Talk:Modularity theorem

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Hi!

I find much of the math articles here on Wikipedia are inaccessible to anyone less than a masters in mathematics. I suspect that the problem stems from the articles themselves and not me, but if you agree or disagree please comment here. --ShaunMacPherson 03:45, 21 Jun 2004 (UTC)

  • The Taniyama-Shimura theorem is certainly inaccessible to anyone with less than a masters in mathematics. But it's a bit of a special case. I generally don't find the Wikipedia articles to be more complex than necessary to describe the maths involved in each article. -- David Hopwood
  • Actually, I think the definition given in the article would be accessible to someone with only an undergraduate degree in mathematics :-). Though the full proof wouldn't be, of course. Anyway, this is one badass, seriously abstract theorem we're talking about here: there is no way to understand it without mathematical training, and this isn't wikipedia's fault. Some math is just that complicated. --Shibboleth 08:39, 26 Aug 2004 (UTC)
  • The article was pretty much of a mess, so I've fixed it. I hope it isn't any more inaccessible than it ever was. If you follow the link to classical modular curve, you end up with a definition which isn't too highbrow. Gene Ward Smith 04:15, 25 May 2006 (UTC)

Contents

[edit] Move/redirect request

Since it's called "Taniyama–Shimura theorem" everywhere on the page, the article should have that title. 62.136.152.161 11:58, 9 November 2006 (UTC)

The term "Taniyama–Shimura Theorem" is abominable -- it looks like what a layperson would think mathematicians would call this result, especially noting the common wikipedia mis-phrase "Conjecture X became a Theorem". The correct terminology is "Conjecture X was proven" or "establised". The Theorem is properly called the Modularity Theorem and should exist under that heading. The statement is alternatively known as the "Taniyama-Shimura Conjecture". WLior 06:07, 7 January 2007 (UTC)

[edit] Article starts right out with fundamental lack of clarity

The article begins:

In mathematics, the modularity theorem establishes an important connection, between elliptic curves over the field of rational numbers and modular forms, certain analytic functions introduced in 19th century mathematics. It was proved, for all elliptic curves over the rationals whose conductor (see definition below) was not a multiple of 27, in fundamental work of Andrew Wiles and Richard Taylor. The result had previously been called the Taniyama–Shimura–Weil conjecture, or related names. The great interest in the theorem was that it was already known to imply Fermat's Last Theorem, a celebrated unsolved problem on diophantine equations.

The remaining cases of the modularity theorem (of elliptic curve not with semistable reduction) were subsequently settled by Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor.

Does the statement of the "Taniyama-Shimura-Weil" conjecture include conductors that are a multiple of 27, or not?

Are elliptic curves without semistable reduction the same as elliptic curves over the rationals whose conductors are a multiple of 27, or not?

Should people who are unfamiliar with the word "without" be contributing to English-language Wikipedia, or not?

Inquiring minds want to know.Daqu 10:51, 10 June 2007 (UTC)

[edit] changes in the History

The history section had overused "modularity theorem", and I replaced most of these with the conjecture name instead: Taniyama-Shimura-Weil" conjecture. I felt this was more accurate instead of calling it the "modularity theorem" as it wasn't called that at any time during these historic events. Nobody ever proved the theorem, they proved the conjecture they knew at the time, and it later became the theorem. I also introduced the epsilon conjecture into the historical timeline, as it was key to Andrew Wiles getting started on proving FLT, where he focused his efforts on proving Taniyama-Shimura for the semistable elliptic curves, and this was the starting point for others to develop the full proof of the conjecture. Brianonn 19:07, 4 November 2007 (UTC)


[edit] good article

Just to say, this is a nice article. 137.205.56.18 (talk) 11:21, 20 February 2008 (UTC)