Modularity theorem: Difference between revisions

Revision as of 16:59, 12 March 2025

Revision as of 16:59, 12 March 2025

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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'''.

==Statement==