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\def\foilhead#1{\bigbreak\hrule\vskip2pt\hrule\medskip\centerline{\bf #1}\medskip}
\centerline{\bf Update and Overview on Fermat's Last Theorem}
\vskip2pt
\centerline{AMS-MAA Invited Hour Talk, January 13, 1994}
\vskip2pt
\centerline{Professor Kenneth A. Ribet, UC Berkeley}

\foilhead{Elliptic Curves}

\noindent
Elliptic curves over~$\Q$ are
non-singular plane cubic curves such as
$$ \cases{
y^2+y = x^3 -x^2, \cr y^2 = x^3 -x,\cr
  y^2 = x(x-3)(x+32),\cr
  y^2+xy = x^3+x^2 -23x +33\cr} $$
which are given by equations with rational coordinates.

\newfoil
We view them as projective curves with a distinguished
point~$O$.  This point is the ``point at infinity,"
the one point on the curve not visible in the 
affine plane.
\newpage
We can easily view the set of real points of an
elliptic curve [there was a picture of an elliptic
curve here]:


The complex points form a complex
torus\dash---a doughnut.
\newpage
Elliptic curves are groups\dash---give me
two points on an elliptic curve and I'll hand you a third.
This is because (in general) straight lines
intersect an elliptic curve in three points.

The origin of the group law is the point that we
know about\dash---the point at infinity.
In other words, $O=0$.

The process of adding points together is called
the ``chord and tangent" process.  If we add a
point to itself, we have to draw a tangent instead
of a chord, or secant line.
\foilhead{Points mod $p$}
In number theory, it is often easier to study equations
by replacing ``$=$" by ``$\equiv$ mod~$p$," where $p$
is a varying prime.
If you take an equation like $y^2=x^3-x$ and reduce it
mod~$p$, you get a cubic curve, but it might be singular.
If not, then the original curve
has {\it good reduction\/} at~$p$ and we get an elliptic
curve over the field $\F_p$ with $p$ elements.

\newfoil
For example, the equation $$y^2+y=x^3-x^2$$ defines an
elliptic curve over~$\F_2$, so the elliptic curve
$y^2+y=x^3-x^2$ has {\it good reduction\/} at~2.


An important activity has been to count the number of points
on an elliptic curve over~$\Fp$.  
For example, Gauss found a formula for the number of
points on $y^2=x^3-x$ over~$\Fp$.

\newfoil
There are at most $p^2+1$ points on a curve
{\it a priori\/} because there are $p$ values each
for $x$ and~$y$ and we have to add one point at infinity.
A theorem of Hasse states that the number of points
differs from $p+1$ by a number smaller than $2\sqrt p$
in absolute value.


It is traditional to define
$$a_p \seteq p{+}1 - \#E(\F_p).$$
Here the notation suggests correctly that $p$ varies
and that $E$ is a {\it fixed\/} elliptic curve over~$\Q$.

\newfoil

For example, take $p=2$ and look at $$y^2+y=x^3-x.$$
There are five points on the curve and so $$a_2=-2.$$
\foilhead{The Taniyama-Shimura Conjecture}
The elliptic curve $y^2+y=x^3-x^2$ has good reduction
at all $p\ne11$.  It is a fact that the $a_p$ for this
curve are the $p$th coefficients of the formal series
$\sum c_n q^n$
obtained by developing the product expansion
$$ q \prod_{n=1}^\infty (1-q^n)^2 (1-q^{11n})^2.$$

\newfoil
The series $\sum c_n e^{2\pi i nz}$ is a 
weight-two
cusp form
%of weight~two
on a congruence subgroup of~$\SL\Z$
called $\Gamma_0(11)$.  Here $z$ is the ``variable,"
a complex number with positive imaginary part.

\newfoil

The Taniyama-Shimura Conjecture says that all
elliptic curves over~$\Q$ are like this.  In
other words, for each $E$ over~$\Q$ there is
to be a weight-two cusp form $f=\sum c_n e^{2\pi i nz}$
on an appropriate congruence subgroup of~$\SL\Z$
with the property that $a_p(E) = c_p$ for all good
primes~$p$.  The previous example is thus supposed
to be typical,
except that there is usually no cute product expansion
giving the coefficients.

