On asymptotic Fermat over $\mathbb{Z}_p$-extensions of $\mathbb{Q}$

Abstract

Let $p$ be a prime and let ${\mathbb{Q}}_{n,p}$ denote the $n$-th layer of the cyclotomic ${\mathbb{Z}}_p$-extension of $\mathbb{Q}$. We prove the effective asymptotic FLT over ${\mathbb{Q}}_{n,p}$ for all $n\ge 1$ and all primes $p\ge 5$ that are non-Wieferich, i.e., $2^{p-1}\not\equiv 1\left(mod{p}^2\right)$. The effectivity in our result builds on recent work of Thorne proving modularity of elliptic curves over ${\mathbb{Q}}_{n,p}$.