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Concept
Yutaka Taniyama
Summary
was a Japanese mathematician known for the Taniyama–Shimura conjecture.
Taniyama was best known for conjecturing, in modern language, automorphic properties of L-functions of elliptic curves over any number field. A partial and refined case of this conjecture for elliptic curves over rationals is called the Taniyama–Shimura conjecture or the modularity theorem whose statement he subsequently refined in collaboration with Goro Shimura. The names Taniyama, Shimura and Weil have all been attached to this conjecture, but the idea is essentially due to Taniyama.
“Taniyama's interests were in algebraic number theory and his fame is mainly due to two problems posed by him at the symposium on Algebraic Number Theory held in Tokyo and Nikko in 1955. His meeting with André Weil at this symposium was to have a major influence on Taniyama's work. These problems form the basis of a conjecture: every elliptic curve defined over the rational field is a factor of the Jacobian of a modular function field. This conjecture proved to be a major component in the proof of Fermat's Last Theorem by Andrew Wiles.”
In 1986 Ken Ribet proved that if the Taniyama–Shimura conjecture held, then so would Fermat's Last Theorem, which inspired Andrew Wiles to work for a number of years in secrecy on it, and to prove enough of it to prove Fermat's Last Theorem. Owing to the pioneering contribution of Wiles and the efforts of a number of mathematicians, the Taniyama–Shimura conjecture was finally proven in 1999. The original Taniyama conjecture for elliptic curves over arbitrary number fields remains open.
In an episode of Nova (American TV program) on the proof of Fermat's Last Theorem, reflecting on Taniyama's work, Goro Shimura stated:
Taniyama was not a very careful person as a mathematician. He made a lot of mistakes. But he made mistakes in a good direction and so eventually he got right answers. I tried to imitate him, but I found out that it is very difficult to make good mistakes.
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