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

TL;DR

This work addresses the problem of describing the image of the $p$-adic logarithm on principal units in the cyclotomic extension $K=\mathbb{Q}_p(\zeta_p)$. It provides a self-contained analytic proof that the image of the annulus $(1+\mathfrak{m}_K)\setminus(1+\mathfrak{m}_K^2)$ under $\log_p$ is exactly $\mathfrak{m}_K^2$, and that the map is onto with $p-1$ preimages per element, using explicit $\pi$-adic expansions and an inductive coefficient construction. As a consequence, the local index $[\mathfrak{m}_K:\log_p(1+\mathfrak{m}_K)]=p$ is established in the cyclotomic case, clarifying the local contribution to $p$-adic regulators and enabling explicit regulator computations. The method combines a precise choice of uniformizer $\pi$ with $\pi^{p-1}=-p$ and the relation $\pi^p/p=-\pi$, yielding an explicit, convergent expansion of $\log_p(1+x)$ and a constructive way to realize all elements of $\mathfrak{m}_K^2$ as logarithms of units.

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 $K=\mathbb{Q}_p(ζ_p)$ 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.

Theorems & Definitions (6)

• Theorem 2.1
• proof
• Theorem 3.1
• proof
• Lemma A.1
• proof