On the Image of the $p$-adic Logarithm on Annuli of Principal Units

TL;DR

This paper provides an explicit analytic proof describing the image of certain principal units under the $p$-adic logarithm in cyclotomic extensions, clarifying their structure in the context of number theory.

Contribution

It offers a self-contained analytic proof for the image of a specific annulus of principal units under the $p$-adic logarithm in cyclotomic extensions.

Findings

The image of the annulus $(1+rak{m}_K) ackslash (1+rak{m}_K^2)$ under $p$-adic logarithm is exactly $rak{m}_K^2$.

Provides explicit $p$-adic logarithmic expansions for the proof.

Clarifies the structure of principal units' images in cyclotomic extensions.

Abstract

Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $\mathfrak{m}_K$ be its maximal ideal. The image of the group of principal units $1+\mathfrak{m}_K$ under $p$-adic logarithm plays important role in several areas of number theory. In general, when the ramification index of $K/\mathbb{Q}_p$ is greater or equal to $p-1$, the precise description of this image is not known. For the cyclotomic extension $K=\mathbb{Q}_p(\zeta_p)$ of degree $p-1$, it was previously proved in \cite{MAS} that the image of the annulus region $(1+\mathfrak{m}_K) \setminus (1+\mathfrak{m}_K^2)$ by $p$-adic logarithm is exactly $\mathfrak{m}_K^2$. In this paper, we give a self-contained analytic proof of this result based on explicit $p$-adic logarithmic expansions.