Modularity theorem: Difference between revisions

Revision as of 23:20, 6 February 2003

Revision as of 08:54, 16 February 2003

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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]].

If ''p'' is a [[prime number]] and ''E'' is an elliptic curve over '''Q''', 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 Taniyma-Shimura theorem states:

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

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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.

Several theorems in number theory similar to Fermat's last theorem follow from the Taniyama-Shimura theorem. For example: no cube can be written as a sum of two [[relatively prime]] ''n''-th powers, ''n''  ≥  3. (The case ''n''  =  3 was already known by [[Euler]].)

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.