% AMS-TeX
\documentstyle{amsppt}
\nologo  % sorry -- delete this if you wish
\magnification=\magstep1
\leftheadtext{Kenneth A. Ribet}
\rightheadtext{Progress in Mathematics}

\newcount\sectCount

% We make a non-outer version of \head:
\edef\nonouterhead{\noexpand\head}

\def\section#1\endhead{\advance\sectCount by 1%
\nonouterhead\the\sectCount. #1\endhead\noindent}

% As usual, we automatically number the references

\newcount\refCount
\def\newref#1 {\advance\refCount by 1
\expandafter\edef\csname#1\endcsname{\the\refCount}}
  
  \newref  AL   %Atkin-Lehner
  \newref  BOSMA %Bosma-Lenstra
  \newref  CAREARLY
  \newref  COX
  \newref  DIAMONDREF
  \newref  FALT
  \newref  FLACH
  \newref  FREYONE
  \newref  FREYTWO
  \newref  GELBART
  \newref  LABESSE %actually by Gerardin and Labesse
  \newref  GOUCONTROL
  \newref  GOUVEA
  \newref  GOUSURVEY
  \newref  HAYESRIBET
  \newref  HEARSTRIBET
  \newref  HINDRY
  \newref  AXJ % Allyn Jackson
  \newref  LANGBOOK
  \newref  LANG
  \newref  LANGLANDS
  \newref  HWL
  \newref  MATSUMURA
  \newref  EIS  % Mazur eisenstein ideal paper
  \newref  PRIMEDEGREE
  \newref  MAZUR  %deformation theory
  \newref  GAD
  \newref  MAZURNOTES
  \newref  MAZURTILOUINE
  \newref  MIYAKE
  \newref  OESTERLE
  \newref  PRASAD
  \newref  RIBETSURVEY
  \newref  FERMAT
  \newref  TOULOUSE
  \newref  KOREA
  \newref  NOTICES
  \newref  MOTIVES
  \newref  RUBINSILVER
  \newref  RSNOTE
  \newref  SERRETAU
  \newref  SERREOLD
  \newref  SERRECOURS
  \newref  SERREIM
  \newref  ARCATA
  \newref  DUKE
  \newref  ST
  \newref  SHIMURAOLD
  \newref  SHCRELLE
  \newref  SHIMURABOOK
  \newref  SHIMURAPINK
  \newref  TUNNELL
  \newref  WILES





% some of my math macros, most not used

\let\phi\varphi
\def\U#1{\Z/#1\Z} % integers mod #1 without ( )
\def\UU#1{(\Z/#1\Z)^\ast} %units mod #1

\def\Q{{\bold Q}}
\def\Z{{\bold Z}}
\def\C{{\bold C}}
\def\Qbar{\overline{\Q}}
\def\F{{\bold F}}
\def\Fbar{\overline{\F}}
\def\Fq{\F_q}
\def\T{{\Bbb T}}
\def\m{{\goth m}}
\def\jo#1{J_0(#1)}
\def\go#1{\Gamma_0(#1)}
\def\ssigma{{}^\sigma\mkern-\thinmuskip}
\def\hlinefill{\leaders\hrule height 3pt depth -2.5pt\hfill}
\def\emrule{\thinspace\hbox to 0.75em{\hlinefill}\thinspace}
\def\GalQ{\Gal(\Qbar/\Q)}
\def\seteq{\mathrel{:=}}
\def\SL#1{{\bold{SL}}(2,#1)}
\def\GL#1{{\bold{GL}}(2,#1)}
\def\GLTWO{{\bold{GL}}(2)}

% provisional definition -- can one do better?
\def\notdivide{\setbox1=\hbox{$|$\llap{%\raise2pt -- nah!
\hbox{/}\kern-0.75pt}}\mathrel{\box1}}

%% Guillemets \`a la fran\c{c}aise
%% by jsv@cs.brown.edu and brouard@frined51
%% modified by bnb@math.ams.com  on friday, 10 November 1989
\def\<<{\leavevmode
  \raise0.28ex\hbox{$\scriptscriptstyle\langle\!\langle$}\nobreak
  \hskip -.6pt plus.3pt minus.2pt$\,$}
\def\>>{$\,$\nobreak\hskip -.6pt plus.3pt minus.2pt
  \raise0.28ex\hbox{$\scriptscriptstyle\rangle\!\rangle$}}

% make the \circ a little smaller
\let\tempcirc=\circ
\def\circ{\mathord{\raise0.25ex\hbox{$\scriptscriptstyle\tempcirc$}}}

% make some mathops
\def\newop#1
{\expandafter\def\csname #1\endcsname{\mathop{\roman{#1}}\nolimits}}
\newop Aut
\newop Gal 
\newop End 
\newop Frob
\newop tr


\topmatter

\title Galois representations and modular forms\endtitle
\author Kenneth A. Ribet\endauthor
\affil University of California, Berkeley\endaffil
\address{UC Mathematics Department,
  Berkeley, CA  94720-3840 USA}\endaddress
\email ribet\@math.berkeley.edu \endemail
\thanks 
This work was
to some extent
supported by NSF Grant \#DMS 93-06898.
I am grateful to
F.\,Q.~Gouv\^ea,
B.~Mazur,
C.~O'Neil,
R.~Taylor
and A.~Wilkinson
for comments and suggestions on drafts of these
notes.  A special thanks to S.~Ribet for moral support.
\endthanks

\endtopmatter

\document
Last fall, the organizers of 
this Mathfest asked me to discuss
``the mathematics behind Andrew Wiles's solution of the
Fermat conjecture.''
At the time it was anticipated that Fermat's Last Theorem
would be proved by the methods which Wiles
outlined
in his lectures last summer.  
The situation changed dramatically in December when
Wiles posted a message to 
the USENET news group
{\tt sci.math} stating
that
``the final calculation of a precise upper  
bound for the Selmer group\dots\
is not yet  
complete as it stands'' (see \cite\AXJ).
At this writing, the required upper bound has not been
obtained\emrule
Fermat's conjecture is not a theorem.
Nevertheless, it is my opinion
that the methods introduced
by Wiles are significant and continue to
merit our attention.
The most striking consequence of Wiles's work is that one
now has tools for exhibiting infinite families of
elliptic curves over the rational field which are attached
to modular forms~\cite\RSNOTE.  No such families were known
before last summer.


Because of the intense publicity surrounding Fermat's Last
Theorem, most of the background material in these notes
has been discussed in news and
expository articles which were
written on the heels of Wiles's announcement.
Among these are the author's news item in
the Notices~\cite\NOTICES\ and his article
with Brian Hayes in American Scientist~\cite\HAYESRIBET,
two pieces in the American Mathematical Monthly
\cite{\COX, \GOUVEA}, the report by K.~Rubin
and A.~Silverberg in the AMS Bulletin~\cite\RUBINSILVER,
and unpublished
notes by various authors which can
be obtained by gopher from {\tt e-math.ams.org}.
In view of the large literature in this
subject, I imagined
these notes more as
an annotated bibliography than as a survey.
In the end, it seems that I have written an
abbreviated survey which has a relatively large
list of~references.
I have tried to cite some papers which
are still in preparation, or have not yet appeared.
\medskip
I hope that these notes will be of interest to 
Mathfest participants.  
In all likelihood, a corrected version of this text
will appear in the AMS Bulletin.  So
if you find misprints, ambiguities or errors,
or if you have suggestions for improvements,
please let me know about them either in Minneapolis or by
e-mail.
My oral presentation on August 17 will provide a
general overview, slanted toward developments
which have emerged in the last year.

