Bases for some modules of cyclotomic units

TL;DR

This paper constructs explicit bases for groups of cyclotomic units in certain abelian number fields and their cyclotomic towers, advancing understanding in Iwasawa theory.

Contribution

It provides new explicit bases for Washington's cyclotomic units and their inverse limits in specific abelian fields where the field equals its narrow genus field.

Findings

Established a $\Lambda$-basis for the inverse limit of cyclotomic units in the tower.

Derived a $\mathbb{Z}[1/2]$-basis for the cyclotomic units under the same conditions.

Extended the understanding of cyclotomic units in the context of Iwasawa theory.

Abstract

Let $\mathbf{Was}(\mathbb{K})$ denote the group of Washington's cyclotomic units of any abelian number field $\mathbb{K}$. If $\mathbb{K}$ coincides with its genus field in the narrow sense, we give a $\Lambda$-basis of $\lim\limits\_{\xleftarrow{}} \mathbf{Was}(\mathbb{K}\_k^{+})$ where $(\mathbb{K}\_k)\_{k \geqslant 0}$ denotes the cyclotomic $\mathbb{Z}\_p$-tower of $\mathbb{K}$ and $\Lambda$ denotes the Iwasawa's algebra. This results from a $\mathbb{Z} [1/2]$-basis of $\mathbf{Was}(\mathbb{K}) \otimes\_{\mathbb{Z}} \mathbb{Z} [1/2]$ that we give under the same hypothesis.