transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
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
(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.
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:
Last revised on June 4, 2024 at 07:50:46. See the history of this page for a list of all contributions to it.