From the journal of Steven H. Cullinane

Sunday, November 30, 2003

The Proof and the Lie

A mathematical lie has been circulating on the Internet.

It concerns the background of Andrew Wiles's recent work on mathematics related to Fermat's last theorem, which involves the earlier work of a mathematician named Taniyama.

This lie states that at the time of a conjecture by Taniyama in 1955, there was no known relationship between the two areas of mathematics known as "elliptic curves" and "modular forms."

The lie, due to Harvard mathematician Barry Mazur, was broadcast in a TV program, "The Proof," in October 1997 and repeated in a book based on the program and in a Scientific American article, "Fermat's Last Stand," by Simon Singh and Kenneth Ribet, in November 1997.

"... elliptic curves and modular forms... are from opposite ends of the mathematical spectrum, and had previously been studied in isolation."

-- Site on Simon Singh's 1997 book Fermat's Last Theorem

"JOHN CONWAY: What the Taniyama-Shimura conjecture says, it says that every rational elliptic curve is modular, and that's so hard to explain.

BARRY MAZUR: So, let me explain.  Over here, you have the elliptic world, the elliptic curves, these doughnuts.  And over here, you have the modular world, modular forms with their many, many symmetries.  The Shimura-Taniyama conjecture makes a bridge between these two worlds.  These worlds live on different planets.  It's a bridge.  It's more than a bridge; it's really a dictionary, a dictionary where questions, intuitions, insights, theorems in the one world get translated to questions, intuitions in the other world.

KEN RIBET: I think that when Shimura and Taniyama first started talking about the relationship between elliptic curves and modular forms, people were very incredulous...."

-- Transcript of NOVA program, "The Proof," October 1997

The lie spread to other popular accounts, such as the column of Ivars Peterson published by the Mathematical Association of America:

"Elliptic curves and modular forms are mathematically so different that mathematicians initially couldn't believe that the two are related."

-- Ivars Peterson, "Curving Beyond Fermat," November 1999 

The lie has now contaminated university mathematics courses, as well as popular accounts:

"Elliptic curves and modular forms are completely separate topics in mathematics, and they had never before been studied together."

-- Site on Fermat's last theorem by undergraduate K. V. Binns

Authors like Singh who wrote about Wiles's work despite their ignorance of higher mathematics should have consulted the excellent website of Charles Daney on Fermat's last theorem.

A 1996 page in Daney's site shows that Mazur, Ribet, Singh, and Peterson were wrong about the history of the known relationships between elliptic curves and modular forms.  Singh and Peterson knew no better, but there is no excuse for Mazur and Ribet.

Here is what Daney says:

"We might add a few more words about the j-invariant. It is a complex number that characterizes elliptic curves up to isomorphism: two curves are isomorphic if and only if they have the same j-invariant. Not only that, but for any non-zero complex value, there actually exists an elliptic curve with a j-invariant equal to that value. So there is a 1-1 correspondence between (isomorphism classes of) elliptic curves and C*.

Now, we have already seen that an elliptic curve as a complex torus is essentially determined by the period lattice of the [Weierstrass] p-function that parameterizes the curve. More precisely, two tori are isomorphic if and only if their corresponding lattices are 'similar,' that is, if and only if one is obtained from the other by a 'homothety,' i. e. multiplication by a non-zero complex number.

But there is another way to characterize similar lattices. Suppose we have two lattices. Each has a Z-basis of the form {omega1, omega2}. Applying a homothety, we can just consider the period ratios and assume the two bases are {1, tau}, {1, tau'}, with both tau and tau' in the upper half plane H = {z | Im(z) > 0}. These define the same lattice if and only if they are related by a transformation in SL2(Z). This latter is essentially what is known as the modular group Gamma. So there is a 1:1 correspondence of similar lattices and elements of H/Gamma.

In summary, there are 1:1 correspondences between each of the following: Returning to the j-invariant, it is the 1:1 map betweem isomorphism classes of elliptic curves and C*. But by the above it can also be viewed as a 1:1 map j:H/Gamma -> C.  j is therefore an example of what is called a modular function. We'll see a lot more of modular functions and the modular group. These facts, which have been known for a long time, are the first hints of the deep relationship between elliptic curves and modular functions."

"Copyright © 1996 by Charles Daney,
All Rights Reserved.
Last updated: March 28, 1996"

Update of Dec. 2, 2003

For the relationship between modular functions and modular forms, see (for instance) Modular Function at MathWorld and Modular Form at Wikipedia.

