I know that for an abelian variety $A$ over an algebraically closed field $k=\overline k$ and an ample line bundle $L$ on $A$, one can define an isogeny $\phi_L \colon A \rightarrow \mathrm{Pic}^0(A)$ by $x \mapsto \tau_x^* L \otimes L^{-1}$, where $\tau_x$ denotes the translation by $x$.
First, I assume that this goes through verbatim over a non algebraically closed field, assuming that $L$ is defined over $k$. Is it right, or am I missing something?
Second, does the above construction work if $A$ is just a principal homogeneous space over an abelian variety? That is, $k$ is not algebraically closed, $A$ has no $k$-rational points, but $A(\overline{k})$ is an abelian variety. In this situation, do I still get a finite dominant morphism $\phi \colon A \rightarrow \mathrm{Pic}^0(A)$? My guess is that the same construction should still work, again provided that $L$ is a polarization defined over $k$.
As I am not very comfortable with non-closed fields, I would really appreciate some feedback on what I sketched above.