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Topic: Tautologies, math, and Wiles's work
Replies: 59   Last Post: Jul 27, 2003 5:48 AM

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 tchow@lsa.umich.edu Posts: 1,133 Registered: 12/6/04
Re: Tautologies, math, and Wiles's work
Posted: Jul 4, 2003 5:30 PM

James Harris <jstevh@msn.com> wrote:
>http://mathworld.wolfram.com/Taniyama-ShimuraConjecture.html
>"In effect, the conjecture says that every rational elliptic curve is
>a modular form in disguise."

The words "in disguise" should tip you off to the fact that this sentence is
not intended to be a strictly accurate mathematical statement, but is an
informal statement in which the writer has taken some poetic license.

>"Equivalently, for every elliptic curve, there is a modular form with
>the same Dirichlet L-series."

Ah, well, this quotation helps explain where you got your impression of
what Wiles was claiming. You are perfectly correct in analyzing this
sentence by pointing out that on the one hand we have elliptic curves,
and on the other hand we have modular forms, and there is no "superset"
of objects of which elliptic curves and modular forms are both members.
So just because you get this L-series thingy from an elliptic curve,
and you can get the same L-series thingies from modular forms, how can
this possibly imply anything like "all elliptic curves are modular"?

The answer is that you are right, it *doesn't* immediately imply that
all elliptic curves are modular. It has to be combined with other
theorems in a way that isn't fully explained on the website (which
after all is just a sketchy overview). So before you can justly
criticize the proof, you need to look up the other theorems and see
how they are used together with the L-series statement to deduce that
every elliptic curve is modular.

>Besides even if what you claim is correct, it'd mean that modular
>forms are Weil curves. Is that your assertion?

No. You need to have the courage of your convictions! You have correctly
noted that there is no "superset" that contains both elliptic curves and
modular forms. Saying that an elliptic curve *is* a modular form is
therefore, as you correctly point out, simply wrong, and even Wiles has
never claimed that. Similarly, saying that modular forms are Weil curves
is a mistake of the same type, and I don't make that assertion; neither
does Wiles make that assertion. It may follow from the claim that you
are erroneously *attributing* to Wiles, but that's irrelevant.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences