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

TL;DR

This paper investigates minimal relative units in the cyclotomic $Z_2$-extension, proposing conjectures and partial results on their trace properties, with connections to class number problems.

Contribution

It introduces a new conjecture on minimal trace values of units in cyclotomic extensions and proves it for small cases, advancing understanding of unit structures.

Findings

Confirmed the conjecture for $n \\leq 6$.

Established an upper bound for even $n$.

Observed links to class number problems.

Abstract

Let $\mathbb B_n:=\mathbb Q(\cos(\pi/2^{n+1}))$. 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 $\mathrm{Tr} \epsilon^2 \geq 2^n(2^{n+1}-1)$ for $\epsilon \in RE_n -\{\pm 1\}$. Morisawa and Okazaki showed that it holds for $\epsilon \in RE_n -RE_n^+$. In this paper we study the case $\epsilon \in RE_n^+$. We conjecture that $\min \{\mathrm{Tr} \epsilon^2 \mid \epsilon \in RE_n^+-\{\pm 1\}\}= 2^n(1+8c_n)$, 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: $\min \{\mathrm{Tr} \epsilon^2 \mid \epsilon \in RE_n^+-\{\pm 1\}\} \leq 2^n(1+8c_n)$ holds for even $n$. We also observe a relation to the class number problem.