#### Vol. 14, No. 9, 2020

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

### Nuno Freitas, Alain Kraus and Samir Siksek

Vol. 14 (2020), No. 9, 2571–2574
##### Abstract

Let $p$ be a prime and let ${ℚ}_{n,p}$ denote the $n$-th layer of the cyclotomic ${ℤ}_{p}$-extension of $ℚ$. We prove the effective asymptotic FLT over ${ℚ}_{n,p}$ for all $n\ge 1$ and all primes $p\ge 5$ that are non-Wieferich, i.e., ${2}^{p-1}\not\equiv 1\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}{p}^{2}\right)$. The effectivity in our result builds on recent work of Thorne proving modularity of elliptic curves over ${ℚ}_{n,p}$.

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##### Keywords
Fermat, unit equation, $\mathbb{Z}_p$-extensions
Primary: 11D41
Secondary: 11R23
##### Milestones
Received: 2 April 2020
Accepted: 11 May 2020
Published: 13 October 2020
##### Authors
 Nuno Freitas Department de Matemàtiques i Informàtica Universitat de Barcelona Barcelona Spain Alain Kraus Institut de Mathématiques de Jussieu - Paris Rive Gauche Sorbonne Université UMR 7586 CNRS - Paris Diderot Paris France Samir Siksek Mathematics Institute University of Warwick Coventry United Kingdom