$p$-adic regulator

Let $K$, $n$, $r_1$, $r_2$, $\{{\epsilon }_n,\mathrm{\dots },{\epsilon }_{r-1}\}$, and $||\cdot |{|}_i$ be as in the entry regulator, but with $K$ taken to be a CM field.

Define the $p$-adic logarithm ${\log }_p:{ℂ}_p^{\times }\to {ℂ}_p$ by

${\log }_p(x)=-{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}{\displaystyle \frac{{(1-x)}^k}{k}}$

Let $A_{K,p}$ be the $(r-1)\times (r-1)$ matrix with general entry given by $a_{i,j}={\log }_p{||{\epsilon }_j||}_i$. The absolute value of the determinant of this matrix is again independent of your choice of basis for the units and of the ordering of the embeddings. This value is called the $p$-adic regulator of $K$, and is denoted by $R_{p,K}$, or $R_p(K)$.