\documentstyle[12pt,oldlfont]{article}
\vfuzz 10pt
%\nofiles
\font\datefont=cmdunh10
\setcounter{secnumdepth}{1}
\makeatletter
     \def\@begintheorem#1#2{\sl \trivlist \item[\hskip \labelsep{\bf #1\ #2}]}
   \def\@opargbegintheorem#1#2#3{\sl \trivlist
            \item[\hskip \labelsep{\bf #1\ #2\ (#3)}]}


\def\maketitle{\par
 \begingroup
 \def\thefootnote{\fnsymbol{footnote}}
 \def\@makefnmark{\hbox
 to 0pt{$^{\@thefnmark}$\hss}}
 \if@twocolumn
 \twocolumn[\@maketitle]
 \else \newpage
 \global\@topnum\z@ \@maketitle \fi\thispagestyle{plain}\@thanks
 \endgroup
 \setcounter{footnote}{0}
 \let\maketitle\relax
 \let\@maketitle\relax
 \gdef\@thanks{}\gdef\@author{}\gdef\@title{}\let\thanks\relax}
\def\@maketitle{\newpage
 \null
 \vskip 2em \begin{center}
 {\large \@title \par} \vskip 1.5em {\large \lineskip .5em
\begin{tabular}[t]{c}\@author
 \end{tabular}\par}
 \vskip 1em {\large \@date} \end{center}
 \par
 \vskip 1.5em}



\makeatother

\def\doublemap{\mathrel{\null _{\raise.2ex\hbox{$\textstyle\rightarrow$}}
  ^{\lower.2ex\hbox{$\textstyle\rightarrow$}}}}
\newcommand{\End}{\mathop{\rm End}\nolimits} 


\newcommand{\V}{{\cal V}}

\newcommand{\J}{{\cal J}}
\newcommand{\Hom}{\mathop{\rm Hom}\nolimits}
\newcommand{\Aut}{\mathop{\rm Aut}\nolimits}
\newcommand{\card}{\mathop{\rm card}\nolimits}
\newcommand{\finite}{^{\sf f}}
\newcommand{\toric}{^{\sf t}}
\newcommand{\Pico}{{\rm Pic}^o}
\newcommand{\trace}{\mathop{\rm trace}\nolimits}
\newcommand{\Pic}{\mbox{\rm Pic}}
\newcommand{\Frob}{{\sf Frob}}
\newcommand{\Alb}{\mbox{\rm Alb}}
\newcommand{\Zp}{{\bf Z}_p}
\newcommand{\Gal}{{\rm Gal}}
\newcommand{\GalQ}{{\rm Gal}(\Qbar/\Q)}
\newcommand{\GalQp}{{\rm Gal}(\Qpbar/\Qp)}
\newcommand{\Fell}{{\bf F}_\ell}
\newcommand{\Zq}{{{\bf Z}_q}}
\newcommand{\Qq}{{{\bf Q}_q}}
\newcommand{\Mtwoql}{{\rm M}(2,{\bf Q}_\ell)}
\newcommand{\Mtwozl}{{\rm M}(2,{\bf Z}_\ell)}
\newcommand{\GLtwoql}{{\bf GL}(2,{\bf Q}_\ell)}
\newcommand{\SLtwoZ}{{\bf SL}(2,\Z)}
\newcommand{\Fq}{{{\bf F}_q}}
\newcommand{\Fqbar}{{{\overline{\bf F}}_q}}
\newcommand{\Fbar}{{{\overline{\bf F}}}}
\newcommand{\Fpbar}{{{\overline{\bf F}}_p}}
\newcommand{\Qpbar}{{{\overline{\bf Q}}_p}}
\newcommand{\Qqbar}{{{\overline{\bf Q}}_q}}
\newcommand{\Tbar}{{{\overline{\bf T}}}}
\newcommand{\TNbar}{\Tbar_N}
\newcommand{\Tunderline}{{\underline{T}}}
\newcommand{\Z}{{\bf Z}}
\newcommand{\C}{{\bf C}}
\newcommand{\Q}{{\bf Q}}
\newcommand{\Qbar}{\overline{{\bf Q}}}
\newcommand{\GL}{{\bf GL}}  
\newcommand{\SL}{{\bf SL}}  
\newcommand{\Qp}{{{\bf Q}_p}}
\newcommand{\Fp}{{\bf F}_p}  
\newcommand{\Gm}{{\bf G}_{\rm m}}
\newcommand{\F}{{\bf F}}
\newcommand{\JtJ}{J_{o}(M)  \times  J_{o}(M) }
\newcommand{\T}{{\bf T}}
\newcommand{\m}{{\sf m}}

\newtheorem{theorem}{{\sc Theorem}}[section]
\newtheorem{lemma}[theorem]{{\sc Lemma}}
\newtheorem{remark}[theorem]{Remark}  %Always used with {\rm ...}
\newtheorem{remarks}[theorem]{Remarks} %like \remark
\newtheorem{prop}[theorem]{{\sc Proposition}}
\newtheorem{variant}[theorem]{{\sc Variant}}
\newtheorem{cor}{{\sc Corollary}}\let\thecor\relax  

\newenvironment{proof}{
\medskip
\noindent {\em Proof\/}. }{\nolinebreak[2]\rule{.45em}{.9em}\medskip}

\title{From the Taniyama-Shimura Conjecture to Fermat's Last Theorem%
\thanks{To appear in the {\sl Annales de la Facult\'e des Sciences
         de Toulouse}.}}
\author{Kenneth A. Ribet\thanks{Supported by the N.S.F.}}

\date{\null}  

\begin{document} 
\maketitle
\section{Introduction}

In this article I outline a proof 
of the theorem (proved in \cite{Fermat}):
  \[
\hbox{Conjecture of Taniyama-Shimura }
      \Longrightarrow \hbox{ Fermat's Last Theorem.}
  \] 
My aim is to summarize the main ideas
of \cite{Fermat} for a relatively wide audience and to communicate
the structure of the proof to non-specialists.
The discussion is inevitably technical at points, however, since
a large amount of machinery from arithmetical algebraic geometry is
required. The reader interested in a genuinely
non-technical overview may prefer to begin with
Mazur's delightful introduction
\cite{Gadfly} to the Taniyama-Shimura conjecture, and to relations with
Fermat's Last Theorem and similar problems.  Another excellent
alternative source is the
 Bourbaki seminar of
Oesterl\'e \cite{Oesterle}.
This seminar 
 discusses the relation between elliptic
curves and Fermat's Last Theorem from several points of view, but
gives fewer details about the argument of \cite{Fermat} than
the present summary.  See also \cite{FreyUlm}.



The proof sketched here differs from that of \cite{Fermat} in
two ways.  First of all, we exploit a suggestion of
B.~Edixhoven which allows us to prove the theorem without
introducing an auxiliary prime   in the proof.
Such a prime is necessary to prove the general result of
\cite{Fermat}, but turns out to be superfluous in the case of Galois
representations which are ``semistable,'' but not finite, at some prime.  
(In the 
application to Fermat, the relevant representations are
semistable and non-finite at
the prime~2.)
Secondly, we give a uniform argument
to lower the level, avoiding  a preliminary use of Mazur's Theorem
(\cite{Fermat}, \S6), but appealing to the result of
\cite{Mazur-Ribet} instead.




This article is the written version of a June, 1989
lecture given at the
Universit\'e Paul Sabatier (Toulouse) in connection 
with the {\em Prix Fermat}.   I wish to thank Matra Espace,
the sponsor of the prize, as well as the
mathematical
community of Toulouse, for their warm welcome.  
Special thanks are due Professor J.B.~Hiriart-Urruty, who 
devised the prize,
arranged its  sponsorship, and organized its administration.
The author also thanks B.~Edixhoven, S.~Ling and R.~Taylor 
for helpful conversations,
and the I.H.E.S. for hospitality during the preparation of this
article.

\section{Elliptic curves over $\Q$}\label{sectionone}
An elliptic curve over the rational field $\Q$ is a curve of genus~1
over $\Q$ which is furnished with a distinguished rational point $O$.
(For background on elliptic curves, the reader is invited to consult
\cite{Husemoller}, \cite{Silverman}, or \cite{MG}, Chapter~IV.)
The curves which will interest us are  
those arising from an affine equation
\begin{equation}\label{Freycurve}
y^2 =  x(x-A)(x+B),
\end{equation}
where $A$ and $B$ are non-zero relatively prime integers with $A+B$
non-zero.  The curve $E$ given by this affine equation is understood
to be the curve in the projective plane ${\bf P}^2$ given by the
homogenized form of (\ref{Freycurve}), namely
\[ Y^2Z = X^3 - (A-B)X^2Z - ABXZ^2;\]
the point $O$ is then the unique ``point at infinity,'' which has
homogeneous coordinates $(0,1,0)$

It is frequently essential to view $E$ as a commutative algebraic group
over $\Q$.  In particular, given two points $P$ and $Q$ on $E$ with
coordinates in a field $K \supseteq \Q$ (we write $P,Q \in E(K)$),
a sum $P+Q$ is defined, and is rational over $K$.  The coordinates of
$P+Q$ are rational functions in the coordinates of $P$ and $Q$, with
coefficients in $\Q$; these coefficients depend on the defining equation
of $E$.  The identity element of this group is the distinguished
point $O$.  Further, three distinct points sum to $O$ in the group
if and only if they are collinear.

