Principal homogeneous space (reviewed)

A principal homogeneous space (or torsor) of an abelian variety $A$ over a field $k$ is a variety $X$ equipped with an $A$-action such that ($X_{\overline{k}}$ with the induced $A_{\overline{k}}$-action) is isomorphic to ($A_{\overline{k}}$ with $A_{\overline{k}}$-action given by translation).

The set of isomorphism classes of principal homogeneous spaces of $A$ is in bijection with the Galois cohomology group $H^1(k,A) = H^1(G_k,A(\overline{k}))$, where $\overline{k}$ is a separable closure of $k$, and $G_k$ is the absolute Galois group $\Gal(\overline{k}/k)$, so it has the structure of an abelian group.

For a principal homogeneous space $X$, the following are equivalent:

1. $X$ has a rational point;
2. there is an $A$-equivariant isomorphism $X \to A$;
3. the class of $X$ in $H^1(k,A)$ is $0$.