Some other relevant quotations:

From J. S. Milne, Modular Functions and Modular Forms:

"The definition of modular form may seem strange, but we have seen that such functions arise naturally in the [nineteenth-century] theory of elliptic functions."

The next quote, also in a nineteenth-century context, relates elliptic functions to elliptic curves.

From Elliptic Functions, a course syllabus:

"Elliptic functions parametrize elliptic curves."

Putting the quotes together, we have yet another description of the close relationship, well known in the nineteenth century (long before Taniyama's 1955 conjecture), between elliptic curves and modular forms.

Another quote from Milne, to summarize:

"From this [a discussion of nineteenth-century mathematics], one sees that arithmetic facts about elliptic curves correspond to arithmetic facts about special values of modular functions and modular forms."

Serge Lang apparently agrees:

"Elliptic functions parametrize elliptic curves, and the intermingling of the analytic and algebraic-arithmetic theory has been at the center of mathematics since the early part of the nineteenth century."

-- Editorial description of Lang's Elliptic Functions (second edition, 1987)

Update of Dec. 3, 2003

"The theory of modular functions and modular forms, defined on the upper half-plane H and subject to appropriate tranformation laws with respect to the group Gamma = SL(2, Z) of fractional linear transformations, is closely related to the theory of elliptic curves, because the family of all isomorphism classes of elliptic curves over C can be parametrized by the quotient Gamma\H. This is an important, although formal, relation that assures that this and related quotients have a natural structure as algebraic curves X over Q. The relation between these curves and elliptic curves predicted by the Taniyama-Weil conjecture is, on the other hand, far from formal."

-- Robert P. Langlands, review of Elliptic Curves, by Anthony W. Knapp.  (The review appeared in Bulletin of the American Mathematical Society, January 1994.)

Update of Dec. 30, 2003

As the above remarks indicate, a nineteenth-century concept that connects elliptic curves with modular forms is the modular group Gamma. For a brief but informative introduction to this group, click here.

Update of April 15-18, 2004

A concise statement of the historical relationship between elliptic curves and modular forms is as follows:

"2.2 Wiles' theorem

As we have seen, elliptic curves and modular forms are related to each other. This fact was already clear in the 19th century - the moduli of elliptic curves are expressible in terms of modular forms of the parameter tau in the upper half plane. However, in the 1950's a new sort of relation between elliptic curves and modular forms was perceived, first by Taniyama...."

This is from "Topics in Elliptic Curves and Modular Forms," by J. William Hoffman, Sept. 28, 2001, p. 14.

(Web page at http://www.math.lsu.edu/~hoffman/papers/elmod.pdf)

The home page of Professor Hoffman is at http://www.math.lsu.edu/~hoffman/.

The above-cited expository lecture of Professor Hoffman, who was educated at Princeton and Harvard, should be compared with the more amusing, but less informative, web page http://www.crank.net/maths.html, which contains a link to the page you are now reading.

The background of Professor Hoffman should then be compared with the background of the editor of crank.net, Erik Max Francis, a programmer who has studied mathematics "up to some linear algebra and differential equations."

Concluding observation:

At the Amazon.com site of Mathematical Cranks, by Underwood Dudley (MAA, 1992), we have the following definition.

"From Book News, Inc.
By 'cranks' the author means people who think they've done something that's impossible--like trisecting an angle with only a straight edge and compass, or proving Fermat's Last Theorem--or who have just plain odd ideas...."

This definition uses perhaps too broad a brush, since it includes Ribet, Wiles, and much of the human race.

It certainly includes Richard Taylor, who worked with Andrew Wiles on proving part of the Taniyama conjecture.  Combined with earlier work of Ribet, the work of Wiles and Taylor is said to constitute a proof of Fermat's last theorem.  Consider, in light of the above material on modular forms in the 19th century, the following remarks by Taylor in a 1996 Harvard Math Club interview:

"HOW OLD ARE MODULAR FORMS?
Certainly about thirty years. Sort of in this period the ideas have been becoming more and more fixed. The first indications maybe go back to maybe the late fifties. But the ideas didn't really start becoming definite until maybe 1970. These dates are very rough."

Yes, they are.

For further background on modular forms that shows more respect for history than is apparent at Harvard, consult (for instance)

Automorphic Forms on GL(2) and Its Inner Forms (pdf), by Haruzo Hida, which begins as follows:

"We know traditionally from the time of Gauss [1777-1855] and Eisenstein [1823-1852] that modular forms on a congruence subgroup Gamma of SL2(Z) contain an amazing amount of arithmetic information."

