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

Abstract

Let F be a number field and O F its ring of integers. We use Chevalley's ambiguous class number formula to give a criterion for the nonexistence of solutions to the unit equation λ + µ = 1, λ, µ ∈ O × F . This is then used to strengthen a criterion for the asymptotic Fermat's Last Theorem due to Freitas and Siksek.

Local criteria for the unit equation and the asymptotic Fermat’s Last Theorem

“…6. Proof : Suppose that m λ,µ ≥ 2 ordq (2). By Lemma 2.1, there is a solution (λ , µ ) to the S -unit equation Eq.…”

On asymptotic Fermat over the ℤ 2 -extension of ℚ

“…In the proof of Theorem 1.1, Theorem 1.3 (with ℓ = 2) plays a similar rôle to the absence of newforms of weight 2 and level 2. For the deduction of Theorem 1.1 from Theorem 1.3 we refer to [1]. The proof of Theorem 1.1 in [1] makes heavy use of the theory of 𝑝-groups and 𝑝-extensions.…”

“…For the deduction of Theorem 1.1 from Theorem 1.3 we refer to [1]. The proof of Theorem 1.1 in [1] makes heavy use of the theory of 𝑝-groups and 𝑝-extensions. In the present note we give a simpler proof of Theorem 1.3, which uses nothing beyond basic facts about elliptic curves, Tate curves and Tate modules.…”

“…If 𝐵 𝐾 is effectively computable, we shall refer to this as the effective asymptotic Fermat's Last Theorem over 𝐾. In [1] the following two theorems are established.…”

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

“…Let p be a prime, n a positive integer and write Q n,p for the n-th layer of the cyclotomic Z p -extension. In [3], the authors established the following theorem.…”