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

Abstract

Let $K$ be a totally real number field of odd degree in which $2$ is inert. Let $l \geq 5$ 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., $2^{l-1} \not\equiv 1 \pmod{l^2}$, 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 $Ax^p+By^p+Cz^p=0$ has no asymptotic solution over each $n$-th layer $K_{n,l}$ when $A,B,C \in \{u2^r : u\in \mathcal{O}_K^\times,\ r \in \mathbb{Z}_{\geq 0}\}$. For any odd prime $d$, we also prove that if $A,B,C \in \{\pm 2^r d^s : r,s \in \mathbb{Z}_{\geq 0}\}$ and $h_{\mathbb{Q}_{n,l}}^+$ is odd, then the generalized Fermat equation $Ax^p+By^p+Cz^p=0$ 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 $\mathbb{Q}_{n,l}$.