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I would like to refer you to Exercise $2.30(c)$ in Silverman's Advanced Topics in Elliptic Curves.

Question: Let $E/L$ be an EC with CM by $K$. Assume that $K\nsubseteq L$, and let $L'=LK$ and let $\mathfrak{P}$ be a prime in $L$. Assume $E$ has good reduction at $\mathfrak{P}$ and let $p$ be the residue characteristic of $\mathfrak{P}$. Prove that

  • $\tilde E$ is ordinary if $\mathfrak{P}$ splits in $L'$ and $p$ splits in $K$
  • $\tilde E$ is supersingular if $\mathfrak{P}$ is inert in $L'$ and $p$ doesn't split in $K$

I'm having trouble proving that $\tilde E$ is ordinary if $\mathfrak{P}$ splits in $L'$ and $p$ splits in $K$.

My approach is to consider $a_{\mathfrak{P}}=\psi(\mathfrak{P}')+\psi(\mathfrak{P''})=\psi(\mathfrak{P}')+\overline{\psi(\mathfrak{P'})}=2\mathfrak{Re}\,\psi(\mathfrak{P'})$ and try to show that $\psi(\mathfrak{P'})$ is never purely imaginary. (Showing $a_{\mathfrak{P}}\neq0$ is sufficient to show that reduction is ordinary.) But I'm stuck, I'm not even sure if this is the right way forward. Any help is appreciated, and thanks in advanced!

Edit: Actually it maybe that we need to show that $a_{\mathfrak{P}}\not\equiv0 \text{ mod } p$. I'm not too sure, can someone put me straight? Thanks!

Second Edit: I think I've found a proof in Lang's Elliptic Functions [$13\S4$ Thm $12$]. However, I can't quite understand a few points.

  1. How is it that we are able to choose the embedding?
  2. How does $\mu'\notin\mathfrak{p}$ and $\theta$ a normalised embedding and reduction of $\mu'\omega$ mod $\mathfrak{P}$ not zero imply that $\overline{\theta(\mu')}$ is separable?
  3. How does $\theta(\mu')$ having degree a power of $p$ imply the reduction mod $\mathfrak{P}$ has degree power of $p$, and also that $\overline{A}$ has non-trivial point of order $p$?
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