Path: math.ohio-state.edu!gateway
From: rubin@math.harvard.edu (Karl Rubin)
Newsgroups: math.announce
Subject: update on Fermat's Last Theorem
Date: 25 Oct 1994 10:29:18 -0400
Organization: The Ohio State University, Department of Mathematics
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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