Etude locale des torseurs sous une courbe elliptique

This article concerns the geometry of torsors under an elliptic curve. Let ${\text{OO}}_K$ 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 ${\text{OO}}_K$, and $S=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}({\text{OO}}_K)$. 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 $X_K\mathrm{/}K$ a torsor of order $d$ under $J_K$, let $X$ be the $S$-minimal regular proper model. Then there is an invertible id?al $\mathcal{I}\subset {\text{OO}}_K$ such that ${\mathcal{I}}^d=\pi {\text{OO}}_X\subset {\text{OO}}_X$. Moreover, there exists a canonical morphism $q:{\text{Pic}}_{X\mathrm{/}S}^{\circ }\to J$ which induces a surjective map $q(S):{\text{Pic}}^{\circ }(X)\to J(S)$. The purpose of the article is to prove this last morphism $q(S)$ is compatible with respect to the $\mathcal{I}$-adic filtration on ${\text{Pic}}^{\circ }(X)$, 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})