% This is the version of the New Item for the Notices of
% the AMS prepared by Ken Ribet (University of California,
% Berkeley)
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\def\startproof{\noindent{\sl Proof.}\enspace}
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% math macros

\def\GL#1{\mathop{\bold{GL}}(2,#1)}
\def\SL#1{\mathop{\bold{SL}}(2,#1)}
\def\PGL#1{\mathop{\bold{PGL}}(2,#1)}
\def\PSL#1{\mathop{\bold{PSL}}(2,#1)}
\def\T{{\Bbb T}}
\def\C{{\Bbb C}}
\def\F{{\Bbb F}}
\def\Fp{\F_p}
\def\Z{{\Bbb Z}}

\def\Q{{\Bbb Q}}


\def\Qbar{\overline{\Q}}
\def\Gbar{\overline{G}}

\def\Gal{\mathop{\roman{Gal}}\nolimits}
\def\GalQ{\Gal(\Qbar/\Q)}
\def\rhobar{\overline{\rho}}
\def\F{{\Bbb F}}

% Numbering references
\newcount\refCount
\def\newref#1 {\advance\refCount by 1
\expandafter\edef\csname#1\endcsname{\the\refCount}}

\newref FLACH %Flach
\newref FREY  % Frey
\newref FREYULM % another Frey
\newref BASECHNG % langlands
\newref EIS %Mazur Eisenstein ideal
\newref DEF %Mazur deformations
\newref	GAD %Mazur gadfly
\newref FERMAT % Ribet IM
\newref TOULOUSE %Ribet in Toulouse
\newref LETTER %Serre to Mestre
\newref DUKE %Serre Duke
\newref SILV % Silverman's book
\newref TUNN % Tunnell BAMS


\topmatter

% \title News Item for the Notices \endtitle

% \author Kenneth A. Ribet  \endauthor

% \affil University of California, Berkeley \endaffil
\rightheadtext{Ribet's article for the AMS Notices}
%\address{UC Mathematics Department, Berkeley, CA  94720 USA} \endaddress

% \email ribet\@math.berkeley.edu \endemail

\endtopmatter


\document
{\sl
Suppose that $p$, $u$, $v$, and~$w$ are integers, with $p>1$.
If $u^p+v^p+w^p=0$, then $uvw=0$.}

Professor Andrew Wiles of Princeton University
deduced this form of Fermat's Last Theorem
at the
conclusion of a series of three lectures during
the June, 1993 workshop on Iwasawa theory, autmorphic
forms, and $p$-adic representations
at the Isaac Newton Institute for Mathematical Sciences
in Cambridge, UK\null.  Wiles had given his series a
suggestive, but ambiguous, title---``Elliptic curves,
modular forms, and Galois representations"---so that
the audience had little inkling how the lectures would
conclude.  Persistent rumors had been circulating for days;
the tension mounted as the series proceeded.
The third lecture was attended by
more than
sixty mathematicians, a fair number
of them carrying cameras to record the event.

In this last lecture, Wiles announced that he had proved
Taniyama's conjecture---an
enormously important conjecture in arithmetical 
algebraic geometry---for 
a large class of
elliptic curves over~$\Q$.
These are the so-called ``semistable" elliptic curves,
those with square-free conductor.
Most people in the audience knew
that Fermat's Last Theorem would be
a consequence of this result.
Although Fermat's Last Theorem holds great fascination
for amateurs and
professionals alike,
the Taniyama conjecture is
ultimately of much greater significance for modern
mathematics.

Yutaka
Taniyama's conjecture, to the effect that every elliptic curve
over~$\Q$ is modular, 
was first proposed in somewhat tentative form at the Tokyo-Nikko
conference in the mid 1950s.  Its statement was refined 
through the efforts of G.~Shimura and A.~Weil; it has been
known, variously, as Weil's conjecture, the Shimura-Taniyama
conjecture, and so on.  
In its usual formulation,
this conjecture associates
objects
of representation theory (modular forms) to
objects of
algebraic geometry (elliptic curves). It states
that the $L$-series of an elliptic curve
over~$\Q$, which measures the behavior of the curve
mod~$p$ for all primes~$p$, can be identified with an
integral transform of the Fourier series derived from
a modular form.
Taniyama's conjecture is a particular case of
the ``Langlands philosophy," a web of interrelated
conjectures made by R.~P.~Langlands and his colleagues.

Although the Langlands conjectures 
require a substantial background in automorphic
forms,
Taniyama's conjecture has been rephrased in such a way
that only complex-analytic
maps appear~\cite\GAD.
 One considers elliptic curves over~$\Q$
up to $\Qbar$-isomorphism: they are
those compact Riemann surfaces
of genus~one
which may be defined by polynomial equations with
{\it rational\/} coefficients.  
Taniyama's conjecture states that
for each such surface~$S$, there is a congruence
subgroup $\Gamma$ of~$\SL\Z$ and a
non-constant analytic map
$\Gamma\backslash\Cal H \to S$, where
$\Cal H$ is the complex upper half-plane.


