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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Source Modularity Theorem (Wikipedia)
Sources
1. Modularity Theorem (Wikipedia)
Wikimedia FoundationLead section
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.
Proved By: Andrew Wiles, Lead
Andrew Wiles and Richard Taylor proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's Last Theorem (FLT).
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