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 n,p denote the n-th layer of the cyclotomic p-extension of . We prove the effective asymptotic FLT over n,p for all n 1 and all primes p 5 that are non-Wieferich, i.e., 2p11(modp2). The effectivity in our result builds on recent work of Thorne proving modularity of elliptic curves over n,p.

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