Papers › On Darmon's program for the Generalized Fermat equation, II
On Darmon's program for the Generalized Fermat equation, II
Nicolas Billerey, Imin Chen, Luis Dielefait, Nuno Freitas
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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 ℚ due to Kraus, and ideas from the Darmon program to give a complete resolution of the generalized Fermat equation x⁷ + y⁷ = 3 zⁿ for all integers n ≥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 x⁷ + y⁷ = zⁿ 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 ≥2.
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