The problem
Every elliptic curve E over ℚ is modular: there exists a weight-2 newform f with the same L-series (equivalently, E admits a parametrisation by the modular curve \(X_{0}\)(N)).
The Modularity Theorem
historicReferences
History & significance
Taniyama sketched the correspondence at the 1955 Tokyo–Nikko conference ('interesting problems'); Shimura developed it into a precise conjecture; Weil's 1967 remarks supplied the converse-direction evidence. André Weil and others doubted generality for years. Fontaine famously argued in 1999 that the general case might be FALSE — the same year it became provable.
The resolution (human proof)
Wiles & Taylor (1995) proved the semistable case — enough for Fermat's Last Theorem. Breuil, Conrad, Diamond and Taylor (2001) lifted it to ALL elliptic curves over ℚ, published as 'On the modularity of elliptic curves over Q: wild 3-adic exercises'. The theorem now anchors the Langlands programme's arithmetic side: every rational elliptic curve sings a modular song. Cross-reference: our Fermat entry tells the 1637→1995 half of this story; this is the 2001 completion.