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From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz)
Subject: sci.math FAQ: Status of FLT
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Summary: Part 5 of many, New version,
Originator: alopez-o@neumann.uwaterloo.ca
Keywords: Fermat Last Theorem
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Date: Tue, 25 Apr 1995 17:41:22 GMT
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Archive-Name: sci-math-faq/FLT/status
Last-modified: December 8, 1994
Version: 6.2
What is the current status of FLT?
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 was found to be wanting. In summer 1994, 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.
In summary:
Both manuscripts have been accepted for publication, according to
Taylor. Hundreds of people have a preprint. Faltings has simplified
the argument already. Diamond has generalised it. People can read it.
The immensely complicated geometry has mostly been replaced by simpler
algebra. The proof is now generally accepted. There was a gap in this
second proof as well, but it has been filled since October.
You may also peruse the AMS site on Fermat's Last Theorem at:
gopher://e-math.ams.org/11/lists/fermat
_________________________________________________________________
alopez-o@barrow.uwaterloo.ca
Tue Apr 04 17:26:57 EDT 1995