nLab Wiles' proof of Fermat's last theorem

Idea

In Frey82 Gerhard Frey? showed that, if the equation $a^{p}+b^{p}=c^{p}$ had any nontrivial integer solutions $(a,b,c)$ for primes $p\gt 2$, the elliptic curve (now known as the Frey-Hellegouarch curve after Frey and Yves Hellegouarch?)

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

would exhibit certain unusual properties. In particular in Serer87 Jean-Pierre Serre 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 Ken Ribet‘s proof of the epsilon conjecture (now known as Ribet's theorem) in Ribet90 this implied a proof of Fermat's last theorem.

References

• Gerhard Frey?, 1982, Rationale Punkte auf Fermatkurven und getwisteten Modulkurven, J. reine angew. Math., 331: 185–191

• Jean-Pierre Serre, 1987, Sur les représentations modulaires de degré 2 de Gal(Q/Q), Duke Mathematical Journal, 54 (1): 179–230

• Ken Ribet, 1990, On modular representations of Gal(Q/Q) arising from modular forms, Inventiones Mathematicae. 100 (2): 431–476.

See also the references at modularity theorem (here)

See also:

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