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


Authors: Nicolas Billerey, Imin Chen, Luis Dieulefait and Nuno Freitas
Journal: Proc. Amer. Math. Soc. 145 (2017), 4111-4117
MSC (2010): Primary 11D41
DOI: https://doi.org/10.1090/proc/13475
Published electronically: June 16, 2017
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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. More precisely, for $ r \ge 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 $ \operatorname {GL}_2$-type over totally real fields.


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Additional Information

Nicolas Billerey
Affiliation: Laboratoire de Mathématiques, Université Clermont Auvergne, Université Blaise Pascal, BP 10448, F-63000 Clermont-Ferrand, France – and – CNRS, UMR 6620, LM, F-63171 Aubière, France
Email: Nicolas.Billerey@math.univ-bpclermont.fr

Imin Chen
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email: ichen@sfu.ca

Luis Dieulefait
Affiliation: Departament d’Algebra i Geometria, Universitat de Barcelona, G.V. de les Corts Catalanes 585, 08007 Barcelona, Spain
Email: ldieulefait@ub.edu

Nuno Freitas
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2
Email: nunobfreitas@gmail.com

DOI: https://doi.org/10.1090/proc/13475
Keywords: Fermat equations, Frey abelian varieties, irreducibility
Received by editor(s): May 7, 2016
Received by editor(s) in revised form: August 19, 2016
Published electronically: June 16, 2017
Additional Notes: The first author acknowledges the financial support of CNRS and ANR-14-CE-25-0015 Gardio, the second author acknowledges the financial support of an NSERC Discovery Grant, the third author acknowledges the financial support of the MEC project MTM2015-66716-P and the fourth author acknowledges financial support from from the grant Proyecto RSME-FBBVA $2015$ José Luis Rubio de Francia
Communicated by: Romyar T. Sharifi
Article copyright: © Copyright 2017 American Mathematical Society