Etude locale des torseurs sous une courbe elliptique

This article concerns the geometry of torsors under an elliptic curve. Let $\OO_K$ be a complete discrete valuation ring with algebraically closed residue field and function field $K$. Let $\pi$ be a generator of the maximal ideal of $\OO_K$, and $S=\mathrm{Spec}(\OO_K)$. Suppose that we are given $J_K$ an elliptic curve over $K$, with $J$ the connected component of the $S$-N?ron model of $J_K$. Given $X_K/K$ a torsor of order $d$ under $J_K$, let $X$ be the $S$-minimal regular proper model. Then there is an invertible id?al $\mathcal{I}\subset \OO_K$ such that $\mathcal{I}^{d}=\pi\OO_X\subset \OO_X$. Moreover, there exists a canonical morphism $q:\Pic^{\circ}_{X/S}\rightarrow J$ which induces a surjective map $q(S):\Pic^{\circ}(X)\rightarrow J(S)$. The purpose of the article is to prove this last morphism $q(S)$ is compatible with respect to the $\mathcal{I}$-adic filtration on $\Pic^{\circ}(X)$, and the $\pi$-adic filtration on $J(S)$. As a byproduct, we obtain {\textquotedblleft Herbrand functions\textquotedblright}, similar to those Serre used in his description of local class fields (\cite{Serre})