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

Let ${\mathbb{B}}_n:=\mathbb{Q}(\cos (\pi \mathrm{/}{2}^{n+1}))$. For the relative norm map $$\mathrm{N}${n/n-1} \colon \mathcal O{\mathbb B_n}^\times \rightarrow \mathcal O_{\mathbb B_{n-1}}^\times$ontheunitsgroup,wedefine$RE_n:=$\mathrm{N}${n/n-1}^{-1}({\pm 1})$,$RE_n^+:=$\mathrm{N}${n/n-1}^{-1}({1})$\mathrm{.}Komatsuconjecturedthat$$\mathrm{T}\mathrm{r}$ \epsilon^2 \geq 2^n(2^{n+1}-1)$for$\epsilon \in RE_n -{\pm 1}$. Morisawa and Okazaki showed that it holds for $ϵ\in R{E}_n-R{E}_n^{+}$. In this paper we study the case $ϵ\in R{E}_n^{+}$. We conjecture that $\min {$\mathrm{T}\mathrm{r}$ \epsilon^2 \mid \epsilon \in RE_n^±{\pm 1}}= 2^n(1+8c_n)$,where$c_1:=2$and$c_n:=2\cdot $\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}$(2^n/5)$($n\geq 2$). We show that this holds for $n\le 6$ and that a “half” of this: $\min {$\mathrm{T}\mathrm{r}$ \epsilon^2 \mid \epsilon \in RE_n^±{\pm 1}} \leq 2^n(1+8c_n)$holdsforeven$n$. We also observe a relation to the class number problem.