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I am reading Number Theory vol. 1 by Henri Cohen (among other things) and I am curious about the Iwasawa logarithm. Let $\mathbb{C}_p$ be the completion of the algebraic closure of $\mathbb{Q}_p$.

Let $\log_p$ denote the function on the unit ball around 1 (in $\mathbb{C}_p$) given by the standard power series. I understand that it is not possible to analytically continue this power series anywhere outside the ball; however, it is possible to uniquely extend $\log_p$ to $\mathbb{C}_p - \{0\}$ by 'filling in' values subject to the constraints $\log_p(xy) = \log_p(x) + \log_p(y)$ and $\log_p(p) = 0$. Similarly $\exp_p$ can be extended to all of $\mathbb{C}_p$ (although not uniquely). This technique does not seem very generalizable to arbitrary power series over $\mathbb{C}_p$.

However, I have read also that Tate developed a theory of Rigid Analytic Spaces, where apparently one can do analytic continuation of power series over non-Archimedean domains. Certainly I may as well ask whether analytic continuation of (some form of) the logarithm or Iwasawa logarithm, or exponential, makes sense through Tate's theory, and whether it agrees with the extension above.

More generally, in the classical case over $\mathbb{C}$ one can use analytic continuation to 'figure out' what kind of singularity occurs on the radius of convergence of a power series; for example, in the case of $\log$, a branch point at 0 prevents the power series converging outside the unit ball around 1. Can similar methods here characterize singularities on the radius of convergence of $\log_p$, if given suitable tools from rigid analytic spaces?

My knowledge of this area is slim at best and most resources on Tate's theory seem beyond my grasp. Thanks for any clarification that can be provided.

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    $\begingroup$ The lightest amendment to your definition of the $p$-adic logarithm: since the series is convergent only on the open disk $v_p(1+x)>0$, you must also specify that $\log(\zeta)=0$ for all roots of unity $\zeta$. This is automatic for $p^m$-th roots of unity, since the series gives a homomorphism (unlike the complex log series), but needs to be specified for $m$-th roots of unity with $m$ prime to $p$. $\endgroup$ – Lubin Jun 28 '14 at 22:43
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Naively, no. Look at the functions $\exp(z)$ and $\exp(z+\frac{z^p}{p})$ (in a neighborhood of the origin). The power series expansion of the latter about $z=0$ has a larger radius of convergence than the former, but there is no easy explanation of this in terms of singularities.

If there is a more subtle analysis that explains this phenomenon in some kind of geometric terms, I'd be very interested to hear about it. But I think the nonarchimedean theory is not nearly as clean as the complex theory in this respect.

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  • $\begingroup$ Thanks. Is there any notion of singularity or complete analytic function at all? For example one could take the the polynomial $y^2 - x$ and (as far as I can tell) use Hensel's lemma to expand a power series $p(x) = \sum c_i (x-\alpha)^i$ around every point $\alpha \in \mathbb{C}_p - \{0\}$, in fact, a pair of power series, such that $(x,p(x))$ is identically zero on $y^2 - x$ It seems reasonable to assume that such power series bear some relationship to each other, that an arbitrary pair of power series do not. $\endgroup$ – Joe Bebel Jul 8 '14 at 10:16

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