nLab Wiles' proof of Fermat's last theorem

Idea

Frey 1982 showed that, if the equation $a^{p}+b^{p}=c^{p}$ had any nontrivial integer solutions $(a,b,c)$ for primes $p\gt 2$, then the elliptic curve give by

$$y^{2} \;=\; x(x-a^{p})(x-b^{p})$$

(now known as the Frey-Hellegouarch curve after Gerhard Frey and Yves Hellegouarch)

would exhibit certain unusual properties. In particular, Serre 1987 showed, assuming the “epsilon conjecture”, that if such a solution existed then the Frey-Hellegouarch curve would not be modular.

Andrew Wiles with Richard Taylor proved the modularity theorem stating that all elliptic curves over $\mathbb{Q}$ are modular. By the proof of the epsilon conjecture due to Ribet 1990 (now known as Ribet's theorem) this finally implied a proof of Fermat's last theorem.

References

See also the references at modularity theorem (here).

• Gerhard Frey, Rationale Punkte auf Fermatkurven und getwisteten Modulkurven, J. reine angew. Math. 331 (1982) 185–191 [doi:10.1515/crll.1982.331.185, eudml:152424]

• Jean-Pierre Serre, Sur les représentations modulaires de degré 2 de $Gal(\bar{Q}/Q)$, Duke Mathematical Journal, 54 1 (1987) 179–230 [pdf]

• Ken Ribet, On modular representations of $Gal(\bar{Q}/Q)$ arising from modular forms, Inventiones Mathematicae. 100 2 (1990) 431–476 [pdf]

See also:

• Wikipedia, Wiles's proof of Fermat's last theorem