In Memoriam: Yutaka Taniyama
Today (11-17-2008) marks the 50th year of the passing of a great, albeit troubled, mathematical genius Yutaka Taniyama. He is linked to one of the most famous problems of mathematics, Fermat’s last theorem, which states that there are no whole number solutions for the equation for . This problem remained unsolved for nearly 350 years and it took centuries of mathematical advancement to crack it’s innocent looking statement. Significant mathematical bridge building, between branches of mathematics that operated as islands, was crucial to finding a solution. One of the first steps in this bridge building was taken in 1955 when Taniyama started playing an inspirational role in connecting the two key branches of mathematics – topology and number theory. Loosely speaking, an elliptic curve is a type of cubic curve whose solutions are confined to a region of space that is topologically equivalent to a torus. A modular form is the most symmetrical of mathematical objects – it can be subject to transformations in infinite number of ways and still remains unchanged. Taniyama hypothesized that every modular form could be matched with an elliptic equation. He was creative and ahead of his time, but his ideas were criticized as unsubstantiated. Shortly after his 31st birthday, lacking confidence in his future, he committed suicide. He left a detailed note, stating among other things, the books he had borrowed from library and friends, how far he had reached in the courses he was teaching and concluded with an apology to his colleagues for the inconveniences this act would cause. Second act of the tragedy occurred few weeks later when his fiancee also took her own life. Goro Shimura and Andre Weil developed Taniyama’s work and formulated Taniyama-Shimura-Weil conjecture within an year of Taniyama’s passing. Establishment of link between Taniyama-Shimura-Weil conjecture and Fermat’s Last Theorem took another three decades. In 1986 Frey and Ribet established that if Fermat’s last theorem were false, then Taniyama-Shimura-Weil conjecture would also be false. So to prove Fermat’s Last Theorem it was enough to prove Taniyama-Shimura-Weil conjecture. Andrew Wiles proved Taniyama-Shimura-Weil conjecture for special classes of elliptic curves sufficient enough to prove Fermat’s Last Theorem. Full Taniyama-Shimura-Weil conjecture was proved in 1999.
Simon Singh’s Fermat’s Enigma provides the best popular account of the history of the problem and the solution. If you are interested in learning more of the number theory around this problem Paulo Ribenboim’s Fermat’s Last Theorem for Amateurs is an excellent resource.