Generalized Fermat equation over cyclotomic ${\mathbb{Z}}_l$-extensions of totally real fields

Let $K$ be a totally real number field of odd degree in which $2$ is inert. Let $l\ge 5$ be a prime with $l\nmid [K:\mathbb{Q}]$ and $\gcd (\frac{l-1}{2},[K:\mathbb{Q}])=1$. We prove that if $2$ is inert in $K$, $l$ is non-Wieferich, i.e., $2^{l-1}\equiv ̸1(\mathrm{m}\mathrm{o}\mathrm{d}{l}^2)$, and $l$ is totally ramified in $K$, then the asymptotic Fermat’s Last Theorem holds over each $n$-th layer $K_{n,l}$ of the cyclotomic ${\mathbb{Z}}_l$-extension of $K$. We then prove that the generalized Fermat equation $A{x}^p+B{y}^p+C{z}^p=0$ has no asymptotic solution over each $n$-th layer $K_{n,l}$ when $A,B,C\in \{u{2}^r:u\in {\mathcal{O}}_K^{\times },r\in {\mathbb{Z}}_{\ge 0}\}$. For any odd prime $d$, we also prove that if $A,B,C\in \{\pm 2^r{d}^s:r,s\in {\mathbb{Z}}_{\ge 0}\}$ and $h_{{\mathbb{Q}}_{n,l}}^{+}$ is odd, then the generalized Fermat equation $A{x}^p+B{y}^p+C{z}^p=0$ has no effective asymptotic solution $(a,b,c)\in {\mathcal{O}}_{{\mathbb{Q}}_{n,l}}^3$ with $2\mid abc$. The effectivity in the case of ${\mathbb{Q}}_{n,l}$ follows from a result of Thorne proving the modularity of elliptic curves over ${\mathbb{Q}}_{n,l}$.