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?
> 
> 
> 



Kernel of powers of Frobenius on supersingular elliptic curves
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


<http://projecteuclid.org/euclid.dmj/1077296358>


