Minimal relative units of the cyclotomic $\mathbb Z_2$-extension

Abstract

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