Let $E$ be an elliptic curve over an algebraically closed field $k$ of characteristic $p$. Is there any nice computation for the group $H^1(E,\alpha_p)$ and $H^1(E,\mathbb{G}_a)$? The cohomology is taken in the flat topology.
When these groups are trivial? and is there any way to describe them? I am not asking for the description of $H^1$ using flat torsors on $E$, I looking for a description using which I could (at least) determine the groups are trivial or not.
$\mathbb G_a$ is a smooth group scheme, so the flat cohomology is the same as the etale cohomology. It is also a quasicoherent sheaf, so the etale cohomology is the same as the Zariski cohomology, which is a $1$-dimensional vector space over $k$.
For $\alpha_p$, you can use the exact sequence $0 \to \alpha_p \to \mathbb G_a \to \mathbb G_a \to 0$ and take cohomology:
$H^0(E,\mathbb G_a)\to H^0(E,\mathbb G_a) \to H^1(E,\alpha_p) \to H^1(E,\mathbb G_a)\to H^1(E,\mathbb G_a)$
The map $H^0(E,\mathbb G_a)\to H^0(E,\mathbb G_a)$ given by taking $p$th powers is surjective since $k$ is algebraically closed. If it is only separably closed and not perfect than this is false. The map $H^1(E,\mathbb G_a)\to H^1(E,\mathbb G_a)$ given by taking $p$th powers is injective because it's a field, which has no nilpotent elements. So $H^1(E,\alpha_p)$ is trivial.
