On Darmon's program for the Generalized Fermat equation, II

TL;DR

This paper advances Darmon's program for the generalized Fermat equation by applying multi-Frey methods and higher-dimensional abelian varieties to completely solve specific cases like x^7 + y^7 = 3z^n and analyze dependencies on Darmon's conjecture.

Contribution

It introduces a multi-Frey approach combining elliptic and hyperelliptic curves, and demonstrates how higher-dimensional Frey abelian varieties improve proof efficiency and connect to Darmon's conjecture.

Findings

Complete resolution of x^7 + y^7 = 3z^n for all n ≥ 2.

Solutions to x^7 + y^7 = z^n with specific 2 or 7-adic conditions.

Dependence of the full resolution on Darmon's big image conjecture.

Abstract

We obtain additional Diophantine applications of the methods surrounding Darmon's program for the generalized Fermat equation developed in the first part of this series of papers. As a first application, we use a multi-Frey approach combining two Frey elliptic curves over totally real fields, a Frey hyperelliptic curve over $\mathbb{Q}$ due to Kraus, and ideas from the Darmon program to give a complete resolution of the generalized Fermat equation $$x^7 + y^7 = 3 z^n$$ for all integers $n \ge 2$. Moreover, we explain how the use of higher dimensional Frey abelian varieties allows a more efficient proof of this result due to additional structures that they afford, compared to using only Frey elliptic curves. As a second application, we use some of these additional structures that Frey abelian varieties possess to show that a full resolution of the generalized Fermat equation $x^7 + y^7 = z^n$ depends only on the Cartan case of Darmon's big image conjecture. In the process, we solve the previous equation for solutions $(a,b,c)$ such that $a$ and $b$ satisfy certain $2$ or $7$-adic conditions and all $n \ge 2$.