A result on the equation $x^p + y^p = z^r$ using Frey abelian varieties
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- by Nicolas Billerey, Imin Chen, Luis Dieulefait and Nuno Freitas PDF
- Proc. Amer. Math. Soc. 145 (2017), 4111-4117 Request permission
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.
References
- Nicolas Billerey, Critères d’irréductibilité pour les représentations des courbes elliptiques, Int. J. Number Theory 7 (2011), no. 4, 1001–1032 (French, with English and French summaries). MR 2812649, DOI 10.1142/S1793042111004538
- Henri Darmon, Rigid local systems, Hilbert modular forms, and Fermat’s last theorem, Duke Math. J. 102 (2000), no. 3, 413–449. MR 1756104, DOI 10.1215/S0012-7094-00-10233-5
- Agnès David, Caractère d’isogénie et critéres d’irréductibilité, arXiv:1103.3892 (2012).
- Luis Dieulefait and Nuno Freitas, Fermat-type equations of signature $(13,13,p)$ via Hilbert cuspforms, Math. Ann. 357 (2013), no. 3, 987–1004. MR 3118622, DOI 10.1007/s00208-013-0920-7
- Nuno Freitas and Samir Siksek, Criteria for irreducibility of $\textrm {mod}\, p$ representations of Frey curves, J. Théor. Nombres Bordeaux 27 (2015), no. 1, 67–76 (English, with English and French summaries). MR 3346965
- Alexander Grothendieck, Modèles de Néron et monodromie., In Séminaire de Géométrie Algébrique 7, Exposé 9 (1967–1969).
- Alain Kraus, Courbes elliptiques semi-stables et corps quadratiques, J. Number Theory 60 (1996), no. 2, 245–253 (French, with French summary). MR 1412962, DOI 10.1006/jnth.1996.0122
- Alain Kraus, Courbes elliptiques semi-stables sur les corps de nombres, Int. J. Number Theory 3 (2007), no. 4, 611–633 (French, with English summary). MR 2371778, DOI 10.1142/S1793042107001127
- Eric Larson and Dmitry Vaintrob, Determinants of subquotients of Galois representations associated with abelian varieties, J. Inst. Math. Jussieu 13 (2014), no. 3, 517–559. With an appendix by Brian Conrad. MR 3211798, DOI 10.1017/S1474748013000182
- Kenneth A. Ribet, Galois action on division points of Abelian varieties with real multiplications, Amer. J. Math. 98 (1976), no. 3, 751–804. MR 457455, DOI 10.2307/2373815
- Jean-Pierre Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331 (French). MR 387283, DOI 10.1007/BF01405086
- Walter Tautz, Jaap Top, and Alain Verberkmoes, Explicit hyperelliptic curves with real multiplication and permutation polynomials, Canad. J. Math. 43 (1991), no. 5, 1055–1064. MR 1138583, DOI 10.4153/CJM-1991-061-x
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
- MR Author ID: 823614
- Email: Nicolas.Billerey@math.univ-bpclermont.fr
- Imin Chen
- Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
- MR Author ID: 609304
- 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
- MR Author ID: 671876
- Email: ldieulefait@ub.edu
- Nuno Freitas
- Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2
- MR Author ID: 1044711
- Email: nunobfreitas@gmail.com
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 4111-4117
- MSC (2010): Primary 11D41
- DOI: https://doi.org/10.1090/proc/13475
- MathSciNet review: 3690598