Modularity theorem
An elliptic curve and a modular form are very different objects, and the theorem says they carry the same data. Wiles and Taylor proved the semistable case in 1995, which was enough for Fermat. The full statement followed six years later. It is the template for every later modularity result.
Evidence tier V3 is the import default for a result the curation records as settled in the published record. No formal proof or certificate is attached to this page yet.
Posed 1950s–60s, Taniyama, Shimura and Weil. Closed 2001, Breuil, Conrad, Diamond and Taylor, completing Wiles.
Sources
Relations to this page
- related to Birch and Swinnerton-Dyer conjecture open
The original curation links this problem to Modularity theorem; the argument for the link has not been written yet.
- used by Fermat's Last Theorem solved · V3
Progress here drew on Modularity theorem, whose resolution the original curation lists as unlocking it; the argument for the link has not been written yet.
- related to Langlands reciprocity (general) open
The original curation links this problem to Modularity theorem; the argument for the link has not been written yet.
- used by Sato–Tate conjecture solved · V3
Progress here drew on Modularity theorem, whose resolution the original curation lists as unlocking it; the argument for the link has not been written yet.
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