Modularity theorem
Relates rational elliptic curves to modular forms / From Wikipedia, the free encyclopedia
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The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Shimura-Weil conjecture or modularity conjecture for elliptic curves) 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. 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.
Field | Number theory |
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Conjectured by | Yutaka Taniyama Goro Shimura |
Conjectured in | 1957 |
First proof by | Christophe Breuil Brian Conrad Fred Diamond Richard Taylor |
First proof in | 2001 |
Consequences | Fermat's Last Theorem |