\newfoil
This amazing conjecture links the algebraic world of elliptic
curves with the analytic world of modular forms.

If $E$ satisfies the conjecture, we say that it is
``modular"\dash---it is a modular elliptic curve.

\newfoil
To an elliptic curve over~$\Q$ we can attach a rational
number\dash---its $j$-invariant.  Every rational number
is the $j$-invariant of some elliptic curve over~$\Q$.
Two elliptic curves with the same $j$-invariant are
isomorphic over~$\C$, but not necessarily over~$\Q$.
Nevertheless, if $E$ and~$E'$ have the same $j$-invariant,
they are either both modular or both non-modular.

In fact, the modularity of an elliptic curve can be
seen over~$\C$.  This leads to a description
involving hyperbolic geometry$\ldots$.

\newfoil

Before Wiles's work, we could exhibit only finite sets of
numbers which were the $j$-invariants of modular elliptic
curves over~$\Q$.  For example, Shimura proved
in 1971 that elliptic curves with complex multiplication
are modular.
One year ago,
proving that all elliptic curves are
modular seemed to be completely out of reach.

In June, Wiles announced a proof that two disjoint classes
of elliptic curves over~$\Q$ were modular.  Both classes
correspond to infinite numbers of $j$-invariants.

One of the classes has an application to Fermat's Last Theorem;
this class is the one occluded by the famous ``gap":

\vfill\eject
\hrule\vskip2pt\hrule\medskip
{\ssq\baselineskip10pt
\advance\leftskip1.5em\advance\rightskip1.5em
	In view of the speculation on the status of my work on the  
Taniyama-Shimura conjecture and Fermat's Last Theorem I will give a  
brief account of the situation. During the review process a number of  
problems emerged, most of which have been resolved, but one in  
particular I have not yet settled. The key reduction of (most cases  
of ) the Taniyama-Shimura conjecture to the calculation of the Selmer  
group is correct. However the final calculation of a precise upper  
bound for the Selmer group in the semistable case (of the symmetric  
square representation associated to a modular form) is not yet  
complete as it stands. I believe that I will be able to finish this  
in the near future using the ideas explained in my Cambridge  
lectures.

	The fact that a lot of work remains to be done on the  
manuscript makes it still unsuitable for release as a preprint . In  
my course in Princeton beginning in February I will give a full  
account of this work.

Andrew Wiles.\par}

\newfoil
\noindent
According to Karl Rubin and Alice Silverberg, the
second class includes the infinite set of $j$-invariants
obtained by plugging into the formula
$$
\left({{12 (1 + 18 t^2 - 27 t^4)} \over {(1 - 18 t^2 - 27 t^4)}}\right)^3
$$
all rational numbers $t$ with no~3 in their denominators.

The idea of Wiles is that being modular is a contagious
disease.  The elliptic curves in the Rubin-Silverberg example
contracted their modularity from the curve $y^2=x^3-x$.

%which had already known to be modular.  The contagion
%results from a congruence mod~3.

\foilhead{Taniyama-Shimura $\Longrightarrow$ Fermat}

In 1985, G. Frey initiated the construction of elliptic
curves whose coefficients depend on the $a$, $b$ and~$c$
occurring in a Fermat counterexample
$ a^\ell+b^\ell = c^\ell$.
(Here $a$, $b$ and $c$ are non-zero and relatively prime\dash---%
and $\ell$ is a prime $\ge5$.)

Frey proposed that
$$ y^2 = x(x-a^\ell)(x+b^\ell)$$
could not be modular.  

\newfoil
Serre laid out a
specific strategy for showing this, and in 1987 I proved
the conjectures of Serre
which justified this strategy.

\newfoil
\noindent
Frey's curves belong to the class of {\it semi\-stable\/} elliptic 
curves.  The curve for $(a,b,c)$ has bad reduction when
two of the roots of
$x(x-a^\ell)(x+b^\ell)=0$ mod~$p$ are the same.
This happens precisely when $p$ divides $abc$.
The semistability stems from the fact that the
three roots can never all be equal.

\newfoil

In June, Wiles announced that he could prove that
all semistable elliptic curves over~$\Q$ are
modular.  The proof of this statement depends on
an inequality involving Selmer groups, which was
to be obtained by the methods of Kolyvagin and
Flach.

