Editing Modularity theorem (section)

{{Short description|Relates rational elliptic curves to modular forms}}
{{Distinguish|Serre's modularity conjecture}}
{{Infobox mathematical statement
| name = Modularity theorem
| image = 
| caption = 
| field = [[Number theory]]
| conjectured by = [[Yutaka Taniyama]]<br>[[Goro Shimura]]
| conjecture date = 1957
| first proof by = [[Christophe Breuil]]<br>[[Brian Conrad]]<br>[[Fred Diamond]]<br>[[Richard Taylor (mathematician)|Richard Taylor]]
| first proof date = 2001
| open problem = 
| known cases =
| implied by = 
| generalizations = 
| consequences = [[Fermat's Last Theorem]]
}}

In [[number theory]], the '''modularity theorem''' states that [[elliptic curve]]s over the field of [[rational number]]s are related to  [[modular form]]s in a particular way.  [[Andrew Wiles]] and [[Richard Taylor (mathematician)|Richard Taylor]] proved the modularity theorem for [[semistable elliptic curve]]s, which was enough to imply [[Fermat's Last Theorem]]. Later, a series of papers by Wiles's former students [[Brian Conrad]], [[Fred Diamond]] and Richard Taylor, culminating in a joint paper with [[Christophe Breuil]], extended Wiles's techniques to prove the full modularity theorem in 2001. Before that, the statement was known as the '''Taniyama–Shimura conjecture''', '''Taniyama–Shimura–Weil conjecture''', or the '''modularity conjecture for elliptic curves'''.