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From: bs@gauss.mitre.org (Robert D. Silverman)
Subject: Hear Ye!
Message-ID: <1993Jun23.171131.670@linus.mitre.org>
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Date: Wed, 23 Jun 93 11:11:31 GMT+7:00
Lines: 27


I have news regarding the proof of FLT by Professor Wiles.

He has apparently proved the Taniyama-Weil conjecture for elliptic
curves over Q with semi-stable reduction.

It is known that the Frey curves (the elliptic curves associated with
FLT) are of this type.

It is also known, by the work of Ken Ribet that Taniyama-Weil implies
FLT.

The method of proof did not involve anything new. It somehow used
Galois representations on torsion points. The key step in the proof
was to show that if there is a modular representation, and the modular
function is deformed [in some way], then the representation still remains
modular.

This is all I know.

I will post more when I learn more.

--
Bob Silverman
These are my opinions and not MITRE's.
Mitre Corporation, Bedford, MA 01730
"You can lead a horse's ass to knowledge, but you can't make him think"

Snippets from the (British) Daily Telegraph Jun 24 front page
story "Number's up for Fermat's Last Theorem"

  Prof Andrew Granville of University of Georgia describes the
proof as "rich and profound and likely to be correct". Says "Prof
Wiles's proof draws heavily on the Shimura-Taniama-Wei conjecture,
which took 15 years to formulate. I can only say it involves
elliptical curves".
  Asked whether the proof would have practical use, Prof
Granville said: "I cannot see any, but it would be stupid and
dogmatic to assert that it would not. A parallel might be
John von Neumann's apparently useless theories in the 1930's
about automata which led to modern computer software."
  Also says: "Fermat was a rather obnoxious character
who enjoyed issueing such challenges."
  Tom Stoppard is a bit pissed off, as his play "Arcadia"
about FLT is now ruined.

From: wjcastre@math.ohio-state.edu (W. Jose Castrellon G.)
Newsgroups: sci.math
Subject: Fermat's Last Theorem. Sketch of proof
Date: 24 Jun 1993 17:06:54 -0400
Organization: The Ohio State University, Math.Dept.(student)
Lines: 125


Hello netters, local number theorist  [ no crank... he's the most 
recent recipient of the AMS Cole Prize in Number theory ]
Prof. Karl Rubin was present at the Wiles lectures in Cambridge. 
He has posted the following outline of the proof to our math newsgroup:

============================================================================
>From K.C.Rubin@newton.cam.ac.uk Thu Jun 24 14:50:52 EDT 1993
Article: 535 of math.announce
Path: math.ohio-state.edu!gateway
From: K.C.Rubin@newton.cam.ac.uk
Newsgroups: math.announce
Subject: sketch of Fermat
Date: 24 Jun 1993 09:19:10 -0400
Organization: The Ohio State University, Department of Mathematics
Lines: 103
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Several people have asked for more details about Andrew's proof.
Here is a lengthy sketch.  Enjoy.

Karl
____________________________________________________________


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}.)

    .-1
    
    no wonder fermat could not fit the proof in the margin of his book! :)
    even if he might have done short cuts such as this one below:
    
From: baez@guitar.ucr.edu (john baez)
Newsgroups: sci.math
Subject: Fermat's last theorem rumors ctd.
Date: 30 Oct 1993 18:12:08 GMT
    
I got some email from someone who got it from someone who got it from
someone who knows a lot about automorphic forms.  It says first that,
yes, Wiles proof makes use of a result by Faltings that the latter had
proved rather sloppily (as in "it can easily be seen").  Also, the chief
reviewer of Wiles' proof discovered an "amazing leap of logic" in which
something "obvious" turned out to require 10 pages of difficult proof.
(But it was true.).  I hope it all works out.
 
Moral: don't write "it can easily be seen" - prove it!
 
 
 
 

    i thought this this letter gives the basic idea behind FLT Wiles solution 
    approach in nice way.
    ---------------------------------------------------------------------
    
From: stimpson@panix.com (S. Joel Katz)
Subject: Open Letter to Marilyn Vos Savant -- FLT
Date: 24 Nov 1993 15:35:17 -0500
 
 
        Dear Marilyn:
 
        I must admit, I was quite puzzled by your response to Joseph McGriff
about Fermat's Last Theorem. I find your position indefensible and logically
untenable. I would have simply assumed that you were uninformed about the
subject and made some reasonable sounding but unfounded inferences, but I
noticed that you have a book out on the subject. I have not read the book, 
but I will.
 
        Imagine if it were a famous unsolved problem in construction to build
a usable shelter with only one wall and no additional supports. Suppose
someone claimed to have solved it, with their solution being to find or
make a circular depression in curved ground and simply use a flat wall to 
cover it, making a pit with a roof.
 
        This 'solution' actually shows us that we had a hidden assumption in
the problem. We had never considered that ground could be anything other than
flat. Now that we are aware that curved ground exists, we can rephrase the
problem to require that the ground be flat. The problem has not been solved,
but it's requirements are now more precisely understood.
 
        This is exactly what happened when Bolyai squared the circle using
hyperbolic geometry. He used a new type of "ground" on which squaring circles
is possible. Circles embedded in Euclidean space are not similarly squarable.
We reject Bolyai's solution because the problem was posed for *any* circle
and, at the time, that meant a circle of arbitrary size embedded in a 
Euclidean plane.
 
