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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 $(p) = \mathfrak{p}^e$?

Tob Ernack
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The main goal is to construct a continuous function $log_p: \mathbf C_p ^* \to \mathbf C_p$ s.t. $log_p (xy) = log_p (x) + log_p (y)$. 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).

it's a hire car baby
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nguyen quang do
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    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). – Lubin Jul 11 '17 at 14:00
  • Thank you for the answer. Since I haven't learned about CFT yet, could you elaborate on the meaning of the claim "most of the ramification problems in CFT are concentrated in $U_1$"? – Tob Ernack Jul 11 '17 at 14:58
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    @Lubin Right, I modify my last sentence. – nguyen quang do Jul 11 '17 at 15:20
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    @Tob Ernack Let $K$ be a finite extension of $\mathbf Q_p$. Local CFT establishes the existence of a "reciprocity homomorphism" $rec_K: K^* \to G_K^{ab}$ (= the Galois group of the maximal abelian extension of $K$) with the following properties : (1) The restriction of $rec_K$ to $U_K$ (= the unit group of $K$) induces an isomorphism onto the inertia subgroup $I_K$ (2) Since the maximal abelian unramified extension of $K$ (= fixed by $I_K$) is simply the extension obtained by adding all roots of 1 of order prime to $p$, it remains "only " to describe $I_K$. – nguyen quang do Jul 11 '17 at 15:39
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    First one describes the filtration of $U_K$ provided by the subgroups $U_K ^{n} = 1+ P^n$, the successive quotients $U_K^{n} / U_K^{n+1}$ being "well known" . Then one transfers this filtration to $I_K$, knowing by CFT that $rec_K$ induces a surjective homomorphism of $U_K^{v}$ onto $I_K^{v}$. Here $v$ is a positive real and $I_K^{v}$ is the ramification subgroup of index $v$ in the upper enumeration (I don't recall the definitions). – nguyen quang do Jul 11 '17 at 15:50
  • While I do understand that setting $log(p) = 0$ is the simplest and most natural thing to do, I do not understand what the reciprocity map of CFT really has to do with that. Could you clarify? – Torsten Schoeneberg Jun 20 '18 at 23:54
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    @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... – nguyen quang do Jul 28 '18 at 11:04
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    ...$log (a)$ 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. – nguyen quang do Jul 28 '18 at 11:08
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    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. – nguyen quang do Jul 28 '18 at 11:08