\section Modular forms\endhead
We begin with some background concerning
modular forms.  
Among reference books in the subject, one might
cite \cite\LANGBOOK, \cite\MIYAKE, and~\cite\SHIMURABOOK.
Also,
many books on elliptic curves contain substantial 
material on modular forms.
For a first introduction, a good
starting point is~\cite{\SERRECOURS, Ch.~VII}.

The modular forms we will consider are holomorphic
functions of one variable, defined
on the upper half-plane of complex numbers
with positive imaginary part.  These functions
are usually presented as converging Fourier series
$\sum_{n=1}^\infty a_n q^n$, where $q=e^{2\pi i z}$.
For the forms which 
most interest us, the coefficients $a_n$,
which are a priori complex numbers, are in fact algebraic
integers and often even ordinary integers.

Suppose that $N$ is a positive integer.
Let $\go N$ be
the group of integer matrices
with determinant~1 which are upper-triangular mod~$N$.
The weight-two cusp forms on~$\go N$
are holomorphic functions $f(z)$
on the complex upper half-plane which
obey the functional equation
$$f\left({az+b\over cz+d}\right)= (cz+d)^2 f(z)$$
for all
$\pmatrix a & b\cr c& d\cr\endpmatrix\in\go N$.
This equation implies, of course, that $f(z)$ is
invariant under integer translation.
In addition to the holomorphy and the functional
equation, one imposes a
suitable condition at infinity \cite{\SHIMURABOOK, Ch.~2}.
The cusp forms of weight two on~$\go N$ form
a complex vector space $S(N)$.
The dimension of~$S(N)$ is finite and may be calculated
easily using the Riemann-Roch theorem. 

The dimension
is~zero for $N\le10$ and is~1 for~$N=11$.
A generator for~$S(11)$ may be described in
terms of the formal power series 
with integral coefficients
$\sum a_n X^n$
defined by the identity
$$ X \prod_{m=1}^\infty (1-X^m)^2(1-X^{11m})^2 =
\sum_{n=1}^\infty a_nX^n.$$
The holomorphic function $\sum a_n q^n $
(with $q=e^{2\pi i z}$)
is a cusp form $h$
of weight two on~$\go{11}$, i.e., a non-zero
element of~$S(11)$.
Anticipating somewhat, we refer the reader to
Shimura's article~\cite\SHCRELLE,
which discusses the relationship between $h$ and a
certain elliptic curve, and introduces the Galois
groups $G_n$ which appear below.


For each integer $n\ge1$, the $n$th {\it Hecke operator\/}
on~$S(N)$ is an endomorphism
$T_n$ of~$S(N)$, whose action is generally written
$f\mapsto f|T_n$. 
The various $T_n$ commute with each other and are interrelated
by identities which express a given $T_n$ in terms of the
Hecke operators indexed by the prime factors of~$n$.
If $p$ is a prime, the operator $T_p$ may be described
easily from the perspective of Fourier coefficients.
Namely, if if $p$ is not a divisor of~$N$ and $f=\sum a_n q^n$,
then $f|T_p$ has the Fourier expansion 
$$ \sum_{n=1}^\infty a_{np} q^n + p \sum_{n=1}^\infty a_nq^{pn}.$$
If $p|N$, then
$f|T_p$ is given by the simpler formula
$\sum_{n=1}^\infty a_{np} q^n$.  



The elements of~$S(N)$ having special
arithmetic interest are the {\it normalized
eigenforms\/} in~$S(N)$; these are the non-zero
cusp forms $f=\sum a_nq^n$
which are eigenvectors for all the $T_n$ and which
satisfy the normalizing condition $a_1=1$.
If $f$ is such an eigenform, its Fourier coefficients
and its eigenvalues coincide: One has
$f|T_n = a_n f$ for all $n\ge1$.  The
normalized eigenforms are
``arithmetic'' because the~$a_n$ belong to the realm
of algebraic number theory:
If $f$ is a normalized
eigenform, then the subfield of~$\C$
generated by the $a_n$ is a finite (algebraic) extension $E$
of~$\Q$ in~$\C$, and the elements $a_n$ of~$E$ are
algebraic integers.

It is a mildly complicating
fact that the normalized eigenforms in~$S(N)$
do not always form a basis of~$S(N)$.  In other words,
the operators $T_n$ are not necessarily diagonalizable.
This problem arises from those $n$ which have a
common factor with~$N$, and can be repaired by the
introduction of {\it newforms\/}~\cite\AL.  Briefly, a newform
is a normalized eigenform $f=\sum a_n q^n$ for which
the space $\{\, g\in S(N) \, | \, g|T_n = a_n g \hbox{ for
all $n$ prime to $N$}\,\}$ contains only $f$ and its multiples.
Atkin and Lehner have shown
that $S(N)$ has a basis built out of 
suitable transforms of
the newforms
in the spaces $S(M)$,
where $M$ runs over the positive divisors of~$N$.  Thus $S(N)$
may be described purely in terms of newforms provided that
one includes newforms whose ``levels'' $M$ run over the
divisors of~$N$.

\section The Taniyama-Shimura conjecture\endhead
The Taniyama-Shimura conjecture relates elliptic curves
over~$\Q$ and certain modular forms.
Here is a compact description of the conjecture;
another is provided in the recent article
by K.~Rubin and A.~Silverberg~\cite{\RUBINSILVER, \S1}.
To describe the T-S conjecture, we need to allude to
some fundamental concepts pertaining to
elliptic curves.  For a serious treatment of these concepts,
the reader may consult a large number
of textbooks and
monographs which focus on 
elliptic curves.  (Several of these
books
discuss modular forms as well.)
Rather than list these books here,
we refer the reader to the bibliography of a
book review by W.~Hearst~III
and the author~\cite\HEARSTRIBET.

To describe the conjecture, we
begin with a construction due to Shimura.
Suppose 
that $f\in S(N)$ is a normalized eigenform and let
$E$ be the field generated by the coefficients of~$f$.
Shimura has associated to~$f$ an abelian variety $A_f$
over~$\Q$
whose dimension is the degree $[E\:\Q]$ and whose arithmetic
incorporates the eigenvalues $a_p$ for $p$ prime to~$N$
\cite{\SHIMURABOOK, Th.~7.14}.   %check this
The construction $f\mapsto A_f$ is sufficiently flabby
that it associates to $f$ a ``clump'' of {\it isogenous\/}
abelian varieties, rather than a specific abelian variety
which is singled out up to isomorphism.  (By definition,
two abelian varieties
are isogenous if and only if they are obtained from each other
by dividing by a finite group.  
A theorem of Faltings~\cite{\FALT, \S5} justifies
the vague statement that
two
abelian varieties are isogenous if and only if they are
equivalent arithmetically.)

