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

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

Keywords

$2$‑adic logarithm, units, class groups, pure quartic fields

2010 Mathematics Subject Classification

11R27, 11R29

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The author is supported by the Fundamental Research Funds for the Central Universities (No. WK3470000020) and by the Anhui Initiative in Quantum Information Technologies (Grant No. AHY150200)

Received 5 April 2020

Accepted 11 June 2020

Published 15 October 2021