ef{cr})) the formula \begin{equation} \label{equation2} U_p \circ \alpha = \alpha \circ \pmatrix{T_p & p\cr -1 & 0\cr }, \end{equation} in which the matrix refers to the natural left action of ${\rm M}(2,\T_N)$ on the product $\JtJ$. Concerning the behavior of $\Sh$ and $\Sigma$ under Hecke operators, the following (easy) result is noted briefly in \cite{ICM} and proved in detail in \cite{Bordeaux}. \begin{prop}\label{propICM} The Shimura subgroup $\Sh$ of $J_0(N)$ is annihilated by the endomorphisms \[ \eta _r = T_r - (r+1) \] of $J_0(N)$ for all primes $r \not| N$. \end{prop} \begin{cor} The subgroup $\Sigma$ of $\JtJ$ lies in the kernel of the endomorphism $\pmatrix{1+p & T_p \cr T_p & 1+p \cr}$ of $\JtJ$. It is annihilated by the operators $T_r - (r+1)$ for all prime numbers $r$ not dividing $pN$. \end{cor} The significance of the endomorphism introduced in the corollary appears when we note the formula $\beta\circ\alpha = \pmatrix{1+p & T_p \cr T_p & 1+p \cr}$, in which $\beta : J_0(Np) \to \JtJ$ is the map induced by the two degeneracy maps $X_0(Np) \doublemap X_0(N)$ and Albanese functoriality of the Jacobian. (The map $\beta$ becomes the dual of $\alpha$ when we use ``autoduality of the Jacobian'' to identify the Jacobians with their own duals.) The formula results from the fact that the two degeneracy maps are each of degree $p+1$, and from the usual definition of $T_p$ as a correspondence in terms of degeneracy maps. Let $\Delta \subset \JtJ$ be the kernel of $\pmatrix{1+p & T_p \cr T_p & 1+p \cr}$. Then $\Delta$ is a finite subgroup of $\JtJ$. Indeed, $\Delta$ differs only by 2-torsion from the direct sums of the kernels of $T_p \pm (p+1)$ on $J_0(N)$. These latter kernels are finite because neither number $\pm(p+1)$ can be an eigenvalue of $T_p$ on $S(N)$, in view of Weil's Riemann hypothesis, which bounds $T_p$'s eigenvalues by $2\sqrt p$. Further, the group $\Delta$ comes equipped with a perfect $\Gm$-valued skew-symmetric pairing, in view of its interpretation as the kernel $K(L)$ of a polarization map \[\phi _L: \JtJ \to (\JtJ)\dual.\] (One takes $L$ to be the pullback by $\alpha$ of the ``theta divisor'' on the Jacobian $J_0(pN)$.) The subgroup $\Sigma$ of $\Delta$ is self-orthogonal under the pairing on $\Delta$. In other words, if we let $\Sigma^\perp$ be the annihilator of $\Sigma$ in the pairing, we have a chain of groups \[ \Delta \supset \Sigma^\perp \supset \Sigma .\] Note also that $\Delta / \Sigma$ is naturally a subgroup of the abelian variety $A$, since $A$ and $\Sigma$ are the image and kernel of $\alpha$, respectively. Thus the subquotient $\Sigma^\perp / \Sigma$ of $\Delta$ is in particular a subgroup of $A$. On the other hand, the quotient $\Delta/\Sigma^\perp$ is canonically the Cartier (i.e., $\Gm$) dual $\Sigma^*$ of $\Sigma$. It is naturally a subgroup of $A\dual$. Indeed, $\Sigma$ is the kernel of the isogeny $\JtJ\to A$ induced by $\alpha$. The kernel of the dual homomorphism $A\dual\to(\JtJ)\dual$ may be identified with $\Sigma^*$. To state the final result that we need, we introduce the $p$-new abelian subvariety $B$ of $J_0(pN)$. To define it, consider the map \[ J_0(pN)\dual \to A\dual \] which is dual to the inclusion $A\hookrightarrow J_0(pN)$. Its kernel is an abelian subvariety $Z$ of $J_0(pN)\dual$. Using the autoduality of $J_0(pN)$ to transport $Z$ back to $J_0(pN)$, we obtain $B$. This subvariety of $J_0(pN)$ is a complement to $A$ in the sense that $J_0(pN) = A + B$ and $A\cap B$ is finite. It is $p$-new in that $\T_{pN}$ stabilizes $B$ and acts on $B$ through its $p$-new quotient $\Tbar _{pN}$ (which acts faithfully on $B$). The following main result of \cite{ICM} is a formal consequence of Proposition~\ref{propICM}: \begin{theorem}\label{theoremICM} The finite groups $A\cap B$ and $\Sigma^\perp /\Sigma$ are equal. \end{theorem} In the notation of \cite{ICM}, $A\cap B$ is the group $\Omega$, which can be described directly in terms of $\Delta$ and the kernel of $\alpha$ (\cite{ICM}, pp.