According to the Weierstrass theory (\cite{Husemoller}, Ch.~9),
we may describe $E$ over $\C$ as
the quotient $\C/L$, where $L$ is a suitable period lattice for $E$.
To do this, we fix a non-zero holomorphic differential $\omega$ of
$E$ over $\C$ and construct $L$ as
\[ \left\{ \, \int_\gamma \omega \,  \Bigm| \, \gamma\in H_1\left(
    E(\C),\Z\right) \, \right \}. \]
 From this description, we see easily for each $n\ge1$ that the group
\[ E[n]  =  \{ \, P\in E(\C) \, |\, n\cdot P =O \, \} \]
is a free $\Z/n \Z$-module of rank~2, and in particular has order $n^2$.
Indeed, this follows from the isomorphism $E[n]\approx {1\over n}L/L$
and the fact that $L$ is free of rank~2 over $\Z$.
The points of $E[n]$ are those whose
coordinates   satisfy certain algebraic equations with rational coefficients,
depending again on the defining equation of $E$.  In
particular, the finite group
$E[n]$ consists of points of $E(\Qbar)$ and is stable under conjugation
by elements of $\GalQ$.  Thus $E[n]$ may be viewed as a free rank-2
$\Z/n\Z$-module which is furnished with an action of $\GalQ$.
This action amounts to a homomorphism
\[ \rho_n : \GalQ \to \Aut(E[n]), \]
where $\Aut(E[n])$ is the group of automorphisms of $E[n]$ as a
$\Z/n\Z$-module, so that $\Aut(E[n])\approx \GL(2,\Z/n\Z)$.
Notice that $\GalQ$ acts on $E[n]$ through group automorphisms
because conjugation is compatible with the addition law on $E$.
To see this, we note again that the addition law is given by
formulas with coefficients in $\Q$.

The kernel of $\rho_n$ is an open subgroup $H_n$ of $\GalQ$.
Let $K_n=\Qbar^{H_n}$.  Then $K_n$ is concretely the finite extension
of $\Q$ obtained by adjoining to $\Q$ the coordinates of all points
in $E[n]$.  This field is a Galois extension of $\Q$, whose Galois
group $G_n={\rm Gal}(K_n/\Q)$ is isomorphic to the image of $\rho_n$,
and especially is a subgroup of $\Aut(E[n])$.  A great deal is known
about the family of $G_n$ as $E/K$ remains fixed and $n$ varies.
For example, J-P.~Serre proved in \cite{SerreIM} that the index of
$G_n$ in $\Aut(E[n])$ remains bounded as $n\to\infty$, provided that $E$
is not an elliptic curve with ``complex multiplication.''  

The curves excluded by this theorem of Serre, i.e., those with
complex multiplication, form an extremely interesting class of
elliptic curves, for which a variety of  problems can be treated.
For example, Shimura \cite{Shimurapink} proved   the Taniyama-Shimura
Conjecture (described below) for such elliptic curves.  However,
this class is essentially disjoint from the class of  curves that
we are going to consider.

Indeed, suppose that $E$ is given by (\ref{Freycurve}), where the
integers $A$ and $B$ satisfy the mild conditions $32|B$, $4|(A+1)$.
Then as Serre has indicated (\cite{Duke}, \S4.1), $E$ is a {\em
semistable\/} elliptic curve, having bad reduction precisely at the
primes dividing $ABC$, where $C=-(A+B)$.  By a well known theorem
of Serre-Tate \cite{Serre-Tate}, $E$ does not have complex multiplication.

The concepts   ``semistability'' and ``bad reduction'' can be explained 
in a rough way
as follows (cf., e.g., \cite{Husemoller}, Ch.~5).
Given an equation for $E$ with integer
coefficients, plus a prime number $p$, we reduce the equation mod $p$,
thus obtaining an equation over the finite field $\Fp$.  If this equation
defines an elliptic curve $\tilde E$
over $\Fp$ (as opposed to a singular cubic curve),
 then $E$ has {\em good reduction\/} at $p$,
and the elliptic curve $\tilde E$ over $\Fp$ is the reduction of $E$ mod
$p$.  For example,  we get
an elliptic curve mod $p$ from the equation
(\ref{Freycurve}) if and only if the three numbers $A$, $B$, $A+B$ remain
non-zero mod $p$.  An elliptic curve $E$ has {\em bad reduction\/} mod $p$
if no equation for $E$ can be found which reduces to an elliptic curve
over $\Fp$.  There is, in fact,  a ``best possible equation'' for $E$: its
minimal Weierstrass model (\cite{Husemoller}, Ch.~5, \S2).  
The curve $E$ has good reduction at~$p$
if and only if this model defines an elliptic curve mod~$p$.
In the case of bad reduction, the minimal Weierstrass model is singular
mod $p$; one says that the bad reduction is {\em multiplicative\/} if
the only singularity of the minimal Weierstrass model mod~$p$ is an
ordinary double point.  Finally, $E$ is said to be semistable if, for
every prime number~$p$, $E$ has either good or multiplicative reduction
at~$p$.

The {\em conductor\/} of an elliptic curve $E$ over $\Q$ is a positive
integer $N$ which is divisible precisely by the primes $p$ at which
$E$ has bad reduction (\cite{Husemoller}, Ch.~14).  
To say that $E$ is semistable is equivalent to
say that $N$ is square free.  The conductor $N$ is then the product 
$\prod\limits_{p\in\cal B} p^1$, where $\cal B$ is the set of primes of
bad reduction.  For instance, assume again that $E$ is given by
(\ref{Freycurve}) and that the conditions  
\begin{equation}\label{conditions}
 32|B ,\quad  4|(A+1)  \end{equation}
are satisfied.  Then 
  \begin{equation}\label{NFrey}
         N  =  \prod_{p|(ABC)} p   .
  \end{equation}
Serre has suggested that $N$ be termed the ``radical'' of $ABC$ in this
case.

A second integer associated with a curve $E$ over $\Q$ is its
{\em minimal discriminant}, a non-zero integer $\Delta=\Delta_E$
which is roughly the discriminant of the minimal Weierstrass equation
for $E$.  It is given in terms of the coefficients of this equation
by a well known formula (see for example
\cite{Husemoller}, p.~68).
If $E$ is given by  (\ref{Freycurve}) and the conditions
(\ref{conditions}) are
satisfied, then we have
\begin{equation}\label{DeltaFrey}
    \Delta = {1\over 2^8}\cdot (ABC)^2.
\end{equation}

Next, let $E$ be an elliptic curve over $\Q$, and let $p$ be a prime of
good reduction for $E$.  Write again $\tilde E$ for the reduction of
$E$ mod $p$, so that $\tilde E$ is an elliptic curve over $\Fp$.
Let $\tilde E(\Fp)$ denote the group of rational points of $\tilde E$,
i.e., the group of points on $\tilde E$ with coordinates in the base
field $\Fp$.  We define $a_p = a_p(E)$ to be the difference
  \[ p+1 \, - \, \#\tilde E(\Fp). \]
To motivate consideration of this integer, we  recall that $p+1$
represents the number of points on the projective line ${\bf P}^1$ over
$\Fp$.

\section{$\ell$-division points}
In this \S, we suppose that $E$ is the elliptic curve given by
an equation (\ref{Freycurve}), where
$A$ and $B$ are as usual non-zero relatively prime integers 
such that $A+B$ is
non-zero.  We suppose in addition that the divisibilities
(\ref{conditions}) hold.  We fix a prime number $\ell\ge5$, and let
$\rho=\rho_\ell$ be the representation of $\GalQ$ giving the action
of $\GalQ$ on the group $E[\ell]$ of $\ell$-division points of
$E$.  We let $K=K_\ell$ be the Galois extension of $\Q$ gotten by
adjoining to $\Q$ the (coordinates of the) points in $E[\ell]$.

\begin{prop}[Mazur]\label{irred}
The representation $\rho$ is an irreducible two-di\-men\-sional
representation of $\GalQ$.
\end{prop}

This proposition is proved as Proposition~6 in \cite{Duke}, \S4.1.
The proof given by Serre in \cite{Duke} is based on results of
Mazur \cite{MazurE}, and relies on the fact that the 2-division
points of $E$ are rational over $\Q$ (i.e., $K_2=\Q$).  This latter
fact is evident from the 
 fact that the right-hand side of (\ref{Freycurve})
factors over $\Q$.

The global property of $\rho$ furnished by Proposition~\ref{irred}
is complemented by a second, much more elementary, global property.
Let $\chi=\chi_\ell$ be the mod~$\ell$ cyclotomic character
$\GalQ\to{\bf F}^*_\ell$ giving the action of $\GalQ$ on the group
$\mu_\ell$
of $\ell^{\rm th}$ roots of unity in $\Qbar$.  Then we have
\begin{prop} \label{Weilpairing}
The determinant of $\rho$ is the character $\chi$.
\end{prop}

The proposition follows from the isomorphism $\mathop{\stackrel{2}{\Lambda}}
E[\ell]
\mathrel{\stackrel{\sim}{\to}}\mu_\ell$ given by the $e_\ell$-pairing
of Weil.

We now discuss some local properties of $\rho$.  Let $p$ be a prime
number.  Choose a place $v$ of $\Qbar$ lying over $p$, and let
$D_v\subset\GalQ$ be the corresponding decomposition group.  This
group is isomorphic to the Galois group $\GalQp$, where $\Qpbar$ is the
algebraic closure of $\Qp$ in the completion of $\Qbar$ at $v$.
Let $I_v$
be the inertia subgroup of $D_v$.  The quotient $D_v/I_v$ may be identified
canonically with
the Galois group of the residue field of $v$, which is
an algebraic closure $\Fpbar$ of $\Fp$.  Hence $D_v/I_v$ is isomorphic
to the pro-cyclic group $\hat\Z$, with the chosen ``generator'' of the 
group being the Frobenius substitution $x\mapsto x^p$.  Lifting this
generator back to $D_v$, we obtain an element $\Frob_v$ of $D_v$ which
is well defined modulo $I_v$.  Any other place $v'$ of $\Qbar$ over $p$
is of the form $g\cdot v$ for some $g\in\GalQ$.  Replacing $v$ by $v'$
has the effect of conjugating the triple $(D_v,I_v,\Frob_v)$ by $g$.

