Let $H/k$ be a genus $2$ curve. Consider the function field of $H$ given by $k(H)$. Take the base extension of $H$ to $k(H)$, namely $\hat H:=H\otimes_k k(H)$. Consider the Jacobian $J$ of the curve $H$.

I want to know if $\text{Pic}^0(\hat H)\cong J(k(H))$

This is because I want to work with divisor classes having $k(H)$-rational points in their support.

Is well known that $\text{Pic}^0(H)\cong J(k)$ when $k$ is perfect. When is not perfect, when do I have this isomorphism?

In this paper it seems to be defined for Jacobians of genus $2$ curves according to Propositions 3.1,3.2 and 3.3

http://www-math.mit.edu/~poonen/papers/descent.pdf

But I am not sure, can anybody help me to understand?.