# Kernel of powers of Frobenius on supersingular elliptic curves

I am trying to understand some things related to elliptic curves and finite flat group schemes but I am a little bit confused.

Let $$A$$ be a supersingular elliptic curve over an algebraically closed field $$K$$ of characteristic $$p$$. Let $$F: A \rightarrow A^{(p)}$$ be the Frobenius isogeny. Then $$\ker F$$ is as a finite flat group scheme over $$K$$ isomorphic to $$\alpha_p$$, the kernel of the Frobenius on $$\mathbb{G}_a$$.

Now lets look at $$F^2: A \rightarrow A^{(p^2)}$$.

Is $$\ker F^2: A \rightarrow A^{(p^2)}$$ isomorphic (as a $$K$$-group scheme) to any one of the following group schemes:

$$\ker F^2:\mathbb{G}_a \rightarrow \mathbb{G}_a$$

$$\ker F : W_2 \rightarrow W_2$$ where $$W_2$$ is the group scheme of truncated Witt vectors

$$A[p]$$

$$\alpha_p \oplus \alpha_p$$

If $$\ker F^2: A \rightarrow A^{(p^2)}$$ is not isomorphic to one of them, I have an other question:

Is $$\ker F^2: A \rightarrow A^{(p^2)}$$ an extension of $$\alpha_p$$ over $$\alpha_p$$ and why not?

And a last one:

Is $$\ker F^2: A \rightarrow A^{(p^2)}$$ isomorphic to an other well-known group scheme?

The kernel of $F^2$ is the same as the kernel of $[p]$, once $A^{(p^2)}$ and $A$ are identified, and is a non-trivial extension of $\alpha_p$ by $\alpha_p$, whose class can be described in terms of the supersingular modular form $B$. See Ulmer, p-descent in characteristic p. Duke Math. J. 62 (1991), 237–265, section 4, especially Prop 4.1
• The old-fashioned way I see this, and Felipe can slap me down if I’m wrong, is that the transcendence-degree-one field of functions on $A$ has only one purely inseparable subfield of each possible degree $p^r$. Nov 29 '12 at 16:37