not a scalar because
of the $-1$ in the lower-left corner.  (The element $\tau$ of $\T_N$
is not in $R$; however $\tau$ preserves $\jo N[\m]$ and acts on $\jo N[\m]$
as a scalar, in view of the corollary to Proposition~2.) The Theorem is
therefore proved.
%
%
\medskip\noindent
{\bf Remark:} The example of level~23 mentioned above shows that Theorem~3
becomes false if the prime $p=2$ is not excluded.  Indeed, suppose that
$p=2$ and that $N=23$.  Let $A$ again be the $p$-old subvariety
of $\jo{pN}=\jo{46}$.  The group $A[2]$ is naturally isomorphic to
$\jo{23}[2] \oplus \jo{23}[2]$, since the kernel of the degeneracy
map $d$ has order $11=\mathop{\rm num}\left({\textstyle 22 \over \textstyle 12}
   \right)$,
which is prime to~2.  
For $n$ odd, the action of
$T_n$ on
$A[2]$ is the diagonal action of $T_n$ coming from the action of 
$\T_{23}/2\T_{23}$ on $\jo{23}[2]$.  The ring $\T_{23}/2\T_{23}$ is the
field with 4~elements, because
2 remains prime in $\Q(\sqrt{5})$.
The $T_n$ with $n$ odd map to elements of the prime field $\F_2$,
since they lie in the order of $\T_{23}$ having index~2 in $\T_{23}$.
The element $T_2$ of $\T_{46}$ induces on $\jo{23}[2] \oplus \jo{23}[2]$
the map $(x,y)\mapsto (\tau x,-x)$, where $\tau$ is the automorphism
of $\jo{23}[2]$ coming from the Hecke operator $T_2$ of~$\T_{23}$.
It follows that all $T_n\in\T_{46}$ act as elements of $\F_2$
on $0 \oplus \jo{23}[2]$, with $T_2$ acting as~0.  If $\wp\subseteq
\T_{46}$ is the kernel of the map $\T_{46}\to\F_2$ representing this
action, then $A[\wp]$ contains $0 \oplus \jo{23}[2]$, and therefore
has dimension~$\ge4$ over $\T_{46}/\wp=\F_2$.
The associated
2-dimensional representation $\rho_\wp$ of $\GalQ$ is irreducible
as follows from [\MAZE], Chapter~II, Proposition~14.1.  This 
representation is an $\F_2$-model for $\jo{23}[2]$, considered
as a representation of $\GalQ$ over~$\F_4$.
%
%
\beginsection 3.  Study of $B[\wp]$.\par
%
In this \S, $N$ is again a positive integer, and $p$ a prime which is
prime to~$N$.  We do not make the assumption $p>2$.
The $p$-old
and $p$-new
subvarieties of $J=\jo{Np}$ are again denoted $A$ and~$B$, respectively.
Their intersection is a finite subgroup $\Delta$ of $J$.  
Consider a maximal ideal $\wp$ of $\T=\T_{Np}$ for which
$\rho_\wp$ is irreducible.  
(We do not assume that the residue characteristic of $\wp$ is~$p$.)
%
\proclaim Theorem 4.  If $B[\wp] \ne 0$, then $B[\wp]=J[\wp]$.
\par\noindent
%
We have an exact sequence
$$ 0 \to \Delta \to A \times B \to J \to 0,$$
where $\Delta$ is embedded ``diagonally" in $A\times B$, and where this
product is mapped to $J$ by $(x,y) \mapsto x-y$.  View $J[\wp]$
as the group ${\cal R}/\Delta$, where
$$ {\cal R} =  \{ \,  (x,y) \in A(\Qbar) \times B(\Qbar) 
  \, | \, (\lambda x,\lambda y) \in\Delta \hbox{ for all } 
     \lambda\in\wp \, \}.$$
Let $\pi\ac{\cal R}\to A$ be the   projection 
$(x,y)\mapsto x$.  We may identify $\ker\pi$ with $B[\wp]$.  Further,
since $\cal R$ contains the diagonal image of $\Delta$, the 
image $\pi({\cal R})$ of
$\pi$ contains $\Delta$, regarded as a subgroup of~$A$.  From this perspective,
it is clear that the theorem amounts to the statement that $\pi({\cal R})$
coincides with $\Delta$ in $A$.

