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

Abstract

Let $K$ be a real abelian field of odd class number in which $5$ is unramified. Let $S_5$ be the set of places of $K$ above $5$. Suppose for every nonempty proper subset $S\subset S_5$ there is a totally positive unit $u\in {\mathcal{O}}_K$ such that

$$\prod _{\mathfrak{q}\in S}{Norm}_{{\mathbb{F}}_{\mathfrak{q}}∕{\mathbb{F}}_5}\left(umod\mathfrak{q}\right)\ne \bar{1}.$$

We prove that every semistable elliptic curve over $K$ is modular, using a combination of several powerful modularity theorems and class field theory. 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 nontrivial primitive solution of the generalized Fermat equation $x^{2\ell }+y^{2m}=z^p$ we associate a Frey elliptic curve defined over $\mathbb{Q}{\left({\zeta }_p\right)}^{+}$, and study its mod $\ell$ representation with the help of level lowering and our modularity result. We deduce the nonexistence of nontrivial primitive solutions if $p\le 11$, or if $p=13$ and $\ell ,m\ne 7$.