Hypergeometric motives and the generalized Fermat equation

2024/12/11 by Franco Golfieri Madriaga, Madriaga, Franco Golfieri, Ariel Pacetti +1 · 2 citations
Computer Science · Mathematics · #11D41 #11F80 #33C05 #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2412.08804

openalex publication_date 2024/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• On the generalized Fermat equation of signature (5,p,3)
• A note on conductors of Frey representations at 2

Related

• On Darmon's program for the Generalized Fermat equation, II
• Effective Darmon's program for the generalised Fermat equation
• A modular approach to Fermat equations of signature (p,p,5) using Frey hyperelliptic curves
• On the conductor of a family of Frey hyperelliptic curves
• Hypergeometric Motives