Component groups of abelian varieties and Grothendieck's duality conjecture
type: Item
instance of: scholarly article
title: Component groups of abelian varieties and Grothendieck's duality conjecture
zbMATH Open document ID: 0919.14026
author: Siegfried Bosch
publication date: 1997-12-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75263
review text: \(K\) is the field of fractions of a complete discrete valuation ring \(R\) and \(A_K\) an abelian variety over \(K\). Write \(A'_K\) for its dual, \(A\) and \(A'\) for the Néron models of \(A_K\) and \(A'_K\) and \(\phi _A\), \(\phi _{A'}\) for the corresponding component groups. Grothendieck has constructed a pairing \(\phi _A\times \phi _{A'}\rightarrow {\mathbb Q}/{\mathbb Z}\) and conjectured that this pairing is perfect. This conjecture is already known in some cases, e.g., semi-stable reduction. The paper proves the conjecture in the case that the residue field of \(R\) is perfect and \(A\) has potentially multiplicative reduction. There is a hint for a possible proof in a more general situation. The highly technical proof uses arithmetic theory of abelian varieties, sheaves for the étale or smooth topology and a rigid analytic description of the groups \(\phi _A\), \(\phi _{A'}\) obtained from rigid uniformization of the abelian variety \(A_K\).
Mathematics Subject Classification ID: 14K15
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 13F30
zbMATH DE Number: 1092694
zbMATH Keywords: abelian varieties
zbMATH Keywords: Néron model
zbMATH Keywords: component group
zbMATH Keywords: Grothendieck's pairing
zbMATH Keywords: complete discrete valuation ring
zbMATH Keywords: rigid uniformization
zbMATH Keywords: dual variety
MaRDI profile type: Publication
OpenAlex ID: W1999581032
Wikidata QID: Q123279367
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DOI: 10.5802/AIF.1599
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1092694