On Darmon’s program for the generalized Fermat equation, II

Billerey, Nicolas; Chen, Imin; Dieulefait, Luis; Freitas, Nuno

doi:10.1090/mcom/4012

On Darmon’s program for the generalized Fermat equation, II
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by Nicolas Billerey, Imin Chen, Luis Dieulefait and Nuno Freitas;

Math. Comp. 94 (2025), 1977-2003

DOI: https://doi.org/10.1090/mcom/4012

Published electronically: September 6, 2024

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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 \begin{equation*} x^7 + y^7 = 3 z^n \end{equation*} 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 $n \ge 2$.

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On Darmon’s program for the generalized Fermat equation, II
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Abstract: