Talk:Modularity theorem

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 Field:  Number theory

Article is inaccessible?[edit]


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)

Move/redirect request[edit]

Since it's called "Taniyama–Shimura theorem" everywhere on the page, the article should have that title. 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)

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

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)

changes in the History[edit]

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)

Good article[edit]

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


This article states that the modularity theorem was previously called the Taniyama-Shimura-Weil conjecture. However, I have found several book sources that refer to it simply as the Taniyama-Shimura conjecture:

--Latiosoital (talk) 22:15, 2 March 2009 (UTC)

Taniyama-Shimura-Weil is more accurate. We should leave the original version. Only the last two of your book sources are real math books. The connection with Weil is e.g. mentioned in Wiles' article, in Diamond, Modular forms and modular curves, Dolgachev, Lectures on Modular Forms. Ringspectrum (talk) 08:29, 3 March 2009 (UTC)
Taniyama and Shimura made the conjecture in 1955. However, it remained relatively unknown until André Weil rediscovered it in 1967. Yes, he rediscovered it, but he didn't create it. Just as theorems are named after the person who proved it, conjectures are named after the person (or people) who first conjectured it. Since Taniyama and Shimura (NOT Weil) were the first people to make the conjecture, its correct name is the Taniyama-Shimura conjecture. --Latiosoital (talk) 13:19, 3 March 2009 (UTC)
"Just as theorems are named after the person who proved it" -- well, this isn't true, see e.g. Burnside's lemma. We should stick to the names used in real world. There are no "correct" names, just names used by people. e.g. Faltings calls it "Taniyama-Weil" conjecture, Husemoller writes "For this reason the modular curve conjecture goes under the name of the Shimura–Taniyama–Weil conjecture or the earlier name of the Taniyama–Weil conjecture.". To include all possible and used names, we should use the full name Taniyama-Shimura-Weil. Ringspectrum (talk) 17:36, 3 March 2009 (UTC)

Weight or Level?[edit]

This is a quote from the current version of the article (prior to my edit): "Some modular forms of level two, in turn, correspond to holomorphic differentials for an elliptic curve." Surely this should say "weight two", instead of "level two"; I will make the change. Hopefully someone can confirm this change is correct, or make a case for changing it back to "level". (talk) 04:43, 9 December 2010 (UTC)

Weil's involvement[edit]

The history section currently contains the phrase, "The conjecture was rediscovered by André Weil in 1967." Does this mean that (a) Weil came to the conjecture independently, or (b) he discovered the 1956 Shimura & Taniyama paper that originally presented the conjecture?

Simon Singh ('Fermat's Last Theorem', p210) seems to imply the second case; he phrases it as follows: "Andre Weil... was to adopt the conjecture and publicise it in the west".

Either way, I think the original sentence could benefit from clarification. Manning (talk) 03:03, 14 February 2011 (UTC)

is 'tau' complex?[edit]

If 'tau' is complex, the series will contain real constants (if the exponentials converge) not normally associated with the Fourier series. The coefficients contain any real part of the cyclic bases. — Preceding unsigned comment added by (talk) 15:09, 6 March 2015 (UTC)

Is it clear that Weil "rediscovered" the Taniyama-Shimura conjecture[edit]

Under History, this passage appears:

"Taniyama (1956) stated a preliminary (slightly incorrect) version of the conjecture at the 1955 international symposium on algebraic number theory in Tokyo and Nikko. Goro Shimura and Taniyama worked on improving its rigor until 1957. Weil (1967) rediscovered the conjecture, and showed that it would follow from the (conjectured) functional equations for some twisted L-series of the elliptic curve; this was the first serious evidence that the conjecture might be true. "

But is it clear that Weil rediscovered the conjecture? He was present at the 1955 interntional symposium on algebraic number theory in Tokyo and Nikko, as shown by the conference proceedings, online at

In fact, in the September 12 paper session from 9 a.m. to 12 p.m., Shimura, Taniyama, and Weil each presented their own paper, in that order (and excluding "short" communications, were the only presenters in that session).

So it seems likely that Weil would have been aware of the conjecture at that time.

What is the evidence that Weil rediscovered the conjecture in 1967 rather than knowing of it since 1955?Daqu (talk) 18:37, 1 September 2016 (UTC)