Talk:Modularity theorem

Article is inaccessible?

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

Good article

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

Name

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)[reply]

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)[reply]

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)[reply]

"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)[reply]

Weight or Level?

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". 140.114.81.55 (talk) 04:43, 9 December 2010 (UTC)[reply]

Weil's involvement

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)[reply]

is 'tau' complex?

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 71.83.154.30 (talk) 15:09, 6 March 2015 (UTC)[reply]

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

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 http://www.jmilne.org/math/Documents/TokyoNikko1955.pdf.

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)[reply]

name

I've seen it's also called Weil-Taniyama-Shimura conjecture [Andre Weil]. Setenzatsu (talk) 17:09, 7 July 2019 (UTC)[reply]

Truly terrible "explanation" of what the theorem says

This is the article's truly bad, bad, bad "explanation" of the content of the theorem:

"The theorem states that any elliptic curve over Q can be obtained via a rational map with integer coefficients from the classical modular curve $X_{0}(N)$ for some integer N; this is a curve with integer coefficients with an explicit definition."

Does anyone really think this is clear? It is not.

The essence of the theorem lies in the 100% unclear words "obtained via".

What does "obtained via" mean? Is that too much to ask from someone writing for Wikipedia?

It matters not at all if practitioners in algebraic geometry would immediately know what is meant. The relevant fact is that nobody else knows what this means.

Can someone knowledgeable on this subject who is also able to write clearly please rewrite the "explanation" so it is more clear than mud?2600:1700:E1C0:F340:DD38:C78:3F8B:9257 (talk) 15:12, 25 September 2019 (UTC)[reply]

Article is truly bad

The closest this article comes to defining its subject is the Statement section:

"The theorem states that any elliptic curve over Q can be obtained via a rational map with integer coefficients from the classical modular curve $X_{0}(N)$ for some integer N; this is a curve with integer coefficients with an explicit definition. This mapping is called a modular parametrization of level N. If N is the smallest integer for which such a parametrization can be found (which by the modularity theorem itself is now known to be a number called the conductor), then the parametrization may be defined in terms of a mapping generated by a particular kind of modular form of weight two and level N, a normalized newform with integer q-expansion, followed if need be by an isogeny."

There is nothing wrong with discussing the statement using just the English language. The phrase "can be obtained via a rational map" is far, far, far too vague. This calls for a clear statement in mathematical language.

But the article lacks any clear mathematical statement of its own subject. In other words, this article does not say what it is about. I hope that someone knowledgeable about the Modularity Theorem can fix this serious problem. 98.255.224.144 (talk) 13:22, 20 March 2021 (UTC)[reply]

I think that the clarity you are saying is needed, is just spread out over several articles.

https://en.wikipedia.org/wiki/Modular_elliptic_curve

https://en.wikipedia.org/wiki/Base_change_lifting

Etc, etc.

I do agree with you that there could be some redundancy in this article, where paragraphs where lifted from the other articles, and hyperlinked. However, I may be in the minority, and it is just considered that Wikipedia to answer a query, that the onus is on the reader finding the right page through trial and error.

Signed - Brahmagupta — Preceding unsigned comment added by 49.185.41.142 (talk) 22:53, 2 May 2022 (UTC)[reply]