Informally, the Taniyama-Shimura theorem states:
- "all elliptic curves are modular"
This theorem was first conjectured by Yutaka Taniyama in September 1955. With Goro Shimura he improved its rigor until 1957. Taniyama died in 1958. In the 1960s it became associated with the Langlands program of unifying conjectures in mathematics, and was a key component thereof. The conjecture was picked up and promoted by André Weil in the 1970's, and Weil's name was associated with it in some quarters. Despite the interest, some considered it beyond proving.
It attracted considerable interest in the 1980's when Gerhard Frey proposed that the Taniyama-Shimura conjecture implies Fermat's last theorem. In 1995, Andrew Wiles proved a special case of the Taniyama-Shimura theorem which was strong enough to yield a proof of Fermat's Last Theorem.
The full Taniyama-Shimura theorem was finally proved in 1997 by a team of a half-dozen mathematicians who, building on Wiles's work, incrementally chipped away at the remaining cases until the full result was proved.
In March 1996 Wiles shared the Wolf Prize with Robert Langlands. Although neither of them had originated nor finished the proof of the full theorem that had enabled their achievements, they were recognized as having had the decisive influences that led to its finally being proven.