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Suppose we have a finite extension $K / \mathbb{Q}_p$ with valuation ring $\mathcal{O}$ and maximal ideal $\mathfrak{p}$.

One can define the $\mathfrak{p}$-adic logarithm on the group of principal units $U^{(1)}$ of the local field $K$ using the power series expansion $$\log(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ldots.$$

One can then extend the definition to a map $\log\colon K^\ast \rightarrow K$ satisfying the properties $\log(xy) = \log(x) + \log(y)$ and $\log(p) = 0$.

My question is, why do we want $\log(p) = 0$?

With the usual logarithm over $\mathbb{R}$, the kernel of $\log$ is $\{1\}$, and I don't see an analogy where $p$ could correspond to something in $\mathbb{R}$. So what makes this particular choice of $\log(p)$ desirable, over for example some other choice like $\log(p) = e$, where $(p) = \mathfrak{p}^e$?

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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).

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    $\begingroup$ Perfect answer. But the logarithmic series already vanishes at the $p$-primary roots of unity, no matter what you define $\log(p)$ to be, so I would recommend deletion of your last sentence (unless I have misunderstood your intention here). $\endgroup$ – Lubin Jul 11 '17 at 14:00
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    $\begingroup$ @Lubin Right, I modify my last sentence. $\endgroup$ – nguyen quang do Jul 11 '17 at 15:20
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    $\begingroup$ @Torsten Schoeneberg. Sorry for my late answer, but here is the (somewhat technical) point. After the establishment of the main theoretical properties of CFT, a natural continuation was to give "explicit" descriptions of the reciprocity map, in particular the local Artin symbol. In the classical first results obtained for the cyclotomic fields $\mathbf Q_p (\zeta_{p^n})$ by computing the Hilbert symbol $<a,b>_n:=\zeta_{p^n}^{[a,b]_n}$ defined by combining CFT and Kummer theory (Hasse, Shafarevich, Iwasawa, Brückner, Vostokov...), extended later to Lubin-Tate towers by Coates and Wiles... $\endgroup$ – nguyen quang do Jul 28 '18 at 11:04
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    $\begingroup$ ...$log (a)$ appears in the company of the logarithmic derivative of $g_b$ at $1-\zeta_{p^n}$, where $g_b$ denotes the Coleman power series of $b$. In their work on the special values of $L$-functions, Bloch and Kato later extended and "explained" these explicit formulas in the framework of $p$-adic representations "à la" Fontaine (with the mysterious Fontaine rings $B_{dR} , B_{crys}$, etc., introducing the Bloch-Kato exponential and dual exponential maps. $\endgroup$ – nguyen quang do Jul 28 '18 at 11:08
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    $\begingroup$ In view of a general (partly conjectural) process to construct $p$-adic $L$-functions starting from so called "Euler systems", Perrin-Riou generalized the B-K exponential map to crystalline representations, and Colmez constructed the inverse function for de Rham representations, now called "P-R log". The story is still going on. For an introduction, see chapter 4 of the book "The Bloch-Kato conjecture for the Riemann Zeta Function", ed. Coates & al., Cambridge, 2015. $\endgroup$ – nguyen quang do Jul 28 '18 at 11:08

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