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Article Dans Une Revue Proceedings of the American Mathematical Society Année : 2017

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

Résumé

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. More precisely, for $r \geq 5$ a regular prime we prove that there exists a constant $C(r)$ such that for every prime number $p > C(r)$ the equation $x^p + y^p = z^r$ has no non-trivial primitive integer solutions $(a,b,c)$ satisfying $r \mid ab$ and $2 \nmid ab$. For the proof, we complement Darmon's ideas in a particular case by providing an irreducibility criterion for the mod $\mathfrak{p}$ representations attached to certain families of abelian varieties of $\mathrm{GL}_2$-type over totally real fields.
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Dates et versions

hal-04616585 , version 1 (18-06-2024)

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Citer

Nicolas Billerey, Imin Chen, Luis Dieulefait, Nuno Freitas. A result on the equation $x^p + y^p = z^r$ using Frey abelian varieties. Proceedings of the American Mathematical Society, 2017, 145 (10), pp.4111-4117. ⟨10.1090/proc/13475⟩. ⟨hal-04616585⟩
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