On asymptotic Fermat over the Z_2 extension of Q

TL;DR

This paper provides an elementary proof of Kraus' conjecture, which is a key step in establishing the asymptotic Fermat's Last Theorem over certain cyclotomic fields, using basic elliptic curve theory.

Contribution

It offers a new, elementary proof of Kraus' conjecture, simplifying the understanding of elliptic curves over specific number fields.

Findings

Elementary proof of Kraus' conjecture.

Validation of the conjecture for fields with odd narrow class number.

Supports the asymptotic Fermat's Last Theorem over these fields.

Abstract

In a recent work the authors prove the effective asymptotic Fermat's Last Theorem for the infinite family of fields $\mathbb{Q}(\zeta_{2^{r+2}})^+$ where $r \ge 0$. A crucial step in their proof is the following conjecture of Kraus. Let $K$ be a number field having odd narrow class number and a unique prime $\lambda$ above $2$. Then there are no elliptic curves defined over $K$ with conductor $\lambda$ and a $K$-rational point of order $2$. In this note we give a new elementary proof of Kraus' conjecture that makes use only of basic facts about elliptic curves, Tate curves and Tate modules.