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arXiv · 1804.02849

On asymptotic Fermat over the Z_2 extension of Q

Nuno Freitas·Alain Kraus·Samir Siksek
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Abstract

In a recent work the authors prove the effective asymptotic Fermat's Last Theorem for the infinite family of fields $\mathbb{Q}(ζ_{2^{r+2}})^+$ 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 $λ$ above $2$. Then there are no elliptic curves defined over $K$ with conductor $λ$ 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.

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Nuno Freitas, Alain Kraus, Samir Siksek. 2020-12-06. On asymptotic Fermat over the Z_2 extension of Q. https://arxiv.org/abs/1804.02849

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