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Modularity theorem
Relates rational elliptic curves to modular forms
Modularity theorem
In number theory, the modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms in a particular way. Andrew Wiles and Richard Taylor proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's Last Theorem (FLT). 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.
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