The modularity conjecture for elliptic curves over totally real fields

The conjecture asserts that every elliptic curve over every totally real field is modular. Public sources record substantial special cases, but no result settling the conjecture for arbitrary fields of degree $[F:\mathbb{Q}] \ge 4$.

Known results

• All elliptic curves over $\mathbb{Q}$ are modular; Wiles and Taylor proved the semistable case in 1995, and Diamond completed the general case in 2001.
• Freitas, Le Hung, and Siksek proved modularity for every elliptic curve over a real quadratic field.
• For any totally real field $K$, only finitely many elliptic curves over $K$ can be non-modular, up to isomorphism over $\overline{K}$; for real quadratic $K$, none are non-modular.
• If $K$ is totally real and abelian over $\mathbb{Q}$ and unramified above $3$, $5$, and $7$, every elliptic curve over $K$ is modular.

Current status (as of September 2026): Modularity is settled for $\mathbb{Q}$, all real quadratic fields, and further special families, while the conjecture remains open for general totally real fields, especially when $[F:\mathbb{Q}] \ge 4$.