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arXiv · 1605.02198

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

Nicolas Billerey·Imin Chen·Luis Dieulefait·Nuno Freitas
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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.

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Nicolas Billerey, Imin Chen, Luis Dieulefait, Nuno Freitas. 2016-05-07. A result on the equation $x^p + y^p = z^r$ using Frey abelian varieties. https://arxiv.org/abs/1605.02198

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