Printer Friendly

Curving beyond Fermat's last theorem.

When Andrew Wiles of Princeton University proved Fermat's last theorem several years ago, he relied on recently discovered links between Pierre de Fermat's centuries-old conjecture concerning whole numbers and the theory of so-called elliptic curves (SN: 11/5/94, p. 295). Establishing the validity of Fermat's last theorem involved proving aspects of the Taniyama-Shimura conjecture, which focuses on properties of elliptic equations. Now, four mathematicians have extended this aspect of Wiles' work, offering a proof of the Taniyama-Shimura conjecture for all elliptic curves rather than just particular types.

The Taniyama-Shimura theorem "is one of the major results of 20th-century mathematics," says Joe P. Buhler of the Mathematical Sciences Research Institute in Berkeley, Calif. "It verifies a truly surprising connection between disparate objects and, along the way, has all sorts of consequences in number theory."

An elliptic curve is not an ellipse. It is a solution of the equation [y.sup.2] = [x.sup.3] + [ax.sup.2] + bx + c (where a, b, and c are constants), which can be plotted as a curve. In general, values of x have corresponding values of y. Number theorists are interested in the specific instances when x and y are both fractions, or rational numbers. In the 1950s, Japanese mathematician Yutaka Taniyama 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. His 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 (SN: 6/20/87, p. 397).

News that Brian Conrad and Richard Taylor of Harvard University, along with Christophe Breuil of the Universite Paris-Sud and Fred Diamond of Rutgers University in New Brunswick, N.J., had tackled the Taniyama-Shimura conjecture for all elliptic curves appeared earlier this summer. "The proof is complete," Conrad now says. Parts involving intricate computations and various technical details have already been independently checked, and a lengthy paper describing the proof is nearly ready for distribution.
COPYRIGHT 1999 Science Service, Inc.
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 1999, Gale Group. All rights reserved. Gale Group is a Thomson Corporation Company.

Article Details
Printer friendly Cite/link Email Feedback
Title Annotation:mathematicians offer proof of Taniyama-Shimura theorem
Publication:Science News
Article Type:Brief Article
Geographic Code:1USA
Date:Oct 2, 1999
Previous Article:Cancers pick up GLUT of vitamin C.
Next Article:Crunching Internet security codes.

Related Articles
A shortage of small numbers.
The curious power of large numbers.
Fermat's last theorem: a promising approach.
Doubts about Fermat solution.
A curvy path leads to Fermat's last theorem.
Fermat's famous theorum: proved at last?
The Power of Partitions.
Honors for connecting number theory, geometry, and algebra. (Math Prizes).
Drama in numbers: putting a passion for mathematics on stage.
All square: a surprising, far-reaching overhaul for theories about quadratic expressions.

Terms of use | Privacy policy | Copyright © 2020 Farlex, Inc. | Feedback | For webmasters