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

TL;DR

This paper proves asymptotic Fermat's Last Theorem and non-existence of solutions for generalized Fermat equations over certain layers of cyclotomic $ ext{Z}_l$-extensions of totally real fields under specific ramification and inertness conditions.

Contribution

It establishes new cases of the asymptotic Fermat's Last Theorem and generalized Fermat equations over cyclotomic extensions with explicit conditions on ramification and inertness.

Findings

Asymptotic FLT holds over each layer $K_{n,l}$ under specified conditions.

No asymptotic solutions for certain generalized Fermat equations with specific coefficients.

Proves non-existence of solutions with even product of variables under additional class number conditions.

Abstract

Let $K$ be a totally real number field of odd degree. 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 Throne proving the modularity of elliptic curves over $\mathbb{Q}_{n,l}$.