If $f$ has integral Fourier coefficients,
then $E=\Q$ and $A_f$ has dimension~1.  This means that $A_f$
is an {\it elliptic curve\/} over~$\Q$, which we can regard
as being defined up to isogeny.  
The eigenvalues $a_p$ can be extracted from the
elliptic curve $A=A_f$
 via the process of
``reduction mod~$p$.''
More precisely, it is possible to
find cubic polynomials with integral
coefficients which define~$A$.  To each such polynomial,
one associates a non-zero discriminant.
There is an essentially unique defining equation
whose discriminant has smallest absolute value;
the discriminant of this equation
is said to be the
(minimal) discriminant $\Delta$ of~$A$.
The prime numbers which divide $\Delta$
are the primes at which $A$ has {\it bad reduction}; the
others are the primes at which $A$ has {\it good
reduction}.
For good primes~$p$,
the minimal equation, when viewed
mod~$p$, yields an elliptic curve $\tilde{A}_p$
over the
finite field $\Z/p\Z$.
The reduced curve
$\tilde A_p$ over~$\Z/p\Z$.
is a projective plane curve over~$\Z/p\Z$; 
its 
{\it rational points}, i.e., points with values in~$\Z/p\Z$,
form a finite abelian group which we can call~$A(\Z/p\Z)$.
A theorem of H.~Hasse states that the integer $$b_p\seteq
p{+}1-\#(A(\Z/p\Z))$$ is bounded in absolute value by~$2\sqrt p$.
The relation  between $f$ and $A=A_f$ is as follows.
Firstly, if $p$ does not divide~$N$, then $A$ has good reduction
at~$p$.  Further, for such $p$,
the $p$th
Fourier coefficient of~$f$ coincides with the quantity
$p{+}1-\#(A(\Z/p\Z))$.  In other words, we have
$b_p=a_p$ for all $p\notdivide N$.
(If $A_f$ has dimension greater than~1, the relation
between $f$ and $A_f$ is a bit more complicated to
describe, but involves no essentially new ideas.)

Incidentally, each elliptic curve $A$ over~$\Q$ has a
{\it conductor}, which is a positive
integer divisible precisely by the
primes at which $A$ has bad reduction.
(For the definition, see for example~\cite{\SERREOLD, \S2}.)
The conductor and the minimal discriminant of~$A$
are divisible by the same set of
prime numbers.  
Nevertheless, these integers have a completely
different feel.
For one thing, the conductor is always positive, while 
the minimal discriminant
may be positive or negative.  Further,
while the latter number may be
divisible by a large power of a given prime,
the conductor of an elliptic curve tends to be divisible
only by low powers of its prime divisors. 
(In fact,
a prime $p\ge5$ can divide the conductor
 at most to the second
power.)
Finally, the conductor of an elliptic curve
depends only on its $\Q$-isogeny class; the
discriminant changes under isogeny.

It happens quite frequently that
the conductor is square free, i.e., divisible
only by the first powers of its prime divisors. The
elliptic curve $A$ is then
said to be {\it semistable}.

Suppose again that $f$ is an eigenform in~$S(N)$ 
with integral Fourier coefficients
and
that $A$ is the elliptic curve~$A_f$.
It is natural to ask how the conductor of~$A$
is related to~$N$.  The answer begins with the fact that 
$f$ is necessarily built from a newform in~$S(M)$, for
some divisor $M$ of~$N$.  It is then true that
the conductor of~$A$
is precisely this divisor $M$.  This theorem was proved
by H.~Carayol in~\cite\CAREARLY, following work of
Shimura, Igusa, Deligne and Langlands.


The Taniyama-Shimura conjecture has several equivalent
formulations.  In one form, it states that
every elliptic curve over~$\Q$ is isogenous to a some
curve $A_f$.  A more precise statement runs as
follows:
{\it Let $A$ be given over~$\Q$
and suppose that $N$ is the conductor of~$A$.
Define 
integers $b_p$ for $p\notdivide N$ as above.
Then there is a newform $f\in S(N)$
whose Fourier coefficients $a_n$ satisfy the identity $a_p=b_p$
for $p\notdivide N$.}  

The theory of newforms shows that
$f$ is unique, if it exists.
Also, if $f$ exists, we may apply ``Tate's conjecture"\emrule
now a
theorem of Faltings~\cite{\FALT, \S5}\emrule to
show that $A$ and~$A_f$
are isogenous over~$\Q$.

If the Taniyama-Shimura conjecture is true for~$A$,
we say that $A$ is {\it modular}.

Mazur's article~\cite\GAD\ formulates
the Taniyama-Shimura conjecture purely as a statement
about the Riemann surface associated with an elliptic curve
over~$\Q$.
If $A$ is such an elliptic curve, write $A(\C)$ for the
space of points of~$A$ with complex coordinates, so
that $A(\C)$ is a complex torus.
For each integer~$N$,
consider the group $\Gamma(N)$ consisting of
integer matrices with determinant~1 which are congruent to
the identity matrix modulo~$N$.
This group acts naturally on
the complex upper half-plane $\Cal H$ by fractional linear
transformations, with
$\pmatrix a & b\cr
c& d\cr\endpmatrix$ giving rise
to $$z\mapsto {az+b\over cz+d}.$$  The quotient
$\Gamma(N)\backslash\Cal H$
has a natural compactification called~$X(N)$.
Mazur shows that an elliptic curve $A$ over~$\Q$ is modular
if and only if one can find a non-constant holomorphic
map from $X(N)$ to~$A(\C)$, for some positive integer~$N$.

In~\cite\KOREA, I consider the set of elliptic curves $A$ over~$\C$
for which there is a non-constant holomorphic
map $X(N)\to A(\C)$ for some positive integer~$N$.
Answering a question of Serre, I show that the Taniyama-Shimura
conjecture may be generalized to a natural conjectural
characterization of such elliptic curves.


\section Galois representations attached to elliptic curves\endhead
Let $A$ be an elliptic curve over~$\Q$.
We
view $A$ as a commutative {\it algebraic group\/} over~$\Q$.
Concretely, one models $A$ as the projective plane
curve defined by an affine cubic equation of the form $y^2=x^3+ax+b$
with $a,b\in\Q$.
If $K$ is a field containing~$\Q$, the set of points of~$A$
with values in~$K$ is then a subset $A(K)$ of the projective plane
over~$K$ with a classical group law, the so-called
``chord and tangent operation.''  There is a unique point $O$
of~$A$ outside the affine plane; this is taken to be the 
identity element
in the group. Three distinct points on $A$ then
sum to~$O$ if and only
if they are colinear.  The composition law on~$A(K)$ can
be described in terms of coordinates by a family of polynomial
equations with coefficients in~$\Q$.  
(For a recent discussion concerning families of such
equations, the reader may consult~\cite\BOSMA.)

A model for $A(K)$ when $K=\C$
is given by Weierstra\ss\ theory: the group $A(\C)$ is
the complex torus~$\C/L$,
where $L$ is the lattice of periods associated to the given
cubic equation.  (Explicitly, $L$ is
obtained by integrating the differential 
${\displaystyle dx\over\displaystyle y}$ 
on~$A$ over the
free abelian group $H_1(A(\C),\Z)$ of rank~two.)  Let $n$
be a positive integer, and let $A[n]$ be the group of
elements of~$A(\C)$ whose order divides~$n$.  This group
of {\it $n$-division points\/} on~$A$ may be modeled as 
${1\over n}L/L$; it is therefore a free module of rank two
over $\U n$, since $L$ is free of rank two over~$\Z$.

