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3 Articles on the proof of Fermat's Last Theorem

 Newsweek, New York Times, and [e-mail].
 =========

                       Internet Amateur Mathematics Society
                                  Newsletter 5

                [e-mail]

Fermat's Last Theorem Proved

If you haven't heard it yet, Dr. Andrew Wiles had claimed that he
proved the famous Fermat's Last Theorem.  Here I have three articles,
one from Newsweek, one from New York Time, and the other one from
[e-mail].



Brief Summary of Dr. Wiles' Proof, From [e-mail]


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's Theorem 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) \to \GL_2(\Z/p\Z)

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, ie 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:

    1. \rho_p is modular
    2. R = T.

It was proved by Tunnell that \rho_3 is modular for every elliptic
curve.  This is because \PGL_2(\Z/3\Z) = S_4.  So (1) 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.
There are 2 cases:

    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.

    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 1, 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 \bmod 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}.)




New Answer for an Old Question, The proof's in the putting

By Sharon Begley with Joshua Copper Ramo

Newsweek, July 5th 1993, Pg 52
Copyright 1993 NEWSWEEK, INC.: 444 MADISON AVENUE, N.Y, N.Y, 10022 All
Rights Reserved.

"I have found a truly wonderful proof which this margin[of my
notebook] is too small to contain."  So asserted French mathematician
Pierre de Fermat in 1637, in what became the biggest historical dare
in mathematics.  He was referring to a beguilingly simple assertion
about numbers that had intrigued mathematicians since Roman times.
Fermat died 18 years later, never having gotten around to writing down
his "admirable proof".  That should have been a tip-off.  The greatest
minds of the next four centuries tried to find this proof and failed
so abysmally that Fermat's last Theorem, as the assertion was known,
became the top unsolved challenge in all mathematics.  Now it may have
fallen.  Last week, in a lecture at Cambridge University in England,
Andrew Wiles of Princeton University announced that he had proved
Fermat's Last.  Within an hour number theorists were spreading the
word from London to Boulder to Berkeley through a joyous hail of
E-mail.  "There is a sort of euphoria", exalted Princeton math
department chairman Simon Kochen.  "Euphoria because we lived to see
this."

Pure mathematics is a sucker's game.  It lures the curious and
confident with its seeming simplicity only to make them look like
fools.  Consider the equation 9+16=25.  It can be written
3^2+4^2=5^2.  More generally, one can write that a number squared
(multiplied by itself) plus a second number squared equals a third
number squared.  Now the inquisitive are hooked like rubes in
three-card monte.  Try to find three whole numbers that fit the
equation x^3+y^3=z^3.  That is, pick numbers for x and y, multiply
each by itself twice (like 3\times 3\times 3), and find a third
number z which, when multiplied by itself twice, equals to the sum of
the x^3 and y^3.  There is no such z.  That's what Fermat's Last
states.  Nor are there any numbers that fit the equation where the
exponents are anything greater than 2.  That's what Fermat claimed to
have proved.

Fermat did, in fact, prove his assertion for exponents of 4.  Leonard
Euler, the great Swiss mathematician, proved it for exponents of 3 in
the 1790s.  France's Adrien Legendre proved it for exponents of 5 in
1823.  A few years ago a computer proved it for everything less than
30,000.  Things were looking pretty good for Fermat's last, but none
of this constituted a persuasive proof.  What if the number right
after the last number checked turned out to falsify the theorem?  Only
a rigorous proof would do.  Their inability to find one, especially in
the face of Fermat's taunt, drove mathematicians crazy.

Too shy: Wiles approached Fermat's last Theorem the way one would a
skittish horse -- obliquely.  He started with a 1984 finding that if
there are any numbers for which Fermat's equations holds, then the
solutions can be fashioned into something called an elliptic curve.
Then wiles noted a 1987 proof by Ken Ribet of the University of
California, Berkeley, that any such elliptic curves could not be of a
certain type.  (Those who can't balance their checkbooks can drop out
here.) When Wiles proved the contrary -- that the relevant elliptic
curves are of this type -- he had shown the "if" he started with to be
wrong: there are no numbers that make Fermat's equation work.  Just as
the Frenchman said 356 years ago.  This all took Wiles 200 pages.
Wiles (who describes himself as too shy to talk to the press) combined
ideas from number theory, topology and other disparate fields and
basically "three the kitchen sink" at Fermat's Last, says Kochen.
"Among theorists there is often a sense that something just looks
right.  It's the general feeling that [Wiles's proof] looks right," he
says.

Mathematicians will not know for sure until they check every line, a
process that could take years.  Thousands of other claims to have
proved Fermat's Last Theorem have fizzled.  But if Wiles has triumphed
over the historical dare, his proof would promise a huge advance in
number theory.  It is a field of almost pristine irrelevance to
everything except the wondrous demonstration that pure numbers, no more
substantial than Plato's shadows, conceal magical laws and orders that
the human mind can discover after all.