The Fermat-Taniyama connection grew out of a
1985 Oberwolfach lecture
by G.~Frey, who pointed out
that a non-trivial solution to $a^p+b^p=c^p$
(with $p$ an odd prime)
permits one to write down a
semistable elliptic curve
which does not appear to satisfy Taniyama's
conjecture~\cite{\FREY, \FREYULM}.
Frey's curve is the elliptic curve $E$ given by the 
deceptively simple cubic equation
$ y^2 = x(x-a^p)(x+b^p)$. (It might be necessary
to 
effect a preliminary
adjustment of~$(a,b,c)$
before writing down the curve.)
In a manuscript which he distributed in~Oberwolfach,
Frey outlined an incomplete proof that his curve
was not modular, i.e., that one has the implication
``Taniyama $\Rightarrow$ Fermat."
He expected that his proof would be completed by
experts in the theory of modular curves.

Frey begins with the observation that
once $E$ is modular, so
is its group $E[p]$ of $p$-division
points.  This means that $E[p]$,
viewed as an algebraic group over~$\Q$, can be embedded
in the Jacobian of the algebraic curve over~$\Q$ 
canonically
associated
with an appropriate quotient~$\Gamma\backslash\Cal H$.
A pair of conjectures, which Serre formulated
after learning of 
Frey's construction,
imply then
that $E[p]$ is associated with a specific
congruence
subgroup $\Gamma_0(2)$
of~$\SL\Z$ (see~\cite{\LETTER, \DUKE}).
This is absurd because the Jacobian 
of~$\Gamma_0(2)\backslash\Cal H$ is~zero.


In Serre's conjectures, I recognized a 
generalization of a
problem
that I had formulated
while reading
B.~Mazur's article~\cite\EIS.
I succeeded in proving
the conjectures in July, 1986, approximately
one year after they were
made~\cite{\FERMAT, \TOULOUSE}.
My announcement that I had proved
``Taniyama $\Rightarrow$ Fermat"
convinced the mathematical community that
Fermat's Last Theorem
must be true: we all 
expected
that Taniyama's conjecture would someday
be a theorem.
It was generally accepted, however, that a proof of
Taniyama's
conjecture was far from imminent.

Oblivious to the received idea that Taniyama's
conjecture was inaccessible, Wiles began
working on his proof as soon as he learned that
Fermat was a consequence of the conjecture.
The proof would ultimately incorporate results
and techniques
from his previous works (including joint
articles with J.~Coates and with Mazur),
and from the publications of
G.~Faltings,
R.~Greenberg,
H.~Hida,
V.~Kolyvagin,
Mazur,
K.~Ribet,
K.~Rubin, 
J.~Tilouine,
to cite just a few names.
%New techniques and results were incorporated as they appeared in
%the literature.
A major stumbling block for Wiles
was removed after he received
a preprint by M.~Flach (see~\cite\FLACH).

The following paragraphs
outline the proof that Wiles sketched in his
Cambridge lectures.  The details of the proof are
contained in a 200-page manuscript, which
Wiles intends to release
to the mathematical public in the coming weeks.

To show that a semisimple elliptic curve
$E/\Q$ is modular, Wiles fixes an odd prime~$\ell$,
which in practice is taken to be 3 or~5.
Associated to~$E$ is the $\ell$-adic representation
$\rho_\ell\:\GalQ\to\GL{\Z_\ell}$ gotten by considering the
action of~$\GalQ$ on the $\ell$-power division points of~$E$.
(For background, the reader may consult any of
the recent texts on elliptic curves, such as~\cite\SILV.)
The elliptic curve $E$ satisfies Taniyama's conjecture
if and only if $\rho_\ell$ is ``modular" in the sense that
it is associated to a weight-two cuspidal eigenform in the
usual way.
The representation $\rho_\ell$ ``looks and feels" modular in
that it has the right determinant
and
satisfies some necessary local conditions
at~$\ell$ and other ramified primes.

Roughly speaking,
Wiles proves that a representation like
$\rho_\ell$ is modular if it ``looks and feels" modular
and reduces mod~$\ell$ to a representation
$\rhobar_\ell\:\GalQ\to\GL{\F_\ell}$
which is (1) surjective and (2) itself modular.
Condition~(2) means that $\rhobar$ lifts to
{\it some\/} representation which is modular;
in other words, we want $\rho_\ell$ to be congruent
to some modular representation.
(In many cases, we can replace ``surjective" by
``irreducible" in studying $\rhobar_\ell$.)

Wiles's argument is couched in the language of
Mazur's deformation theory \cite\DEF.
Wiles considers deformations of 
a representation $\rhobar$ satisfying (1) and~(2),
restricting his attention to those deformations
that could plausibly be related to cusp forms of
weight~two.
(He requires the determinant of the deformation to
be the cyclotomic character, and imposes a local
condition at the prime~$\ell$.  For example, if
$\rhobar$ is supersingular, he demands that the
deformation be associated with a Barsotti-Tate group,
locally at~$\ell$.)
Wiles shows that the universal such deformation is
modular, thereby
verifying a conjecture of Mazur.
To do this, he must show
that a certain structural map $\varphi$ of
local rings, a priori a surjection, is 
in fact an isomorphism.
It is here that Wiles uses the ideas of Mazur, Hida,
Tilouine, 
Flach, Kolyvagin and others.
To prove the
injectivity of~$\varphi$,
Wiles was led to study the analogue of the classical
Selmer group for the symmetric square of a modular
lift $\rho$ of~$\rhobar$, bounding it by techniques
derived from those of Kolyvagin and~Flach.
(In many cases, Wiles calculates precisely
the order of this Selmer group.)