Further analysis shows that $A[n]$ in fact lies in~$A(\Qbar)$,
where $\Qbar$ is the subfield of~$\C$ consisting of
all algebraic numbers.  Indeed, a point of~$A(\C)$
has order
dividing~$n$ if and only if its coordinates satisfy a 
certain set
of polynomial equations with rational coefficients; it follows
that the coordinates are algebraic numbers.
Moreover, let
$\GalQ$ be the group of automorphisms
of~$\Qbar$.  Then the same reasoning shows that
$A[n]$ is stable under the natural action of~$\GalQ$
on~$A(\Qbar)$.  Thus
$A[n]$ comes equipped with a canonical action of the
Galois group $\GalQ$.

It is important to observe, for each $\sigma\in\GalQ$, that
the automorphism $P\mapsto\ssigma P$ is a {\it group\/} automorphism
of~$A[n]$: We have $\ssigma(P+Q)=\ssigma P+\ssigma Q$ for
$P,Q\in A[n]$. 
This follows from the fact that the composition law which
expresses $P+Q$ in terms of $P$ and~$Q$ involves polynomials
with rational coefficients.
Our observation means simply that
the action of~$\GalQ$ on~$A[n]$ may be viewed
as a (continuous) homomorphism
$$ \rho_{A,n}\:\GalQ\to\Aut(A[n])$$
in which $\Aut(A[n])$ stands for the group
of automorphisms of~$A[n]$ as an {\it abelian group}.
Since $A[n]$ is isomorphic
to the group $(\U n)^2$, $\Aut(A[n])$ is isomorphic to
the group $\GL{\U n}$ of two-by-two invertible matrices with
coefficients
in~$\U n$.  While there is no canonical isomorphism
$\Aut(A[n])\approx \GL{\U n}$, each choice of basis
$A[n]\approx (\U n)^2$ determines such an isomorphism,
and the various isomorphisms obtained in this way
differ by inner automorphisms of~$\GL{\U n}$.  Therefore,
each element of $\Aut(A[n])$ has a well-defined trace
and determinant in~$\U n$.  

It is often fruitful to fix a choice of basis 
$A[n]\approx (\U n)^2$ and to view $\rho_{A,n}$
as taking values in the matrix group $\GL{\U n}$.
The kernel of $\rho_{A,n}$ corresponds, via Galois
theory, to a finite Galois extension $K$ of~$\Q$ in~$\Qbar$.
Concretely, $K$ is the extension of~$\Q$ obtained by
adjoining to the rational field the coordinates of the
various points in~$A[n]$.  The Galois group $\Gal(K/\Q)$
is the image of $\rho_{A,n}$; it is therefore embedded
in the target group $\GL{\U n}$.  The elliptic curve $A$
and the positive integer $n$ have given rise to a finite
Galois extension $K/\Q$ whose Galois group is a subgroup $G_n$
of the group of two-by-two invertible matrices with
coefficients in~$\U n$.

It is natural to ask for a description of this~$G_n$ as a
subgroup of~$\GL{\U n}$.  There is a 
(relatively rare) special case to
consider: that where $A$ has {\it complex multiplication\/} (over~$\C$).
Viewing $A$ as~$\C/L$, we may define
the complex multiplication
case to be that for which there is a non-integral complex
number $\alpha$ such that $\alpha L\subseteq L$.
The group $G_n$ then has an abelian subgroup
of index~$\le2$, so it is quite far from the ambient group
$\GL{\U n}$.  In the much more common case where $A$ has no
complex multiplication, Serre has shown that the index of~$G_n$
in~$\GL{\U n}$ is bounded as a function of~$n$.
An essentially equivalent piece of information is that
one has the equality 
$G_p = \GL{\U p}$ for all but finitely many primes~$p$
(see \cite\SERREIM).

To orient the reader, we might point out that the Taniyama-Shimura
conjecture was proved for complex multiplication elliptic
curves over~$\Q$ by Shimura in 
1971 \cite\SHIMURAPINK.  On the other hand,
it is relatively easy to show that semistable elliptic curves
over~$\Q$ {\it never\/} have complex multiplication.
As we shall see below, the elliptic curves which appear in
connection with Fermat's Last Theorem are semistable elliptic
curves.  Accordingly, the result of~\cite\SHIMURAPINK\ sheds
no light on the question of whether or not they are modular.

A key piece of
information
about the extension $K/\Q$ (which depends on~$n$
and~$A$) is that its discriminant
is divisible only by those prime numbers
which divide the product of~$n$ and the conductor of~$A$.
In other words, if $p\notdivide n$ is a prime number at
which $A$ has good reduction, then $K/\Q$ is unramified
at~$p$.  In this case, one can introduce a Frobenius element
$\sigma_p$ in~$G_n$ which is well defined up to conjugation.
Since $G_n$ is a group of matrices,
we may view $\sigma_p$ as a matrix whose
trace and determinant
are well-defined elements of~$\U n$.
It is
easy to prove
that $\det \sigma_p$ is the number $p$ mod~$n$.  On the
other hand, one has the striking congruence
$$ \tr(\sigma_p) \equiv b_p \hbox{ mod }n,$$
where $b_p$ is the number $p{+}1-\#(A(\Z/p\Z))$ introduced
above.  This means that the representation $\rho_{A,n}$
encapsulates information about the numbers $b_p$ (for $p$
prime to~$n$); more precisely, it determines the
numbers $b_p$ mod~$n$.


\section Galois representations attached to modular forms\endhead
Suppose that $f\in S(N)$ is a normalized eigenform.  If the
coefficients $a_n$ of~$f$ happen to be integers,
the abelian variety $A=A_f$ is an elliptic curve.
By considering the
family $\rho_{A,n}$, we obtain a series of
representations of the Galois group~$\GalQ$.  These representations
are related to~$f$ by the congruence 
$ \tr(\sigma_p) \equiv a_p \hbox{ mod }n$, valid for the $n$th
representation and all primes $p\notdivide nN$.  We are especially
interested in the case where $n$ is a prime number~$\ell$;
the ring $\U n$ is then the {\it finite field}~$\F_\ell$.

The representations
$\rho$ are associated to~$A$, which in turn arises from~$f$.
Hence we are tempted to write $\rho_{f,\ell}$
for the representations $\rho_{A,\ell}$.  The obstacle to
doing this arises from the circumstance
that $A$ is determined
up to isogeny, but not always
up to isomorphism.  If we replace $A$
by an isogenous elliptic curve, the representations $\rho_{A,\ell}$
may change!  To circumvent this difficulty, we introduce the
``semisimplifications'' of the~$\rho_{A,\ell}$.

The general definition runs as follows.
If $\rho$ is a two-dimensional
representation of a group over a field, $\rho$ is either
irreducible, or else ``upper-triangular,'' i.e., an extension
of a one-dimensional representation
$\alpha$ by another, $\beta$.  In the case
where $\rho$ is irreducible, we declare its semisimplification
to be $\rho$ itself.  In the reducible case, the semisimplification
of~$\rho$ is the {\it direct sum\/} of the two one-dimensional
representations $\alpha$ and~$\beta$.
Clearly, the trace and determinant are the same
for~$\rho$ and for its semisimplification.