        However, Fermat's Last Theorem is a problem in pure number theory. The
numbers can be used to represent the measurement of any quantity one wishes
without destroying the validity of the proof when one's results are translated
back into pure numbers.
 
        What Dr. Wiles did, in simplest terms, was to show that any solution
in integral numbers to Fermat's Last Theorem would correspond to a specific
type of entity in hyperbolic space. He then showed that no such entity could 
exist. Absent a formal error in the specifics, this reasoning is absolutely 
ironclad.
 
        As a simple analogy, consider Katz's First Theorem -- that there are
no solutions in positive integers to X^2 + Y^2 = -1. I can show that any
solution to the equation X^2 + Y^2 = N corresponds to a circle centered at
the origin in Euclidean space. Now all I need to show is that a circle with
an N of -1 could not exist, and I will have proven this theorem -- whether or
not one had originally intended any correspondence between the problem and
Euclidean space.
 
        It is as if our mythical solution to the one-wall construction 
problem required one to draw the plans on curved ground. So long as the plans 
are translatable into a house that can be built on flat ground, the solution 
is valid. 
 
        Dr. Wiles went from numbers, to hyperbolic space, and back to numbers. 
Unless he made an error of a different sort, his proof is valid and should be 
accepted as such.
 
        Sincerely,
        S. Joel Katz
        Stimpson@Panix.COM
 
 

Article 59844 of sci.math:
From: wiles@rugola.Princeton.EDU (Andrew Wiles)
Newsgroups: sci.math
Subject: Fermat status
Organization: Princeton University
Lines: 21


	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.

    could some one explain what is meant by finite integers and infinite
    integers that LP is talking about below? (please see the last 3 lines
    in this note).

    thanks
    \nasser
    ---------------------------------------------------------------------

From: Ludwig.Plutonium@dartmouth.edu (Ludwig Plutonium)
Subject: Re: Fermat status
Organization: Dartmouth College, Hanover, NH
 
               PROOF OF FERMAT'S LAST THEOREM
    FERMAT'S LAST THEOREM IS FALSE. Proof: 
For exp3 in 10 adics these are three counterexamples
 
1
10
....52979382777667001
 
1
20
....4437336001
 
1
30
....4919009001
 
The expression a^n+b^n=c^n is true for all n given the following values
 
a= ...9977392256259918212890625
b=...0022607743740081787109376
c= 1 
Because a,b,c are idempotents.   ATOM,ATOM
 
   The reason such a simply stated math problem, FLT was not solved by
some of the world's greatest mathematicians Gauss, Riemann, Galois, . .
(not those dime a dozen professors of math-- just regurgitators of the
subject.) Was because it is false. And new math is created, more
realistically---uncovered to what was already laying there within the
Peano Axioms, by solving FLT. Galois photons saw "Group theory" to
solve the quintic problem. I photon saw the unearthing of infinite
integers from the endless adding of 1 in the Peano Axioms. The
mathematics world on Earth will never be the same after this year 0053.

        That is referring to the p-adic numbers.
        
        Dan

Well, in a Ludwig Plutonium posting it could refer to anything; in this case it
probably refers either to the inseam of the Virgin Mary's pantyhose or to
something he ate for breakfast. Ludwig is connected to the rest of the universe
through a semipermeable membrane made of Silly Putty... 

    Re: .0 .. What's the current verdict ?  Proof holds ?
    
    Thanks,
    John.

Article 50816 of sci.math:
From: wgd@zurich.ai.mit.edu (William G. Dubuque)
Newsgroups: sci.math
Subject: FLT proven: Wiles closes the gap!
Organization: M.I.T. Artificial Intelligence Lab.
Lines: 42

Following is a message describing Wiles announcement
of the completion of his proof of Fermat's Last Theorem.
--------
Date: Tue, 25 Oct 94 10:24:46 EDT
From: rubin@math.harvard.edu (Karl Rubin)
Message-Id: <9410251424.AA03857@math.harvard.edu>
Subject: update on Fermat's Last Theorem

As of this morning, two manuscripts have been released
 
  Modular elliptic curves and Fermat's Last Theorem, 
      by Andrew Wiles
 
  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.
 
As most of you know, the argument described by Wiles in his Cambridge 
lectures turned out to have a serious gap, namely the construction of an 
Euler system.  After trying unsuccessfully to repair that construction, 
Wiles went back to a different approach, which 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 overall outline of the argument is similar to the one Wiles described
in Cambridge.  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.)
 
Versions of these manuscripts have been in the hands of a small number
of people for (in some cases) a few weeks.  While it is wise to be 
cautious for a little while longer, there is certainly reason for
optimism.
 
Karl Rubin

At the Bourbaki Seminary organized at the Henri Poincare Institute in Paris
on Saturday, June 17, 1995, there were two lectures, the first given by
Serre and the second by Oeterle, concerning Wiles's (and Taylor's) results
aand Fermat's Last Theorem. It appears that the specialists community
have already concluded that Wiles's amended proof is correct. This proof
was presented entirely by the two lecturers.

Mihai.