From: kubo@brauer.harvard.edu (Tal Kubo)
Newsgroups: sci.math
Subject: Re: Fermat Proof
Date: 4 Feb 1995 02:49:52 GMT
In article <1995Feb3.041740.21038@Princeton.EDU>
conrad@math.Princeton.EDU (Brian Conrad) writes:
>
>Or hand, if one is going to worry about axiom of choice,
>then one basically has to chuck the foundations of modern algebraic
>geometry as given by Grothendeick (esp fancier aspects like
>etale cohomology, etc) out the window, and so one might as well
>kiss goodbye to many of the deepest theorems in number theory
>(Weil Conjectures, Faltings' proof of Tate conjectures, etc.)
>Take your pick.
The main use of AC in the Weil conjectures is to lighten the notation (by
referring to "isomorphisms" between algebraic closures of various p-adic
fields, "embedding" them into C, etc.). Deligne, in a footnote early on in
his paper, apologizes for the use of this "repugnant" (his term) axiom, and
explains how to get rid of it wherever it appears in the paper.
>In particular, Wiles' proof makes implicit use of such things too,
>so axiom of choice is certainly crucial, but not quite for
>the reason you seem to have inferred from the NYT article.
AC hidden in the language as you point out, is usually eliminable.
The real question is to what extent the proof is constructive, e.g whether
the methods of Wiles et al help you find modular parametrizations of
particular curves. Wiles' Selmer group inequality can be reduced to a
finite computation for any given conductor, so the original proof (without
the AC-dependent fix) does give a method for showing that all elliptic
curves with given squarefree conductor are modular. This was actually
done, prior to the Wiles/Taylor repairs, for prime conductor up to 3000;
the person who did that work reads this newsgroup, so I'll leave it
to him to say more if he wants to.