When $A$ is fixed, $\rho_{A,\ell}$ can be reducible only
for a finite number of~$\ell$.  In fact, a theorem of
Mazur\cite\PRIMEDEGREE\ shows that $\rho_{A,\ell}$
is irreducible for all $\ell$ not in the set
$\{\,2, 3, 5, 7, 13, 11, 17, 19, 37, 43, 67, 163\,\}$.  
Hence the replacement
of~$\rho_{A,\ell}$ by its semisimplification can be
thought of as ``fine tuning'' which affects only a
small number of the representations.
One shows
easily for all~$\ell$
that the semisimplification of~$\rho_{A,\ell}$ 
depends only on~$f$ and on~$\ell$ (but not on the choice of~$A$).
Introducing
$$ \rho_{f,\ell} \seteq \hbox{semisimplification of }\rho_{A,\ell},$$
we obtain a sequence of semisimple representations of~$\GalQ$
which are well defined up to isomorphism.  The characteristic
property of~$\rho_{f,\ell}$ may be summarized in terms of
Frobenius elements $\Frob_p$ in the Galois group $\GalQ$:
If $p$ is a prime number not dividing $\ell N$, then
$\rho_{f,\ell}(\Frob_p)$ has trace $a_p$ mod~$\ell$ and determinant
$p$ mod~$\ell$.

Our aim will be to reverse this process: Given
a
two-dimensional representation of~$\GalQ$ over~$\Z/\ell\Z$,
we would like to show in some circumstances that this
representation arises from some eigenform~$f$.  
Although this program may seem quite natural, examples
show that
it is unreasonable to expect $f$ to have integral
coefficients;
we must confront the situation where the coefficients of~$f$
are algebraic integers but not ordinary integers.

Suppose then that $f=\sum a_n q^n$ is a normalized eigenform,
and let $E$ be the field
generated by the~$a_n$.  Let $\Cal O$ be the ring
of integers of~$E$, so that $a_n\in\Cal O$ for all $n\ge1$.  
Using the abelian variety $A_f$, one constructs
representations indexed not by the prime {\it numbers}, but
rather by the non-zero prime {\it ideals\/} of~$\Cal O$.
(For details, see~\cite{\SHIMURABOOK, Ch.~7}.)
If $\lambda$ is such a prime, its residue field $\F_\lambda$ is
a finite field, say of characteristic~$\ell$.  The prime
field $\F_\ell=\U\ell$ is then canonically embedded in~$\F_\lambda$.
For each $\lambda$, one finds a semisimple representation
$\rho_{f,\lambda}\:\GalQ\to\GL{\F_\lambda}$
which is characterized up to isomorphism
by the following property.  If $p$ is
a prime number not dividing $\ell N$, then 
$\rho_{f,\lambda}(\Frob_p)$ has trace $a_p$ mod~$\lambda$ and
determinant $p$ mod~$\lambda$.

The assertion concerning the determinant 
of the matrices~$\rho_{f,\lambda}(\Frob_p)$ 
may be rephrased in terms
of the
determinant of
the representation~$\rho_{f,\lambda}$.  The latter
determinant is
a continuous homomorphism $\GalQ\to\F^\ast_\lambda$.
This homomorphism turns out to be the composite of
the ``cyclotomic character'' $\chi_\ell\:\GalQ\to\F_\ell^\ast$
and the natural inclusion $\F_\ell^\ast\subseteq\F_\lambda^\ast$.
The cyclotomic character may be defined by considering
the group $\mu_\ell$ of $\ell$th roots of unity in~$\Qbar$.
The Galois group $\GalQ$ acts on~$\mu_\ell$ by conjugation,
giving rise to a continuous homomorphism
$$ \GalQ\to\Aut(\mu_\ell).$$
Since $\mu_\ell$ is a cyclic group of order~$\ell$,
its group of automorphisms is the group 
$\UU\ell=\F_\ell^\ast$.  We emerge with a 
map $\GalQ\to\F_\ell^\ast$,
which is the character in question.

Suppose now that $c\in\GalQ$ is the
automorphism ``complex conjugation.''  Then the determinant
of~$\rho_{f,\lambda}(c)$ is~$\chi_\ell(c)$.  Now $c$
operates on roots of unity by the map $\zeta\mapsto\zeta^{-1}$,
since roots of unity have absolute value~1.  Thus we have 
$\det(\rho_{f,\lambda}(c))=-1$; one says that $\rho_{f,\lambda}$
is {\it odd}. 

This parity observation generalizes to modular forms of
weights other than~two.  Here is a quick synopsis of the
situation; some relevant references are provided
in~\cite\RIBETSURVEY.  For integers $k\ge2$,
$N\ge1$ and characters $\epsilon\:\UU N\to\C^\ast$, one
considers the space $S_k(N,\epsilon)$ of weight-$k$ cusp
forms with character $\epsilon$ on~$\go N$; we have
$S(N)=S_2(N,1)$.  
This space is automatically zero unless $\epsilon(-1)=(-1)^k$,
so we will assume that this parity condition is satisfied.
The space $S_k(N,\epsilon)$ admits an
operation of Hecke operators $T_n$, and we again have the
concept of a normalized eigenform in~$S_k(N,\epsilon)$.
If $f=\sum a_nq^n$ is such a form, the numbers $a_n$ ($n\ge1$)
and the values of~$\epsilon$ all lie in a single integer 
ring~$\Cal O$. For each non-zero prime ideal $\lambda$ of~$\Cal O$,
one constructs a semisimple representation
$$\rho_{f,\lambda}\:\GalQ\to\GL{\F_\lambda}.$$
Let $\ell$ again denote the characteristic of~$\F_\lambda$.
Then for all $p\notdivide \ell N$, the trace 
of~$\rho_{f,\lambda}(\Frob_p)$ is again $a_p$ mod~$\lambda$.
The determinant of this matrix is 
$p^{k-1}\epsilon(p)$ mod~$\lambda$.

Once the proper definition is made,
the determinant of the {\it map\/} $\rho_{f,\lambda}$
becomes the product $\chi_\ell^{k-1}\epsilon$, where
$\chi_\ell$ again denotes the mod~$\ell$ cyclotomic
character.  
For this, we regard $\epsilon$
as a map $\GalQ\to\F_\lambda^\ast$.  To do this,
we compose the
natural map $\GalQ\to\UU N$ which gives the action of~$\GalQ$
on the $N$th roots of unity with the character $\epsilon$,
thus obtaining a map $\GalQ\to{\Cal O}^\ast$.  On reducing
this homomorphism mod~$\lambda$, we obtain the desired
variant of~$\epsilon$.
Evaluating the determinant formula on the ``complex
conjugation'' element of~$\GalQ$, we
find
$$\det(\rho_{f,\lambda}(c)) = (-1)^{k-1}\epsilon(c) = -1.$$
In these equalities, we exploit the fact that
$\epsilon(c)$ is another name for~$\epsilon(-1)$ and
remember the parity condition
$\epsilon(-1)=(-1)^k$.  The upshot of this
is that the representations
$\rho_{f,\lambda}$ are always odd, even in the generalized
set-up.

