# 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?

## 1 Answer

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

• 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 $p^r$. – Lubin Nov 29 '12 at 16:37