Let $I\subseteq\T$ be the annihilator of $B$ in $\T$, i.e., the kernel
of the natural map $\T\to\End B$.  The hypothesis $B[\wp] \ne 0$ 
is equivalent to the statement
that $\wp$ contains~$I$. For $\lambda\in I$ and $(x,y)\in\cal R$, we
then have $\lambda x=0$, since $(\lambda x,\lambda y)$ is necessarily
the 0-element of $\Delta$.  In other words, we have 
$\Delta\subseteq\image\subseteq A[I].$  To conclude the proof,
we will show that the quotient $A[I]/\Delta$ involves only ``Eisenstein
primes" of $\T$.  This gives the desired equality $\Delta=\image$,
since $\image/\Delta$ is killed by $\wp$ and since $\rho_\wp$ is irreducible.

Consider the element $\gamma\seteq T_p^2-1$ of $\T$.  By (3.7) and~(3.10)
of [\FERMAT], $T_p$ coincides with the negative of the Atkin-Lehner
involution on $B$. Therefore, $\gamma=0$ on~$B$, so that
$\gamma\in I$.  Accordingly, we have $A[I]\subseteq A[\gamma]$.
Let $\tau$ again denote the $p^{\rm th}$ Hecke operator on $\jo N$.
Since the restriction to~$A$ of~$T_p$ is induced by the endomorphism
$\pmatrix{\hfill\tau&p\cr-1&0\cr}$ of $\foom$,
$\gamma$ 
restricts to the endomorphism of~$A$ induced by the product
$\pmatrix{-1&\hfill\tau\cr\hfill 0&-1\cr}\pmatrix{1+p&\tau\cr\tau&1+p\cr}$.
Abusing notation, we will again refer to this product as~$\gamma$.

The main theorem of [\ICM] relates $\Delta$ to
  the kernel $\Omega$
of $\pmatrix{1+p&\tau\cr\tau&1+p\cr}$ on $\foom$.
Firstly, $\Omega$ contains the
kernel $\Sigma$ of the quotient $\foom\to A$.  
Secondly, we have
$\Delta$ = $\tilde\Delta/\Sigma$, where $\tilde\Delta$ is a subgroup
of $\Omega$ such that the quotient
$\Sigma^*\seteq\Omega/\tilde\Delta$ is naturally dual to $\Sigma$.
Both groups $\Sigma$ and $\Sigma^*$ are Eisenstein in the
sense that the relation $T_l= 1+l$ holds on each of them, for all primes
$l$ prime to $pN$.  The group
$\Omega$ is also the kernel of $\gamma$ on $\foom$, since
$\pmatrix{-1&\hfill\tau\cr\hfill 0&-1\cr}$ is visibly an automorphism of
$\foom$.

We claim that $Q\seteq A[\gamma]/\Delta$ 
is an extension of $\Sigma$
by $\Sigma^*$.  To see this, apply the Snake Lemma to the diagram
$$
\matrix{0 &\to& \Sigma\hfill& \to &\foom& \to& A\hfill& 
   \to& 0 \cr
&& \uparrow 0 && \uparrow
  \gamma &&\uparrow \gamma&&\cr
0 &\to& \Sigma\hfill& \to &\foom& \to& A
   \hfill& \to& 0 \cr}$$
to obtain a 4-term exact sequence
$ 0 \to \Sigma \to \Omega \to A[\gamma]\to\Sigma\to 0$.  Equivalently,
this gives a 3-term exact sequence $0\to \Omega/\Sigma\to A[\gamma]\to\Sigma
\to 0$.  Then $Q$ maps to $\Sigma=A[\gamma]/(\Omega/\Sigma)$ with
kernel $(\Omega/\Sigma)/(\tilde\Delta/\Sigma)$.  The latter group is
$\Omega/\tilde\Delta=\Sigma^*$.

