| This is all of the USENET traffic that I have seen about this
(at least I think I have it all in here).
Dan
**********
Newsgroups: sci.math
Path: decwrl!decvax!mcnc!gatech!udel!princeton!phoenix!mjlarsen
Subject: Fermat's Last Theorem
Posted: 2 Mar 88 18:01:17 GMT
Organization: Princeton University, NJ
I just heard a rumor (relayed by a friend at the I.H.E.S.) that a
Japanese mathematician has proved Fermat's conjecture. Has anybody
heard any details?
-Michael Larsen
**********
Newsgroups: sci.math
Path: decwrl!decvax!mcnc!gatech!udel!princeton!phoenix!mjlarsen
Subject: Re: Fermat's Last Theorem
Posted: 3 Mar 88 16:20:00 GMT
Organization: Princeton University, NJ
More rumors. This one comes independently from a friend at the IAS and
a friend at Harvard (who got it from the Max Planck Institute):
Fermat's conjecture has been claimed by Miyaoka. Some time ago Miyaoka
proved the inequality (c_1)^2 < 3 c_2 for the Chern classes of an
algebraic surface of general type. Parshin showed that the same
inequality for arithmetic surfaces (where Chern classes have to be
interpreted in the sense of Arakelov theory) implies Fermat.
Miyaoka then figured out how to prove the desired inequality.
Apparently the audience at the Max Planck Institute was not completely
convinced by the proof.
-Michael Larsen
**********
Newsgroups: sci.math
Path: decwrl!labrea!agate!pasteur!ames!ll-xn!mit-eddie!uw-beaver!cornell!rochester!udel!princeton!phoenix!mjlarsen
Subject: Re: Fermat's Last Theorem
Posted: 4 Mar 88 20:41:51 GMT
Organization: Princeton University, NJ
Rumors III. This just in from the IHES. Miyaoka's proof only works
(if it works at all) for exponents n "sufficiently large." It remains
to be seen whether this bound can be made effective, let alone whether
it is small enough to make a computer aided proof possible.
-Michael Larsen
**********
Newsgroups: sci.math
Path: decwrl!decvax!purdue!gatech!linus!bs
Subject: FLT
Posted: 7 Mar 88 22:24:06 GMT
Organization: The MITRE Corporation, Bedford MA
I have some additional news and a letter from Prof. Larry Washington
of U. Maryland regarding the purported proof of FLT.
The method was to prove an inequality involving Chern Classes on
arithmetical surfaces. A result of Parshin implies a conjecture of Szpiro
(involving diophantine inequalities) which in turn implies FLT. The
letter below, from L. Washington, indicates that the proof does seem
to be pretty good.
The proof also gives effective bounds on the work of Faltings regarding
Mordell's conjecture (for which he got the Fields medal). Miyaoka's
result seems to give effect bounds on the number of solutions to
diophantine equations of genus > 2. Faltings work established that there
we a finite number.
It may also be that this work of Miyaoka implies the well-known
ABC conjecture, but the jury is still out on that.
Bob Silverman
<<<< letter follows >>>>
"Lawrence C. Washington" <lcw%julia.umd.edu@eneevax.umd.edu>
From: "Lawrence C. Washington" <lcw%julia.umd.edu@eneevax.umd.edu>
Subject: FLT
Comments: To: MCCURLEY@uscvm, NMBRTHRY%NDSUVM1.BITNET@CUNYVM.CUNY.EDU,
Kevin@julia.umd.edu, List@julia.umd.edu, McCurley@julia.umd.edu,
Number@julia.umd.edu, Theory@julia.umd.edu
To: Bob Silverman <bs@linus>
Status: R
I talked to Don Zagier in Bonn and here is what I learned. Yoichi Miyaoka gave
a talk at Max Planck in Bonn last Friday in which he gave a proof of a
conjecture of
Parshin, which in turn implies FLT (for sufficiently large exponents).
Don was at the talk and said it looks reasonable, though there are a lot
of details that need to be checked by the experts. He regards it as a
serious attempt. Miyaoka was previously known for a certain inequality
involving Chern classes on Kahler surfaces (later strengthened by Yau).
Parshin apparently showed a year or two ago (but still unpublished, as far as
I know), that the analogue of this result for arithmetic surfaces (the type of
object Faltings works with) implies FLT. It seems that Miyaoka has proved this.
-Larry Washington
|
| Hi. I enjoyed the Kaiku in .9. Anyway, here is the latest
that I have gotten from usenet sci.math.
If anyone understands what on earth they are talking about,
please let us know.
