Modular elliptic curves over real abelian fields and the generalized Fermat equation $x^{2\ell}+y^{2m}=z^p$

TL;DR

This paper proves a new modularity theorem for semistable elliptic curves over certain real abelian fields and applies it to show the non-existence of solutions to a generalized Fermat equation under specific conditions.

Contribution

It establishes a modularity result for elliptic curves over real abelian fields with small conductors and uses it to solve a class of generalized Fermat equations.

Findings

All semistable elliptic curves over specified real abelian fields are modular.

No non-trivial primitive solutions exist for the generalized Fermat equation when p ≤ 11.

Solutions are also excluded for p=13 if certain conditions on ll and m are met.

Abstract

Using a combination of several powerful modularity theorems and class field theory we derive a new modularity theorem for semistable elliptic curves over certain real abelian fields. We deduce that if $K$ is a real abelian field of conductor $n<100$, with $5 \nmid n$ and $n \ne 29$, $87$, $89$, then every semistable elliptic curve $E$ over $K$ is modular. Let $\ell$, $m$, $p$ be prime, with $\ell$, $m \ge 5$ and $p \ge 3$.To a putative non-trivial primitive solution of the generalized Fermat $x^{2\ell}+y^{2m}=z^p$ we associate a Frey elliptic curve defined over $\mathbb{Q}(\zeta_p)^+$, and study its mod $\ell$ representation with the help of level lowering and our modularity result. We deduce the non-existence of non-trivial primitive solutions if $p \le 11$, or if $p=13$ and $\ell$, $m \ne 7$.