Fermat-type equations of signature (13,13,p) via Hilbert cuspforms

TL;DR

This paper proves the non-existence of certain primitive solutions to Fermat-type equations with signature (13,13,p) for large primes p, using modularity of elliptic curves over a specific number field and Hilbert cusp forms.

Contribution

It introduces a novel approach linking solutions to equations over $\\Q(\sqrt{13})$ with modularity results for elliptic curves over totally real fields.

Findings

No non-trivial primitive solutions for large p with $13 mid c$

Modularity of Frey-curves over $\\Q(\sqrt{13})$ established

Modularity results for elliptic curves over certain totally real fields

Abstract

In this paper we prove that equations of the form $x^{13} + y^{13} = Cz^{p}$ have no non-trivial primitive solutions (a,b,c) such that $13 \nmid c$ if $p > 4992539$ for an infinite family of values for $C$. Our method consists in relating a solution (a,b,c) to the previous equation to a solution (a,b,c_1) of another Diophantine equation with coefficients in $\Q(\sqrt{13})$. We then construct Frey-curves associated with (a,b,c_1) and we prove modularity of them in order to apply the modular approach via Hilbert cusp forms over $\Q(\sqrt{13})$. We also prove a modularity result for elliptic curves over totally real cyclic number fields of interest by itself.