We say that $\rho$ is {\em unramified\/} at $p$ if $\rho(I_v)=\{1\}$.
This condition is visibly independent of the choice of $v$ as a place
lying over $p$.  It amounts to the statement that the extension $K/\Q$
is unramified at the prime $p$, or equivalently to the condition that
$p$ be prime to the discriminant of $K$.  If $\rho$ is unramified at
$p$, then $\rho(\Frob_v)$ is an   element of the image
of $\rho$ which is well defined, i.e., independent of the choice of
$\Frob_v$ as a lifting of the Frobenius substitution.  Further, the
{\em conjugacy class\/} of $\rho(\Frob_v)$ in the image depends only
on $p$.  Note that the representation $\rho$ is ramified
(i.e., not unramified) at the prime $p=\ell$.  This follows from
Proposition~\ref{Weilpairing}, since $\chi$ is ramified at $\ell$.
Equivalently, Proposition~\ref{Weilpairing} guarantees that $K$
contains the cyclotomic field $\Q(\mu_\ell)$, and this field is
already
ramified at~$\ell$.

The following result is standard (cf.~\cite{MG}, Ch.~4, \S1.3).

\begin{prop}\label{merguez}
Suppose that $p$ is prime to $\ell N$, where $N$ is the conductor
of $E$.  Then $\rho$ is unramified at $p$.  Moreover, for each
$v$ dividing $p$, we have the congruence
\begin{equation}\label{tracerho}
    \trace\Bigl(\rho(\Frob_v)\Bigr) \equiv a_p \pmod\ell.
\end{equation}
\end{prop}


Let $\Delta=\Delta_E$ again be the minimal discriminant of $E$.
For each prime number $p$, let $v_p(\Delta)$ be the {\em valuation\/}
of $\Delta$ at $p$, i.e., the exponent of the highest power of
$p$ which divides $\Delta$.

\begin{prop}\label{beatsme}
Suppose that $p\ne\ell$ and that $p$ divides $N$.  Then $\rho$ is
unramified at $p$ if and only if $v_p(\Delta)\equiv 0\hbox{\rm\ mod }\ell$.
\end{prop}

This result, noted by Frey  \cite{Frey} and
Serre \cite{Duke}, \S4, is an easy consequence
of the theory of the Tate curve, as exposed in \cite{MG}.  In concrete
terms, one checks the ramification of $\rho$ at $p$ by viewing $E$
over the maximal unramified extension ${\bf Q}_p^{\rm nr}$ of $\Qp$
in $\Qpbar$.  The representation $\rho$ is unramified at $p$ if
and only if all points of $E[\ell]$ are defined over ${\bf Q}_p^{\rm nr}$.
By the theory of the Tate curve, we can construct the extension
${\bf Q}_p^{\rm nr}(E[\ell])$ of ${\bf Q}_p^{\rm nr}$ by adjoining to
${\bf Q}_p^{\rm nr}$ the $\ell^{\rm th}$ roots of the Tate parameter
$q$ attached to $E$ over $\Qp$.  Since $\ell\ne p$, $q$ is an $\ell^{\rm th}$
power in ${\bf Q}_p^{\rm nr}$ if and only if the valuation of $q$ is
divisible by $\ell$.  This valuation coincides with the valuation of
$\Delta$.

\medskip

Proposition~\ref{beatsme} remains true if the hypothesis
$p|N$ is suppressed.  Indeed, if $p$ is prime to $N$, then $v_p(\Delta)=0$,
so the congruence $v_p(\Delta)\equiv 0\hbox{ mod }\ell$ is
satisfied {\em a fortiori\/}.  At the same time, the representation $\rho$ is
unramified at $p$, as stated in Proposition~\ref{merguez}.

In order to remove the hypothesis $p\ne\ell$ in the two previous 
Propositions, we need to introduce the technical notion of
{\em finiteness\/} (\cite{Duke}, p.\ 189).  Let $p$ be a prime number, and
choose a place $v|p$.  The decomposition group $D_v$ is
then the Galois group $\GalQp$, as noted above.  By restricting
to $D_v$
the action of $\GalQ$ on $E[\ell]$, we may view $E[\ell]$ as a
$\GalQp$-module.
We say that $\rho$ is {\em finite\/}
at $p$ if there is a finite flat group scheme $\cal V$ of type $(\ell,\ell)$
over $\Zp$ such that the $\GalQp$-modules ${\cal V}(\Qpbar)$
and $E[\ell]$ are isomorphic.  If $p\ne\ell$, this means simply that
$\rho$ is unramified at $p$.  In general, we may paraphrase this
property of $\rho$ by saying that $E[\ell]$ has ``good reduction at
$p$.''

\begin{prop}
The representation $\rho$ is finite at a prime number $p$ if and
only if $v_p(\Delta)\equiv 0\hbox{\rm\ mod }\ell$.
\end{prop}

The only new information provided by this Proposition is the criterion
for finiteness in the case $p=\ell$.  For the proof, see \cite{Duke},
p.\ 201.

\section{Modular elliptic curves}
Let $E$ be an elliptic curve over $\Q$.  The
 {\em Taniyama-Shimura Conjecture\/} (also known as the Weil-Taniyama
Conjecture)
states that $E$ is {\em modular}.  One can define this concept in
several equivalent ways, either by connecting up the
integers $a_p$ of \S\ref{sectionone} with the coefficients of modular
forms, or else by considering the geometry of modular curves. 

We begin with the latter.  (See \cite{Sh},
  \cite{StLouis}, or \cite{Silverman}, Ch.~11, \S3 
for background
on modular curves.)  Let $N$ be a positive integer, and consider
the group
\[ 
\Gamma _{o}(N) = \left\{ \, 
\pmatrix
{a&b \cr c&d\cr} \in \SLtwoZ \Bigm| c\equiv 0 \hbox{ mod } N \right\}.\]
Recall that the group $\SLtwoZ$ acts on the
Poincar\'e upper half plane
\[ {\cal H}  =  \{ \, \tau\in\C \,\mid\, \Im(\tau) > 0\,\} \]
by fractional linear transformations 
\[ \pmatrix
{a&b \cr c&d\cr}: \tau \mapsto \frac{a\tau +b}{c\tau + d}.\]
The quotient $Y_o(N)=\Gamma_o(N)\backslash\cal H$ is an open
Riemann surface, which has a standard compactification $X_o(N)$.
This {\em modular curve\/} has a canonical model over $\Q$, which we
denote simply $X_o(N)$ to lighten notation.  

Although there is no question here of explaining the origin of
the canonical model, we should mention at least that this model
is connected intimately with the interpretation of $X_o(N)$ and
$Y_o(N)$ as {\em modular varieties}.  The idea is that to each
$\tau\in\cal H$ we can associate the lattice $L_\tau=\Z + \Z\tau$ in $\C$,
and then the elliptic curve $E_\tau = \C/L_\tau$.
Two elliptic curves $E_\tau$ and $E_{\tau'}$ are isomorphic
if and only if the elements $\tau$ and $\tau'$ of $\cal H$ have
the same image in $\SL(2,\Z)\backslash\cal H$.  More generally, given $N\ge1$,
we associate to $\tau\in\cal H$ the pair $(E_\tau,C_N)$ consisting of
the elliptic curve $E_\tau$ and the cyclic subgroup $C_N$ of $E$ generated
by the image of ${1\over N}\in\C$ on $E_\tau$.  Then two $\tau$'s give
isomorphic pairs if and only if they map to the same element of
$Y_o(N)$.  Further, every pair consisting of an elliptic curve $E$
over $\C$, together with a cyclic subgroup of order $N$ arises in
this way from some $\tau$.  Hence the curve $Y_o(N)$ parametrizes isomorphism
classes of data
``elliptic curves with cyclic subgroups of order $N$.''  It is via this
interpretation that one defines $Y_o(N)$ and $X_o(N)$ over $\Q$.
In fact, with a more sophisticated definition of the data to be classified,
one arrives at a good definition over $\Z$ \cite{KM}.

The Jacobian of $X_o(N)$
is the abelian variety $J_o(N)$; this is a complete group variety
whose dimension is the genus $g(N)$ of the curve $X_o(N)$.
This genus is given by a well known formula (see, e.g., \cite{SMF},
p.~455).  The integer $g(N)$ is 0 for all $N\le 10$, and we have
for example
$g(11)=1$, $g(23)=2$.

We let $S(N)$ be the complex vector space of weight-2 holomorphic cusp
forms for $\Gamma_o(N)$.  Such a form is a holomorphic function $f(\tau)$
on $\cal H$ for which the differential $f(\tau)\, d\tau$ is invariant under
the group $\Gamma_o(N)$.  Thus $f(\tau)\, d\tau$ may be viewed as a regular
differential on the open curve $Y_o(N)$, and the condition that $f(\tau)$
belong to $S(N)$ is the condition that $f(\tau)\, d\tau$ extend to a
holomorphic differential on $X_o(N)$ over $\C$.  The space $S(N)$ may
thereby be identified with the space of holomorphic differentials on
the curve $X_o(N)$, a space of dimension~$g(N)$.

Each $f(\tau)\in S(N)$ may be expanded as a Fourier series
$\sum\limits_{n\ge 1} a_n q^n$, where $q=e^{2\pi i\tau}$.  The sum
$\sum a_n q^n$ is viewed often as a formal series in $q$, and is
known as the ``$q$-expansion'' of the cusp form $f$.