In fact, it is possible in this situation to find an eigenform
$f'$ of weight~two with some character~$\epsilon'$, along with
a maximal ideal $\lambda'$ of the integer ring for~$f'$,
so that the representations $\rho_{f,\lambda}$
and~$\rho_{f',\lambda'}$ are isomorphic.  (To compare
these representations, we need to embed the residue fields
of~$\lambda$ and of~$\lambda'$ in a common finite field
of characteristic~$\ell$.)  Hence the ``generalized set-up''
can be reduced to the case of weight~two, although one
needs to consider forms for which the associated characters
may not be trivial.  The idea of reduction to weight two
is a very powerful one in the theory, since the representations
arising from forms of this weight are constructed directly
from points of finite order on suitable abelian varieties.
To the best of my knowledge, this idea originated with an
unpublished 1968 manuscript of Shimura~\cite\SHIMURAOLD.

We said above that we would like to reverse the process
of associating representations to modular forms.
The idea that this might be possible was raised by~Serre,
initially
for modular forms of level~1 (i.e., on~$\SL\Z$) in the 1970s,
and then more generally in the mid-1980s.
After computer calculations
by J-F.~Mestre convinced Serre
that the statement was likely to  be true,
Serre conjectured in~\cite\DUKE\
that all two-dimensional
odd irreducible representations of~$\GalQ$ over a finite
field are 
essentially of the form~$\rho_{f,\lambda}$.
To give meaning to this statement,
we choose a prime number $\ell$
and let $\F$ be an algebraic closure of the prime field
$\F_\ell$.  Suppose that $\rho\:\GalQ\to\GL\F$ is an odd
continuous irreducible representation.  
We shall say that $\rho$ is {\it modular\/} if
one can find: (i) an eigenform $f$ in
some space $S_k(N,\epsilon)$; (ii) a prime $\lambda$
dividing $\ell$ in the ring of integers $\Cal O$ associated
to~$f$; and (iii) an embedding $\F_\lambda\hookrightarrow\F$
such that $\rho$ is isomorphic to the representation
obtained by composing $\rho_{f,\lambda}$ with the
inclusion $\GL{\F_\lambda}\hookrightarrow\GL\F$ associated
with~(iii).  Serre has conjectured that every two-dimensional
odd irreducible representation of~$\GalQ$ is modular.
It has emerged that this
seemingly imprecise conjecture is equivalent to an
intricate statement (also given in~\cite\DUKE), which
immediately implies
the Taniyama-Shimura
conjecture, Fermat's Last Theorem, and a host of other
assertions! 



\section Frey's construction\endhead
Almost all recent work on Fermat's Last Theorem begins with 
the connection between Fermat's equation
and elliptic curves.  Although Y.~Hellegouarch and
others had noted such connections,
a decisive step
was taken by G.~Frey in 
an unpublished 1985 manuscript entitled ``Modular
elliptic curves and Fermat's conjecture.''

Frey's idea runs as follows.
Suppose that we have a non-trivial
solution to Fermat's equation
$X^\ell+Y^\ell=Z^\ell$.  We can assume that the exponent
is
a prime number different from 2 and~3 and that
the solution is given by a triple of 
relatively prime integers.
Changing the sign in the value of~$Z$, we obtain
a 
triple $(a,b,c)$
which
satisfies the symmetric equation
$a^\ell+b^\ell+c^\ell=0$.
The equation
$y^2=x(x-a^\ell)(x+b^\ell)$
defines an elliptic curve $A$
with unexpected properties.

These properties are catalogued in~\S4.1 of Serre's
article~\cite\DUKE.
After 
a possible permutation of the three integers
$a$, $b$ and~$c$ defining~$A$,
one finds that the conductor of~$A$
is the product of the prime
numbers which divide $abc$
(each occurring to the first power), 
so that $A$ is semistable.
On the other hand, the minimal discriminant
$\Delta$ of~$A$
is the quotient of~$(abc)^{2\ell}$ by the
trivial factor $2^8$.  From our point of view,
the  main ``unexpected'' property of~$A$ is that
$\Delta$ is the product of a 
power of~2 and a
perfect $\ell$th power, where $\ell$ is a prime~$\ge5$.
(From the point of view of 
Szpiro's conjecture and
the $abc$ conjecture, the surprising feature
of~$A$ is that its discriminant is larger than a
high power of its conductor.)


This property
persuaded Frey
that $A$ was unlikely to be modular.
In other
words,
Frey  concluded
heuristically that the existence
of~$A$ was incompatible
with the Taniyama-Shimura conjecture, which asserts that
all elliptic curves over~$\Q$ are modular. 

Frey's construction spawned several lines of inquiry,
in which mathematicians sought either to prove Fermat's
Last Theorem outright or to link it to established conjectures.
Much of what emerged is off the main topic of these
notes\emrule we will not discuss the $abc$ conjecture or
Szpiro's conjecture, for instance.  Note, however, that
these conjectures are treated by
such articles as
\cite\FREYONE, \cite\FREYTWO, \cite\LANG,
\cite\HINDRY, and~\cite\OESTERLE.


A first step toward justifying
Frey's heuristic conclusion
was taken in August, 1985 by
Serre in a letter to J-F.~Mestre~\cite\ARCATA.
In this letter,
Serre formulated two related conjectures about
modular forms, which he called
$C_1$ and~$C_2$.  He pointed out
that Fermat's Last Theorem is a consequence
of the Taniyama-Shimura conjecture {\it together with\/}
the two new conjectures.
In other words, Serre exhibited the implication
$$ C_1 + C_2 + \hbox{Taniyama-Shimura} 
    \Longrightarrow \hbox{Fermat}.$$
In my article~\cite\FERMAT, I proved Serre's conjectures
$C_1$ and~$C_2$.
My theorem established the
implication
  $$\hbox{Conjecture of Taniyama-Shimura }
       \Longrightarrow\hbox{ Fermat}$$
which was the goal of Frey's construction.
More generally, let $\ell\ge5$ be a prime number and
suppose that $A$ is a semistable elliptic
curve over~$\Q$ whose discriminant is the product of a
power of~2 and a perfect $\ell$th power.  Then 
the result of~\cite\FERMAT\ shows that
$A$ cannot
be a modular elliptic curve.
(See \cite\PRASAD\ and \cite\TOULOUSE\
for expository accounts of this work.)

After writing \cite\ARCATA, Serre
presented in~\cite\DUKE\ a broad group
of conjectures linking mod~$p$ Galois
representations and mod~$p$ modular forms.
These conjectures constitute the ``precise statement''
that was mentioned briefly at the end of the
last section.  As was alluded to above,
Serre's conjectures imply Fermat's Last Theorem by a
simple direct argument, and they imply a host of other
statements about Diophantine equations and elliptic curves.
There seems no hope of proving Serre's
conjectures in the near term.
Nevertheless, certain consequences of the conjectures
for the mod~$p$ Galois representations associated to
modular forms have
seemed
quite accessible.
A group of these consequences
have become known as the ``weak Serre
conjecture''; this conjecture is an intricate
generalization of the two
statements $C_1$ and~$C_2$ formulated in~\cite\ARCATA.
Thanks to the work of a large group of mathematicians, the
weak Serre conjecture is now a theorem, at least for $p\ge3$;
see \cite\MOTIVES\ and~\cite\DIAMONDREF\ for details.

The relationship between the weak Serre conjecture and
the
full group of conjectures of~\cite\DUKE\ 
is that the conjectures of~\cite\DUKE\ amount to
the conjunction of the weak conjecture and the 
single supplementary
statement which 
was introduced at the end of the previous section.
This statement, to the effect that two-dimensional
continuous
irreducible odd
representations of~$\GalQ$ are modular,
is the one which appears to be intractable.