The claim implies that the support of $Q$ contains only primes which
divide the ideal generated by all $T_l - (1+l)$.  It follows that $\wp$
is not in the support of $Q$, cf.~[\FERMAT], 5.2c.  {\it A fortiori},
$\wp$ is not in the support of $A[I]/\Delta$.  As noted above,
this proves that $\image=\Delta$, and concludes the proof of the Theorem.
%
%
%
\vskip -\parskip
\beginsection References.\par
\vskip -\parskip
\frenchspacing
\baselineskip 17pt
\catcode`\?=\active
\def?{.\hskip 0.1667em\relax}
\item{\BLR.} N. Boston, H?W. Lenstra, Jr.,  K?A. Ribet,
   {\sl Quotients of group rings arising from two-dimensional
   representations}, CRAS (Paris).  To appear. 
\item{\DOI.} K. Doi, H. Naganuma, {\sl On the jacobian varieties of
   the fields of elliptic functions~II}, J.~Math. Kyoto Univ.
    {\bf 6} (1967), 177--185.
\item{\KATZ.} N?M. Katz, {\sl A result on modular forms
in characteristic~$p$}, Lecture Notes in Math. {\bf 601}
(1977), 53--61.
\item{\MAZE.}
    B. Mazur, {\sl Modular curves and the Eisenstein ideal},
   Publ.   Math. IHES {\bf 47} (1977) 33--186.
\item{\MAZR.}
   B. Mazur, K?A. Ribet, {\sl Two-dimensional representations 
  in the arithmetic of modular curves}, Ast\'erisque.  To appear.
\item{\RAYN.}
  M. Raynaud, {\sl Sch\'emas en groupes de type $(p,...,p)$},  Bull  
  SMF {\bf 102} (1974) 241--280.
\item{\HECKE.}
K?A. Ribet, {\sl  Mod~$p$ Hecke operators and congruences between
 modular forms}, Invent. Math. {\bf 71} (1983), 193--205.
\item{\ICM.}   K?A. Ribet, {\sl  Congruence relations between modular
   forms},  
   Proc. Int. Cong. of Mathematicians (1983) 503--514.
\item{\BX.} K?A. Ribet, {\sl
On the Component Groups and the Shimura Subgroup
   of $J_o(N)$},  S\'em.\ Th.\ Nombres, Universit\'e Bordeaux
   (1987--88) expos\'e 6.
\item{\FERMAT.}
   K?A. Ribet, {\sl On modular representations of
  $\GalQ$ arising from modular forms},  Invent. Math. {\bf 100} (1990) 431--476.
\item{\JPS}
  J-P. Serre, {\sl  Congruences et formes modulaires [d'apr\`es 
   H?P?F. Swinnerton-Dyer]}, S\'em. Bourbaki $\rm n^o~416$ (1971/72),
    Lecture Notes in Math. {\bf 317} (1973), 319--338.
\item{\DUKE.}
  J-P. Serre, {\sl  Sur les repr\'esentations modulaires de degr\'e  
   2 de Gal$(\Qbar /\Q )$},  Duke Math. J. {\bf 54} (1987) 179--230.
\item{\SHIM.}
     G. Shimura, 
   {\sl Introduction to the arithmetic theory of automorphic functions},
   Princeton University Press, Princeton 1971.
\item{\HPF.} H?P?F. Swinnerton-Dyer,
  {\sl On $l$-adic representations and congruences for modular forms},
  Lecture Notes in Math. {\bf 350} (1973), 1--55.
\bye
