# Additive characters of $\mathbb{C}_p$

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!

• Do you mean that you want $a$ to take values outside of $\Bbb Z_p$, or do you mean that you want to take $|x|\ge1$ ? – Lubin Mar 20 at 2:07
• @Lubin I mean when $a$ takes values outside Z_p – andres Mar 20 at 7:47
• 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. – Lubin Mar 20 at 23:00
• @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. – andres Mar 21 at 8:55

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