English

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

Number Theory 2016-05-10 v1

Abstract

We prove a diophantine result on generalized Fermat equations of the form xp+yp=zrx^p + y^p = z^r which for the first time requires the use of Frey abelian varieties of dimension 2\geq 2 in Darmon's program. For that, we provide an irreducibility criterion for the mod p\mathfrak{p} representations attached to certain abelian varieties of GL2\text{GL}_2-type over totally real fields.

Keywords

Cite

@article{arxiv.1605.02198,
  title  = {A result on the equation $x^p + y^p = z^r$ using Frey abelian varieties},
  author = {Nicolas Billerey and Imin Chen and Luis Dieulefait and Nuno Freitas},
  journal= {arXiv preprint arXiv:1605.02198},
  year   = {2016}
}