~508--509). Once this kernel is identified, the description of Theorem~\ref{theoremICM} is immediate. \section{Proof of Theorem~\protect\ref{theorem1}} We assume from now on that $\rho$ is modular of level $N$, and choose an ideal $\m$ of $\T_N$, plus an embedding $\omega\colon\T_N/\m \hookrightarrow \F$ as in the definition of ``modular of level $N$.'' Assuming that one of the two congruences (\ref{equation1}) is satisfied, we will construct \begin{enumerate} \item A maximal ideal $\cal M$ of $\Tbar _{pN}$, and \item An isomorphism $\T_N/\m \approx \Tbar _{pN} /{\cal M}$ which takes $T_r$ to $T_r$ for all primes $r \ne p$. \end{enumerate} This is enough to prove the theorem, since the representations $\rho _\m$ and $\rho_{\cal M}$ will necessarily be isomorphic, in view of the $T_r$-compatible isomorphism between the residue fields of $\m$ and $\cal M$. Our procedure is to construct $\cal M$ first as a maximal ideal of $\T_{Np}$ and then to verify that $\cal M$ in fact arises by pullback from a maximal ideal of $\Tbar _{pN}$. \par\bigskip It might be worth pointing out explicitly that our construction of $\cal M$ depends on the sign $\pm$ in (\ref{equation1}). If $p \not\equiv -1$ (mod $\ell$), then there is a unique sign $\pm$ which makes (\ref{equation1}) true, under the hypothesis of the theorem, and our construction proceeds in a mechanical way. In case $p \equiv -1$ (mod $\ell$), both congruences (\ref{equation1}) are satisfied under the hypothesis of the theorem, and the construction requires us to decide whether (\ref{equation1}) should read $0\equiv +0$ or $0 \equiv -0$. The two choices of sign lead to different ideals $\cal M$, at least when $\ell$ is odd, since our construction shows that $U_p \equiv \pm 1$ (mod $\cal M$), with the same sign $\pm$ as in (\ref{equation1}).\par\bigskip Before beginning the construction, we introduce the following abbreviations: \[ R = \T_N, \qquad k = \T_N/\m, \qquad \T = \T_{pN}, \qquad \Tbar = \Tbar _{pN}.\] Also, let \[ V = J_0(N)[\m] \] be the kernel of $\m$ on $J_0(N)$, i.e., the intersection of the kernels on $J_0(N)$ of the various elements of $\m$. This group is a finite $k$-vector space which is easily seen to be non-zero (cf.~\cite{MazurE}, or \cite{Fermat}, Theorem~5.2). The group $V\times V$ is then a finite subgroup of $\JtJ$. This subgroup has zero intersection with $\Sh \times \Sh$, in view of the irreducibility of $\rho_m$, Proposition~\ref{propICM} above, and \cite{Fermat}, Theorem~5.2(c). In particular, $\alpha$ maps $V\times V$ isomorphically into $A$. Therefore, we can (and will) regard $V\times V$ as a subgroup of that abelian variety. We now assume that one of the two congruences (\ref{equation1}) is satisfied. To fix ideas we will treat only the case \[ \tr \rho(\Frob_p) \equiv - (p+1) \pmod{\ell}.\] Using the isomorphism between $\rho$ and $\rho_\m\otimes_\omega\F$, we restate this congruence in the form \begin{equation}\label{equation3} T_p \equiv -(p+1) \pmod{\m}. \end{equation} (The left-hand side of (\ref{equation3}) is the trace of $\rho_\m(\Frob_p)$.) We embed $V$ in $V\times V$ via the {\em diagonal\/} embedding; the antidiagonal embedding would be used instead if $T_p$ were $p+1$ modulo $\m$. We have \[ V \hookrightarrow V \times V \hookrightarrow A.