Modularity theorem: Difference between revisions

Revision as of 08:05, 29 October 2004

Revision as of 16:17, 16 November 2004

Line 1:

[[Category:Algebraic curves]] [[Category:Riemann surfaces]][[Category:Modular forms]][[Category:Theorems]]

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 ''[[Conjectureconjecture]]'', 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