Date: Mon, 4 Apr 94 16:03:53 CDT
From: rusin (Dave Rusin)
To: [Permission pending]
Subject: seeking clarification of taniyama-weil conj.

Hi,
	I find myself attempting to explain the significance of last
year's news on FLT to grad students. I enjoyed reading a popular account
you and silverberg wrote up on this. As a group theorist turned 
algebraic topologist I'm a little out of my league particularly on the
more analytic aspects of elliptic curves. I understand that a geometric
statement of the conjecture is that every elliptic curve is a surjective 
image of one of the X_0(N) = H / \Gamma_0(N).  (+cusps). I guess I
would expect to see the "dual" (algebraic) version that the function
field of the elliptic curve embeds in that of some X_0(N), but that
version seems not to appear in what I have been reading of elliptic
curves. Inasmuch as this would make the conjecture sound a lot like
the Kronecker-Weber theorem, hence possibly more believable, I  would
have expected to see this statement.
	So, before I fill young minds with rubbish, can I ask you to
tell me (1) if I've grossly misunderstood something (2) (assuming I'm
not totally off-base so far) where I might find some pointers in the
literature to the connection with the K-W theorem, and (3) any
recommendations of easily-understood descriptions of the X_0(N)
(e.g., is there a quick way to write them as subvarieties of P^2(C)?)
	Thanks for your attention.
	dave rusin@math.niu.edu.

==============================================================================

Date: Thu, 7 Apr 1994 15:28:10 -0400
From: [Permission pending]
To: rusin@math.niu.edu
Subject: Re:  seeking clarification of taniyama-shimura conj.

> 	So, before I fill young minds with rubbish, can I ask you to
> tell me (1) if I've grossly misunderstood something

I don't think so.

>                                                     (2) (assuming I'm
> not totally off-base so far) where I might find some pointers in the
> literature to the connection with the K-W theorem, and 

I'm not sure, but I am not aware of any such references.  The problem may
be that there has never been any progress in attacking the conjecture 
from this direction.   

>                                                        (3) any
> recommendations of easily-understood descriptions of the X_0(N)
> (e.g., is there a quick way to write them as subvarieties of P^2(C)?)

The map z -> [j(z),j(Nz),1] gives an embedding of X_0(N) into P^2.
I think you can find in Shimura's book a discussion of the polynomial
satisfied by j(z) and j(Nz).

Hope this helps.

[sig deleted -- djr]
==============================================================================

From: prezky@apple.com (Michael Press)
Newsgroups: sci.math
Subject: Re: What is: "Fermat's Last Theorm" ?
Date: Fri, 18 Dec 1998 17:07:27 -0800

In article <366EC6DE.CF1DA947@rlemail.dseg.ti.com>, jlr1@ti.com wrote:
[...]
>
>The theorem was proven by Andrew Wiles in (I think) 1994.  Actually
>Wiles proved the Taniyaka-Shimura conjecture which says that all
>elliptic equations have an equivalent modular form.  

Really?

I have a preprint of a paper from November 1993 by K. Rubin and A. Silverberg,
"Wiles' Proof of Fermat's Last Theorem".  In the paper they state

"Wiles reduces the proof of the Taniyama-Shimura
Conjecture to what we call the Modular Lifting
Conjecture (which can be viewed as a weak form of the
Taniyama-Shimura Conjecture), by using a theorem of
Langlands and Tunnell. ... Although he does not prove
the full Mazur Conjecture (and thus does not prove the
full Taniyama-Shimura Conjecture), Wiles result
(Theorem 5.3) implies enough of the Modular Lifting
Conjecture to prove Fermat's Last Theorem."

Has the Taniyama-Shimura Conjecture been proven since?

Historical note:  This paper was published before Wiles'
ditched Euler systems. In the margin of the preprint next
to the section "Wiles' geometric Euler System" these
words appear

"Unfortunately, space considerations (and our incomplete
understanding) make it impossible to give the details of
Wile's truly marvelous construction."

Their "incomplete understanding" is now understandable.  

[...]

-- 
Michael Press
prezky@apple.com