Every rational elliptic curve is modular, meaning it arises from a modular form. Formerly the Taniyama-Shimura-Weil conjecture, its proof for semistable curves by Andrew Wiles (with Richard Taylor) supplied the last step needed to prove Fermat's Last Theorem; the full theorem was completed shortly after.
Facts
StatementEvery elliptic curve defined over the rational numbers is modular: it arises from a modular form, meaning it can be obtained via a rational map with integer coefficients from a classical modular curve X0(N), where N is the curve's conductor. 1 Proof YearWiles and Taylor proved the semistable case in 1994-1995, sufficient to imply Fermat's Last Theorem; Breuil, Conrad, Diamond and Taylor extended this to the full theorem in 2001. Classification
Statement FormCharacterization Theorem 1 Connections
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Sources
1. Modularity Theorem (Wikipedia)
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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.
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