A result on the equation $x^p + y^p = z^r$ using Frey abelian varieties

TL;DR

This paper advances the understanding of the generalized Fermat equation $x^p + y^p = z^r$ by employing Frey abelian varieties of dimension at least 2, introducing new irreducibility criteria for related Galois representations.

Contribution

It is the first to utilize Frey abelian varieties of dimension ≥ 2 in Darmon's program for this equation, providing a new irreducibility criterion for mod $rak{p}$ representations.

Findings

Proved a diophantine result on $x^p + y^p = z^r$ using higher-dimensional Frey abelian varieties.

Established an irreducibility criterion for Galois representations attached to abelian varieties of $ ext{GL}_2$-type.

Extended Darmon's approach to cases involving abelian varieties of dimension ≥ 2.

Abstract

We prove a diophantine result on generalized Fermat equations of the form $x^p + y^p = z^r$ which for the first time requires the use of Frey abelian varieties of dimension $\geq 2$ in Darmon's program. For that, we provide an irreducibility criterion for the mod $\mathfrak{p}$ representations attached to certain abelian varieties of $\text{GL}_2$-type over totally real fields.