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Consider $x \in \mathbb{C}_p$ with $|x|<1$ then for $a \in \mathbb{Z}_p$ we have the characters $$ a \mapsto (1+x)^a $$ where $(1+x)^a= \exp(a\log_p(1+x))$ My question is : it's possible to extend these characters to locally analytic characters in all $\mathbb{C}_p$?

Thanks for references!

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  • $\begingroup$ Do you mean that you want $a$ to take values outside of $\Bbb Z_p$, or do you mean that you want to take $|x|\ge1$ ? $\endgroup$ – Lubin Mar 20 at 2:07
  • $\begingroup$ @Lubin I mean when $a$ takes values outside Z_p $\endgroup$ – andres Mar 20 at 7:47
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    $\begingroup$ Well, the binomial formula for $(1+x)^a$ that I have given below shows that when $a\notin\Bbb Z_p$, some coefficients will be nonintegral (i.e. have absolute value greater than $1$) and in that case the whole series will presumably fail to be convergent, except on a smaller disk. This could be worth a closer examination. $\endgroup$ – Lubin Mar 20 at 23:00
  • $\begingroup$ @Lubin So it seems difficult to extend these characters, but exists a sort of classification of such characters $(\mathbb{C}_p,+)\to(\mathbb{C}_p,\times) $? I saw that $exp_p$ could be extended in a non-canonical way like the classical logarithm to obtain, for example, the Iwasawa logarithm. So maybe this could give some results. But really thanks for the answer. $\endgroup$ – andres Mar 21 at 8:55
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This is a comment, not in any sense an answer.

But you simply can not define $(1+x)^a$ that way, because for $|x|$ very nearly $1$ (in the language of the additive valuation $v(x)=-\log_p(|x|)$: for $v(x)$ very small but positive), $v(\log(1+x))$ will be negative, in other words $\log(x)$ is not even in the ring of integers of the field $\Bbb Q_p(x)$, and certainly not in the domain of any $p$-adic exponential function.

One perfectly satisfactory way of defining $(1+x)^a$ is: $$ (1+x)^a=1+\sum_{k=1}^\infty\binom akx^k\,,\\ \text{where }\binom ak=\frac{a(a-1)\cdots(a-k+1)}{k!}\,. $$ It’s a satisfying exercise to show that if $a\in\Bbb Z_p$, then so is $\binom ak$ for every positive integer $k$.

Another perfectly satisfactory way of defining $(1+x)^a$ is: $$ (1+x)^a=\lim_{|n-a|\to0}(1+x)^n\,, $$ where the values allowed for $n$ are positive integers, and the limit is taken $p$-adically.

Please note: You have written “$\log_p$” for the $p$-adic logarithm, but I have called this simply “log”, while when I wrote “$\log_p$”, I meant the real logarithm to base $p$, as you learn to do in high-school.

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  • $\begingroup$ Yes I forgot to recall it, my notation for Iwasawa logarithm is with $ \log_p$ $\endgroup$ – andres Mar 20 at 7:50

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