Modularity theorem: Difference between revisions

Revision as of 17:53, 3 March 2005

Revision as of 15:59, 4 March 2005

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:"All elliptic curves over '''Q''' are modular."

This [[theorem]] was first [[conjecture]]d 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 [[1970s]], 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 [[1980s]] when [[Gerhard Frey]] suggested[[conjecture]]d that the '''Taniyama-Shimura conjecture''' (as it was then called) implies [[Fermat's last theorem]]. He did this by attempting to show that any counterexample to Fermat's last theorem would give rise to a non-modular elliptic curve. [[Ken Ribet]] later proved this result. In [[1995]], [[Andrew Wiles]] and [[Richard Taylor (mathematician)|Richard Taylor]] proved a special case of the Taniyama-Shimura theorem (the case of [[semistable elliptic curve]]s) which was strong enough to yield a proof of Fermat's Last Theorem.

The full Taniyama-Shimura theorem was finally proved in [[1999]] by Breuil, Conrad, Diamond, and Taylor who, building on Wiles' work, incrementally chipped away at the remaining cases until the full result was proved.