Some extensions of the modular method and Fermat equations of signature (13,13,n)

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Nicolas Billerey
Université Clermont Auvergne
Imin Chen
Simon Fraser University. Department of Mathematics
https://orcid.org/0000-0002-5916-7265
Lassina Dembélé
University of Luxembourg. Department of Mathematics
https://orcid.org/0000-0001-9001-5035
Luis Dieulefait
Universitat de Barcelona. Departament d'Algebra i Geometria
Nuno Freitas
Instituto de Ciencias Matemáticas (Madrid, Espanya)

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.

Paraules clau
Fermat equations, abelian surfaces, modularity, Galois representations

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Billerey, Nicolas et al. “Some extensions of the modular method and Fermat equations of signature (13,13,n)”. Publicacions Matemàtiques, vol.VOL 67, no. 2, pp. 715–741, https://raco.cat/index.php/PublicacionsMatematiques/article/view/418428.
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