The Taniyama–Shimura conjecture for elliptic curves over the rationals

The conjecture asserts that every elliptic curve over $\mathbb{Q}$ is modular, meaning it receives a nonconstant map from some modular curve $X_0(N)$. It is named for Taniyama and Shimura; the scanned sources give no posing date.

Known results

• Wiles announced modularity for a sufficiently large semistable family in 1993; this was enough, with Ribet’s theorem, to imply Fermat’s Last Theorem (Wiles, 1994).
• Wiles and Taylor established modularity for semistable elliptic curves over $\mathbb{Q}$ (1995).
• Breuil, Conrad, Diamond, and Taylor extended the result to all elliptic curves over $\mathbb{Q}$; later accounts describe this as the full theorem.

Late 1990s completion

The full Taniyama–Shimura conjecture was established by Breuil, Conrad, Diamond, and Taylor, rather than merely announced. Sources date completion variously to 1997, 1999, or 2001, but agree that every elliptic curve over $\mathbb{Q}$ is modular.

Current status (as of August 2026): The conjecture is resolved; modularity is proved for every elliptic curve over $\mathbb{Q}$, and no unresolved objection or competing counterexample was found.