The space $S(N)$ comes equipped with a commuting family of endomorphisms,
the {\em Hecke operators\/} $T_n$ ($n\ge1$).  (See \cite{Sh}, Chapter~3.)
These operators are traditionally written on the right, so that
$f|T_n$ denotes the image of $f$ under $T_n$.  The $T_n$
may   be expressed as polynomials in the operators $T_p$ where
$p$ is prime.  (For example, they are multiplicative: $T_{nm} = T_n \circ T_m$
when $n$ and $m$ are relatively prime.)  The operators $T_p$ are
given by the following well known formula, expressing
the
$q$-expansion of $f|T_p$ in terms of that of $f$:
\[ T_p : f= \sum a_{n}q^{n} \,\mapsto\,
\cases{ \sum a_{pn}q^{n}+p\sum a_{n}q^{pn} & if $p$ is prime to $N$; \cr
\sum a_{pn}q^{n} & if $p$ divides $N$.\cr}
\] 

An {\em eigenform\/} in $S(N)$ is a non-zero $f\in S(N)$ which is an
eigenvector for each of the operators $T_n$.  The eigenvalues $(\lambda_n)$
coming from a given eigenform $f$ are known to be algebraic integers
which all lie in a finite extension of $\Q$ which is independent of $n$.

\proclaim{\sc Taniyama-Shimura Conjecture}. 
Let $E$ be an elliptic curve over $\Q$,
and let $N$ be its conductor.  Then there is an eigenform $f\in S(N)$
with the following property: for each prime $p$ not dividing $N$,
the integer $a_p$ of \S\ref{sectionone} is the eigenvalue of $f$ for the
operator $T_p$.\par

If $f$ exists for a given $E$, one says that $E$ is {\em modular}.  
Because of our detailed knowledge of the arithmetic of modular forms
and abelian varieties (the author is especially thinking of \cite{Carayol,%
Faltings}), the condition that $E$ be modular can be reformulated in
a variety of ways.  For example, B.~Mazur \cite{Gadfly} has recently proved 
\begin{prop}
Suppose that $E$ is an elliptic curve over $\Q$ with the following
property: there is a non-zero homomorphism of abelian varieties
$h\colon E \to J_o(M)$, defined over $\C$, for some $M\ge1$.
Then $E$ is modular.
\end{prop}

The novelty of this statement is that $h$ is not assumed to be
defined over $\Q$.

Another remark which may serve to orient the reader is that the form
$f$ in the Taniyama-Shimura Conjecture is (up to multiplication by
a constant)
the unique non-zero element of $S(N)$ having the property \[
 f|T_p = a_p \cdot f\hbox{ for all but finitely many $p$.}\]
More precisely, $f$ is a constant multiple of a newform (in the sense
of \cite{AL}) of level $N$.  All coefficients of $f$ are
rational integers.  

Finally, A.~Weil proved in \cite{Weil} that an elliptic curve $E$ over
$\Q$ is modular provided that its $L$-function $L(E,s)$ has the analytic
properties one expects of it.  This $L$-function is defined as an
infinite product $\prod_p L_p(E,s)$ ($s\in\C$, $\Re s > 3/2$), over
the set of prime numbers $p$ (Euler product).  
The factor $L_p(E,s)$, for $p$ prime to $N$, is 
\[ (1 - a_p p^{-s}+ p^{1-2s})^{-1}.\]

\section{Modular representations of $\GalQ$}
Let $N$ be a positive integer, and let $\T=\T_N$ be the subring (i.e.,
$\Z$-algebra) of $\End_\C\bigl(S(N)\bigr)$ generated by the operators
$T_n$ for $n\ge1$.  As is well known (\cite{Sh}, Ch.~3), this ring
is a free $\Z$-module of rank equal to the dimension $g(N)$ of $S(N)$.
In particular, if $\m$ is a maximal ideal of $\T$, then the residue
field $k_\m = \T/\m$ is a finite field, say of residue characteristic
$\ell$.  One can show (see \cite{DS}, Th.~6.7 or \cite{Fermat}, \S5)
that there is a semisimple continuous homomorphism
\[ \rho_\m : \GalQ  \to  \GL(2,k_\m)\]
having the properties:
\begin{enumerate}
\item $\det\rho_\m = \chi_\ell : \GalQ \to {\bf F}_\ell^* \subseteq k_\m^*$
\item $\rho_\m$ is unramified at all primes $p$ not dividing $\ell N$
\item $\trace \rho_\m(\Frob_v) = T_p \hbox{ mod } \m$ for all $p\not|\ell N$.
\end{enumerate}
(Here, $\chi_\ell$ is again the mod~$\ell$ cyclotomic character of
$\GalQ$  and, 
in the last property, $\Frob_v$ is a Frobenius element in a decomposition
group $D_v$ for a prime of $\Qbar$ dividing $p$.)

The representation $\rho_\m$ is unique up to isomorphism.  This follows
from the Cebotarev Density Theory, which states that all elements of
the image of $\rho_\m$ are conjugate to Frobenius elements $\rho_\m(\Frob_v)$,
plus the fact that a two-dimensional 
semisimple representation is determined
by its trace and determinant.

\medskip
\noindent {\em Example\/}.  Let $E$ be a modular elliptic curve of
conductor $N$, and
let $f\in S(N)$ be an eigenform whose eigenvalue under $T_p$ is $a_p$
for each $p$ prime to $N$.  The action of $\T$ on $f$ is given by
the homomorphism $\varphi:\T\to\C$ which takes each $T\in\T$ to the eigenvalue
of $f$ under $T$.  This homomorphism is in fact $\Z$-valued, as remarked
above.  If $\ell$ is a prime number, and $\m=\varphi^{-1}\Bigl((\ell)
\Bigr)$, then $\rho_\m:\GalQ\to\GL(2,\Fell)$ is the {\em semisimplification\/}
$\rho^{\rm ss}$
of the representation $\rho:\GalQ\to\Aut(E[\ell])$.   This semisimplification
  is defined to be
the direct sum of the 
Jordan-H\"older factors of $\rho$.  In other words, $\rho^{\rm ss}$ is the 
direct sum of two one-dimensional representations in case $\rho$ is
not simple, while $\rho^{\rm ss}$ is $\rho$ if $\rho$ is already
semisimple.  Hence, we have $\rho_\m \approx \rho$ in this latter case.

\medskip

Now let $\F$ be a finite field and suppose that
\[ \sigma : \GalQ \to \GL(2,\F) \]
is a continuous semisimple representation.  We say that $\sigma$
is {\em modular of level $N$} if there is a maximal ideal $\m$ of
$\T$ and an embedding $\iota:\T/\m\subset\Fbar$ such that the $\Fbar$-%
representations
\begin{displaymath}\matrix{
\GalQ \stackrel{\sigma}{\longrightarrow} \GL(2,\F)\subset\GL(2,\Fbar)\hfill\cr
\cr
\GalQ \stackrel{\rho_\m}{\longrightarrow}\GL(2,\T/\m)
   \stackrel{\iota}{\hookrightarrow}\GL(2,\Fbar) \hfill\cr}
\end{displaymath}
are isomorphic.  Equivalently, one requires a homomorphism
\[ \omega:\T \to \Fbar \]
such that
\[ \trace\Bigl(\sigma(\Frob_p)\Bigr) = \omega(T_p), \qquad
   \det\Bigl(\sigma(\Frob_p)\Bigr) = \bar p \]
for all but finitely many primes $p$.  Here, $\Frob_p$ denotes
$\Frob_v$ for some $v|p$ and $\bar p$ is the image of $p$ in $\F$.
(Thus $\bar p$ is $p$ mod $\ell$, if $\ell$ is the characteristic of
$\F$.)  The term ``modular of level $N$'' is a shorthand phrase used
to indicate that the representation $\sigma$ arises from the space
$S(N)$; this space could be described more verbosely as the
space of weight-2 cusp forms on $\Gamma_o(N)$ with trivial character.

Assuming that $\rho$ is modular of level $N$, we say that $N$ is
{\em minimal\/} for $\rho$ if there is no divisor $M$ of $N$,
with $M<N$, such that $\rho$ is modular of level $M$.  Regarding $\rho$
  as fixed, and possible levels $N$ as varying, we note that if $\rho$
is modular of some level $N$, then $\rho$ is modular of some minimal
level $N_o|N$.  (The question of the uniqueness of $N_o$ is   more subtle,
 but not an issue here.) Also, it is essentially
evident that once $\rho$ is modular of level $N$, it is modular of
all levels $N'$ which are multiples of $N$.