Dan
Newsgroups: sci.math
Path: decwrl!hplabs!hp-sdd!ucsdhub!ucrmath!marek
Subject: Re: Fermat's LT - Faltings' refutation - request for posting
Posted: 3 Apr 88 02:14:07 GMT
Organization: University of California, Riverside
In article <1317@gvax.cs.cornell.edu>, regan@gvax.cs.cornell.edu (Ken Regan) writes:
> This is a vote of encouragement to anyone who might have some details on
> just what part of Miyaoka's argument breaks down, or might know of a
> published article or communication on this, if any has appeared within the
> past two weeks.
Below is a file circulated through a network among many people, containing
some information about recent events regarding FLT. It's a bit chaotic,
and hard to understand unless someone knows well number theory. I don't,
but it's fun, anyway.
**************************************************************************
**************************************************************************
================================================================
Date: 29 Mar 1988 13:51:40
To: TGBS distribution list
From: HAFNER at ALMVMA
FERMAT News
Below is a copy of a file forwarded through many people and last edited
by Sol Friedberg, concerning the dialogue surrounding the "proof"
of FLT. It got to him via David Wright.
========================================================================
To: COG%hbunos.bitnet@RELAY.CS.NET,
bump%gauss.stanford.edu@RELAY.CS.NET,
coleman%ucbcartan.berkeley.edu@RELAY.CS.NET,
dats%huma1@HARVARD.HARVARD.EDU, friedbe@ucscc,
dhayes%umass.bitnet@RELAY.CS.NET, wmc@AMETHYST.MA.ARIZONA.EDU,
ribet%ucbcartan.berkeley.edu@RELAY.CS.NET,
ma408000%brownvm.bitnet@RELAY.CS.NET, jhs@bu-cs.bu.edu,
mike%mssun8.msi.cornell.edu@RELAY.CS.NET,
rswitze1%dgogwdg1.bitnet%nemo.math.okstate.edu@RELAY.CS.NET,
teitelbaum@UB.CC.UMICH.EDU
I have only the following information:
1) The proposed proof starts from Miyaoka's inequality for complex algebraic
surfaces.
2) Parshin proved that Miyaoka's inequality for arithmetic algebraic surfaces
implies Fermat's last theorem.
3) Miyaoka now claims to have extended his inequality to the arithmetic
case.
4) He lectured on his proof at the Max Planck Institute in Bonn on February
29. I have received (through David Cox) a copy of notes by Alan
Durfee from this lecture.
5) Some people believe the proof will succeed for large exponent. Some people
believe the upper bound on exponents where FLT may fail is effective.
LATEST ON FLT #2
The following documents may be had from Jim Cogdell in Israel
COG@HBUNOS.BITNET
1. A preprint of Parshin where he shows that the Bogumolov-Miyaoka-Yau
inequality for arithmetic surfaces implies some version of Szpiro's
conjecture, which implies Fermat for sufficiently large exponents.
LATEST ON FLT #4 (3/16) (and maybe the end....)
Summary of events so far:
1) Starting point: Bogumolov-Miyaoka-Yau inequality for complex algebraic
surfaces (c_1)2 <= 3 c_2 (Miyaoka spent last year at Columbia, in contact
with Szpiro; to what extent this instigated work on the present developments
I have no further details about.)
2) Parshin's lecture notes (or preprint?) proving that the analogue of the
above inequality for arithmetic surfaces implies Szpiro's conjecture bounding
minimal discriminants of elliptic curves over Q in terms of the conductor.
3) Frey's result that Szpiro's conjecture implies the Masser-Oesterl\'e
"abc" conjecture about integral solutions of a+b+c=0, which in turn implies
Fermat's Last Theorem for large exponents.
( For e>0, there is a K>0 such that for all pairwise coprime
integer solutions to a+b+c=0,
max{ |a|, |b|, |c| } <= K {\prod_{p|abc} p}(1+e) )
4) Miyaoka, in a lecture at the Max Planck Institute on February 26, claimed
to have established the arithmetic version of his inequality.
Before a few days ago, there had been some very positive feedback. From IAS,
there now come some very pessimistic reports. Below is a direct quote from
GOSS@IASSNS.BITNET:
" I have talked with Faltings at some length and Bombieri at
great length: Faltings is extremely dubious and feels that there is
no hope. He himself has thought along these lines and feels that
Miyaoka did not treat any of the difficulties that he himself
encountered. He has seen Zagier's notes and has found them to
have many mistakes-so it appears hopeless. Bombieri was, at first,
very excited. Then he realized that Miyaoka had "proved" something
about chern numbers in a way that should hold for surfaces. However,
this fact is FALSE for algebraic surfaces...... So he began to doubt."
...... " I think Fermat has survived this time also. "
LATEST NEWS ON FLT #5 (3/17)
Selected cryptic quotes:
From: IN%"GOSS%IASSNS.BITNET@cunyvm.cuny.EDU" 16-MAR-1988 20:34
"I just had a little lecture from E. Bombieri. He has been thinking
about the notes on Miyaoka. He believes that M used the wrong
extensionn of sheafs to begin with-you should use an extension of
a positive sheaf; whereas M uses an extension of a negative one. With
this positive extension one might get a reasonable result. The problem
though is this result seems to have nothing to do with chern numbers
and thus will not give what Parshin needs!"