At Last, Shout of `Eureka!' In Age-Old Math Mystery

By Gina Kolata

The New York Times, Thursday, June 24, 1993
Copyright 1993 The New York Times

More than 350 years ago, a French mathematician wrote a deceptively
simple theorem in the margins of a book, adding that he had discovered
a marvelous proof of it but lacked space to include it in the margin.
He died without ever offering his proof, and mathematicians have been
trying eversince to supply it.  Now, after thousands of claims of
success that proved untrue, mathematicians say the daunting challenge,
perhaps the most famous of unsolved mathematical problems, has at last
been surmounted.

The problem is known as Fermat's last theorem, and its apparent
conqueror is Dr. Andrew Wiles, a 40-year-old English mathematician who
works at Princeton University.  Dr. Wiles announced the result
yesterday at the last of three lectures given over three days at
Cambridge University in England.

Within a few minutes of the conclusion of his final lecture, computer
mail messages were winging around the world as mathematicians alerted
each other to the startling and almost wholly unexpected result.

Dr. Leonard Adelman of the University of Southern California said he
received a message about an hour after Dr. Wiles's announcement.  The
frenzy is justified, he said.  "It's the most exciting thing that's
happened in -- geez -- maybe every, in mathematics."

Mathematicians present at the lecture said they felt "an elation",
said Dr. Kenneth Ribet of the University of California at Berkeley, in
a telephone interview from Cambridge.

Impossible Is Possible

The theorem, an overarching startment about what solutions are
possible for certain simple equations, was stated in 1673 by Pierre de
Fermat, a 17th century French mathematician and physicist.  Many of
the brightest minds in mathematics have struggled to find the proof
ever since, and many have concluded that Fermat, contrary to his
tantalizing claim, had probably failed to develop one despite his
considerable mathematical ability.

With Dr. Wiles' result, Dr. Ribet said, "the mathematical landscape
has changed."  He explained: "Your discover that things that seemed
completely impossible are more of a reality.  This changes the way you
approach problems, what you think is possible."

Dr. Barry Mazur, a Harvard University mathematician, also reached by
telephone in Cambridge, said: "A lot more is proved than Fermat's last
theorem.  One could envision proof of a problem, no matter how
celebrated, that had no implications.  But this is just the reverse.
This is the emergence of a technique that is visibly powerful.  It's
going to prove a lot more."

Remember Pythagoras?

Fermat's last theroem has to do with equations of the form
x^n+y^n=z^n.  The case where n is 2 is familiar as the Pythagorean
theorem that the squares of the lengths of two sides of a right angled
triangle equal the square of the length of the hypotenuse.  One such
equation is 3^2+4^2=5^2, since 9+16=25.

Fermat's last theorem states that there are no solutions to such
equations when n is a whole number greater than 2.  This means, for
instance, that it would be impossible to find any whole numbers x, y
and z such that x^3+y^3=z^3.  Thus 3^3+4^3, (27+64)=91, which is
not the cube of any whole number.

Mathematicians in the United States said that the stature of Dr. Wiles
and the imprimatur of the experts who heard his lectures, especially
Dr. Ribet and Dr. Mazur, convinced them that the new proof was very
likely to be right.  In addition, they said, the logic of the proof is
persuasive because it is is built on a carefully developed edifice of
mathematics that goes back more than 30 years and is well accepted by
researchers.

Experts cautioned that Dr. Wiles could of course have made some subtle
misstep.  Famous and not-so-famous mathematicians have claimed proofs
in the past, only to be tripped up by errors.  Dr. Harold M. Edwards,
a mathematician at the Courant Institute of Mathematical Sciences in
New York, said that until the proof was published in a mathematical
journal, which could take a year, and until it is checked many times,
there is always a chance it is wrong.  The author of a book on
Fermat's last theorem, Dr. Edwards noted that "even good
mathematicians have had false proofs."

Luring the World's `Cranks'

But even he said that Dr. Wiles's proof sounds like the real thing and
"has to be taken very seriously."

Despite the apparent simplicity of the theorem, proving it was so hard
that in 1815 and in again 1860, the French Academy of Science offered
a gold medal and 300 francs to anyone who could solve it.  In 1908,
the German Academy of Science offered a prize of 100,000 marks for a
proof that the theorem was correct.  The prize, which still stands
though has been reduced to 7,500 marks, about $4,385, has attracted
the world's "cranks", Dr. Edwards said.  When the Germans said the
proof had to be published, "the cranks began publishing their
solutions in the vanity press," he said, yielding thousands of
booklets.  The Germans told him they would even award the prie for a
proof that the theorem was not true, Dr. Edwards added, saying that
they "would be so overjoyed that they wouldn't have to read through
these submissions."

