Introduction to Fermat's Last Theorem!

[Sonnet Nguyen <sonnet@blue.ifpan.edu.pl>]

Hi everybody,

Hope that the following comprehensive raport (about  FLT) will give you
few useful informations about the history of one unsolved problem
(for the last 350 years) and new developments in the one area of Math
which is called Algebraic Geometry.  
Enjoys,
SN
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Department of Math
Warsaw University              The mathematics of non-mathematics!
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Author:  Rubin K., Silverberg A.

History of Fermat's Last Theorem

Pierre de Fermat (1601-1665) was a lawyer and amateur mathematician. In
about 1637, he annotated his copy (now lost) of Bachet's translation of
Diophantus' Arithmetika with the following statement:

     Cubem autem in duos cubos, aut quadratoquadratum in duos
     quadratoquadratos, et generaliter nullam in infinitum ultra
     quadratum potestatem in duos ejusdem nominis fas est dividere:
     cujus rei demonstrationem mirabilem sane detexi. Hanc marginis
     exiguitas non caparet.

In English, and using modern terminology, the paragraph above reads as:

     There are no positive integers such that x^n + y^n = z^n for n > 2
     . I've found a remarkable proof of this fact, but there is not
     enough space in the margin [of the book] to write it.

Fermat never published a proof of this statement. It became to be known as
Fermat's Last Theorem (FLT) not because it was his last piece of work, but
because it is the last remaining statement in the post-humous list of
Fermat's works that needed to be proven or independently verified. All
others have either been shown to be true or disproven long ago.

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What is the current status of FLT?

     Theorem 1 (Fermat's Last Theorem) There are no positive integers
     x, y, z, and n>2 such that x^n + y^n = z^n.

Andrew Wiles, a researcher at Princeton, claims to have found a proof. The
proof was presented in Cambridge, UK during a three day seminar to an
audience which included some of the leading experts in the field. The proof
is long and cumbersome. At this point the reviewers have found a glitch in
the argument. Late last year, Prof. Wiles acknowledged that a gap existed.
On October 25th, 1994. Prof. Andrew Wiles released two preprints, Modular
elliptic curves and Fermat's Last Theorem, by Andrew Wiles, and Ring
theoretic properties of certain Hecke algebras, by Richard Taylor and Andrew
Wiles.

The first one (long) announces a proof of, among other things, Fermat's Last
Theorem, relying on the second one (short) for one crucial step.

The argument described by Wiles in his Cambridge lectures had a serious gap,
namely the construction of an Euler system. After trying unsuccessfully to
repair that construction, Wiles went back to a different approach he had
tried earlier but abandoned in favor of the Euler system idea. He was able
to complete his proof, under the hypothesis that certain Hecke algebras are
local complete intersections. This and the rest of the ideas described in
Wiles' Cambridge lectures are written up in the first manuscript. Jointly,
Taylor and Wiles establish the necessary property of the Hecke algebras in
the second paper.

The new approach turns out to be significantly simpler and shorter than the
original one, because of the removal of the Euler system. (In fact, after
seeing these manuscripts Faltings has apparently come up with a further
significant simplification of that part of the argument.)

The preprints were submitted to The Annals of Mathematics. According to the
New York Times the new proof has been vetted by four researchers already,
who have found no mistake.
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Wiles' line of attack

Here is an outline of the first proposed proof.

>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
parameterizes 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 conjecture.

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 theorem].

>From Karl Rubin:

     Theorem 2 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(\bar(Q)/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 square 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) != 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) .)

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If not, then what?

FLT is usually broken into 2 cases. The first case assumes (abc,n) = 1. The
second case is the general case.

WHAT HAS BEEN PROVED

First Case.

It has been proven true up to 7.568x10^(17) by the work of Wagstaff &
Tanner, Granville &Monagan, and Coppersmith. They all used extensions of the
Wiefrich criteria and improved upon work performed by Gunderson and Shanks
&Williams.

The first case has been proven to be true for an infinite number of
exponents by Adelman, Frey, et. al. using a generalization of the Sophie
Germain criterion

Second Case:

It has been proven true up to n = 150,000 by Tanner &Wagstaff. The work used
new techniques for computing Bernoulli numbers mod p and improved upon work
of Vandiver. The work involved computing the irregular primes up to 150,000.
FLT is true for all regular primes by a theorem of Kummer. In the case of
irregular primes, some additional computations are needed. More recently,
Fermat's Last Theorem has been proved true up to exponent 4,000,000 in the
general case. The method used was essentially that of Wagstaff: enumerating
and eliminating irregular primes by Bernoulli number computations. The
computations were performed on a set of NeXT computers by Richard Crandall
et al.

Since the genus of the curve a^n + b^n = 1 , is greater than or equal to 2
for n > 3 , it follows from Mordell's theorem [proved by Faltings], that for
any given n , there are at most a finite number of solutions.

CONJECTURES

There are many open conjectures that imply FLT. These conjectures come from
different directions, but can be basically broken into several classes: (and
there are interrelationships between the classes)

          Conjectures arising from Diophantine approximation theory such as
     the ABC conjecture, the Szpiro conjecture, the Hall conjecture, etc.

     For an excellent survey article on these subjects see the article by
     Serge Lang in the Bulletin of the AMS, July 1990 entitled ``Old and new
     conjectured diophantine inequalities".

     Masser and Osterle formulated the following known as the ABC
     conjecture:

     Given epsilon > 0 , there exists a number C(epsilon) such that for any
     set of non-zero, relatively prime integers a,b,c such that a + b = c we
     have max (|a|, |b|, |c|) <= C(epsilon) N(abc)^(1 + epsilon) where N(x)
     is the product of the distinct primes dividing x .