The ``gap" occurs in the application of those
methods to Wiles's problem.

\foilhead{Elements of Wiles's Work}

Wiles fixes a prime $p$; we will take it to be~3.
Let $E$ be a semistable elliptic curve and
consider the group of 3-division points
$$ E[3] = \{\, P\in E(\C) \, | \,P+P+P =0\,\},$$
where ``$+$" is the group operation.
By thinking about the complex torus $E(\C)$, we realize
that $E[3]$ is the direct sum of two cyclic groups of
order~3:
$E[3] \approx (\Z/3\Z)^2$.

\newfoil
Now $E[3]$ is stable by all automorphisms of~$\C$ (which
act on the coordinates of points), so its points have
algebraic coordinates.  Conjugation is compatible with
addition, so conjugations are described by $2\times2$
matrices with entries in~$\F_3$.

\newfoil
But $E[3]$ is modular, according to a hard theorem
of Langlands and Tunnell (Saito-Shintani lifting).
This means that there is an $f=\sum c_n q^n$
for which the $a_p$ (with $p\ne3$) are congruent to
the $c_p$ mod 3 (or, actually, modulo an ideal
dividing~3 in a ring of algebraic integers).

\newfoil
The contagion metaphor refers to the principle that
the modularity of~$E[3]$ should propagate out to
the modularity of~$E$.  (In the process, $f$
changes.)
This doesn't work if $E[3]$ is reducible in
the sense that it has a non-zero cyclic subgroup
stable under conjugation.  In this case,
Wiles has an ingenious argument for switching from
3 to~5.  (Ingenuity is required because there is
no Langlands-Tunnell theorem for~5.)

\foilhead{Deformations}
Let $G$ be the Galois group of the field of algebraic
complex numbers.  The ``conjugation" on~$E[3]$
is a continuous homomorphism
$\rho\:G\to\GL{\F_3}$.
It comes equipped with a lift
$$\rho_E\:G\to\GL{\Z_3}$$
gotten by considering all 3-power division points
instead of 3-division points.  To prove that $E$
is modular is to prove that $\rho_E$ is modular.

\newfoil

That $E[3]$ is already modular implies that there is a
second lift
$$\rho_f\:G\to\GL{{\cal O}}$$
coming from the modular form~$f$.
Here $\cal O$ is the integer ring of a 3-adic field.
We can think: ${\cal O}=\Z_3$ without loss of
generality.

\newfoil
The lifts of~$\rho$ are called ``deformations."
A conjecture of Mazur states that all deformations
with suitable
local
properties
should be modular.
Using commutative algebra, Wiles reduces
Mazur's conjecture to what has been termed a 
``count," 
a single inequality.
We want to prove something along the lines:
$$ \#\hbox{(lifts)} \,\buildrel?\over\le\, \#\hbox{(forms congruent to $f$).}
$$
We'll call this inequality ``$(\ast)$".

\newfoil
In mathematical language, the left-hand side turns
into a Selmer group; this is a subgroup of the
cohomology group
$${\rm H}^1(G, \Ad(\rho_f)\otimes_{\Z_3}\Q_3/\Z_3).$$

The right-hand side can be expressed in a number of ways.
Through work of H.~Hida, it can be related to the
``algebraic part" of the value at $s=2$ of the symmetric
square $L$-function made from~$f$.  

\newfoil
The ``fundamental
inequality" $(\ast)$ then strongly resembles an equality
which was conjectured by Bloch and Kato (in a very 
general context) several years ago.

\newfoil
There are many possible inequalities $(\ast)$, obtained as
we tighten or relax the local conditions that we impose
on lifts.  Wiles proves that $(\ast)$ for a given problem
implies the analogous inequality for all looser problems.
One can also show that $f$ may be chosen so that
$\rho_f$ obeys all the constraints suggested by~$\rho$.
This allows Wiles to assume that the deformation problem
is as tight as possible (``minimal problem").

\newfoil
Even if the ``gap" does not get filled in the coming
weeks, it appears likely that the Taniyama-Shimura Conjecture
for $E$ will have been reduced to a single numerical
inequality: $(\ast)$ for the minimal deformation
problem.  
\end{document}
