On torsors under abelian varieties

Let A be an abelian variety over a local field K of mixed characteristic and with algebraically closed residue field. We provide a geometric construction (via the relative Picard functor) of the Shafarevich duality between the group of isomorphism classes of torsors under A and the "fundamental group" of the N\'eron model of the dual abelian variety A'. An analogous construction works over fields of positive characteristic p providing a duality on the prime-to-p parts.