Effective equidistribution of norm one elements in CM-fields

TL;DR

This paper investigates the distribution properties of norm one elements in CM-fields, providing effective equidistribution results that deepen understanding of their arithmetic and geometric structure.

Contribution

It establishes new effective equidistribution theorems for norm one elements in CM-fields, extending previous results and offering quantitative bounds.

Findings

Proves effective equidistribution of norm one elements in CM-fields.

Characterizes the structure of the group al{S}_K in relation to CM-fields.

Provides explicit bounds for distribution rates.

Abstract

For a number field $K$ let $\mathcal{S}_K$ be the maximal subgroup of the multiplicative group $K^\times$ that embeds into the unit circle under each embedding of $K$ into the complex numbers. The group $\mathcal{S}_K$ can be seen as an archimedean counterpart to the group of units $\mathcal{O}_K^\times$ of the ring of integers $\mathcal{O}_K$. If $K=\mathbb{Q}(\mathcal{S}_K)$ is a CM-field then $\mathcal{S}_K/{\mathop{\rm Tor}\nolimits}(K^\times)$ is a free abelian group of infinite rank. If $K=\mathbb{Q}(\mathcal{S}_K)$ is not a CM-field then $\mathcal{S}_K=\{\pm 1\}$. In the former case $\mathcal{S}_K$ is the kernel of the relative norm map from $K^\times$ to the multiplicative subgroup $k^\times$ of the maximal totally real subfield $k$ of $K$.