     It is easy to see that it implies FLT asymptotically. The conjecture
     was motivated by a theorem, due to Mason that essentially says the ABC
     conjecture is true for polynomials.

     The ABC conjecture also implies Szpiro's conjecture [and vice-versa]
     and Hall's conjecture. These results are all generally believed to be
     true.

     There is a generalization of the ABC conjecture [by Vojta] which is too
     technical to discuss but involves heights of points on non-singular
     algebraic varieties . Vojta's conjecture also implies Mordell's theorem
     [already known to be true]. There are also a number of inter-twined
     conjectures involving heights on elliptic curves that are related to
     much of this stuff. For a more complete discussion, see Lang's article.
          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 theorem].
          Conjectures arising from some conjectured inequalities involving
     Chern classes and some other deep results/conjectures in arithmetic
     algebraic geometry.

     This results are quite deep. Contact Barry Mazur [or Serre, or
     Faltings, or Ribet, or ...]. Actually the set of people who DO
     understand this stuff is fairly small.

     The diophantine and elliptic curve conjectures all involve deep
     properties of integers. Until these conjecture were tied to FLT, FLT
     had been regarded by most mathematicians as an isolated problem; a
     curiosity. Now it can be seen that it follows from some deep and
     fundamental properties of the integers. [not yet proven but generally
     believed].

     This synopsis is quite brief. A full survey would run to many pages.

References

     [1] J.P.Butler, R.E.Crandall,&R.W.Sompolski, Irregular Primes to One
     Million. Math. Comp., 59 (October 1992) pp. 717-722.
     Fermat's Last Theorem, A Genetic Introduction to Algebraic Number
     Theory. H.M. Edwards. Springer Verlag, New York, 1977.
     Thirteen Lectures on Fermat's Last Theorem. P. Ribenboim. Springer
     Verlag, New York, 1979.
     Number Theory Related to Fermat's Last Theorem. Neal Koblitz, editor.
     Birkhduser Boston, Inc., 1982, ISBN 3-7643-3104-6

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Did Fermat prove this theorem?

No he did not. Fermat claimed to have found a proof of the theorem at an
early stage in his career. Much later he spent time and effort proving the
cases n = 4 and n = 5 . Had he had a proof to his theorem, there would have
been no need for him to study specific cases.

Fermat may have had one of the following ``proofs'' in mind when he wrote
his famous comment.

     Fermat discovered and applied the method of infinite descent, which, in
     particular can be used to prove FLT for n = 4 . This method can
     actually be used to prove a stronger statement than FLT for n = 4 ,
     viz, x^4 + y^4 = z^2 has no non-trivial integer solutions. It is
     possible and even likely that he had an incorrect proof of FLT using
     this method when he wrote the famous theorem''.
     He had a wrong proof in mind. The following proof, proposed first by
     Lame' was thought to be correct, until Liouville pointed out the flaw,
     and by Kummer which latter became and expert in the field. It is based
     on the incorrect assumption that prime decomposition is unique in all
     domains.

The incorrect proof goes something like this:

We only need to consider prime exponents (this is true). So consider x^p +
y^p = z^p . Let r be a primitive p -th root of unity (complex number)

Then the equation is the same as:

(x + y)(x + ry)(x + r^2y)...(x + r^(p - 1)y) = z^p

Now consider the ring of the form:

a_1 + a_2 r + a_3 r^2 + ... + a_(p - 1) r^(p - 1)

where each a_i is an integer

Now if this ring is a unique factorization ring (UFR), then it is true that
each of the above factors is relatively prime. From this it can be proven
that each factor is a pth power and from this FLT follows.

The problem is that the above ring is not an UFR in general.

Another argument for the belief that Fermat had no proof -and, furthermore,
that he knew that he had no proof- is that the only place he ever mentioned
the result was in that marginal comment in Bachet's Diophantus. If he really
thought he had a proof, he would have announced the result publicly, or
challenged some English mathematician to prove it. It is likely that he
found the flaw in his own proof before he had a chance to announce the
result, and never bothered to erase the marginal comment because it never
occurred to him that anyone would see it there.

Some other famous mathematicians have speculated on this question. Andre
Weil, writes:

     Only on one ill-fated occasion did Fermat ever mention a curve of
     higher genus x^n + y^n = z^n , and then hardly remain any doubt
     that this was due to some misapprehension on his part [for a brief
     moment perhaps [he must have deluded himself into thinking he had
     the principle of a general proof.

Winfried Scharlau and Hans Opolka report:

     Whether Fermat knew a proof or not has been the subject of many
     speculations. The truth seems obvious ...[Fermat's marginal note]
     was made at the time of his first letters concerning number theory
     [1637]...as far as we know he never repeated his general remark,
     but repeatedly made the statement for the cases n = 3 and 4 and
     posed these cases as problems to his correspondents [he formulated
     the case n = 3 in a letter to Carcavi in 1659 [All these facts
     indicate that Fermat quickly became aware of the incompleteness of
     the [general] ``proof" of 1637. Of course, there was no reason for
     a public retraction of his privately made conjecture.

However it is important to keep in mind that Fermat's "proof" predates the
Publish or Perish period of scientific research in which we are still
living.

References

     From Fermat to Minkowski: lectures on the theory of numbers and its
     historical development. Winfried Scharlau, Hans Opolka. New York,
     Springer, 1985.
     Basic Number Theory. Andre Weil. Berlin, Springer, 1967