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

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 $n\mathrm{\ne }29$, $87$, $89$, then every semistable elliptic curve $E$ over $K$ is modular. Let $\mathrm{\ell }$, $m$, $p$ be prime, with $\mathrm{\ell }$, $m\ge 5$ and $p\ge 3$.To a putative non-trivial primitive solution of the generalized Fermat $x^{2\mathrm{\ell }}+y^{2m}=z^p$ we associate a Frey elliptic curve defined over $\mathbb{Q}({\zeta }_p{)}^{+}$, and study its mod $\mathrm{\ell }$ 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 $\mathrm{\ell }$, $m \ne 7$.