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Re: Tautologies, math, and Wiles's work
Posted:
Jul 4, 2003 5:30 PM


In article <3c65f87.0307031909.5e391fac@posting.google.com>, James Harris <jstevh@msn.com> wrote: >http://mathworld.wolfram.com/TaniyamaShimuraConjecture.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 Lseries."
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 Lseries thingy from an elliptic curve, and you can get the same Lseries 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 Lseries 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 tchowatalumdotmitdotedu The range of our projectileseven ... the artilleryhowever 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