The following statement is a variant of the main theorem of \cite{Fermat}.

\begin{theorem}\label{main}
Let $\sigma$ be an irreducible two-dimensional representation of
$\GalQ$ over a finite field of characteristic $\ell>2$.
Assume that $\sigma$ is modular of square free level $N$, and that
there is a prime $q|N$, $q\ne\ell$ at which $\sigma$ is not finite.
Suppose further that $p$ is a divisor of $N$ at which $\sigma$ is
finite.  Then $\sigma$ is modular of level $N/p$.
\end{theorem}

This theorem immediately gives:
\[
      \hbox{Conjecture of Taniyama-Shimura }\Longrightarrow \hbox{ Fermat's
   Last Theorem.} \]
Indeed, suppose that the Taniyama-Shimura Conjecture is true.  To prove
Fermat's Last Theorem under this assumption, it suffices to show
that there is no triple $(a,b,c)$ of non-zero integers which satisfies
the equation
\[ a^\ell + b^\ell + c^\ell = 0,\]
where $\ell\ge5$ is a prime.  Arguing by contradiction, we assume that
there is such a triple.  Dividing out by any common factor, we may
suppose that the three integers $a$, $b$, and $c$ are relatively prime.
Further, after permuting these integers, we may suppose that $b$ is
even and that $a\equiv1$ mod $4$.  Following Frey  \cite{Frey},
we consider the elliptic curve
 $E$ with Weierstrass equation
\[y^2 = x(x-a^\ell)(x+b^\ell).\]
According to our discussion above, the conductor of $E$ is the
radical
$N=\mathop{\rm rad}(abc)$ of $abc$, and its minimal discriminant
is
$\Delta=(abc)^{2\ell}/2^8$.  The group $V=E[\ell]$ provides an irreducible
representation $\sigma$ of $\GalQ$ which is finite at every odd prime
but not finite at the prime 2.  Applying Theorem~\ref{main} inductively,
we deduce that $\sigma$ is modular of level~2.  This is impossible,
since the ring $\T_2 \subseteq \End(S(2))$ is the 0-ring, in view of
the fact that the dimension $g(2)$ of $S(2)$ is~0.

 

To prove the Theorem, we will work with two related abelian varieties over
$\Q$.  First, we have $J_o(N)$, the Jacobian $\Pico(X_o(N))$
of the modular curve $X_o(N)$.  We make use of the fact that the
ring $\T=\T_N$ operates faithfully on $J_o(N)$ as a ring of endomorphisms
which are defined over $\Q$.
This action is determined uniquely by the
  relation between it and the tautological action of $\T$ on $S(N)$.
Namely, $S(N)$ is canonically the space of differentials on the Albanese
variety ${\rm Alb}(J_o(N))$, which in turn is the abelian variety dual to
$J_o(N)$.  Thus $S(N)$ is functorially the cotangent space to the dual
of $J_o(N)$.  This functorial relation translates the action of $\T$ on
$J_o(N)$ as a ring of endomorphisms into the classical action of $\T$
on $S(N)$.

Suppose now that $\sigma$ is given as in the Theorem, and let $\m\subset\T$
be a maximal ideal corresponding to $\sigma$ as in the definition of
``modular of level $N$.''  One checks immediately that we can replace
$\sigma$ by $\rho_\m$ in  the Theorem.  From now on, let us regard
$\m$ as fixed and let us assume that $\sigma$ is $\rho_\m$.
We will write simply $k$ for the residue field $k_\m$ of $\m$
and $\rho$ for $\rho_\m$.


For each endomorphism $\alpha\in\m$, we let $J_o(N)[\alpha]$ be the kernel
of $\alpha$ on the group
 $J_o(N)(\Qbar)$ of
points on $J_o(N)$ with coordinates in
$\Qbar$.   Let $W= J_o(N)[\m]$ be the intersection
\[  \bigcap_{\alpha\in\m} J_o(N)[\alpha].\]
This group is a subgroup of the finite group $J_o(N)[\ell]$ of $\ell$-division
points of $J_o(N)$, and carries natural commuting actions of $k$ and
of $\GalQ$.  

Here is the first key result needed to prove Theorem~\ref{main}:

\begin{prop}\label{firstingredient}
  Assume, if $\ell$ divides $N$, that $\rho$ is not modular of level
$N/\ell$.  Then $\rho$ is equivalent to the $k$-representation $W$
of $\GalQ$.
\end{prop}

Let $V$ be the representation $\rho$, regarded as a 2-dimensional
$k$-vector space with an irreducible
action of $\GalQ$.  Using the Eichler-Shimura
relations for $J_o(N)$,
one shows immediately
(\cite{MazurE}, Chapter~II, Proposition~14.2, or \cite{Fermat},
Theorem~5.2) that the semisimplification of $W$ as a $k[\GalQ]$-module
is isomorphic to a non-empty product of copies of $V$.  Thus the
proposition states, in effect, that the $k$-dimension of $W$ is~2.
This is again easy to prove in case $\ell$ is prime to $N$
{\em (loc.\ cit.)}.  When $\ell|N$, \cite{Mazur-Ribet} proves the
desired statement under the hypothesis that $\rho$ is not modular of level
$N/\ell$. The reader is invited to consult these
articles for details of the proofs.\par\medskip

With this preliminary proposition recorded, it is time to discuss
the strategy of the proof of Theorem~\ref{main}.  As noted above,
the representation $\rho$ occurring in Theorem~\ref{main} will be
modular of some minimal level $D|N$.  This level is divisible by $q$,
because $\rho$ is assumed to have ``bad reduction'' at this prime (i.e.,
assumed not to be finite at $q$).  Further, to prove 
Theorem~\ref{main},
 it suffices 
to show that $D$ is prime
to $p$.  Indeed, if $D$ is prime to $p$, then
$N/p$ is a multiple of $D$, and $\rho$ is certainly modular of
level $N/p$, as desired.  

To prove that $D$ is prime to $p$, we 
 can argue by contradiction,
supposing instead that $D$ is divisible by $p$.  In this situation,
$D$ is of the form $pqL$, with $L$ square free and prime to $pq$.
Thus the pair $(\rho,D)$ has all the properties of the pair $(\rho,N)$,
and the additional property that $D$ is a minimal level for $\rho$.
We are to deduce a contradiction from these data.

Replacing $D$ by $N$, we see that Theorem~\ref{main} will follow 
if we can deduce a contradiction from the following assumptions
about the irreducible representation $\rho$:
\begin{enumerate}
\item $\rho$ is modular of   level $N$, where $N$ is square free
and divisible by the distinct primes $p$ and $q$,
with $q\ne\ell$.
\item $\rho$ is finite at $p$, but not at $q$.
\item The level $N$ is a minimal level for $\rho$.
\end{enumerate}

Note that the last assumption implies in particular the hypothesis
of Proposition~\ref{firstingredient}.  Hence we will be able to use
  in our argument that the Galois module $W=J_o(N)[\m]$ defines $\rho$. 
 

 

Fix now primes of $\Qbar$ dividing $p$ and $q$,  let $D_p$ and $D_q$ be
the corresponding decomposition groups in $\GalQ$, and let $\Fpbar$
and $\Fqbar$ be the corresponding residue fields.  For the moment,
we focus attention on the action of $D_q$ on $J_o(N)[\ell]$ and its
subgroup
$W$.

We require some results about this action which follow from the
fact that $J_o(N)$ has semistable reduction at the prime $q$.
These results are certainly not elementary.  Indeed,
the semistable reduction that we need
follows from the   work of Deligne-Rapoport
\cite{DR}
on the reduction of modular curves (see also \cite{KM} and \cite{Edix}
for recent, related work), together with results of Raynaud \cite{RaynP}
relating the reductions of curves with the reductions of their Jacobians.
The information that is to be deduced about $J_o(N)[\ell]$ is
contained in Grothendieck's article on
N\'eron models and monodromy  \cite{SGA7}.

We need first to consider the modular curve $X_o(N/q)$ over the base fields
$\Fq$ and $\Fqbar$.  For this, it is necessary to invoke the ``modular
interpretation'' of the curves $X_o(N)$ which was discussed above.  
We   view $X_o(N/q)$ over $\Fq$ as classifying pairs
$(F,C)$, where $F$ is an elliptic curve and $C$ a cyclic subgroup of
order $N/q$ on $F$.  The points of the curve $X_o(N/q)$ over $\Fqbar$
(i.e., with coordinates in $\Fqbar$) consist of a finite number of
``cusps'' (points at infinity), together with points representing the
isomorphism classes of pairs $(F,C)$ over $\Fqbar$.
Among these latter points are the {\em supersingular points}, those 
represented by pairs $(F,C)$ for which $F$ is  supersingular
in the sense that the group $F(\Fqbar)[q]$ of points on $F$ of order
dividing $q$ is reduced to the trivial group.  The number of supersingular
points on $X_o(N/q)$ over $\Fqbar$ is finite, and can be calculated
quite easily.  For example, this number is $1+g(q)$ when $N=q$ and so
is $2$ when $N=11$ and $3$ when $N=23$.

Let $\cal S$ be the set of these supersingular points.  Then $\cal S$
carries a natural action of $\Gal(\Fqbar/\Fq)$, coming from the action
of this Galois group, by conjugation, on pairs $(F,C)$ over $\Fqbar$.
We may view this action as an  action of $D_q$ which is  unramified,
i.e., which factors through the quotient of $D_q$ by its inertia
subgroup.  We will interested in the free abelian group $\Z^{\cal S}$
on $\cal S$.  
This group has a commuting family of Hecke operators $T_n \in \End(
{\cal S})$
($n\ge1$).  These can be defined directly in terms of 
 pairs
$(F,C)$ for which $F$ is supersingular.
In particular, suppose that $\ell\ne q$ is a prime, and
let $s\in\cal S$ be the class of $(F,C)$.  For each cyclic subgroup
$D$ of $F$ which has order $\ell$ and which is not contained in
$C$, consider the quotient elliptic curve $F/D$ and its subgroup
$(C\oplus D)/D$, which has order $N$.  The image of $s$ under $T_\ell$
is the sum of the classes of the various quotients 
$(F/D,(C\oplus D)/D)$.
(There are $\ell+1$ possible subgroups $D$ when $\ell$ is prime to $N$,
and $\ell$ such subgroups otherwise.)
On the other hand, the operator $T_q$ is defined on $\cal S$ as the
negative of the operator $(F,C) \mapsto (F,C)^{(q)}$ which takes 
each pair to its conjugate under the generator $x \mapsto x^q$ of the
Galois group  $\Gal(\Fqbar/\Fq)$.  The operators $T_n$ with $n$
not necessarily prime can be expressed in terms of the $T_\ell$
by the same formulas which link together the operators labelled
$T_n$ on $S(N)$.

Let $L$ now be the group of elements of $\Z^{\cal S}$ which have
degree~0, the degree of an element being the sum of its coefficients.
It is immediate that the operators $T_n$ we have just defined map
$L$ into itself.  We get in this way an action of the formal polynomial
ring $\Z[\ldots,T_n,\ldots]$ on $L$.  By transposition, we obtain
an action of this ring on $\Hom(L,\mu_m)$ for every $m\ge1$, where
$\mu_m$ is the group of $m^{\rm th}$ roots of unity in $\Qqbar$.
We have a compatible action of the decomposition group
$D_q= \Gal(\Qqbar/\Qq)$ on $\Hom(L,\mu_m)$ in which $D_q$ acts
both on $L$ (in the manner just explained) and on $\mu_m$ (in the
standard way).

It follows from the work of Grothen\-dieck, Raynaud,
and Deligne-Rapo\-port \cite{SGA7,RaynP,DR}
 that there is a natural inclusion
   \[ \Hom(L,\mu_m) \hookrightarrow J_o(N)(\Qqbar)[m] \]
which is compatible with the actions of $D_q$ and the $T_n$ ($n\ge1$)
on both sides.  The image of this inclusion is known as the {\em toric
part\/} of $J_o(N)(\Qqbar)[m]$, and will be denoted $J_o(N)[m]\toric$.

This terminology requires some explanation.  
Consider  the abelian variety $J_o(N)$ over $\Qq$ and imagine the problem
of ``reducing $J_o(N)$ mod $q$.''  In general,
the variety $J_o(N)$ has bad
reduction at $q$, which is to say that its ``best
reduction'' (N\'eron model) over $\Fq$ is not necessarily an
abelian variety over $\Fq$.  This reduction has, however, a maximal
toric subgroup $\cal T$, and it is the character group 
$\Hom_{\Fqbar}({\cal T},\Gm)$
of $\cal T$
which we have called $L$.  The group $\Hom(L,\mu_m)$ naturally
extends to a finite flat group scheme
over ${\bf Z}_q$ whose reduction mod $q$ is the kernel ${\cal T}[m]$
of multiplication by $m$ on the $\Fq$-torus $\cal T$.

The action of $\Z[\ldots,T_n,\ldots]$ which
we have defined in an {\em ad hoc\/} way is in fact the functorial
action of $\T \subseteq \End\Bigl(J_o(N)\Bigr)$ on 
$\Hom_{\Fqbar}({\cal T},\Gm)$.  In other words, $\Z[\ldots,T_n,\ldots]$
acts on $L$ through its quotient $\T=\T_N$, and the action is just
the functorial one coming from the interpretation of $L$ in terms of
the N\'eron model.

\medskip

We now set $m=\ell$ and return to consideration of $J_o(N)[\ell]$
and its submodule $W$.  We set $W\toric = W \cap J_o(N)[\ell]\toric$.
Equivalently, we have 
\[      W\toric = \Hom(L/\m L, \Gm), \]
since $W$ is the kernel of $\m$ on $J_o(N)[\ell]$.  The quotient
$W/W\toric$ is naturally a submodule of $J_o(N)[\ell]/J_o(N)[\ell]\toric$.
We recall that $\ell\ne q$.

\begin{prop}
\label{unramified}
The $D_q$-modules $W\toric$ and $W/W\toric$ are unramified.
\end{prop}

\begin{proof}
Since $\ell\ne q$, the $D_q$-module $\mu_\ell$ is unramified.
The $D_q$-module $L$ is certainly unramified, since $D_q$ acts
on $L$, by construction, through its quotient $\Gal(\Fqbar/\Fq)$.
Hence $J_o(N)[\ell]\toric$ is unramified, and we may conclude that
its submodule $W\toric$ is unramified.

Similarly, to prove that $W/W\toric$ is unramified, it suffices to
show that $J_o(N)[\ell]/J_o(N)[\ell]\toric$ is unramified.  This latter
fact follows from Grothen\-dieck's theory of semistable reduction (\cite{SGA7},
\S2.3--2.4).  Indeed, let $A$ be a semistable abelian variety over
$\Qq$ (to fix ideas).  Identify the dual $\Hom(A[\ell],\mu_\ell)$
of the group of $\ell$-division points on $A$ with the kernel 
$\skew5\check A[
\ell]$ of multiplication by $\ell$ on the abelian variety 
$\skew5\check A$ dual
to $A$.  Then $A[\ell]/A[\ell]\toric$ is dual to a subgroup of
the largest subgroup $\skew5\check A[\ell]\finite$ of 
$\skew5\check A[\ell]$ which
extends to a finite flat group scheme over ${\bf Z}_q$ (i.e., which
is unramified).   %What a mouthfull!!!!!
\end{proof}

\begin{cor}
The $k$-vector space $L/\m L$ has dimension~1. 
\end{cor}


\begin{proof} The statement to be proved is equivalent to the
statement that $W\toric$ has $k$-dimension~1.  Since $W$ has
dimension~2, we must show that $W$ is isomorphic neither to $W\toric$
nor to $W/W\toric$.  By the Proposition, each of these modules is
unramified.  By hypothesis, however, $W$ is {\em ramified\/} at $q$.
\end{proof}

To prove Theorem~\ref{main}, we will derive the inequality
\[  \dim_k(L/\m L) \ge 2, \]
which contradicts the above Corollary.  To do this, we need  to
find a second interpretation of the lattice $L$, or rather of
a ``primitive'' subgroup of it.  To define this subgroup, we recall
that $L$ is associated to the mod~$q$ reduction of the abelian variety
$J_o(N)$, where
$N$ is a square-free integer divisible by the distinct primes
$p$ and~$q$.  Write $M$ for the quotient $N/(pq)$, and consider the
Jacobian $J_o(Mq)$ of the modular curve $X_o(Mq)$.  Let $X$ be the
analogue of $L$ for $J_o(Mq)$, i.e., the character group of the largest
torus in the mod~$q$ reduction of $J_o(Mq)$.
There are two
natural degeneracy maps \[X_o(N) \doublemap X_o(Mq),\] corresponding to
the two obvious ways of associating to a pair $(F,C)$, with $C$ cyclic
of order $Mpq$, a   pair $(F',C')$ with $C' \subset F'$ cyclic
of order $Mq$.  (We can take $F'=F$ and let $C'$ be the 
unique subgroup of
$C$ having order $Mq$, or else take $F'$ to be the quotient of $F$ by
the subgroup $C''$ of $C$ with order $p$.  In the latter case,
we take $C' = C/C''$.)   These lead to two maps $L \doublemap X$,
or equivalently to a map
\[ \pi : L \to X \oplus X.\]
This map is {\em surjective}, as one sees by applying
theorems of Eichler
on the arithmetic of quaternion algebras over $\Q$.  The kernel of
$\pi$ is the {\em $p$-primitive} subgroup of $L$, and it is this
subgroup that we wish to reinterpret.

The re%
interpretation comes from the second type of modular curve,
  a {\em Shimura curve\/} coming from a rational
quaternion division algebra.    Choose
a quaternion division algebra of discriminant $pq$ over $\Q$, and let
$\cal O$ be a maximal order in this algebra. 
% (The
%ring $\cal O$ is unique up to isomorphism, but it is not quite
%unique up to {\em unique\/} isomorphism: if we choose a second order
%${\cal O}'$, then we cannot find a canonical isomorphism
%${\cal O} \approx {\cal O}'$, even working modulo inner automorphisms
%of these rings.)  

Let $C$ be the algebraic curve over $\Q$ associated to
the problem of classifying abelian varieties $A$ of dimension~2, furnished
with an action of $\cal O$, together with a ``$\Gamma_o(M)$-structure.''
The latter object is an $\cal O$-stable subgroup of $A$ which is
isomorphic to $(\Z/M\Z)^2$.  The curve $C$ is the Shimura curve which
we need.  It is, in some sense, a variant of the curve $X_o(M)$, since the
replacement ${\cal O} \mapsto {\rm M}(2,\Z)$ returns us to the
theory of modular curves which we have already met.  Indeed, to give
an operation of ${\rm M}(2,\Z)$ on a 2-dimensional abelian surface
is essentially the same as to give an elliptic curve: if $E$ is an
elliptic curve, then $E\times E$ has an evident action of ${\rm M}(2,\Z)$,
and one sees easily that any $A$ with an action of this ring is isomorphic
to such a product.  On the other hand, the fact that $p$ and $q$ appear
in the discriminant of $\cal O$ makes $X_o(N)=X_o(pqM)$ a better 
analogue of $C$.

Before we can reinterpret the $p$-primitive subgroup
of $L$, we have two
minor choices to make.  Namely, the ring $\cal O$ has residue fields
at $p$ and at $q$ which are finite fields of cardinality $p^2$ and
$q^2$, respectively.  Choose isomorphisms $\kappa_p$ and $\kappa_q$
between these
fields and the quadratic subfields of $\Fpbar$ and $\Fqbar$, respectively.
(There are two possible choices for each of $p$ and $q$.)
These maps may be viewed as homomorphisms
\[ \kappa_p:{\cal O}\to\Fpbar, \qquad \kappa_q:{\cal O}\to\Fqbar.\]

Let ${\cal T}'$ be the toric part of the mod~$p$
reduction of the N\'eron model of the Jacobian $\Pico(C)$.
One can show that this torus coincides with the connected component
of~$0$ in the reduction.  This means that $\Pico(C)$ has ``purely
multiplicative reduction'' at the prime $p$. To see that this is the
case,
 one uses
results of Grothendieck and Raynaud, cited above, to relate the reduction of
$\Pico(C)$ to the mod~$p$ reduction of $C$.  The latter is studied in
well known articles of
Cerednik and Drinfeld \cite{C,D}.
(Further discussion of the mod~$p$ reduction of $\Pico(C)$ is
found in Kurihara \cite{Kurihara}, articles of Jordan-Livn\'e 
\cite{JL1,JL2}, and \cite{Iwasawa}.)

