On the image of $p$-adic logarithm on principal units

Abstract

The $p$-adic logarithm appears in many places in number theory. Therefore, a comprehensive description of the image of the $p$-adic logarithm would be beneficial. In particular, it is important to figure out the image of $1+{\mathfrak{m}}_K$, where $K$ denotes an algebraic extension of ${\mathbb{Q}}_p$ and ${\mathfrak{m}}_K$ represents its maximal ideal. If the ramification index of $K$ is strictly less than $p-1$, then it is known that the $p$-adic logarithm serves as a bijection from $1+{\mathfrak{m}}_K$ to ${\mathfrak{m}}_K$. If the ramification index is equal to or greater than $p-1$, then the $p$-adic logarithm is no longer a bijection, and the situation is more complicated. Our main result is the computation of ${\log }_p(1+{\mathfrak{m}}_K)$ in two distinct cases: first, when $K={\mathbb{Q}}_p({\zeta }_p)$, a totally ramified $p$-cyclotomic extension of ${\mathbb{Q}}_p$ with a ramification index of $p-1$; and second, when $K$ is a quadratic extension of ${\mathbb{Q}}_2$, characterised by a ramification index of either 1 or 2. As an application, we compute the normalised p-adic regulator of the number field $\mathbb{Q}({\zeta }_p)$ by utilising the image of the $p$-adic logarithm on the units.

arXiv comment31 pages. An Erratum has been accepted by the journal to correct an error identified by an independent reviewer following publication. Comments are welcome