Hypergeometric motives and the generalized Fermat equation
Franco Golfieri Madriaga, Ariel Pacetti
TL;DR
This paper recasts Darmon's modular-program for the generalized Fermat equation in the language of rank-two hypergeometric motives, replacing model curves with HGMs as the primary source of Galois representations. It develops a comprehensive framework: (i) a user-friendly introduction to HGMs and their monodromy, field of definition, specializations, congruences, Galois action, and quadratic twists; (ii) a detailed catalogue of rational rank-two HGMs and their elliptic-curve realizations; (iii) a modularity analysis showing when these motives are modular across several exponent-line families, with unconditional results in the crucial $(2,p,r)$ and $(3,p,r)$ cases; (iv) explicit bounds at wild primes to enable the modular-method steps and a Mazur-type elimination procedure to discard possible newforms. The approach yields concrete nonexistence results for specific exponent families (e.g., $(3,5,p)$) and provides a robust toolkit for extending the modular method to new generalized Fermat equations via hypergeometric motives. Overall, the work integrates hypergeometric-motive technology with modularity lifting and level-lowering to advance the Beal/Fermat-Catalan program without relying on explicit algebro-geometric models.
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.