Let $C$ be a (smooth,proper) curve over a field $k$. Let $\operatorname{Div}_C(k)$ be the free abelian group generated by the closed points of $C/k$ and $k(C)^\times$ be the group of rational functions. We have an injective map: $$i \colon \operatorname{Div}_C(k)/k(C)^\times \to \left(\operatorname{Div}_C(\overline{k})/\overline{k}(C)^\times\right)^{\operatorname{Gal}(\overline{k}/k)}$$

When is this map surjective? I think this is definitely true if the Brauer group $H^2(k,\mathbb G_m) = 0$, as it follows from the four-term exact sequence $$0 \to \overline{k}^\times \to \overline{k}(C)^\times \to \operatorname{Div}_C(\overline{k}) \to \operatorname{Pic}_C(\overline{k})\to 0.$$

**Q1.** Are there other situations where this is true? Also, what are some examples where the map is not injective?

**Q2.** In particular, I saw a comment somewhere that if $k$ is a number field and $C$ has a point locally for every completion $k_v$, then the map is indeed surjective. Is this true and if so, how does one see it?

**Q3.** Is the map surjective if $C$ has a rational point?