Newsgroups: sci.math.research
From: Hoey@AIC.NRL.Navy.Mil (Dan Hoey)
Date: 1998/01/05
Subject: Re: Simon Singh's "Fermat's Last Theorem"

> Lieven Marchand <m...@bewoner.dma.be> wrote:

> >[Aczel] ... accuses Weil of an elaborate plot to steal the credit
> >for the Taniyama-Shimura conjecture from its authors. [ One of the
> >subchapters has as title Intrigues and Treachery, and another one
> >The lie. ] In light of the dubious research on other material in
> >the book, I would very much like to hear some knowledgeable people
> >comment on these allegations.

Aczel is extracting and popularizing Lang's notice "Some History of
the Shimura-Taniyama Conjecture", from the November 1995 _Notices of
the AMS_, also at http://www.ams.org/notices/199511/lang.html.  As in
other parts of the book, Aczel has added his own witty comments to
make the narrative more exciting--and less trustworthy.  Even with the
authoritative version, though, it is unclear to me how much of the
controversy is about lies and treachery and how much is about
exaggeration and sensationalism.

richard.gr...@ox.compsoc.org.uk (Richard Green) writes:

> ... Not only is the historical material somewhat off-topic, but
> the mathematics is often badly explained....  The more technical
> material on elliptic curves is presented in a similarly
> unenlightening way.

Actually, the elliptic curve stuff is even more broken than most of
the rest.  For instance, Aczel's graphs on page 93 are footnoted
  13. Adapted from Kenneth A. Ribet and Brian Hayes, "Fermat's Last
      Theorem and Modern Arithmetic," _American Scientist_, Vol. 82,
      March-April 1994, pp. 144-156.

The figures seem to have been "adapted" by (1) reducing them by half,
so that they are nearly illegible, and (2) removing the captions that
explain that the first two are examples of elliptic curves, but the
second two are not elliptic curves, because they are associated with
singular cubic equations.

Aczel's book has a large collection of disappointing figures.  Page 6
has a picture of the wrong _Arithmetica_, lacking Fermat's comment.
Page 25 was drawn by someone who seems to have lacked both T-square
and protractor (or their computational equivalent).  Pages 82 and 83
are trying to say something, but nothing comes out.  And the top of
page 86 needs just a little more low-res ray-tracing.  An inverse
problem is in the caption on Gerd Faltings's photograph on page 137.
Perhaps "many feared that Faltings would now beat [Wiles] to the true
proof" in 1993, but nothing of this little bit of suspense made it to
the narrative (or even the index).

> I haven't read Singh's book, but it seems to be by far the better work
> from what I've heard other people say about it.

Better, but not by so very far.  Singh doesn't explain the math,
either.  And he misstates the Frey equation--it's supposed to be
  y^2 = x (x - A^N) (x + B^N).
The whole point of the Frey equation (as I understand it) is that the
differences between its (y=0) roots are A^N, B^N, and A^N + B^N, and
it's the differences being Nth powers that makes the curve
non-modular.  Also, as mentioned previously, Singh _never_ mentions
that Wiles proved only the semistable case of Taniyama-Shimura.

If you want to learn some relevant mathematics, I would recommend the
Ribet and Hayes article cited above.  It isn't perfect--particularly
confusing are the parts from p.147 column 3 to p.148 column 2, where
they use the equivalent formulation
 A^N + B^N + C^N = 0, (A B C not zero)
but don't bother to mention the change.  When you read "C" as "-C",
the argument suddenly starts making sense.

There's another error on page 155, where they justify a property by
congruence that should be justified by relative primality, but that's
relatively benign.  And Figure 1 is misleading but unrelated to the
article--someone wanted to show off their Mathematica.

What all three of these lack is a good description of modular (or
automorphic?) forms.  Here Aczel gets the most explicit, but I'm not
sure whether to trust him.  Is it really just any function f:C->C that
has a nontrivial symmetry f(z) = f( (az+b)/(cz+d) ) ?  Does f have to
be continuous?

If I find out what a modular form is, I might--mind you, I said
'might'--want to look up L-series.

Dan
Hoey@AIC.NRL.Navy.Mil