Grothendieck's pairing for Jacobians and base change
type: Item
instance of: scholarly article
title: Grothendieck's pairing for Jacobians and base change
zbMATH Open document ID: 1198.14042
published in: Journal of Number Theory
publication date: 2008-06-09T00:00:00Z
review text: In the following, let \(K\) be the field of fractions of a complete discrete valuation ring with separably closed residue field \(k\) with \(\mathrm{char}(k)=p>0\), and let \(A\) be an abelian variety over \(K\). We denote \(A'=\mathrm{Pic}^0_A\). By Grothendieck's theory, there exists a natural pairing \(\langle \cdot, \cdot \rangle_A: \Phi_{A,K} \times \Phi_{A',K} \rightarrow \mathbb{Q}/\mathbb{Z}\) where \(\Phi_{A,K}\) and \(\Phi_{A',K}\) denote the component groups of the Néron models of \(A\) and \(A'\), respectively. Due to work of Bégueri, Bertapelle, Bosch, McCallum, Lorenzini and Werner, the pairing is known to be perfect in many cases. The pairing is conjectured to be perfect if \(k\) is perfect. There are counter examples which show that the pairing is not perfect in general. The main result of this article extends an earlier result of Bosch and Lorenzini. Let \(C\) be a smooth geometrically connected proper curve of genus \(\geq 1\) over \(K\). We denote the Jacobian of \(C\) by \(J=J(C)\). Let \(\varphi:J \rightarrow J'\) be the canonical principal polarization of \(J\). It is convenient to identify \(J\) with \(J'\) via the morphism \(- \varphi\). This identification has a continuation to Néron models, and hence induces an isomorphism \(\Phi_{J,K} \rightarrow \Phi_{J',K}\). As a consequence, Grothendieck's pairing induces a symmetric pairing \(\langle \cdot, \cdot \rangle_{J,K}:\Phi_{J,K} \times \Phi_{J,K} \rightarrow \mathbb{Q}/\mathbb{Z}\) on the component group \(\Phi_{J,K}\) of the Néron model of \(J(C)\). The main result of Lorenzini reads as follows. Theorem. Assume that \(C\) has a proper flat regular integral model \(\mathcal{C}\) such that the irreducible components of the reduction \(\mathcal{C}_k\) are smooth, \(\mathcal{C}_k^{\mathrm{red}}\) has normal crossings and the multiplicities of the irreducible components have no proper common factor. If there exist two distinct irreducible components of \(\mathcal{C}_k\) with strictly positive int…
reviewed by: Robert Carls
Mathematics Subject Classification ID: 14K15
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 14H25
zbMATH DE Number: 5285724
zbMATH Keywords: Grothendieck pairing
zbMATH Keywords: component groups
zbMATH Keywords: duality
author: Dino Lorenzini
MaRDI profile type: Publication
full work available at URL: j.jnt.2007.12.013
OpenAlex ID: W1969065488
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DOI: 10.1016/J.JNT.2007.12.013
description: scientific article; zbMATH DE number 5285724