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

Abstract

Let p be a prime and let Qn,p denote the n-th layer of the cyclotomic Zp-extension of Q. We prove the effective asymptotic FLT over Qn,p for all n ≥ 1 and all primes p ≥ 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 Qn,p.

Class field theory, Diophantine analysis and the asymptotic Fermat's Last Theorem

“…For example, a recent spectacular result of Triantafillou [9] asserts if 3 totally splits in F and 3 ∤ [F ∶ Q] then (1.1) has no solutions. The authors of the current paper have shown [6] that if F is a Galois p-extension, where p ≥ 5 is prime that totally ramifies in F , then (1.1) has no solutions.…”

$S$-Unit Equations and the Asymptotic Fermat Conjecture over Number Fields

“…In two recent works [FKS20] and [FKS20b], Freitas, Kraus and Siksek prove the asymptotic Fermat conjecture for the layers of various cyclotomic Z ℓ -extension of Q. We first introduce these extensions.…”