  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
$q\colon \mathrm{Pic}^{0}_{X/S}\to J$ 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.
