I am trying to understand Lemme 2.1 page 3 of this paper by Pilloni.
What is says (I think) is that if you have, for a a positive real number $w$, an analytic function $$ f : \mathbf{Z}_p^\times(1+p^w \mathcal{O}_{\mathbf{C}_p}) \longrightarrow \mathbf{C}_p^\times $$ which is homogenous of weight a character : $\kappa : \mathbf{Z}_p^\times \to \mathbf{C}_p^\times$ under $\mathbf{Z}_p^\times$ i.e. $$ f(\lambda x) = \kappa(\lambda)f(x) $$ for all $\lambda \in \mathbf{Z}_p^\times$, such that $f$ is zero on $\mathbf{Z}_p^\times$ then $f$ is zero.
Pilloni says that this is a consequence of the Weierstrass Preparation theorem. I know that Weierstrass preparation implies that if an analytic function has an infinite number of roots of positive valuation then $f$ is zero. But here our zeroes are invertible (i.e. they have $0$ valuation).
I think maybe it should be a consequence of homogeneity or maybe we should use log/exp but I haven't been able to write down the details.
Any ideas ?
For no other reason than to take this question off of the unanswered queue...
As in the comment section above:
By hypothesis, the analytic function $f$ vanishes on $\mathbb Z_p^*$.
Since $\mathbb Z_p^* \supset 1 +p\mathbb Z_p $, $f$ vanishes on the infinite set $\left(1 + p\mathbb Z_p\right) \cap \left(1+p^w\cal O_{\mathbb C_p}\right)$. One can now use the $p$-adic Weierstrass preparation theorem to conclude that $g(z) = f(1+z)$ vanishes on $p^w\cal O_{\mathbb C_p}$.
Therefore, by $\mathbb Z_p^*$-homogeneity, $f$ is identically zero on $\mathbb Z_p^*\left(1+p^w\cal O_{\mathbb C_p}\right) $.