Let $Y = \Hom_\Fpbar({\cal T}',\Gm)$, so that $Y$ is a free abelian
group whose rank is the dimension of $\Pico(C)$.  Then we have

\begin{theorem}[\protect\cite{Fermat,Iwasawa}] \label{thseq}
The group $Y$ is naturally isomorphic to the $p$-primi\-tive part of $L$.
Equivalently, we have an exact sequence
\begin{equation}\label{sequence}
0 \to Y \stackrel{\theta}{\to} L \stackrel{\pi}{\to} X\oplus X \to 0.
\end{equation}
\end{theorem}
The words ``naturally isomorphic'' are intended to convey the information
that the map $\theta$ introduced in (\ref{sequence}) is compatible with
the Hecke operators $T_n \in \End(J_o(N))$ for $n\ge1$
and their analogues
$T_n\in\End(J)$ which were introduced by Shimura.   In connection with
these operators, it might be worth remarking that
the operators $T_p$ and
$T_q$ are each   {\em involutions\/} on the subgroup $Y$ of $L$.  (The
operator $T_q$ is already an involution on $L$.) 
The map $\theta$ is canonical in the sense that it depends only
on the choices of $\kappa_p$ and $\kappa_q$.
  The
effect of a change in  $\kappa_p$ or $\kappa_q$ is to compose $\theta$
(on either the left or right)
with the corresponding Hecke operator $T_p$ or $T_q$.  

The Hecke operators $T_n$ on $J$ form a commutative family, and thus
amount {\em a priori\/} to an action of the formal polynomial
ring $\Z[\ldots,T_n,\ldots]$ on $J$.  One sees from Theorem~\ref{thseq}
that this ring acts on $Y$, at least, through its quotient $\T_N$,
since $\Z[\ldots,T_n,\ldots]$ acts on $L$ through this quotient.
On the other hand, the fact that $J$ has purely multiplicative
reduction at $p$ ensures that $\End(J)$ acts faithfully on $Y$.  Putting
these two facts together, we learn that the $T_n$ may be viewed as
an action of $\T=\T_N$ on $J$.

Next, let us view $X\oplus X$ as a $\T$-module via (\ref{sequence}).
In other words,
we use the degeneracy maps introduced above to
consider  $X \oplus X$ as a quotient of $L$.  Write $(X\oplus X)_\m$
for the localization of this $\T$-module at the maximal ideal $\m$
of $\T$.  

\begin{prop}\label{local0}
The module $(X\oplus X)_\m$ vanishes.
\end{prop}

To prove the proposition, we use
 the third assumption made about $\rho$,
namely the supposed minimality of $N$ as a level for $\rho$.
  Let $\TNbar$ be
the image of $\T_N$ in $\End(X\oplus X)$, and let $\overline\m \subseteq
\TNbar$ be the image of $\m$ in this quotient.  Clearly, the
non-vanishing of $(X\oplus X)_\m$ would
imply that $\overline\m$ is a
maximal ideal of $\TNbar$, rather than the unit ideal $(1)$ of this
ring.  From this, we would be able to deduce the existence of a maximal
ideal $\m_{N/p} \subset \T_{N/p}$ which gives the representation
$\rho=\rho_\m$, contrary to the supposed minimality.
\nolinebreak[2]\rule{.45em}{.9em}\medskip

{\small
Here are some details concerning the construction of $\m_{N/p}$:
Consider the functorial action of $\T_{N/p}$
on $X$ and the resulting diagonal action of this ring on $X\oplus X$.
Let $\overline\T_{N/p}$ be the image of $\T_{N/p}$ in $\End(X\oplus X)$.
The idea is that the rings $\TNbar$ and $\overline\T_{N/p}$ are
essentially identical.  More
precisely, they share a large common subring $R$: the
subring of $\End(X\oplus X)$ generated by the $T_n$ with $n$ prime to
$p$.  (The operator labelled $T_n$ in $\T_N$ and the operator labelled
$T_n$ in $\T_{N/p}$ act in the same way on $X \oplus X$, provided that
$n$ is prime to $p$.)  Moreover, they both lie in the commutative
subring $S$ of $\End(X\oplus X)$ which is generated by $R$ and the
two operators labelled $T_p$: one coming from $\T_N$ and one coming
from $\T_{N/p}$. Suppose
now that $\overline\m$ is a maximal ideal of $\TNbar$.
Then $\overline\m \cap R$ is a maximal ideal of $R$.
 By using Nakayama's lemma (or the going-up
theorem of Cohen-Seidenberg), we may find a maximal ideal $\overline
\m_{N/p}$ of $\overline\T_{N/p}$ whose intersection with $R$ is 
$\overline\m \cap R$ of $R$.  Let $\m_{N/p}$ be the
inverse image of $\overline\m_{N/p}$ in $\T_{N/p}$. Then the
representations $\rho$ and $\rho_{\m_{N/p}}$ are easily seen to 
coincide in the following strong sense: the residue fields of
$\m$, $\overline\m \cap R$, and  $\m_{N/p}$  are all identical,
and the representations $\rho$ and $\rho_{\m_{N/p}}$ are
equivalent representations of $\GalQ$ over this finite field.
\par}

\begin{cor}
The $k$-vector space $Y/\m Y$ has dimension~1.
\end{cor}

This assertion is an immediate
consequence of the corollary to Proposition~\ref{unramified},
the exact sequence (\ref{sequence}), and Proposition~\ref{local0}. \medskip

\begin{prop}
The kernel $J[\m]$ of $\m$ on $J(\Qbar)$ is non-zero.
\end{prop}

\begin{proof}
Indeed, this kernel, viewed ``locally'' as the kernel of $\m$
on $J(\Qpbar)$, contains the toric subgroup
\[ J[\m]\toric = \Hom(Y/\m Y,\mu_\ell). \]
The above corollary shows that this subgroup is of dimension~1,
and in particular that it is non-zero.
\end{proof}

\begin{prop}
The group $J[\m]$, viewed as a $k[\Gal]$-module, contains  a submodule
isomorphic to
the module $V$.
\end{prop}

\begin{proof}
We recall that $V$ is by definition a $k$-vector space of dimension~2
with an action of $\GalQ$ which is equivalent to $\rho$.
By using the Eichler-Shimura relations for $J$, and the   argument
referred to in the proof of Proposition~\ref{firstingredient}, we
may deduce that the semisimplification of $J[\m]$ is isomorphic to
a direct sum of copies of $V$.  The desired result follows from this
fact, plus the non-vanishing of $J[\m]$.
\end{proof}


We again take the ``local'' point of view which amounts to viewing
$J[\ell]$ and its submodules $V$, $J[\m]$, \ldots\ as $D$-modules,
where $D=D_p$ is a
decomposition group for
$p$ in $\GalQ$.  As in \cite{SGA7}, we let $J[\ell]\finite$
be the largest subgroup of $J[\ell]$ which extends to a finite flat
group scheme over $\Zp$. 
% Maybe we don't need this: 
%   We define $V\finite$ and $J[\m]\finite$
%   similarly, or, equivalently, by intersection with $J[\ell]\finite$.
(Note that the case $p=\ell$ is not excluded in our situation, so that
``finite'' cannot automatically be equated with ``unramified.'')
We have $J[\ell]\toric \subseteq J[\ell]\finite$, where 
$J[\ell]\toric$ is a subgroup of $J[\ell]$ isomorphic to
$\Hom(Y/\ell Y,\mu_\ell)$.  The quotient $J[\ell]\finite/J[\ell]\toric$
may be identified with the kernel $\Psi[\ell]$  of multiplication by
$\ell$ on the group $\Psi$ of components in the mod~$p$ reduction of
$J$, since $J$ has purely multiplicative reduction at $p$ 
 (cf.~\cite{SGA7}, 11.6.12).
By hypothesis, $V$ is finite at $p$, so that we have $V\subseteq
J[\ell]\finite$.  Equivalently,
the $k$-vector space $V\finite = V \cap J[\ell]\finite$
is   of dimension~2.  (It   coincides with $V$.)  On the other hand,
the intersection $V\toric=V \cap J[\ell]\toric$ is contained
in the group $\Hom(Y/\m Y,\mu_\ell)$, which is of dimension~1 over $k$
by  the Corollary to Proposition~\ref{local0}.  Therefore, $V\finite /V\toric$
is non-zero.  This gives in particular the statement:
\[ \left( J[\ell]\finite / J[\ell]\toric \right)[\m] \ne 0. \]
Equivalently, we have
\begin{prop}\label{lrg}
The group $\Psi[\m]$ is non-zero.
\end{prop}

This Proposition will lead to the final contradiction, showing the
incompatibility of the three assumptions made above.  
As explained
earlier, this incompatibility implies Theorem~\ref{main}.  The principal
point   is that the assertion of Proposition~%
\ref{lrg} implies that $\rho=\rho_\m$ is modular of level $N/p$,
contrary to the assumption that $N$ is a minimal level for $\rho$.
Indeed, the author shows in \cite{Fermat} that there is a map 
   \[   F:X\oplus X\to\Psi\]
  which
is, for   practical purposes, surjective. Thus, if $\Psi[\m]$ is
non-zero, then $(X\oplus X)_\m$ is non-zero, which contradicts
Proposition~\ref{local0}.
\par\medskip\par
The existence of $F$ is implicit in \cite{JL2}, which exhibits
  a quantitative relation between $\Psi$ and a quotient of
$X\oplus X$.  To construct $F$, the author uses the exact sequence
(\ref{sequence}), plus a description of $\Psi$ in terms of $Y$ which
is due to Grothendieck $\cite{SGA7}$.  Namely, there is a bilinear
pairing
\[ (\, , \, ): Y \times Y \to \Z , \]
the {\em monodromy pairing} for $J$ mod~$p$, such that $\Psi$ is
canonically the cokernel of the map
 \[ Y \to \Hom(Y,\Z), \qquad y \mapsto (y,\cdot). \]
This pairing is defined in terms of the mod~$p$ reduction of the
curve $C$.  An analysis similar to that which establishes (\ref{sequence})
shows that the monodromy pairing on $Y$ is the restriction to $Y$ of
 an analogous pairing on $L$, which in turn is the restriction to
$L$ of an explicit diagonal pairing on $\Z^{\cal S}$, where $\cal S$ is
again the set of supersingular points of $X_o(N/q)$ over $\Fqbar$.
This means, in particular, that $\Psi$ may be calculated in
concrete terms.  

