Date:         Wed, 21 Jul 1993 13:05:13 GMT
From: Alex Lopez-Ortiz <alopez-o@MAYTAG.UWATERLOO.CA>;
Organization: University of Waterloo
Subject:      sci.math: Frequently Asked Questions
To: Multiple recipients of list SCIFAQ-L <SCIFAQ-L@YALEVM.BITNET>;

1Q:  What is the current status of Fermat's last theorem?
    (There are no positive integers x,y,z, and n > 2 such that
    x^n + y^n = z^n)
    I heard that <insert name here> claimed to have proved it but later
    on the proof was found to be wrong. ...
    (wlog we assume x,y,z to be relatively prime)

A:  The status of FLT has remained remarkably constant.  Every few
    years, someone claims to have a proof ... but oh, wait, not quite.


    Andrew Wiles, a researcher at Princeton claims to have found
    a proof.

    The proof was presented in Cambridge during a three day seminar
    to an audience including some of the leading experts in the field.

    The proof is long and cumbersome, but here are some of the first
    few details:

    *From Ken Ribet:

Here is a brief summary of what Wiles said in his three lectures.

The method of Wiles borrows results and techniques from lots and lots
of people.  To mention a few: Mazur, Hida, Flach, Kolyvagin, yours
truly, Wiles himself (older papers by Wiles), Rubin...  The way he does
it is roughly as follows.  Start with a mod p representation of the
Galois group of Q which is known to be modular.  You want to prove that
all its lifts with a certain property are modular.  This means that the
canonical map from Mazur's universal deformation ring to its "maximal
Hecke algebra" quotient is an isomorphism.  To prove a map like this is
an isomorphism, you can give some sufficient conditions based on
commutative algebra.  Most notably, you have to bound the order of a
cohomology group which looks like a Selmer group for Sym^2 of the
representation attached to a modular form.  The techniques for doing
this come from Flach; you also have to use Euler systems a la
Kolyvagin, except in some new geometric guise.

If you take an elliptic curve over Q, you can look at the
representation of Gal on the 3-division points of the curve.  If you're
lucky, this will be known to be modular, because of results of Jerry
Tunnell (on base change).  Thus, if you're lucky, the problem I
described above can be solved (there are most definitely some
hypotheses to check), and then the curve is modular.  Basically, being
lucky means that the image of the representation of Galois on
3-division points is GL(2,Z/3Z).

Suppose that you are unlucky, i.e., that your curve E has a rational
subgroup of order 3.  Basically by inspection, you can prove that if it
has a rational subgroup of order 5 as well, then it can't be
semistable.  (You look at the four non-cuspidal rational points of
X_0(15).)  So you can assume that E[5] is "nice." Then the idea is to
find an E' with the same 5-division structure, for which E'[3] is
modular.  (Then E' is modular, so E'[5] = E[5] is modular.)  You
consider the modular curve X which parametrizes elliptic curves whose
5-division points look like E[5].  This is a "twist" of X(5).  It's
therefore of genus 0, and it has a rational point (namely, E), so it's
a projective line.  Over that you look at the irreducible covering
which corresponds to some desired 3-division structure.  You use
Hilbert irreducibility and the Cebotarev density theorem (in some way
that hasn't yet sunk in) to produce a non-cuspidal rational point of X
over which the covering remains irreducible.  You take E' to be the
curve corresponding to this chosen rational point of X.

    *From the previous version of the FAQ:

    (b) conjectures arising from the study of elliptic curves and
    modular forms. -- The Taniyama-Weil-Shmimura conjecture.

    There is a very important and well known conjecture known as the
    Taniyama-Weil-Shimura conjecture that concerns elliptic curves.
    This conjecture has been shown by the work of Frey, Serre, Ribet,
    et. al. to imply FLT uniformly, not just asymptotically as with the
    ABC conj.

    The conjecture basically states that all elliptic curves can be
    parameterized in terms of modular forms.

    There is new work on the arithmetic of elliptic curves. Sha, the
    Tate-Shafarevich group on elliptic curves of rank 0 or 1. By the way
    an interesting aspect of this work is that there is a close
    connection between Sha, and some of the classical work on FLT. For
    example, there is a classical proof that uses infinite descent to
    prove FLT for n = 4. It can be shown that there is an elliptic curve
    associated with FLT and that for n=4, Sha is trivial. It can also be
    shown that in the cases where Sha is non-trivial, that
    infinite-descent arguments do not work; that in some sense 'Sha
    blocks the descent'. Somewhat more technically, Sha is an
    obstruction to the local-global principle [e.g. the Hasse-Minkowski

    *From Karl Rubin:

Theorem.  If E is a semistable elliptic curve defined over Q,
  then E is modular.

