How does one define the Torelli map $M_g \to A_g$ of moduli **stacks**? On geometric points a curve maps to its principally polarized Jacobian.

So what I am asking is: if I have a curve $C$ over a non-algebraically closed field $k$ such that $C(k)$ is empty, is the Jacobian of C still principally polarized? After base change to $\bar{k}$ one has a theta divisor; does it descend? Also, is the relative Jacobian of a family of curves principally polarized?

The thing I am confused about is that the theta divisor naturally lives on $Pic^{g-1}$ as the image of the map from the symmetric power $C^{g-1}$; this is a torsor under $Pic_0$, but not itself an abelian variety.

Also, the classical Torelli theorem says that this map is an injection on field valued points. Is this actually a locally closed immersion of stacks?