Curving Beyond Fermat

When Andrew Wiles of Princeton University proved Fermat's last theorem several years ago, he took advantage of recently discovered links between Pierre de Fermat's centuries-old conjecture concerning whole numbers and the theory of so-called elliptic curves. Establishing the validity of Fermat's last theorem involved proving parts of the Taniyama-Shimura conjecture.

Four mathematicians have now extended this aspect of Wiles' work, offering a proof of the Taniyama-Shimura conjecture for all elliptic curves rather than just a particular subset of such curves.

Mathematicians regard the resulting Taniyama-Shimura theorem as one of the major results of 20th-century mathematics. It establishes a surprising, profound connection between two very different mathematical worlds and, along the way, has important consequences for number theory.

An elliptic curve is not an ellipse. It is a solution of a cubic equation in two variables of the form y ²= x ³+ax + b (where a and b are fractions, or rational numbers), which can be plotted as a curve made up of one or two pieces.

Examples of elliptic curves: y²= x (x - 1)( x + 5) (purple); y²= x³+ 7 (blue); y²= x(x²- 2) (red).

In the 1950s, Japanese mathematician Yutaka Taniyama (1927-1958) proposed that every rational elliptic curve is a disguised version of a complicated, impossible-to-visualize mathematical object called a modular form. Goro Shimura, now at Princeton, refined the idea.

Elliptic curves and modular forms are mathematically so different that mathematicians initially couldn't believe that the two are related. Wiles verified part of the Taniyama-Shimura conjecture by showing that many types of elliptic curves can indeed be described in terms of modular forms.

Wiles' proof of Fermat's last theorem came as a consequence of this larger effort, since other work had established a link between elliptic curves and Fermat's last theorem.

This fall, Brian Conrad and Richard Taylor of Harvard University, along with Christophe Breuil of the Université Paris-Sud and Fred Diamond of Rutgers University in New Brunswick, N.J., completed a proof of the Taniyama-Shimura conjecture for all elliptic curves.

"The work was collaborative in nature," Conrad says. "Although we. . .worked on different parts of the argument, there really was nontrivial overlap among these parts, with questions and problems in one area leading to questions and problems in other areas."

The conjecture "was widely believed to be unbreachable, until the summer of 1993, when Wiles announced a proof that every semistable elliptic curve is modular," Henri Darmon of McGill University in Montreal remarks in the December Notices of the American Mathematical Society. "The Shimura-Taniyama-Weil conjecture and its subsequent, just-completed proof stand as a crowning achievement of number theory in the 20th century."

Moreover, the Taniyama-Shimura conjecture fits into the so-called Langlands program, formulated by Robert P. Langlands of the Institute for Advanced Study in Princeton, N.J. This vast, visionary program posits a bold, sweeping unification of important areas of mathematics.

The proof of the Taniyama-Shimura conjecture and Wiles' work on Fermat's last theorem provide intriguing insights into the Langlands program. In particular, Darmon notes, novel applications of powerful mathematical techniques pioneered by Wiles promise to keep number theorists busy well into the new millennium.

References:

Darmon, H. 1999. A proof of the full Shimura-Taniyama-Weil conjecture is announced. Notices of the American Mathematical Society 46(December):1397.

Knapp, A.W. 1999. Proof announced of Taniyama-Shimura conjecture. Notices of the American Mathematical Society 46(September):863.

Mackenzie, D. 1999. Fermat's last theorem extended. Science 285(July 9):178.

Peterson, I. 1999. Curving beyond Fermat's last theorem. Science News 156(Oct. 2):221.

Singh, S. 1997. Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. New York: Walker.

A technical description of the Taniyama-Shimura conjecture can be found at http://mathworld.wolfram.com/Taniyama-ShimuraConjecture.html.

A brief biography of Yutaka Taniyama can be found at http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Taniyama.html.

The work of Robert P. Langlands is highlighted at http://sunsite.ubc.ca/DigitalMathArchive/Langlands/intro.html.

The illustration showing various elliptic curves was made using NuCalc 2.0, a powerful, versatile Windows-based version of the Macintosh graphing calculator program. (See http://www.nucalc.com/ for information about the latest version.)

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.