On the $2$-adic logarithm of units of certain totally imaginary quartic fields

TL;DR

This paper investigates the properties of the 2-adic logarithm of fundamental units in specific totally imaginary quartic fields, confirming a conjecture in certain cases and impacting Iwasawa theory.

Contribution

It proves a new result on the 2-adic logarithm of units in fields $Q( oot 4 rom -q)$, confirming a conjecture for $q mod 16=15$ and linking to Iwasawa modules.

Findings

Confirmed Coates-Li conjecture for $q mod 16=15$

Established properties of 2-adic logarithms in these fields

Implications for Iwasawa theory and modules

Abstract

In this paper, we prove a result on the $2$-adic logarithm of the fundamental unit of the field $\mathbb{Q}(\sqrt[4]{-q}) $, where $q\equiv 3\bmod 4$ is a prime. When $q\equiv 15\bmod 16$, this result confirms a speculation of Coates-Li and has consequences for certain Iwasawa modules arising in their work.