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