A profound theorem, stating that elliptic curves over the rational numbers are connected with modular forms. The result was first conjectured by Yutaka Taniyama and Goro Shimura in 1957. Andrew Wiles proved the theorem for an important subset of elliptic curves (the so-called semistable curves) and deduced from this Fermat’s Last Theorem (as Frey’s curve is not modular). The theorem was completely proved by 2001, as a result of papers from Wiles’s students, including Richard Taylor. See also Langlands programme.