Modularity theorem: Difference between revisions

Revision as of 13:17, 16 March 2005

Revision as of 22:43, 20 March 2005

Line 10:

:"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 clear interest, somethe considereddepth it similar toof the [[Reimann Hypothesis]], a hypothesis which if true, would open many doors, butquestion thatwas mightnot inappreciated factuntil belater [[unprovability|unprovable]]developments.

It attracted considerable interest in the [[1980s]] when [[Gerhard Frey]] suggested 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.