On asymptotic Fermat over the Z_2 extension of Q

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.