Minimal relative units of the cyclotomic ${\mathbb{Z}}_2$-extension

Abstract

Let ${\mathbb{B}}_n:=\mathbb{Q}(\cos (\pi /2^{n+1}))$. For the relative norm map ${\mathrm{N}}_{n/n-1}:{\mathcal{O}}_{{\mathbb{B}}_n}^{\times }\to {\mathcal{O}}_{{\mathbb{B}}_{n-1}}^{\times }$ on the units group, we define $R{E}_n:={\mathrm{N}}_{n/n-1}^{-1}(\{\pm 1\})$, $R{E}_n^{+}:={\mathrm{N}}_{n/n-1}^{-1}(\{1\})$. Komatsu conjectured that $\mathrm{Tr}{\epsilon }^2\ge 2^n(2^{n+1}-1)$ for $\epsilon \in R{E}_n-\{\pm 1\}$. Morisawa and Okazaki showed that it holds for $\epsilon \in R{E}_n-R{E}_n^{+}$. In this paper we study the case $\epsilon \in R{E}_n^{+}$. We conjecture that $\min \{\mathrm{Tr}{\epsilon }^2\mid \epsilon \in R{E}_n^{+}-\{\pm 1\}\}=2^n(1+8c_n)$, where $c_1:=2$ and $c_n:=2\cdot \mathrm{round}(2^n/5)$ ($n\ge 2$). We show that this holds for $n\le 6$ and that a "half" of this: $\min \{\mathrm{Tr}{\epsilon }^2\mid \epsilon \in R{E}_n^{+}-\{\pm 1\}\}\le 2^n(1+8c_n)$ holds for even $n$. We also observe a relation to the class number problem.

arXiv comment19 pages, typos corrected

References in corpus (1)

• Generalized Pell's equations and Weber's class number problem