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It is well known that $p$-adic logarithm define an isomorphism from $B(1,p^{-1/p-1})$ to $B(0,p^{-1/p-1})$. I would like to know if is it possible to extend somehow this logarithm to the set of $x$ such that $|x|_p=1$. My idea would be to find a minmal $n$ such that $x^n \in B(1,p^{-1/p-1})$ and define $\log_p(x)=\frac{\log_p(x^n)}{n}$. But I am not able to find such a $n$. So I would like to know if it is indeed possible to extend $\log_p$ to this set.

Kenta S
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  • There is a standard extension of the $p$-adic logarithm to $\Bbb Q_p^*$. – Angina Seng Jun 20 '18 at 02:19
  • Yes, but I would like somehow that the $log_p$ stay an isomorphism on my set. –  Jun 20 '18 at 02:21
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    This is one of my previous questions, which looks related: https://math.stackexchange.com/questions/2353389/why-do-we-define-the-mathfrakp-adic-logarithm-on-a-mathfrakp-adic-numb – Tob Ernack Jun 20 '18 at 02:26
  • Okay, yes it is related, but my question would be how exactly do you extend it to the group of principal unit ? –  Jun 20 '18 at 02:29
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    So you can define $\log_p$ as a power series on the set ${x: |x - 1|_p \lt 1}$, and you can extend its domain to $\mathbb{Q}_p^$ by defining $\log_p(p) = 0$, and in general for $x = up^n$ you would have $\log_p(x) = \log_p(u)$, where $u$ is a unit in $\mathbb{Z}_p^$. Each unit can be written as $u = \zeta u'$ where $\zeta$ is a root of unity and $u'$ is a principal unit. Then $\log_p(u) = \log_p(\zeta) + \log_p(u') = \log_p(u')$ since the log is zero on the roots of unity. – Tob Ernack Jun 20 '18 at 02:35
  • Hmm sure, but by doing so I will no more get an isomorphism right ? –  Jun 20 '18 at 02:51
  • That might work. If $a \in {1, \ldots, p-1}$ is such that $|x - a|_p \lt 1$ then you can let $n = |a|$ (order of $a$ modulo $p$) so that $x^n \in B(1, 1/p)$ (assuming $|x|_p = 1$). – Tob Ernack Jun 20 '18 at 02:55
  • Okay I will try to computate this in details and put the result here after. Thanks ! –  Jun 20 '18 at 02:57
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    I don't think the logarithm usually gives you an isomorphism because it is not injective (any root of unity in $\mathbb{Z}_p$ will be mapped to $0$). It is injective on the principal units though. – Tob Ernack Jun 20 '18 at 02:58
  • But actually it's an isomorphism from the ball I described in my question so the goal is to find the good power for each element of the group of principal unit to make it fall in the ball. –  Jun 20 '18 at 03:01
  • Isomorphism of what? Sets? Topological spaces? Groups? Also, I assume but you have not explicitly stated that the balls are inside $\Bbb Q_p$ (as opposed to some extension thereof) - is that right? – Torsten Schoeneberg Jun 20 '18 at 04:45
  • Of groups would be good, yeah the balls are Inside $\mathbb{Q}_p$. –  Jun 20 '18 at 05:09

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For $p>2,$ it is a standard fact that

$\{x\in \Bbb Q_p: |x|_p=1\} = \Bbb Z_p^\times \simeq \mu_{p-1}(\Bbb Q_p) \times \{x\in \Bbb Q_p: |x-1|_p <1\}$

as topological groups. The second factor on the right hand side are the principal units, often denoted $U_1$, and since the $p$-adic absolute value on $\Bbb Q_p$ takes only integer powers of $p$ as value, it is identical to your $B(1, p^{-\frac{1}{p-1}})$.

To extend the logarithm $log:U_1 \rightarrow \Bbb Q_p$ to the full set $\Bbb Z_p^\times$, you therefore "only" have to define it on $\mu_{p-1}(\Bbb Q_p)$, the $(p-1)$-th roots of unity. Indeed, the only $n$ you have to choose for your proposed extension formula is $n=p-1$. However, if you want the extension to be a a group homomorphism (from the multiplicative group on the left to the additive on the right), you'll have to define it as $log(\zeta) = 0$ for all $\zeta \in \mu_{p-1}(\Bbb Q_p)$, because those $\zeta$'s are torsion, whereas the only torsion element on the right hand side is $0$. (Or, using your extension formula, $log(\zeta) = \frac{log(\zeta^{p-1})}{p-1} = \frac{log(1)}{p-1} = 0$.)

In particular, an extension of the logarithm to $\{x\in \Bbb Q_p: |x|_p=1\}$ cannot be at the same time a group homomorphism and injective.

Torsten Schoeneberg
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  • Thak you very much that's very clear !! –  Jun 20 '18 at 05:43
  • Do you have any references that talk about all this results ? I would be very interested to read them. –  Jun 20 '18 at 05:51
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    Any book about $p$-adics (F. Gouvea, A. Robert) and/or local fields (Serre, or corresponding chapters in Neukirch's Algebraic Number Theory or Class Field Theory). There's probably even dozens of decent lecture notes out there. Also, https://math.stackexchange.com/q/73565/96384 and https://math.stackexchange.com/q/1298669/96384. – Torsten Schoeneberg Jun 20 '18 at 05:59
  • Super ! Thanks ! –  Jun 20 '18 at 06:03