Modularity theorem: Difference between revisions

Revision as of 22:31, 29 July 2004

Revision as of 19:08, 9 October 2004

Line 3:

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."