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Let $E/k$ be a supersingular elliptic curve over an algebraically closed field $k$ of characteristic $p>0$. Then for any positive integer $r$, there is a unique closed subgroup-scheme of $E$ of rank $p^r$ (which can be realized as the kernel of the $r$-fold Froebenius morphism $F^r:E\rightarrow E^{(r)}$).

This statement is made in some point in Katz and Mazur's book. The unicity is important as it allows us to deduce that $\operatorname{Ker}(F^{2r})=\operatorname{Ker}(p^r)$. However, I totally fail to see why it is true. Actually, I fail to relate this to the hypothesis that $E$ is supersingular, as the only definition of this that is given in the book is the fact that the formal completion $\widehat{E}$ of $E$ along its zero section has height $2$.

Could someone please provide a reference or an explanation for this result?

I thank you very much for your help.

EDIT:

I found a proof of this statement in the book "Fermat's Last Theorem: the Proof" by Takeshi Saito. Here, a supersingular elliptic curve is defined as an elliptic curve $E/S$ where $S$ is a scheme over $\mathbb{F}_p$ such that $E[p]=\operatorname{Ker}(F^2)$, where $F^2:E\rightarrow E^{(p^2)}$ is the Froebenius morphism (composed twice).

Given a supersingular elliptic curve $E$ over an algebraically closed field $k$ of characteristic $p$, we have that $E[p](k)=\left(\operatorname{ker}F^2\right)(k)=0$ (as it can be seen, for instance, by considering a description of $E$ by a Weierstraß equation). Hence, the abelian group $ E[p](k)$ is of order $1$.

Then, he writes down the following:

Let $G$ be a closed subgroup scheme of $E$ of order $p^e$. It is then a closed subgroup scheme of $E[p^e]$ of degree $p^e$, thus it is connected. Therefore if $\mathfrak{m}_0$ is the maximal ideam of the local ring $\mathcal{O}_{E,0}$, we must have $\mathbf{G=\operatorname{Spec}(\mathcal{O}_{E,0}/\mathfrak{m}_0^{p^e})=\operatorname{Ker}(F^e)}$.

I am having a bad time understand pretty much every step in this last proof, however... More precisely, I do not understand the parts written in bold letters. Could someone please explain these parts to me?

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  • $\begingroup$ An elliptic curve $E$ has an associated $p$-divisible group $E[p^{\infty}]$ which is basically just the system of finite flat group schemes $E[p^n]$ for all $n$. This $p$-divisible group has height $2$. Saying that an elliptic curve is supersingular is basically the same thing as saying that $E[p^{\infty}]$ is connected, which just means that all the group schemes $E[p^n]$ are connected. $\endgroup$ – user45878 Nov 25 '18 at 20:48

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