But it was not just amateurs whose iamgination was captured by the
enigmatic problem.  Famous mathematicians, too, spent years of their
lives on it.  Others chose never to get near it for fear of being
sucked into a quagmire.  One mathematical genius, David Hilbert, said
in 1920 that he would not work on it because, "before beginning I
should put in three years of intensive study, and I haven't that much
time to spend on a probable failure."

Mathematicians armed with computers have shown that Fermat's theorem
holds true up to very high numbers.  But that falls well short of a
general proof.

Tortuous Path to Proof

Dr. Ribet said that 20th century work on the problem began to grow
ever more divorced from Fermat's quations.  "Over the last 60 years,
people in number theory have forged an incredible number of tools to
deal with simple problems like this," he said.  "As the tools became
more complicated, they took on a life of their own.  People lost
day-to-day contact with the old problems and were preoccupied with the
objects they created."

Dr. Wiles' proof draws on many of these mathematical tools but also
"completes a chain of ideas," said Dr. Nicholas Katz of Princeton
University.  The work leading to the proof began in 1954, when the
late Japanese mathematician Yutaka Taniyama made a conjecture about
mathematical objects called elliptical curves.  That conjecture was
refined by Dr. Goro Shimura of Princeton University a few years later.
But, Dr. Katz said, mathematicians had no perception through the
1950's to 70's that this had any relationship to Fermat's last
theorem.  "They seemed to be on different planets," he said.

In the mid-80's, Dr. Gerhard Frey of the University of the Saarland in
Germany "came up with a very strange, very simple connection between
the Taniyama conjecture and Fermat' last theorem, " Dr. Katz said.
"It gave a sort of rough idea that if you knew Taniyama conjecture you
would in fact know Fermat's last theorem."  He explained.  In 1987,
Dr. Ribet proved the connection.  Now, Dr. Wiles has shown that a form
of the Taniyama conjecture is ture and that this implies tht Fermat's
last theorem must be true.

"One of the things that's most remarkable about the fact that Fermat's
last theorem is proven is the incredibly roundabout path that led to
it," Dr. Katz said.

Arcane to the Arcane

Another remarkable aspect is that such a seemingly simple problem
would require such sophisticated and highly specialized mathematics
for its proof.  Dr. Ribet estimated that a tenth of one percent of
mathematicians could understand Dr. Wiles' work because the
mathematics is so technical.  "You have to know a lot about modular
forms and algebraic geometry," he said.  "You have to have ollowed the
subject very closely."

The general idea behind Dr. Wiles' rpoof was to associate an elliptic
curve, which is a mathematical object that looks something like the
surface of a doughnut, with an equation of Fermat's theorem.  If the
theroem were false and there were indeed solutions to the Fermat
equations, a peculiar curve would result.  The proof hinged on showing
that such a curve could not exist.

Dr. Wiles, who has told mathematicians he is reluctant to speak to the
press, could not be reached yesterday.  Dr. Ribet, who said Dr. Wiles
was shy, said he was asked to speak for him.

Dr. Ribet said it took Dr. Wiles seven years to solve the problem.  He
had a solution for a special case of the conjecture two years ago, Dr.
Ribet said, but told no one.  "It didn't give him enough and he felt
very discouraged by it", he said.

Dr. Wiles presented his results this week at a small conference in
Cambridge, England, his birthplace, on "Padic Galois Representations,
Iwasawa Theory and the Tamagawa Numbers of Motives."  He gave a
lecture a day on Monday, Tuesday and Wednesday with the title "Modular
Forms, Elliptic Curves and Galois Representations."  There was no hint
in the title that Fermat's last theroem would be discussed, Dr. Ribet
said.

"As Wiles began his lectures, thee was more and more speculation about
what it was going to be, " Dr. Ribet said.  The audience of
specialists in these arcane fields swelled from about 40 on the first
day to 60 yesterday.  Finally, at the end of his third lecture, Dr.
Wiles concluded that he had proved a general case of the Tatiyama
cnjecture.  Then, seemingly as an after thought, he noted that that
meant that Fermat;s last theorem was true.  Q.E.D.

People raised their cameras and snapped pictures of this historic
moment, Dr. Ribet said.  Then "there was a warm round of applause,
followed by a couple of questions and another warm round of applause,"
he added.

"I had to give the next lecture, " Dr. Ribet said.  "It was something
incredibly mundane."  Since mathematicians are "a pretty well behaved
bunch," they listened politely.  But, he said, it was hard to
concentrate.  "Most people in the room, including me, were incredibly
shell-shocked," he said.