See, too, references on the Web to "Hilbert's twelfth problem" and Kronecker's "Jugendtraum" and go from there.

Update of April 19, 2004

The reader who follows the above advice to learn more about the Jugendtraum  will encounter material on "class field theory."  This, rather than the famous conjecture of Fermat, is the proper background for the conjecture of Taniyama and the work of Wiles.  See, for instance,

Oration for Andrew Wiles (pdf),

Recent Progress on the Langlands Conjectures,

An Overview of Class Field Theory (pdf),

An Introduction to Class Field Theory (ps), and

Math 254B (Number Theory).  

Update of April 20, 2004

The Ribet-Mazur lie seems to stem from a desire to dramatize their work.  Thus the following material may be relevant. 

Comedy of Errors 

An Introduction to Cartesian Theatre
From the Journal of Steven H. Cullinane

Sunday, April 18, 2004  2:00 AM

Dream of Youth Revisited

For some material related to the entry Dream of Youth of last Dec. 8 (the feast day of St. Hermann Weyl), see the recently updated A Mathematical Lie.

See, too, a "comedy of errors" from

7 rue René Descartes in Strasbourg (pdf)

on what Hilbert reportedly called "the most beautiful part of mathematics."

Monday, April 19, 2004  7:59 PM

Cartesian Theatre

From aldaily.com today:

"If my mind is a tiny theatre I watch in my brain, then there is a tinier mind and theatre inside that mind to see it, and so on forever... more»"

This leads to the dream (or nightmare) of the Cartesian theatre, as pictured by Daniel Dennett.

From websurfing yesterday and today...

The tiny theatre of Ivor Grattan-Guinness:

"... mathematicians often treat history with contempt (unsullied by any practice or even knowledge of it, of course)."

-- The Rainbow of Mathematics

The contempt for history of the Harvard mathematics department (see previous entry) suggests a phrase....

A search on "Harvard sneer" yields, as the first page found, a memorial to an expert practitioner of the Harvard sneer... Robert Harris Chapman, Professor of English Literature, playwright, theatrical consultant, and founding Director of the Loeb Drama Center from 1960 to 1980.

Continuing the Grattan-Guinness rainbow theme in a tinier theatre, we may picture Chapman's reaction to the current Irish Repertory Theatre production of Finian's Rainbow.  Let us hope it is not a Harvard sneer.

In a yet tinier theatre, we may envision a mathematical version of Finian's Rainbow, with Og the leprechaun played by Andrew P. Ogg.  Ogg would, of course, perform a musical version of his remarks on the Jugendtraum:

"Follow the fellow who follows a dream."

Melissa Errico
in Finian's Rainbow

"Give her a song like.... 'Look to the Rainbow,' and her gleaming soprano effortlessly flies it into the stratosphere where such numbers belong. This is the voice of enchantment...."

-- Ben Brantley, today's NY Times

For related philosophical remarks on rainbows, infinite regress, and redheads, see

Loretta's Rainbow and

The Leonardo Code.

Tuesday, April 20, 2004  3:00 PM

Rhetorical Question

Yesterday's Cartesian theatre continues....

Robert Osserman, a professor emeritus of mathematics at Stanford University, is special-projects director at the Mathematical Sciences Research Institute, in Berkeley, Calif.

Osserman at aldaily.com today:

"The past decade has been an exciting one in the world of mathematics and a fabulous one (in the literal sense) for mathematicians, who saw themselves transformed from the frogs of fairy tales -- regarded with a who-would-want-to-kiss-that aversion, when they were noticed at all -- into fascinating royalty, portrayed on stage and screen....

Who bestowed the magic kiss on the mathematical frog?"

Answer:

William Randolph Hearst III.

"Trained as a mathematician at Harvard, he now likes to hang out with Ken Ribet and the other gurus at the University of California, Berkeley's prestigious Mathematical Sciences Research Institute. Two years ago, he moderated a panel of math professors discussing Princeton professor Andrew Wiles's historic proof of Fermat's Last Theorem."

--   Wired magazine, June 1995

See also

Hearst Gift Spurs Math Center Expansion and

Review of Rational points on elliptic curves by Joseph H. Silverman and John T. Tate, Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 2, 248--252,

by William Randolph Hearst III
and Kenneth A. Ribet.

Chet Atkins summarizes:

"And that's the secret of frog kissin',
     and you can do it too if you'll just listen.

Just slow down, turn around, bend down
     and kiss you a frog! Ribet! Ribet!"

 

 Page created Nov. 30, 2003