In \cite{Fermat}, the author inserts $\Psi$ into a commutative diagram
of exact sequences and constructs $F$ by appealing to the Snake Lemma.
The resulting long exact sequence gives a description of the cokernel
of $F$, which turns out to be the a quotient of  $\Phi$, the group
analogous to $\Psi$ 
 which is associated to the mod~$q$ reduction of $J_o(N)$.
One should consider $F$ to be ``essentially surjective'' because
$\Phi$ is a group which can be ignored when studying irreducible two-%
dimensional representations of $\GalQ$.  The point is that $\Phi$ is
{\em Eisenstein\/} in the sense that the relation $T_r = (r+1)$ holds
on $\Phi$, for all primes $r$ which do not divide $N$
(\cite{Fermat,Bordeaux,Bas}).  This implies easily that the support of
$\Phi$ as a $\T$-module is contained in the set of maximal ideals
for which the corresponding two-dimensional representations of
$\GalQ$ are reducible.  In particular, $\Phi_\m = 0$.

\begin{thebibliography}{66}
\bibitem{AL} Atkin, A.O.L. and Lehner, J.
  Hecke operators on $\Gamma _{o}(m)$.  
   Math. Ann. {\bf 185}, 134--160 (1970)
\bibitem{Carayol} Carayol, H.  Sur la mauvaise r\'eduction des courbes de
  Shimura.  
  Compositio Math. {\bf 59}, 151--230 (1986)
\bibitem{C} Cerednik, I.V.  Uniformization of algebraic curves by
  discrete   arithmetic subgroups of ${\rm PGL}_{2}(k_{w}) $ with compact
  quotients   (in Russian).  Mat. Sb. {\bf 100} , 59--88 (1976).  Translation  
  in Math USSR Sb. {\bf 29}, 55--78 (1976)
\bibitem{DR} Deligne, P. and Rapoport, M.  Les sch\'emas de modules de  
    courbes elliptiques.   Lecture Notes in Math. {\bf 349},  
   143--316 (1973)
\bibitem{DS} Deligne, P.\ and Serre, J-P.  Formes modulaires
    de poids 1.  Ann.\ Sci.\ Ec.\ Norm.\ Sup.\ {\bf 7\/}, 507--530 (1974)
\bibitem{D} Drinfeld, V.G.  Coverings of p-adic symmetric regions (in  
   Russian).  Functional. Anal. i Prilozen. {\bf 10}, 29--40  
   (1976).  Translation in Funct. Anal. Appl. {\bf 10}, 107--115  
  (1976)
\bibitem{Edix}  Edixhoven, S. J. Minimal resolution and stable
reduction of $X_o(N)$.  To appear
\bibitem{Bas}  Edixhoven, S. J. L'action de l'alg\`ebre de Hecke
   sur les groupes de composantes des jacobiennes des courbes
   modulaires est ``Eisenstein''.  To appear
\bibitem{Faltings} Faltings, G.  Endlichkeitss\"atze f\"ur abelsche
    Variet\"aten \"uber Zahl\-k\"orpern.  Invent. Math {\bf 73}, 349--366
    (1983).  English translation in: Arithmetic Geometry, G.~Cornell
    and J.~H.~Silverman, eds. New York: Springer-Verlag (1986)
\bibitem{Frey} Frey, G.  Links between stable elliptic curves and certain  
   diophantine equations.  Annales Universitatis Saraviensis  
  {\bf 1}, 1--40 (1986)
\bibitem{FreyUlm} Frey, G. Links between solutions of $A-B=C$ and
   elliptic curves.  Lecture Notes in Math. {\bf 1380}, 31--62 (1989)
\bibitem{SGA7} Grothendieck, A.  SGA7 I, Expos\'e IX.  Lecture Notes in Math.  
  {\bf 288}, 313--523 (1972)
\bibitem{Husemoller} Husem\"oller, D.  Elliptic Curves. Graduate Texts
  in Mathematics {\bf 111}. New York: Springer-Verlag (1987)
\bibitem{JL1} Jordan, B. and Livn\'e, R.  Local diophantine properties  
    of Shimura curves.  Math. Ann. {\bf 270},   235--248 (1985)
\bibitem{JL2} Jordan, B. and Livn\'e, R.  On the N\'eron model of Jacobians  
      of Shimura curves.  Compositio Math.  {\bf 60}, 227--236 (1986)
\bibitem{KM} Katz, N.  M. and Mazur, B.  Arithmetic Moduli of Elliptic  
     Curves.  Annals of Math. Studies {\bf 108}.  Princeton: Princeton  
     University Press, 1985
\bibitem{Kurihara} Kurihara, A.  On some examples of equations defining Shimura  
    curves and the Mumford uniformization.  J. Fac. Sci. Univ.  
   Tokyo, Sec. IA, {\bf 25}, 277--300 (1979)
\bibitem{MazurE} Mazur, B.  Modular curves and the Eisenstein ideal.  Publ.  
   Math. IHES {\bf 47}, 33--186 (1977)
\bibitem{Gadfly} Mazur, B\@.  Number theory as gadfly.  American Math.\
Monthly.  To appear
\bibitem{Mazur-Ribet} Mazur, B. and Ribet, K.  Two-dimensional
   representations in the arithmetic of modular curves.  To appear
\bibitem{Oesterle} Oesterl\'e, J\@.  Nouvelles approches du
  ``th\'eor\`eme'' de Fermat.  S\'eminaire Bourbaki $\rm n^o$ 694
    (1987--88).  Ast\'erisque {\bf 161--162}, 165--186 (1988)
\bibitem{StLouis} Ogg, A\@.  Rational points on certain elliptic
   modular curves.  Proc. Symp. Pure Math. {\bf 24}, 221--231 (1973)
\bibitem{SMF} Ogg, A\@. Hyperelliptic modular curves.  Bull. SMF {\bf 102},
   449--462 (1974)
\bibitem{RaynP} Raynaud, M. Sp\'ecialisation du foncteur de Picard.  Publ.  
  Math. IHES {\bf 38}, 27--76 (1970)
\bibitem{Fermat}  Ribet, K\@.  On modular representations of
   $\GalQ$ arising from modular forms.  Preprint
\bibitem{Iwasawa} Ribet, K\@.  Bimodules and abelian surfaces.
   {\sc Advanced Studies in
        Pure Mathematics}.  In press
\bibitem{Bordeaux} Ribet, K\@.  On the component groups and the Shimura subgroup
   of $J_o(N)$.  S\'em.\ Th.\ Nombres, Universit\'e Bordeaux,
   1987--88
\bibitem{MG} Serre, J-P.  Abelian $\ell$-adic Representations and
Elliptic Curves.  New York: W.A. Benjamin 1968.  (Republished 1989
by Addison-Wesley)
\bibitem{SerreIM} Serre, J-P.  Propri\'et\'es galoisiennes des
   points d'ordre fini des courbes elliptiques.  Invent.\ Math.\
   {\bf 15}, 259--331 (1972)
\bibitem{Duke} Serre, J-P.  Sur les repr\'esentations modulaires de degr\'e  
   2 de Gal$(\Qbar /\Q )$.  Duke Math. J. {\bf 54}, 179--230  (1987)
\bibitem{Serre-Tate} Serre, J-P. and  Tate, J\@. Good reduction of abelian
   varieties.  Ann. of Math. {\bf 88}, 492--517 (1968)
\bibitem{Sh} Shimura, G.  Introduction to the Arithmetic Theory of
  Automorphic   Functions.  Princeton: Princeton University Press 1971
\bibitem{Shimurapink} Shimura, G.  On elliptic curves with complex
   multiplication as factors of the Jacobians of modular function fields.
   Nagoya Math. J. {\bf 43}, 199--208 (1971) 
\bibitem{Silverman} Silverman, J.  The Arithmetic of Elliptic Curves.
   Graduate Texts in Mathematics {\bf 106}. New York: Springer-Verlag (1986)
\bibitem{Weil} Weil, A.  \"Uber die Bestimmung Dirichletscher Reihen
  durch Funktionalgleichungen. Math. Ann. {\bf 168},  165--172 (1967)
\end{thebibliography}
   

\bigskip

{\small \begin{tabular}[t]{c}K. A. Ribet\\
Mathematics Department\\University of
California\\Berkeley CA  94720\\USA
\end{tabular}\par}



\end{document}
