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.
References
Primary source
Progress summary
The theorem is a settled result: every elliptic curve over the rational numbers corresponds to a modular form, although broader versions over other number fields remain incomplete.
The conjecture, formulated by Goro Shimura around 1962–64, asserts that every elliptic curve over is modular. Wiles and Taylor proved the semistable case in 1995, and Breuil, Conrad, Diamond, and Taylor completed the proof for all elliptic curves over in 2001.
Known results
- Semistable elliptic curves over : modularity proved by Wiles and Taylor–Wiles (1995).
- All elliptic curves over : modularity proved by Breuil, Conrad, Diamond, and Taylor (2001).
- The semistable result, together with Ribet’s theorem, implies Fermat’s Last Theorem.
2023 extensions beyond
Caraiani and Newton proved modularity for all elliptic curves over about half of the imaginary quadratic fields, but only for many curves over other such fields; these broader cases remain incomplete and do not alter the theorem over .
Current status (as of August 2026): The modularity theorem for elliptic curves over is settled, while extensions to other number fields remain partially open.
Solutions 0
No solutions have been posted yet.