Modularity theorem

The Taniyama-Shimura theorem states:

"all elliptic curves are modular"

This theorem was first conjectured by Yutaka Taniyama, Goro Shimura, and André Weil in the 1970's. 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.