\] \begin{lemma}\label{lemma1} The subgroup $V$ of $A$ is stable under $\T$. The action of $\T$ on $V$ is summarized by a homomorphism $\gamma:\T \to k$ which takes $T_n$ to $T_n$ modulo $\m$ for $(n,p)=1$ and takes $U_p$ to $-1$. \end{lemma} \begin{proof} That $T_n \in \T$ acts on $V$ in the indicated way, for $n$ prime to $p$, follows from the equivariance of $\alpha$ with respect to such $T_n$. The statement relative to $U_p$ then follows from (\ref{equation2}) and (\ref{equation3}). \end{proof} Define ${\cal M} = \ker \gamma$, so that we have an inclusion $\T/{\cal M} \hookrightarrow k = R/\m$. This map is in fact an {\em isomorphism\/} since $k$ is generated by the images of the $T_n$ with $n$ {\em prime to\/} $p$. Indeed, $T_p$ lies in the prime field $\F_\ell$ of $k$ because of (\ref{equation3}). To conclude our proof of Theorem~\ref{theorem1}, we must show that the maximal ideal $\cal M$ of $\T$ arises by pullback from $\Tbar$. For this, it suffices to show that $\T$ acts on $V$ through its quotient $\Tbar$. This fact follows from \begin{lemma} The subgroup $V$ of $A$ lies in the intersection $A\cap B$. \end{lemma} \begin{proof} We first note that $V$, considered diagonally as a subgroup of $\JtJ$, lies in the group $\Delta$. Indeed, $V \subset J_0(N)$ is killed by $T_p + p + 1$ by virtue of (\ref{equation3}). The isomorphic image of $V$ in $J_0(pN)$ therefore lies in $\Delta /\Sigma$. To prove the lemma, we must show that this image lies in the subgroup $A\cap B = \Sigma^\perp /\Sigma$ of $\Delta /\Sigma$. In other words, we must show that the image of $V$ in $\Delta/\Sigma^\perp$ is 0. A somewhat painless way to see this is to view the varieties $J_0(N)$, $J_0(Np)$, $A$, \ldots\ as being defined over $\Q$. The group $\Delta/\Sigma^\perp$ is canonically the $\Gm$-dual of $\Sigma$, which may identified $\GalQ$-equivariantly with the Shimura subgroup $\Sh$ of $J_0(N)$. This latter group is in turn the $\Gm$-dual of the covering group $\cal G$ introduced above. It follows that the action of $\GalQ$ on $\Delta/\Sigma^\perp$ is trivial. (We note in passing that the action of $\GalQ$ on $\Sh$ is given by the cyclotomic character $\GalQ \to {\hat\Z}^*$.) Hence if $V$ maps non-trivially to $\Delta/\Sigma^\perp$, the semisimplification of $V$ (as a ${\bf F}_\ell [\GalQ]$-module) contains the trivial representation. This semisimplification may be constructed by the following recipe: find the semisimplification $W$ on $V$ as a $k[\GalQ]$-module, and consider $W$ as an ${\bf F}_\ell$-module. (A simple representation over $k$ remains semisimple after ``restriction of scalars'' from $k$ to ${\bf F}_\ell$.) Hence $W$ contains $\GalQ$-invariant vectors, if $V$ maps non-trivially to $\Delta/\Sigma^\perp$. This conclusion is absurd, since $W$ is the direct sum of a number of copies of the $k$-simple 2-dimensional representation $\rho_\m$ (\cite{MazurE}, Chapter~II, Proposition~14.2). \end{proof} \vfill\eject \begin{thebibliography}{66} \bibitem{A-L} Atkin, A.O.L. and Lehner, J. Hecke operators on $\Gamma_{0}(m)$. Math. Ann. {\bf 185}, 134--160 (1970) \bibitem{car} Carayol, H. Sur les repr\'esentations Galoisiennes modulo $\ell$ attach\'ees aux formes modulaires. Preprint \bibitem{D-S} Deligne, P.\ and Serre, J-P. Formes modulaires de poids 1. Ann.\ Sci.\ Ec.\ Norm.\ Sup.\ {\bf 7\/}, 507--530 (1974) \bibitem{Diamond} Diamond, F. I. Congruence primes for cusp forms of weight $k \ge 2$. To appear \bibitem{MazurE} Mazur, B. Modular curves and the Eisenstein ideal. Publ. Math. IHES {\bf 47}, 33--186 (1977) \bibitem{ICM} Ribet, K. Congruence relations between modular forms. Proc. International Congress of Mathematicians 1983, 503--514 \bibitem{Fermat} Ribet, K\@. On modular representations of Gal$(\overline{\Q}/\Q )$ arising from modular forms. Preprint \bibitem{Bordeaux} Ribet, K\@. On the component groups and the Shimura subgroup of $J_0(N)$. S\'eminaire de Th\'eorie des Nombres, Universit\'e de Bordeaux, 1987/88 \end{thebibliography} \bigskip\bigskip \def\baselinestretch{1.0} {\small \begin{verse}K. A. Ribet\\ Mathematics Department\\University of California\\Berkeley CA 94720\\U.S.A. \end{verse}\par} \end{document}