1. Some extensions of the modular method and Fermat equations of signature (13,13,n)
- Author
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Billerey, Nicolas, Chen, Imin, Dembélé, Lassina, Dieulefait, Luis, Freitas, Nuno, Laboratoire de Mathématiques Blaise Pascal (LMBP), Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA), Department of Mathematics Simon Fraser University, University of Luxembourg [Luxembourg], Departament d'Algebra i Geometria, Facultat de Matematiques, Universitat de Barcelona, Instituto de Ciencias Matemàticas [Madrid] (ICMAT), Universidad Carlos III de Madrid [Madrid] (UC3M)-Universidad Complutense de Madrid = Complutense University of Madrid [Madrid] (UCM)-Universidad Autónoma de Madrid (UAM)-Consejo Superior de Investigaciones Científicas [Madrid] (CSIC), and Billerey, Nicolas
- Subjects
Mathematics - Number Theory ,11D41, 11G10, 11F80 ,Galois representations ,Modularity ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Fermat equations ,2022. 2010 Mathematics Subject Classification. Primary 11D41 ,abelian surfaces ,11F80 Fermat equations ,FOS: Mathematics ,Secondary 11G10 ,Number Theory (math.NT) ,Abelian surfaces ,March 7 ,modularity ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - 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 $n \geq 2$, the equation \[ x^{13} + y^{13} = 3 z^n \] has no non-trivial 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 $\ell, m\ge 5$, the only solutions to the equation $x^{2\ell} + y^{2m} = z^{13}$ are the trivial ones., Several modifications after the referees' comments. To appear in Publicacions Matem\`atiques
- Published
- 2023