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

Abstract

Let $p\ge 5$ 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 show that ${\mathbb{Q}}_{n,p}$ has no exceptional units. We use this to prove the effective asymptotic Fermat's Last Theorem over ${\mathbb{Q}}_{n,p}$ for all $n\ge 1$ and all primes $p\ge 5$ that are non-Wieferich, i.e. $2^{p-1}\not\equiv 1(\mathrm{mod}{p}^2)$. The effectivity in our result builds on recent work of Thorne proving modularity of elliptic curves over ${\mathbb{Q}}_{n,p}$.