2023 SOME EXTENSIONS OF THE MODULAR METHOD AND FERMAT EQUATIONS OF SIGNATURE (13,13,n)
Nicolas Billerey, Imin Chen, Lassina Dembélé, Luis Dieulefait, Nuno Freitas
Author Affiliations +
Publ. Mat. 67(2): 715-741 (2023). DOI: 10.5565/PUBLMAT6722309

Abstract

We provide several extensions of the modular method which were motivated by the problem of completing previous work to prove that, for any integer n2, the equation

x13+y13=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 ,m5, the only primitive solutions to the equation x2+y2m=z13 are trivial.

Citation

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Nicolas Billerey. Imin Chen. Lassina Dembélé. Luis Dieulefait. Nuno Freitas. "SOME EXTENSIONS OF THE MODULAR METHOD AND FERMAT EQUATIONS OF SIGNATURE (13,13,n)." Publ. Mat. 67 (2) 715 - 741, 2023. https://doi.org/10.5565/PUBLMAT6722309

Information

Received: 13 May 2021; Revised: 1 March 2022; Published: 2023
First available in Project Euclid: 29 June 2023

MathSciNet: MR4609017
zbMATH: 07720476
Digital Object Identifier: 10.5565/PUBLMAT6722309

Subjects:
Primary: 11D41
Secondary: 11F80 , 11G10

Keywords: abelian surfaces , Fermat equations , Galois representations , modularity

Rights: Copyright © 2023 Universitat Autònoma de Barcelona, Departament de Matemàtiques

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Vol.67 • No. 2 • 2023
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