\section Wiles's strategy\endhead
Suppose that $A$ is an elliptic curve over~$\Q$.
To verify the Taniyama-Shimura conjecture for~$A$
is to link $A$ to modular forms.  For example, we
might consider the representations $\rho_{A,\ell}$ obtained
from the action of~$\GalQ$ on~$A[\ell]$, when $\ell$ is a prime
number. If one shows that an infinite number of these
representations are modular (in the broadest possible
sense), one can go on to prove that $A$ is modular.
The difficulty with this approach, from a philosophical point
of view, is that there is no visible program for manufacturing
modular forms which can be compared with the~$\rho_{A,\ell}$.

Alternatively, one may fix a prime~$\ell$ and consider the
family of spaces $A[\ell^\nu]$ for $\nu=1,2,\ldots$. The
resulting sequence of representations 
$$ \rho_{A,\ell^\nu}\:\GalQ\to\GL{\U{\ell^\nu}}$$
may be re-packaged as a single representation
$$ \rho_{A,\ell^\infty}\:\GalQ\to\GL{\Z_\ell},$$
where $\Z_\ell$ is the ring of $\ell$-adic integers,
i.e., the projective limit of the rings $\U{\ell^\nu}$.
The natural map $\Z\to\Z_\ell$ is injective, so that
$\Z_\ell$ contains $\Z$ as a subring.
To prove that $A$ is a modular elliptic curve, it suffices
to show that $\rho_{A,\ell^\infty}$ is modular in an
appropriate sense.  Indeed, the trace of 
$\rho_{A,\ell^\infty}(\Frob_p)$ coincides with the 
rational integer $b_p$
for all $p\notdivide \ell N$, where $N$ is the conductor of~$A$.
Once one finds an eigenform $f$ in~$S(N)$ whose
Fourier coefficients are related to the
traces of~$\rho_{A,\ell^\infty}(\Frob_p)$, one has essentially
proved that $A$ is modular.

Of course, 
if $\rho_{A,\ell^\infty}$ is modular, then so is
$\rho_{A,\ell}$.  Relating $\rho_{A,\ell^\infty}$ to
modular forms is therefore at least as hard as
the formidable task of
relating
$\rho_{A,\ell}$ to modular forms!
On the other hand, to prove that $A$ is modular by the
$\ell$-adic method, we need only work with a {\it single\/}
prime~$\ell$.  Wiles's approach is based on the lucky
circumstance
that the representations $\rho_{A,\ell}$ are in fact modular
for $\ell\le3$, cf.~\cite{\RUBINSILVER, \S2.3}.
This circumstance arises from the theory of base change
\`a la Saito-Shintani, as developed 
by Langlands in~\cite\LANGLANDS\ and
then applied by~Tunnell in~\cite\TUNNELL. 
(For expositions of the results of Langlands, see
\cite\GELBART\ and \cite\LABESSE.)

Wiles's basic idea is to
prove that that if $\ell$ is a prime for which
$\rho_{A,\ell}$ is modular, then
$\rho_{A,\ell^\infty}$ is automatically modular (and hence $A$
is a modular elliptic curve).
In thinking about the jump from~$\rho_{A,\ell}$
to~$\rho_{A,\ell^\infty}$, one seeks to ignore $A$ as
much as possible\emrule the aim is to prove a theorem
about $\ell$-adic representations with properties
generalizing those of~$\rho_{A,\ell^\infty}$.

Two theorems in this direction are stated
in~\cite\WILES.
In each theorem, the prime $\ell$ is taken to be~odd,
and the representation $\rho_{A,\ell}$ is required to
be irreducible.  One theorem pertains to the case where
$A$ is semistable; this is the result which
would apply to Frey curves arising in
connection with Fermat's Last Theorem, and also the
theorem which is in difficulty.  
That there is difficulty means that the theorem is not
known to be true for all semistable elliptic curves
over~$\Q$.  However,
it is apparently
possible to use the technique of the proof
to exhibit a genuinely infinite class of semistable
elliptic curves over~$\Q$ which are modular.  
As I mentioned at the beginning of these notes, no such
class was known before Wiles's methods were introduced.
The second theorem 
treats the case where the representation~$\rho_{A,\ell}$
arises from a complex multiplication elliptic curve $A'$ over~$\Q$.
Here, there is no ``gap'' in the proof,
and one can again use the theorem to
exhibit an explicit infinite
collection of elliptic curves over~$\Q$ which are 
modular~\cite\RSNOTE.

Wiles seeks to use the first theorem to prove that
all semistable elliptic curves are modular.
His ingenious
argument for doing this is sketched by Rubin and
Silverberg in~\cite{\RUBINSILVER, Prop.~2.4}.
Suppose that $A$ is a semistable elliptic curve over~$\Q$.
If $\rho_{A,3}$ happens to be irreducible, then the
results of Langlands and Tunnell, plus the assertion of Wiles's
first theorem, give the desired result that $A$ is
modular.  The hard case is that where
$\rho_{A,3}$ is
{\it reducible}. 
If $\rho_{A,5}$ is reducible as well, then it is not
hard to show directly that $A$ is modular.  So we might
as well assume that $\rho_{A,5}$ is irreducible.
Wiles shows then that one can find
a second elliptic
curve $A'$
whose mod~5 representation is isomorphic to that of~$A$ and
whose mod~3 representation is irreducible~\cite{\RUBINSILVER,
Appendix~B}.
Two applications of the ``first theorem''
then suffice to show that $A$ is modular.

\section The language of deformations\endhead
Suppose that $A$ is a semistable elliptic curve over~$\Q$
and that $\ell\ge3$ is a prime number for which $\rho_{A,\ell}$
is both modular and
irreducible.  
Choosing a basis of~$A[\ell]$, we regard $\rho_{A,\ell}$
as taking values in~$\GL{\F_\ell}$.
We seek to show that $\rho_{A,\ell^\infty}$ is
modular by showing that {\it all\/} suitable lifts of $\rho_{A,\ell}$
are modular.  The idea of looking at all lifts at once stems
from a fundamental article of Mazur~\cite\MAZUR.
For historical reasons arising from algebraic geometry, isomorphism
classes of~lifts
are called ``deformations.''  (For a recent survey of the
subject of deformations, see~\cite\GOUSURVEY.)

The meaning of ``suitable'' depends on an
auxiliary choice: a finite set of prime numbers $\Sigma$ which
contains $\Sigma_0$, where $\Sigma_0$ is the set
of primes at which $\rho_{A,\ell}$ is ramified.
(Thus, by definition,
$\Sigma_0$ is the set of primes dividing the discriminant of the
number field obtaining by adjoining to~$\Q$ the coordinates
of all points in~$A[\ell]$.) A priori,
$\Sigma_0$ contains~$\ell$
and is contained in the union of~$\{\ell\}$ and the set of
primes at which $A$ has bad reduction. Indeed,
this union is the
set of primes $\Sigma^\ast$ 
at which~$\rho_{A,\ell^\infty}$ is ramified,
according to
the well-known criterion of N\'eron-Ogg-Shafarevich~\cite\ST.