From: Nicholas M. Katz IHES (1)69074326 UIHS003 at FRORS12
"Quick answer to bombieri:
I agree with your analysis of the structure of the argument; the way miyaoka
seems to pull the rabbit out of the hat is by his estimate for the value at
s=1 of a Selberg zeta function. NEITHER I NOR ANYONE I HAVE SPOKEN TO ABOUT IT
CAN GIVE ANY JUSTIFICATION WHATSOEVER FOR SUCH AN ESTIMATE.
so if there's a mistake of an irreparable nature, this could be it. If you
can provide any enlightenment on this topic it will be welcome. Nick"
LATEST NEWS ON FLT #7 (3/18) STOP THE PRESSES!!!
I asked Weil what he thought about today's dispatches. He
said that it did not look too good. "What convinced me was
talking to Faltings who said that he had tried to do things this
way and Miyaoka has it wrong." But, as Faltings said today, "Miyaoka
is hanging tough!" "
From: barry mazur (33)(1)69076101 UIHS012 at FRORS12
To: GOSS at IASSNS
"Dear David, Hi. Nick conveyed to me Bombieri's query, and in case
you haven't heard from him, and in view of the statement that any form
of answer would be appreciated here is something:
I too had been contemplating the upper bound on chi of the symmetric
n-th powers, as described in Zagier's notes,--not trying to make the con-
nection between estimates available for classical algebraic surfaces and
the type of estimates needed in Miyaoka's program for arithmetic surfaces,
as Bombieri did, but just trying to check whether the available literature
for special values of Selberg zeta would yield the type of bounds that
would be useful in Miyaoka's program--and I had some questions about it
(I am far from being an expert in Selberg's zeta-function: in fact I just read
Ray-Singer for the first time last weekend) which led me to try to call
Miyaoka (who wasn't in):-- I got Zagier instead, who tells me that
the actual stated inequality of Miyaoka has changed recently. Therefore
it might be difficult to make direct comparisons with the classical
case until we know with some precision what the new statement is.
I'll be going to Bonn in a few days and will let you know.
Barry"
LAST NEWS ON FLT (3/24)
FERMAT STANDS UNDEFEATED!
Some last minute quotes:
Faltings: "99-1 that it won't work".
" Evidently, Faltings recieved a ms from Miyaoka on Friday. He went
through it and found a "well hidden" mistake. He called Bonn and
was told that another ms was in the works. "
" Lichtenbaum got a message from Gundrin that miyaoka has retracted! "
Date: Thu, 24 Mar 88 13:29 EST
" I guess you know by now that people shouldn't have
taken Faltings up on his proposed bet. Miyaoka now
has withdrawn his claim to have proven
the inequality of Parshin. `The' mistake was
apparently that he was allowing himself to make
certain base extensions; he thought that these
were harmless, but Barry Mazur convinced him that
they change the problem completely. It was on
this point that Miyaoka realized that he had to
`step down.' The other points in contention then
became irrelevant. "
-----------------------------------------------------------------------------
Ed "string theory" Witten:
"Number theory is the one subject I find as interesting as physics".
SOME CLARIFICATION
Date: 27 March 1988, 10:20:12 EDT
From: barry mazur
To: GOSS at IASSNS
" Hi,
I think that your message points up the difficulty of com-
municating mathematical information on e-mail. Concerning
tame base change the problem isn't that Miyaoka's inequality
is not invariant under base change-- it is visibly invariant
under tame base change. This isn't a relief, nor does it
eradicate the gaps in Miyaoka's present argument. Rather, it
is a bit worrisome insofar as it may be leading one to prove
arithmetic results that are too strong. But be that as it may,
the problems surrounding Miyaoka's program at present do not
have--directly-- to do with base change invariance. Two of the
biggest problems are these: Miyaoka would like an upper bound
for the Quillen chi of the n-th symmetric power of his basis
(generic) vector bundle E of rank two. An upper bound of the
form o(n-cubed) would be enough. There are difficulties in his
argument for this. Also, he would need that the zero-th coho-
mology of these n-th symmetric powers vanish. Now his present
argument for this is O.K. for any given n, but requires E to
be "more and more generic" to accomodate "more and more" values
of n. This raises a host of technical questions since one is
trying to keep track of asymptotic estimates in n and must deal
with a basic vector bundle of rank two E that --in some sense--
varies with n. There are a few other issues as well that
complicate the matter.
I certainly hope that Miyaoka fills those gaps. If Faltings
has any ideas that might be helpful to Miyaoka, that would be great. "
|