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

Abstract

The $p$-adic logarithm appears in many places in number theory. Hence having a good description of the image of the $p$-adic logarithm could be useful, and in particular, to figure out the image of $1 + \mathfrak{m}_K$, where $K$ is an algebraic extension of $\mathbb{Q}_p$ and $\mathfrak{m}_K$ its maximal ideal. If the ramification index of $K$ is strictly less than $p-1$ then it is well known that the $p$-adic logarithm is a bijection of $1+\mathfrak{m}_K$ onto $\mathfrak{m}_K$. If the ramification index is equal or greater than $p-1$ than the $p$-adic logarithm is no more a bijection and the situation is more complicated. Our main result is the computation of $\log_p(1+\mathfrak{m}_K)$ in two cases: \begin{enumerate} \item[$\bullet$] for $K=\mathbb{Q}_p(\zeta_p)$, with $\zeta_p^p=1$, totally ramified $p$-cyclotomic extension of $\mathbb{Q}_p$ (ramification index equal $p-1$) \item[$\bullet$] for $K$ a quadratic extension of $\mathbb{Q}_2$ (ramification index equal 1, 2). \end{enumerate}