That $\Sigma_0$ might be strictly smaller
than the union should be regarded as a complicating factor.
The reason for this is that
the so-called ``minimal'' case $\Sigma=\Sigma_0$ is expected to
be the easiest to deal with, while the choice $\Sigma=\Sigma^\ast$
is the one which is relevant to
the representation~$\rho_{A,\ell^\infty}$.

The following description is derived 
from~\S3.4 of~\cite\RUBINSILVER.
Our discussion differs from that of~\cite\RUBINSILVER\
mainly in our writing $\ell$ for the prime called~$p$
in~\cite\RUBINSILVER\ and in our focus on the
auxiliary choice~$\Sigma$.


Having fixed this choice for the moment,
one considers continuous
homomorphisms $\tilde\rho\:\GalQ\to\GL A$, where $A$
is a complete Noetherian $\Z_\ell$-algebra with residue
field~$\F_\ell$.  
(For technical reasons, it might be useful to replace
$\Z_\ell$ by the integer ring of a finite extension of
the $\ell$-adic field~$\Q_\ell$, but we will ignore this
complication.)
One demands a number of properties
of~$\tilde\rho$, which can be summarized as follows:
First, note that the residue map $A\to\F_\ell$ induces
a homomorphism $\GL A\to\GL{\F_\ell}$; we require that 
$\rho_{A,\ell}$ coincide with the
composite of~$\tilde\rho$ and this residue homomorphism.
Second, we demand that $\tilde\rho$ be unramified outside~$\Sigma$.
Third, we ask that $\tilde\rho$ 
have the same qualitative
behavior at the primes in~$\Sigma_0$
as the representation~$\rho_{A,\ell}$.
(The idea of imposing this condition can be traced
back to~\cite\GOUCONTROL.  Note that, locally at~$\ell$,
we ask that $\tilde\rho$
be ``ordinary'' if $A$ has ordinary or multiplicative
reduction at~$\ell$ and that $\tilde\rho$ be ``flat''
if $A$ has supersingular reduction at~$\ell$, 
cf.~\cite\RUBINSILVER.) 
Finally, we insist that the determinant of~$\tilde\rho$
be the composite of the $\ell$-adic cyclotomic character
$\tilde\chi_\ell\:\GalQ\to\Z_\ell^\ast$ and the structural
map $\Z_\ell^\ast\to A^\ast$. 
This latter condition can be removed or modified if one
is willing to consider spaces of modular forms which
are more general than the spaces~$S(N)$.

Representations $\tilde\rho$ with these
properties define {\it deformations\/}
of~$\rho_{A,\ell}$
of type~$\Sigma$.
The deformations are equivalence classes
of representations with respect to a natural identification: Two
representations give the same deformation if they are conjugate
by an element of~$\GL A$ which maps to the identity matrix
in~$\GL{\F_\ell}$.

According to a theorem of Mazur and Ramakrishna, there is a
{\it universal\/} deformation.
This is a deformation
$$\rho_\Sigma\:\GalQ\to\GL{R_\Sigma}$$ of type~$\Sigma$ which is
distinguished by the property that an arbitrary
deformation $\tilde\rho\:\GalQ\to\GL A$
of type~$\Sigma$
may be obtained
from $\rho_\Sigma$ by a unique homomorphism $R_\Sigma\to A$.
Here $R_\Sigma$ should be regarded as a somewhat mysterious
ring, which is not given explicitly, at least at first.
Its construction depends on an abstract representability
theorem, which one applies to the given deformation
property.

Somewhat analogously, one
can define a universal {\it modular\/} deformation
of type~$\Sigma$:
$$\rho'_\Sigma\:\GalQ\to\GL{\T_\Sigma}.$$
The problem is then
to prove that $\rho_\Sigma$ and~$\rho'_\Sigma$
coincide.
In contrast to~$R_\Sigma$,
the ring $\T_\Sigma$ may be constructed
quite directly as an essentially
classical Hecke algebra.
It is free of finite rank over~$\Z_\ell$.
Following a line of inquiry
initiated by~Mazur \cite{\EIS, Ch.~II, \S15}
(and taken up by a fair number of other mathematicians,
including the author),
Wiles proves that $\T_\Sigma$ is a Gorenstein ring.

The Gorenstein property
leads to further speculation as to
the precise nature
of~$\T_\Sigma$ as an object of commutative
ring theory.  
Wiles realized several years ago
that the key to proving that
$\rho_\Sigma$ and~$\rho'_\Sigma$
coincide is to show that $\T_\Sigma$ is a complete intersection
ring in the sense of~\cite\MATSUMURA.
In fact, Wiles shows that if 
$\T_\Sigma$ is a complete intersection
ring, then the universal deformation of type~$\Sigma'$
is modular for all $\Sigma'\supseteq\Sigma$.
(For another perspective on questions of this type,
see~\cite\MAZURNOTES.)

In comparing
$\rho_\Sigma$ and~$\rho'_\Sigma$, one begins with
the canonical
homomorphism $$\phi_\Sigma\: R_\Sigma\to\T_\Sigma$$ which
results from the universality of~$\rho_\Sigma$. 
It is known in most cases, and should be
possible to prove in general, that this
homomorphism is surjective. To
show that all deformations of type~$\Sigma$
are modular
is to prove that $\phi_\Sigma$ is
an {\it isomorphism}.
(Analogous conjectures had been made previously by
~Mazur, cf.~\cite{\MAZURTILOUINE, p.~85}.  For a bit
more detail see~\cite{\RUBINSILVER, \S4.2})

Using techniques from commutative algebra,
Wiles proves a proposition to the effect
that $\phi_\Sigma$ is an isomorphism
whenever a specific numerical inequality
is satisfied~\cite{\RUBINSILVER, Th.~5.2}.
Although the statement of the proposition 
seems to require that $\T_\Sigma$ be Gorenstein,
Lenstra has shown how to re-state the proposition
so that no Gorenstein hypothesis appears~\cite\HWL.
Furthermore, Wiles interprets
the numerical condition as an upper bound for the
order of a Selmer-like group
$S_\Sigma$, whose finiteness is
not known a priori~\cite{\RUBINSILVER, \S5.2}.
The theory of congruences between modular forms
of different levels shows that the
desired bound on~$S_\Sigma$,
if valid for a given set~$\Sigma$,
is automatically verified for all supersets $\Sigma'\supseteq
\Sigma$.
This means that it will be sufficient to verify
the sought-after inequality in the minimal case
$\Sigma=\Sigma_0$.
Incidentally, I do not know whether 
$\phi_\Sigma$ is an isomorphism for
a given set $\Sigma$ if and only if the inequality
is true for~$\Sigma$.
Given simply that
$\phi_\Sigma$ is an isomorphism, can one
conclude that $\phi_{\Sigma'}$ is an isomorphism
for all $\Sigma'\supseteq
\Sigma$?



As the authors of~\cite\RUBINSILVER\
explain,
Wiles has sought to verify his inequality 
for $\Sigma=\Sigma_0$ by 
constructing a ``geometric Euler system,'' thereby
generalizing work of~M.~Flach~\cite\FLACH.
At this juncture,
no satisfactory construction
has been exhibited.
Nevertheless, this area of
inquiry is likely to become extremely active,
and one feels that some progress has been made.
For example, 
Mazur's
1993--94 Harvard graduate course~\cite\MAZURNOTES\
suggests that new arguments in commutative
algebra may uncover further information
which is implicit in Flach's construction.




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\enddocument
