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 $p$.



Pic^0 and H^0(K,Pic^0)
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 $r$.


