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.