Let $k$ be a number field and denote by $H^i(k,-)$ the Galois cohomology functor $H^i(\mathrm{Gal}(\bar{k}/k),-)$. Let $X$ be a smooth geometrically integral curve over $k$. One can easily show that the map $H^1(k,\mathrm{Pic}^0(X_{\bar{k}})) \rightarrow H^1(k,\mathrm{Pic}(X_{\bar{k}}))$ is surjective. Indeed, applying the functor $H^1(k,-)$ to the exact sequence $$0 \rightarrow \mathrm{Pic}^0(X_{\bar{k}}) \rightarrow \mathrm{Pic}(X_{\bar{k}}) \rightarrow \mathrm{NS}(X_{\bar{k}}) \rightarrow 0,$$ we get the exact sequence $$H^1(k,\mathrm{Pic}^0(X_{\bar{k}})) \rightarrow H^1(k,\mathrm{Pic}(X_{\bar{k}})) \rightarrow H^1(k,\mathbb{Z}).$$ The last term is zero because $\mathbb{Z}$ has trivial Galois action and so any 1-cocycle $f \in H^1(k,\mathbb{Z})$ is simply an element of $\mathrm{Hom}(\mathrm{Gal}(\bar{k}/k),\mathbb{Z})$. But $\mathbb{Z}$ is torsion-free and thus all such maps are zero maps.

**Question 0.** Here I did not assume that $X$ is projective, but would all of the above still hold without this assumption? For example, I've never seen the Neron-Severi group of an affine curve discussed in any literature.

Now moving on, we have an exact sequence of Galois modules $$0 \rightarrow \mathrm{Pic}(X_{\bar{k}})_{\mathrm{tor}} \rightarrow \mathrm{Pic}(X_{\bar{k}}) \rightarrow \mathrm{Pic}(X_{\bar{k}})_{\mathrm{free}} \rightarrow 0$$ where $\mathrm{Pic}(X_{\bar{k}})_\mathrm{tor}$ denotes the maximal torsion subgroup of $\mathrm{Pic}(X_{\bar{k}})$ and $\mathrm{Pic}(X_{\bar{k}})_\mathrm{free} = \mathrm{Pic}(X_{\bar{k}})/\mathrm{Pic}(X_{\bar{k}})_\mathrm{tor}$, the maximal free quotient.

**Question 1.** In this case $\mathrm{Pic}(X_{\bar{k}})_\mathrm{free}$ certainly does not have trivial Galois action, so we cannot reduce 1-cocycles to group homomorphisms $\mathrm{Gal}(\bar{k}/k) \rightarrow \mathrm{Pic}(X_{\bar{k}})_\mathrm{free}$. Therefore how do we go about computing $H^1(k,\mathrm{Pic}(X_{\bar{k}})_\mathrm{free})$?

I have thought about first studying $H^1(k,\mathrm{Pic}(X_{\bar{k}}))$ using a wild idea as follows:

We apply the (étale) cohomology functor $H^i(X_{\bar{k}},-)$ to the Kummer sequence $$0 \rightarrow \mu_n \rightarrow \mathbb{G_m} \rightarrow \mathbb{G}_m \rightarrow 0$$ to obtain the long cohomology sequence $$0 \rightarrow \mu_n(\bar{k}) \rightarrow \bar{k}^* \rightarrow \bar{k}^* \rightarrow H^1(X_\bar{k},\mu_n) \rightarrow \mathrm{Pic}(X_{\bar{k}}) \rightarrow \mathrm{Pic}(X_{\bar{k}}) \rightarrow H^2(X_{\bar{k}},\mu_n) \rightarrow H^2(X_{\bar{k}},\mathbb{G}_m)=0.$$

Then we apply $H^1(k,-)$ to $$H^1(X_\bar{k},\mu_n) \rightarrow \mathrm{Pic}(X_{\bar{k}}) \rightarrow \mathrm{Pic}(X_{\bar{k}}) \rightarrow H^2(X_{\bar{k}},\mu_n)$$ and study the result. But I'm not sure if there are any spectral sequences we can use, or if this approach is even feasible.