Modularity theorem: Difference between revisions

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The '''Taniyama-Shimura theorem''' establishes an important connection between [[elliptic curve]]s, which are objects from [[algebraic geometry]], and [[modular form]]s, which are certain periodic [[holomorphic function]]s investigated in [[number theory]]. Despite the name, which was a carry over from the Taniyama-Shimura ''[[Conjecture]]'', the theorem is the work of [[Andrew Wiles]], [[Christophe Breuil]], [[Brian Conrad]], [[Fred Diamond]], and [[Richard Taylor (mathematician)|Richard Taylor]].

If ''p'' is a [[prime number]] and ''E'' is an elliptic curve over '''Q''' (the [[field (mathematics)|field]] of [[rational number]]s), we can reduce the equation defining ''E'' [[modular arithmetic|modulo]] ''p''; for all but finitely many values of ''p'' we will get an elliptic curve over the [[finite field]] '''F'''_''p'', with ''n''_''p'' elements, say. One then considers the sequence

:''a''_''p'' = ''n''_''p'' - ''p'',

which is an important invariant of the elliptic curve ''E''. Every modular form also gives rise to a sequence of numbers, by [[Fourier transform]]. An elliptic curve whose sequence agrees with that from a modular form is called '''modular'''. The Taniyama-Shimura theorem states:

:"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 [[Andre Weil|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 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. Kenneth[[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.