I am studying the chapter on the associated Grössencharacter of a CM ellipic curve in Silverman's Advanced Topics in the Arithmetic of Elliptic Curves (II.9.) and have a question concerning a specific step in the proof of
Theorem 9.1. Let $x \in \mathbf{A}_L^\times$ be an idele of $L$ and let $s = N^L_K(x) \in \mathbf{A}_K^\times$. Then there exists a unique $\alpha = \alpha_{E/L}(x) \in K^\times$ with the following properties:
- $\alpha\mathcal{O}_K = (s)$,
- For any fractional ideal $\mathfrak{a} \subseteq K$ and any complex analytic isomorphism $f \colon \mathbb{C}/ \mathfrak{a} \rightarrow E(\mathbb{C})$ one has
\begin{align} [x,L] \circ f_{|K/\mathfrak{a}} = f \circ \text{mult}_{\alpha s^{-1}}. \end{align}
My problem lies in the following. When showing uniqueness of such an element $\alpha \in K^{\times}$, one finds that (if $\alpha' \in K^{\times}$ is another element satisfying the properties) multiplication by $\alpha' \alpha^{-1}$ is the identity on $K/\mathfrak{a}$. Then Silverman concludes $\alpha' = \alpha$. How is this done? I only see that $\alpha$ and $\alpha'$ have the same image in $K/\mathfrak{a}$ and seem to miss something obvious..
UPDATE: I guess I have it now.. The continuous maps $\text{id}$ and multiplication by $\alpha' \alpha^{-1}$ from $\mathbb{C}/\mathfrak{a}$ to $\mathbb{C}/\mathfrak{a}$ agree when restricted to the dense subset $K/\mathfrak{a}$. As $\mathbb{C}/\mathfrak{a}$ is Hausdorff, one can easily show that this implies $\text{id} = \text{mult}_{\alpha'\alpha^{-1}}$. Hence both the elements $1$ and $\alpha'\alpha^{-1}$ of $\mathcal{O}_K^{\times}$ give the identity on $\mathbb{C}/\mathfrak{a}$ and therefore are equal.