Let $E$ be an elliptic curve defined over a number field $L$, having CM by by the ring of integers $\mathcal{O}_K$ for $K$ quadratic imaginary. If $K \subseteq L$, then (as constructed in Silverman's Advanced Topics in the Arithmetic of Elliptic Curves) we have an associated Grössencharacter $\psi_{E/L}$. If $K \nsubseteq L$, then we take $L' = LK$ (being a quadratic extension of $L$) and consider the Grössencharacter $\psi_{E/L'}$.
I think that Silverman uses the following without mentioning anything about it, so I wonder if there is a short proof of these facts and I am missing something making it totally obvious.
- Assume $K \nsubseteq L$ and let $\mathfrak{P}$ be a prime in $\mathcal{O}_L$ of good reduction for $E$, splitting as $\mathfrak{P'}\mathfrak{P}''$ in $\mathcal{O}_{L'}$. Then we have \begin{align*} \psi_{E/L'}(\mathfrak{P}')\psi_{E/L'}(\mathfrak{P}'') = \sharp(\mathcal{O}_L/\mathfrak{P}). \end{align*}
- Assume $K \nsubseteq L$ and let $\mathfrak{P}$ be a prime in $\mathcal{O}_L$ ramifying in $\mathcal{O}_{L'}$ as $\mathfrak{P'}^2$. Then we have \begin{align*} \psi_{E/L'}(\mathfrak{P}') = 0. \end{align*}
Thanks in advance for any comment.
For part 2, note that $\psi(\mathfrak{P}) = 0$ when $\psi$ is ramified at $\mathfrak{P}$ is a convention. See Silverman's Advanced Topics, page 173, after Conjecture 10.2 (see also page 177).
For part 1, this follows from the fact that $N_{\mathbb{Q}}^K(\psi_{E/L}(\mathfrak{P})) = N_{\mathbb{Q}}^L(\mathfrak{P})$. See Silverman's Advanced Topics, page 175, Corollary 10.4.1, part (a).