Let $K$ be a field of characteristic zero and $X\_K$ a smooth projective curve on $K$. Denote by $\bar{K}$ the algebraic closure of $K$ and $X\_{\bar{K}}$ the base change of $X\_K$ to $\bar{K}$. Under the natural morphism from $X\_{\bar{K}} \to X\_K$, we have an induced morphism $\mbox{Pic}^0(X\_K) \to \mbox{Pic}^0(X\_{\bar{K}})$. Is this morphism surjective? If not true in general, is there any known condition on $K$ under which this holds true?



Jacobian of a curve and field extension
First of all, the image of your homomorphism is invariant under the Galois group $G:=\mathrm{Gal}(\bar{K}/K)$. So the right question is to ask whether the induced homomorphism $\mathrm{Pic}(X\_{K})\rightarrow \mathrm{Pic}(X\_{\bar{K}})^G$ is surjective.


In our answer to [this question](https://mathoverflow.net/questions/180185/injectivity-under-flat-base-change-of-the-picard-group-on-smooth-projective-curv/180189?noredirect=1#comment451772_180189) of yours, we (Daniel Loughran and me) give elements for the answer: the Hochschild-Serre spectral sequence gives an exact sequence
$$0\rightarrow \mathrm{Pic}(X\_K)\rightarrow \mathrm{Pic}(X\_{\bar{K}})^G\rightarrow H^2(G, \bar{K}^\*)\rightarrow H^2(X\_{K},\mathbb{G}\_m)$$
$H^2(G, \bar{K}^\*)$ is the *Brauer group* $\mathrm{Br(K)}$ of $K$. So a sufficient condition is that $\mathrm{Br(K)}=0$. This happens for instance if $K$ is the field of rational functions of a curve over $\mathbb{C}$, or the maximal unramified extension of $\mathbb{Q}\_p$, or an algebraic extension of $\mathbb{Q}$ containing all roots of unity -- see Serre, *Local fields*, ch. X.


