Etude locale des torseurs sous une courbe elliptique

2010/05/04 by Jilong Tong, Tong, Jilong
Mathematics · #14H99 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14H99

paper · pdf · doi:10.48550/arxiv.1005.0462

arxiv created 2010/05/04 · arxiv updated 2010/05/05

Abstract

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 π be a generator of the maximal ideal of \OO_K, and S=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 I⊂ \OO_K such that I^d=π\OO_X⊂ \OO_X. Moreover, there exists a canonical morphism q:\Pic^∘_X/S→ J which induces a surjective map q(S):\Pic^∘(X)→ J(S). The purpose of the article is to prove this last morphism q(S) is compatible with respect to the I-adic filtration on \Pic^∘(X), and the π-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 (\citeSerre)

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