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

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 $n\ge 2$, 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 Matemàtiques

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• Rigorous computation of the endomorphism ring of a Jacobian
• Some extensions of the modular method and Fermat equations of signature $(13,13,n)$
• A modular approach to Fermat equations of signature $(p,p,5)$ using Frey hyperelliptic curves
• An intriguing hyperelliptic Shimura curve quotient of genus 16

Cited by in corpus (2)

• Fermat's Last Theorem and modular curves over real quadratic fields
• Some extensions of the modular method and Fermat equations of signature $(13,13,n)$