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. $2^{ p-1} \not \equiv 1 \pmod{ p^2}$. The effectivity in our result builds on recent work of Thorne proving modularity of elliptic curves over $\mathbb{ Q}_{ n,p}$.
Categories: math.NT