"Fermat Prover Points to Next Challenges," by Barry Cipra.Science, 22 March 1996, pages 1668-1669. Includes sidebar, "AProof to Please Pythagoras."
This article reports on the AMS Colloquium Lectures presented by Andrew Wilesat the Joint Mathematics Meetings in Orlando in January 1996. The proof ofFermat's Last Theorem was completed last year in two papers, one by Wiles andone written jointly with Richard Taylor. With one of the most famous problemsin the history of mathematics settled, Wiles discussed some of the importantopen questions number theorists continue to probe. Elliptic curves stillprovide many mysteries, including the most basic questions about their rationalsolutions. The article discusses the Birch and Swinnerton-Dyer Conjecture,which relates facts about the solutions of elliptic curves to their associatedL-functions. Results by Wiles, John Coates, and Viktor Kolyvagin haveestablished that the conjecture holds in one direction for modular ellipticcurves; the converse remains a tantalizingly difficult open question.