Recipes to Fermat-type equations of the form x^r + y^r = Cz^p

TL;DR

This paper develops a modular method approach over totally real subfields to analyze Fermat-type equations of the form x^r + y^r = Cz^p, establishing irreducibility of associated Galois representations for large primes and proving non-existence of solutions in specific cases.

Contribution

It introduces a new strategy using Frey curves over totally real fields to attack infinitely many Fermat-type equations with fixed prime r and varying prime p, extending modularity results.

Findings

Proves modularity of Frey curves attached to solutions.

Establishes irreducibility of Galois representations for large p.

Shows no non-trivial solutions for x^7 + y^7 = 3z^p when p is large.

Abstract

We describe a strategy to attack infinitely many Fermat-type equations of signature $(r,r,p)$, where $r \geq 7$ is a fixed prime and $p$ is a prime allowed to vary. We use a variant of the modular method over totally real subfields of $\mathbb{Q}(\zeta_r)$. In particular, to a solution $(a,b,c)$ of $x^r + y^r =Cz^p$ we will attach several Frey curves $E=E_{(a,b)}$. We prove modularity of all the Frey curves and the exsitence of a constant constant $M_r$, depending only on $r$, such that for all $p>M_r$ the representations $\bar{\rho}_{E,p}$ are absolutely irreducible. Along the way, we also prove modularity of certain elliptic curves that are semistable at all $v \mid 3$.\par Finally, we illustrate our methods by proving arithmetic statements about equations of signature $(7,7,p)$. Among which we emphasize that, using a multi-Frey technique, we show there is some constant $M$ such that if $p > M$ then the equation $x^7 + y^7 = 3z^p$ has no non-trivial primitive solutions.