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I am trying a cross-post here, as my previous post on stackexchange was not as fruitful as I hoped. The link to the older post is: https://math.stackexchange.com/questions/3327269/values-of-grössencharacter-attached-to-cm-elliptic-curve.

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 for proving the theorem of Deuring (page 175) in the case $K \nsubseteq L$ by following the exercises at the end of the section one has to use the following two points.

  1. 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*}
  2. 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*}

Since Silverman describes a detailed route of the proof within the exercises, but not mentioning anything about these two steps, I wonder if there is a short proof of these facts and I am missing something making it totally obvious.

Thanks in advance for any comment.

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