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.



1st-flat cohomology group for elliptic curves
$\mathbb G\_a$ 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.


