Some extensions of the modular method and Fermat equations of signature (13,13,n)
Billerey, Nicolas (Université Clermont Auvergne)
Chen, Imin (Simon Fraser University. Department of Mathematics)
Dembélé, Lassina (University of Luxembourg. Department of Mathematics)
Dieulefait, Luis (Universitat de Barcelona. Departament d'Algebra i Geometria)
Freitas, Nuno (Instituto de Ciencias Matemáticas (Madrid, Espanya))

Data:	2023
Resum:	We provide several extensions of the modular method which were motivated by the problem of completing previous work to prove that, for any integer n ≥ 2, the equation x13 + y 13 = 3zn has no non-trivial primitive solutions. In particular, we present four elimination techniques which are based on: (1) establishing reducibility of certain residual Galois representations over a totally real field; (2) generalizing image of inertia arguments to the setting of abelian surfaces; (3) establishing congruences of Hilbert modular forms without the use of often impractical Sturm bounds; and (4) a unit sieve argument which combines information from classical descent and the modular method. The extensions are of broader applicability and provide further evidence that it is possible to obtain a complete resolution of a family of generalized Fermat equations by remaining within the framework of the modular method. As a further illustration of this, we complete a theorem of Anni-Siksek to show that, for ', m ≥ 5, the only primitive solutions to the equation x 2' + y 2m = z 13 are trivial.
Nota:	Altres ajuts: We acknowledge the financial support of ANR-14-CE-25-0015 Gardio (N. Billerey), an NSERC Discovery Grant (I. Chen), and the grant Proyecto RSME-FBBVA 2015 José Luis Rubio de Francia (N. Freitas).
Drets:	Tots els drets reservats.
Llengua:	Anglès
Document:	Article ; recerca ; Versió publicada
Matèria:	Fermat equations ; Abelian surfaces ; Modularity ; Galois representations
Publicat a:	Publicacions matemàtiques, Vol. 67 Núm. 2 (2023) , p. 715-741 (Articles) , ISSN 2014-4350

Adreça original: https://raco.cat/index.php/PublicacionsMatematiques/article/view/418428
DOI: 10.5565/PUBLMAT6722309

27 p, 463.9 KB

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