Connected-étale sequence for ordinary CM elliptic curves

Question

Let $E/k$ be an elliptic curve over algebraically closed field of characteristic $p$ with CM, for simplicity, by the maximal order of a quadratic imaginary field $K/\mathbb{Q}$.

Suppose that $p$ is split in $K$ so that $E$ is ordinary, and write $p\cal{O}_K=\frak{p}\overline{\frak{p}}$. I would like to consider two subgroup schemes $E[\frak{p}]$ and $E[\overline{\frak{p}}]$ of $E[p]$ and ask the following two questions:

• Is it true in general that $\text{rank}\ E[\frak{p}]$= $\text{rank}\ E[\overline{\frak{p}}]=p$ ? I think I have an argument that goes through lifting to characteristic zero and the Main Theorem of CM but probably there is a simpler way.

Secondly,

• Is it true that $E[\frak{p}]$ and $E[\overline{\frak{p}}]$ can be seen as connected and étale parts (or viceversa) of $E[p]$? In the setting I have chosen for this question I don't see why this should be true (for example what distinguish $\frak{p}$ from $\overline{\frak{p}} $ ?) but it seems reasonable when I see $E/k$ as the reduction at the appropriate prime (above $\frak{p}$ or $\overline{\frak{p}} $) of a CM elliptic curve over a number field.

Thank you for your help.

Answer 1

Formulating precisely this question helped me to figure it out (probably).

Fix an isomorphism $O_K\cong\text{End}(E)$ identifying complex conjugation with "taking the dual isogeny". Then we can write $\frak{p}=(p,\alpha)$, $\overline{\frak{p}}=(p,\overline{\alpha})$ (by the theory of Dedekind domains) where $\alpha$, $\overline{\alpha}$ are isogenies with $\alpha\circ \overline{\alpha}=\overline{\alpha}\circ\alpha=[p\cdot m]_E$ for some integer $m$. It follows that $E[\frak{p}]$$=E[\alpha]\times_E E[p]$ and $E[\overline{\frak{p}}]=E[\overline{\alpha}]\times_E E[p]$ finite flat group schemes.

The fact that distinguish between the two ideal is the action on the tangent space at the origin of $E$. Indeed $\underline{\omega}_{E/k}$ is one dimensional over $k$ and so the $O_K$-action is through an algebra homomorphism $\iota: O_K\to k$. Suppose that $\frak{p}=\ker\iota$ then $\overline{\alpha}$ is étale because induces an isomorphism between tangent spaces, hence $E[\overline{\alpha}]$ is étale over $\text{Spec}(k)$ and so also $E[\overline{\frak{p}}]$.

Finally, restricting the isogeny $\alpha$ to $E[p]$ gives an exact sequence of finite free group schemes $0\to E[\frak{p}]$$\to E[p]\overset{\alpha}{\to} E[\overline{\frak{p}}]\to 0$ because $p\nmid\alpha$ and $p\nmid\overline{\alpha}$. Moreover it is the connected-ètale sequence for $E[p]$ because $E[\overline{\frak{p}}]$ is ètale.