Birch–Swinnerton-Dyer conjecture for the Picard variety

Let $X$ be a smooth projective geometrically connected variety over a finitely generated field $K$, with a smooth projective model $f:{\mathcal X}\to{\mathcal Y}$ as above, and let $d=\dim X$. Write $\Phi_i(s)$ for the Euler product formed from the degree-$i$ étale cohomology of the fibers, and let $\operatorname{Pic}^0_{X/K}$ be the identity component of the Picard scheme. Birch–Swinnerton-Dyer conjecture. The rank of $\operatorname{Pic}^0_{X/K}(K)$ equals the order of the zero of $\Phi_1(s)$ at $s=\dim({\mathcal Y})$, and, by duality, the order of the zero of $\Phi_{2d-1}(s)$ at $s=\dim({\mathcal X})-1$. This is the function-field Birch–Swinnerton-Dyer-type prediction for the Picard variety and is not known in general.