Hypergeometric motives and the generalized Fermat equation

Abstract

In the beautiful article [11] Darmon proposed a program to study integral solutions of the generalized Fermat equation $Ax^p+By^q=Cz^r$. In the aforementioned article, Darmon proved many steps of the program, by exhibiting models of hyperelliptic/superelliptic curves lifting what he called ''Frey representations'', Galois representations over a finite field of characteristic $p$. The goal of the present article is to show how hypergeometric motives are more natural objects to obtain the global representations constructed by Darmon, allowing to prove most steps of his program without the need of algebraic models.