Definition of principal homogeneous space for elliptic curves

Question

Definition of principal homogeneous space for elliptic curves, for example, in $p322$ of 'The arithmetic of elliptic curves'.

Let $E/K$ be an elliptic curve defined over a field $K$.

Principal homogeneous space for $E/K$ is defined as a 'curve'$C$ which $E/K$ acts simply transitively as algebraic group.

In this definition, is $C$ only projective curve ? What is wrong with if we admit $C$ to be affine curve ?

Answer 1

Let $$ \alpha : E\times C \longrightarrow C $$ be the assumed simple transitive action. Consider the map on points over an algebraic closure $\overline{K}$ of $K$, i.e., $$ \alpha : E(\overline{K})\times C(\overline{K}) \longrightarrow C(\overline{K}). $$ Fix a base point $Q_0\in C(\overline{K})$. Then given a point $Q\in C(\overline{K})$, the simple transitivity of the action says that there is a unique point $\beta(Q)\in E(\overline{K})$ that satisfies $$ \alpha\bigl(\beta(Q),Q_0\bigr)=Q. $$ The map $\beta:C\to E$ is an isomorphism, it's inverse is the map $$ E\longrightarrow C,\quad P \longmapsto \alpha(P,Q_0). $$ So the fact that $E$ is a smooth projective curve implies that $C$ is also a smooth projective currve.