It has been known for some time, by work of Frey and Ribet, that
Fermat follows from this.  If u^q + v^q + w^q = 0, then Frey had
the idea of looking at the (semistable) elliptic curve
y^2 = x(x-a^q)(x+b^q).  If this elliptic curve comes from a modular
form, then the work of Ribet on Serre's conjecture shows that there
would have to exist a modular form of weight 2 on Gamma_0(2).  But
there are no such forms.

To prove the Theorem, start with an elliptic curve E, a prime p and let

     rho_p : Gal(Q^bar/Q) -> GL_2(Z/pZ)

be the representation giving the action of Galois on the p-torsion
E[p].  We wish to show that a _certain_ lift of this representation
to GL_2(Z_p) (namely, the p-adic representation on the Tate module
T_p(E)) is attached to a modular form.  We will do this by using
Mazur's theory of deformations, to show that _every_ lifting which
'looks modular' in a certain precise sense is attached to a modular form.

Fix certain 'lifting data', such as the allowed ramification,
specified local behavior at p, etc. for the lift. This defines a
lifting problem, and Mazur proves that there is a universal
lift, i.e. a local ring R and a representation into GL_2(R) such
that every lift of the appropriate type factors through this one.

Now suppose that rho_p is modular, i.e. there is _some_ lift
of rho_p which is attached to a modular form.  Then there is
also a hecke ring T, which is the maximal quotient of R with the
property that all _modular_ lifts factor through T.  It is a
conjecture of Mazur that R = T, and it would follow from this
that _every_ lift of rho_p which 'looks modular' (in particular the
one we are interested in) is attached to a modular form.

Thus we need to know 2 things:
  (a)  rho_p is modular
  (b)  R = T.

It was proved by Tunnell that rho_3 is modular for every elliptic
curve.  This is because PGL_2(Z/3Z) = S_4.  So (a) will be satisfied
if we take p=3.  This is crucial.

Wiles uses (a) to prove (b) under some restrictions on rho_p.  Using
(a) and some commutative algebra (using the fact that T is Gorenstein,
'basically due to Mazur')  Wiles reduces the statement T = R to
checking an inequality between the sizes of 2 groups.  One of these
is related to the Selmer group of the symmetric sqaure of the given
modular lifting of rho_p, and the other is related (by work of Hida)
to an L-value.  The required inequality, which everyone presumes is
an instance of the Bloch-Kato conjecture, is what Wiles needs to verify.

He does this using a Kolyvagin-type Euler system argument.  This is
the most technically difficult part of the proof, and is responsible
for most of the length of the manuscript.  He uses modular
units to construct what he calls a 'geometric Euler system' of
cohomology classes.  The inspiration for his construction comes
from work of Flach, who came up with what is essentially the
'bottom level' of this Euler system.  But Wiles needed to go much
farther than Flach did.  In the end, _under_certain_hypotheses_ on rho_p
he gets a workable Euler system and proves the desired inequality.
Among other things, it is necessary that rho_p is irreducible.

Suppose now that E is semistable.

Case 1.  rho_3 is irreducible.
Take p=3.  By Tunnell's theorem (a) above is true.  Under these
hypotheses the argument above works for rho_3, so we conclude
that E is modular.

Case 2.  rho_3 is reducible.
Take p=5.  In this case rho_5 must be irreducible, or else E
would correspond to a rational point on X_0(15).  But X_0(15)
has only 4 noncuspidal rational points, and these correspond to
non-semistable curves.  _If_ we knew that rho_5 were modular,
then the computation above would apply and E would be modular.

We will find a new semistable elliptic curve E' such that
rho_{E,5} = rho_{E',5} and rho_{E',3} is irreducible.  Then
by Case I, E' is modular.  Therefore rho_{E,5} = rho_{E',5}
does have a modular lifting and we will be done.

We need to construct such an E'.  Let X denote the modular
curve whose points correspond to pairs (A, C) where A is an
elliptic curve and C is a subgroup of A isomorphic to the group
scheme E[5].  (All such curves will have mod-5 representation
equal to rho_E.)  This X is genus 0, and has one rational point
corresponding to E, so it has infinitely many.  Now Wiles uses a
Hilbert Irreducibility argument to show that not all rational
points can be images of rational points on modular curves
covering X, corresponding to degenerate level 3 structure
(i.e. im(rho_3) not GL_2(Z/3)).  In other words, an E' of the
type we need exists.  (To make sure E' is semistable, choose
it 5-adically close to E.  Then it is semistable at 5, and at
other primes because rho_{E',5} = rho_{E,5}.)