On asymptotic Fermat over ℤp-extensions of ℚ

Freitas, Nuno; Kraus, Alain; Siksek, Samir

doi:10.2140/ant.2020.14.2571

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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

DOI: 10.2140/ant.2020.14.2571

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 $n \geq 1$ and all primes $p \geq 5$ that are non-Wieferich, i.e., $2^{p - 1} ≢ 1 (mod p^{2})$ . 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

Mathematical Subject Classification

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

Vol. 14, No. 9, 2020