Pic^0 and H^0(K,Pic^0) Let $K$ be a field and $C$ a smooth and projective curve over $K$. Then the kernel $Pic^0(C)$ of the degree map injects into $H^0(K,Pic^0_C)$, where $Pic_C^0$ is the connected component of the Picard variety. 
I am wondering if there are examples where this is not an isomorphism for $K$ a global field. I am especially interested if there are elements of order prime to $p$ in the cokernel if $K$ is a global field of characteristic $p$.
 A: By the long exact sequence of low degree terms for the Leray spectral sequence computing $H^r_{\text{et}}(C,\mathbb{G}_m)$ via $H^p_{\text{et}}(\text{Spec}\ K,R^q f_*\mathbb{G}_m)$, the cokernel of the map $$\text{Pic}(f):\text{Pic}(C) \to H^0_{\text{et}}(\text{Spec}\ K,\text{Pic}_{C/K})$$  equals the kernel of the pullback map on Brauer groups, $$\text{Br}(f):\text{Br}(\text{Spec}\ K) \to \text{Br}(C).$$  Of course $\text{Coker}(\text{Pic}(f))$ surjects onto the cokernel of the degree map, $$\text{deg}(f):\text{Pic}(C) \to H^0_{\text{et}}(\text{Spec}\ K,\text{Pic}_{C/K}/\text{Pic}^0_{C/K}) =\mathbb{Z}.$$  By the Snake Lemma, the kernel of the map $$\text{Coker}(\text{Pic}(f))\to \text{Coker}(\text{deg}(f))$$ equals the cokernel of your homomorphism, $$\text{Pic}^0(f):\text{Pic}^0(C)\to H^0_{\text{et}}(\text{Spec}\ K,\text{Pic}^0_{C/K}).$$  The cokernel of $\text{deg}(f)$ is cyclic.  Thus, if $\text{Ker}(\text{Br}(f))$ is not cyclic, then $\text{Coker}(\text{Pic}^0(f))$ is nonzero.  More precisely, for a prime $\ell$, if $\text{Ker}(\text{Br}(f))\otimes \mathbb{Z}/\ell \mathbb{Z}$ has rank $r+1$ as a vector space over  $\mathbb{Z}/\ell\mathbb{Z},$ then $\text{Ker}(\text{Br}(f))\otimes \mathbb{Z}/\ell\mathbb{Z}$ has rank at least $r$, possibly higher rank because of $\text{Tor}_1(\mathbb{Z}/\ell\mathbb{Z},\text{Coker}(\text{deg}(f))).$
The Brauer group of a global field is described by class field theory.  In particular, for any prime $\ell,$ the rank of the $\ell$-torsion subgroup $\text{Br}(K)[\ell]$ is infinite (infinitude of primes).  Let $\alpha_0,\dots,\alpha_r$ be $\mathbb{Z}/\ell\mathbb{Z}$-linearly independent classes in $\text{Br}(K)[\ell].$  Let $P_0,\dots,P_r$ be associated Severi-Brauer $K$-schemes.  Now let $C$ be a general complete intersection curve in the product variety $$P:=P_0\times_{\text{Spec}\ K}\dots \times_{\text{Spec}\ K} P_r.$$  The kernel of the pullback map $\text{Br}(K)\to \text{Br}(P)$ contains the classes $\alpha_0,\dots,\alpha_r.$  Thus, the $\mathbb{Z}/\ell\mathbb{Z}$-vector space $\text{Coker}(\text{Pic}^0(f))\otimes \mathbb{Z}/\ell\mathbb{Z}$ has rank at least $r$.