After proving this key theorem, Wiles shows that $E$
is modular.  
He examines first
the case $\ell=3$.  A theorem of J.~Tunnell~\cite\TUNN,
which incorporates results of H.~Saito-T.~Shintani and
Langlands~\cite\BASECHNG,
shows that $\rhobar_3$ satisfies~(2) whenever
it satisfies~(1).  It follows that $E$ is modular whenever
$\rhobar_3$ is surjective.

A tantalizing problem, raised by Wiles at the close
of his second lecture, is posed by the case where
$\rhobar_3$ is {\it not\/} surjective.
Suppose, for example, that $\rhobar_3$
is reducible:
can we still win the endgame?
Wiles explained his amazing solution to this problem
in the third lecture.  Using the Hilbert irreducibility
theorem and the Cebotarev density theorem, he
constructs an auxiliary semistable
elliptic curve $E'$ whose mod~3 representation satisfies~(1)
and whose mod~5 representation is isomorphic to~$\rhobar_5$.
The construction succeeds because the modular curve $X(5)$
has genus~zero.
Applying his key theorem once, Wiles shows that $E'$ is modular.
Therefore $\rhobar_5$ is modular, since it may be viewed as
coming from~$E'$.  After a second application
of the key theorem, this time to~$\rho_5$, Wiles
deduces that $E$ is modular!

Wiles's proof of Taniyama's conjecture represents an
enormous milestone for modern mathematics.  On the one
hand, it illustrates
dramatically the power of the abstract ``machinery"
we have amassed for dealing with
concrete Diophantine problems.
On the other, it 
brings us significantly closer to the goal
of tying together automorphic representations
and algebraic varieties.

\Refs
\catcode`\?=\active
\def?{.\hskip 0.1667em\relax}

\ref\no\FLACH\by
M. Flach
\paper
A finiteness theorem for the symmetric square of an elliptic
curve
\jour Invent. Math.\vol 109\yr1992\pages 307--327
\endref

\ref\no\FREY\by
G. Frey
\paper Links between stable elliptic curves and certain  
   diophantine equations\jour  Annales Universitatis Saraviensis  
\vol1\yr1986\pages1--40
\endref

\ref\no\FREYULM\bysame
\paper Links between solutions of $A-B=C$ and
	elliptic curves
\jour
Lecture Notes in Math. \vol 1380\yr1989 \pages31--62
\endref

\ref\no\BASECHNG\by R?P. Langlands
\book
Base change for $\bold{GL}(2)$
\bookinfo Annals of Math. Studies, vol. 96
\publ Princeton University Press
\publaddr Princeton
\yr1980
\endref

\ref\no\EIS\by B. Mazur
\paper Modular curves and the Eisenstein ideal
\jour Publ. Math.
\vol47
\yr1977
\pages33--186
\endref

\ref\no\DEF\bysame
\paper
Deforming Galois representations
\inbook
Galois groups over~$\Q$
\bookinfo MSRI Publications., vol. 16
\publ Springer-Verlag\publaddr Berlin and New York \yr1989
\pages385--437\endref


\ref\no\GAD\bysame
   \paper Number theory as gadfly
   \jour Am. Math. Monthly \vol98\yr1991\pages 593--610
\endref

\ref\no\FERMAT\by
K?A. Ribet
\paper
On modular representations of $\GalQ$
   arising from modular forms\jour
    Invent. Math.\vol100\yr1990\pages 431--476\endref

\ref\no\TOULOUSE\bysame
\paper
 From the Taniyama-Shimura Conjecture to Fermat's Last
Theorem
\jour
Annales de la Facult\'e des Sciences de l'Universit\'e
   de Toulouse
\vol 11
\yr1990
\pages 116--139
\endref

\ref\no\LETTER\by
J-P. Serre
\paper
Lettre \`a J-F. Mestre \paperinfo 13 ao\^ut 1985 \jour
  Contemporary  
  Mathematics
\vol 67\yr1987\pages 263--268\endref


\ref\no\DUKE\bysame
\paper
  Sur les repr\'esentations modulaires de degr\'e
2 de $\GalQ$\jour Duke Math. J.\vol 54\yr
 1987\pages 179--230\endref

\ref\no\SILV\by J?H. Silverman
\book The arithmetic of elliptic curves
\bookinfo Graduate Texts in Math., vol. 106
\publ
   Springer-Verlag\publaddr Berlin and New York \yr1986
\endref

\ref\no\TUNN\by
J. Tunnell
\paper
Artin's conjecture for representations
of octahedral type
\jour
  Bull. AMS
(new series)
\vol5
\yr1981
\pages173--175
\endref

\endRefs
\enddocument