An application of the symplectic argument to some Fermat-type equations

Nuno Freitas; Alain Kraus

doi:10.1016/j.crma.2016.06.002

Number theory

An application of the symplectic argument to some Fermat-type equations

Presented by: Jean-Pierre Serre

Nuno Freitas ¹; Alain Kraus ²

¹ University of British Columbia, Department of Mathematics, Vancouver, BC V6T 1Z2, Canada
² Université Pierre-et-Marie-Curie (Paris 6), Institut de mathématiques de Jussieu, 4, place Jussieu, 75005 Paris, France

Comptes Rendus. Mathématique, Volume 354 (2016) no. 8, pp. 751-755.

Abstract
Résumé

Let p be a prime number. In the early 2000s, it was proved that the Fermat equations with coefficients

3 x^{p} + 8 y^{p} + 21 z^{p} = 0 and 3 x^{p} + 4 y^{p} + 5 z^{p} = 0

do not admit non-trivial solutions for a set of exponents p with Dirichlet density 1/4 and 1/8, respectively. In this note, using a recent criterion to decide if two elliptic curves over

Q

with certain types of additive reduction at 2 have symplectically isomorphic p-torsion modules, we improve these densities to 3/8.

Soit p un nombre premier. Au début des années 2000, il a été démontré que les équations de Fermat à coefficients

3 x^{p} + 8 y^{p} + 21 z^{p} = 0 et 3 x^{p} + 4 y^{p} + 5 z^{p} = 0

ne possèdent pas de solutions non triviales pour un ensemble d'exposants p de densité de Dirichlet 1/4 et 1/8, respectivement. Dans cette note, en utilisant un résultat récent permettant de décider si deux courbes elliptiques sur

Q

, ayant un certain type de réduction additive en 2, ont leurs modules des points de p-torsion symplectiquement isomorphes, on améliore ces densités à 3/8.

Received: 2016-03-25
Accepted: 2016-06-08
Published online: 2016-07-12

DOI: 10.1016/j.crma.2016.06.002

Author's affiliations:

Nuno Freitas ¹; Alain Kraus ²

¹ University of British Columbia, Department of Mathematics, Vancouver, BC V6T 1Z2, Canada
² Université Pierre-et-Marie-Curie (Paris 6), Institut de mathématiques de Jussieu, 4, place Jussieu, 75005 Paris, France

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     title = {An application of the symplectic argument to some {Fermat-type} equations},
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Nuno Freitas; Alain Kraus. An application of the symplectic argument to some Fermat-type equations. Comptes Rendus. Mathématique, Volume 354 (2016) no. 8, pp. 751-755. doi : 10.1016/j.crma.2016.06.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.06.002/

References
Cited by

[1] N. Freitas On the Fermat-type equation $x^{3} + y^{3} = z^{p}$ , Comment. Math. Helv. (2016) (in press)

[2] N. Freitas; S. Siksek Fermat's Last Theorem over some small real quadratic fields, Algebra Number Theory, Volume 9 (2015) no. 4, pp. 875-895

[3] E. Halberstadt; A. Kraus Courbes de Fermat : résultats et problèmes, J. Reine Angew. Math., Volume 548 (2002), pp. 167-234

[4] A. Kraus; J. Oesterlé Sur une question de B. Mazur, Math. Ann., Volume 293 (1992), pp. 259-275

[5] J.-P. Serre Sur les représentations modulaires de degré 2 de $Gal (\overline{Q} / Q)$ , Duke Math. J., Volume 54 (1987), pp. 179-230

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