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

Let $K$ be a local field with algebraically closed residue field and $X_K$ a torsor under an elliptic curve $J_K$ over $K$. Let $X$ be a proper minimal regular model of $X_K$ over the ring of integers of $K$ and $J$ the identity component of the N'eron model of $J_K$. We study the canonical morphism $q\text{\hspace{0.17em}⁣}:{\mathrm{P}\mathrm{i}\mathrm{c}}_{X\mathrm{/}S}^0\to J$ which extends the biduality 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.