Plus and minus logarithms and Amice transform

TL;DR

This paper provides a new distribution-based description of Pollack's plus and minus p-adic logarithms, including explicit formulas and connections to earlier distributions, and extends the approach to Loeffler's two-variable analogues.

Contribution

It introduces explicit distribution formulas for plus and minus p-adic logarithms and generalizes the description to Loeffler's two-variable versions.

Findings

Distribution formulas for plus and minus logarithms

Identification of minus distribution with Koblitz's distribution

Extension to Loeffler's two-variable analogues

Abstract

We give a new description of Pollack's plus and minus $p$-adic logarithms $\log_p^\pm$ in terms of distributions. In particular, if $\mu_\pm$ denote the pre-images of $\log_p^\pm$ under the Amice transform, we give explicit formulae for the values $\mu_\pm(a+p^n\mathbb{Z}_p)$ for all $a\in \mathbb{Z}_p$ and all integers $n\ge1$. Our formulae imply that the distribution $\mu_-$ agrees with a distribution studied by Koblitz in 1977. Furthermore, we show that a similar description exists for Loeffler's two-variable analogues of these plus and minus logarithms.