On torsors under elliptic curves and Serre's pro-algebraic structures

Let O_K be a complete discrete valuation ring with algebraically closed residue field of positive characteristic and field of fractions K. Let X_K be a torsor under an elliptic curve A_K over K, X the proper minimal regular model of X_K over S := Spec(O_K), and J the identity component of the Neron model of Pic^0_{X_K/K}. We study the canonical morphism q : Pic^0_{X/S}\to J which extends the natural isomorphism on generic fibres. We show that q is pro-algebraic in nature with a construction that recalls Serre's work on local class field theory. Furthermore, we interpret our results in relation to Shafarevich's duality theory for torsors under abelian varieties.