Why do we define the $\mathfrak{p}$-adic logarithm on a $\mathfrak{p}$-adic number field such that $\log(p) = 0$?

Solution 1:

The main goal is to construct a continuous function $log_p: \mathbf C_p ^* \to \mathbf C_p$ s.t. $log_p (xy) = log_p + log_p (y)$. Since $ \mathbf C_p ^* = p^\mathbf Q \times W \times U_1$, where $U_1$ is the group of principal units and $W$ the group of roots of $1$ of order prime to $p$, it suffices to define $log_p$ on each of the direct factors. On $U_1$ one has already the usual power series $log_p (1+x)$ whose radius of convergence is $1$. On $W$, one must have the nullity of $log_p$, since for any root of unity $w$ of order $n$, necessarily $n.log_p (w)= log_p (1) = 0$. It remains only to adjust the value $log_p (p)$.

The choice is not quite arbitrary, because any $\sigma \in G_\mathbf {Q_p} $can be extended to a continuous automorphism of $\mathbf C_p$, and it follows that $log_p (p) \in \mathbf Q_p$. Your suggested choice $log_p (p)=e$ is not good either because it depends on the ambient field $K$. Actually, most of the ramification problems in CFT are concentrated in $U_1$, as well as most of the calculations about $L_p$-functions , so the definitely most natural (which is also the most simple) choice is $log_p (p)=0$. It follows that Ker $log_p = p^\mathbf Q \times \mu$, where $\mu$ is the group of all roots of unity (of arbitrary order).