Why are $p$-adic characters locally analytic?

Question

By definition, a $p$-adic character is a continuous homomorphism $\chi:\mathbb{Z}_p^\times\rightarrow \mathbb{C}_p^\times$. In several books/papers, I have seen it stated that all $p$-adic characters are locally analytic, in the sense that there is a covering $a+p^n\mathbb{Z}_p$ ($a$ prime to $p$) of $\mathbb{Z}_p^\times$ such that $\chi$ is given by a power series $\sum_{n\geq 0} c_n(x-a)^n$ on the open set $a+p^n\mathbb{Z}_p$. I think this is probably easy and I am just overthinking it, but can someone elucidate this for me? In other words:

Why are $p$-adic characters locally analytic?

I think I can see why this is true for Dirichlet characters of $p$-power conductor, but not in general. A Dirichlet character of conductor $p^n$ for some $n$ can be identified (perhaps after fixing an embedding $\mathbb{C}\rightarrow \mathbb{C}_p$) with a $p$-adic character $\chi$ which is constant on $1+p^n\mathbb{Z}_p$. Hence such characters are locally constant, and therefore locally analytic, represented on $a+p^n\mathbb{Z}_p$ by the constant power series $\chi(a)$. I'm not sure how to proceed for a general character though...

Answer 1

The topological group $\Bbb Z_p^\times$ is isomorphic to the product of $\Bbb Z_p$ with a finite group. In other words, it is locally isomorphic to the additive group $\Bbb Z_p$. The isomorphism is given by the functions $\log$ and $\exp$, which are locally analytic.

Therefore we may just focus on (additive) characters on $\Bbb Z_p$.

This is then implied by the following exercise:

1. If $z$ is an element of $\Bbb C_p$ such that $|z - 1| < 1$, then the function $\chi_z: \Bbb Z \rightarrow \Bbb C_p$ sending any $k\in \Bbb Z$ to $z^k$ extends continuously to a character $\chi_z:\Bbb Z_p \rightarrow \Bbb C_p^\times$, which is given by $$\chi_z(x) = \sum_{n \geq 0} \binom x n (z - 1)^n$$ for every $x \in \Bbb Z_p$.
2. Any continous character $\chi:\Bbb Z_p \rightarrow \Bbb C_p^\times$ is of the form $\chi_z$ for some $z$ as in 1.