Plus and minus logarithms and Amice transform

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 $n\ge 1$. 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.

9 pages