On Néron class groups of abelian varieties
type: Item
instance of: scholarly article
title: On Néron class groups of abelian varieties
zbMATH Open document ID: 1242.14021
DOI: 10.1515/CRELLE.2011.109
author: Cristian D. Gonzales-Aviles
publication date: 2012-04-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/0909.4803
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 11G35
Mathematics Subject Classification ID: 11G10
zbMATH DE Number: 6022594
zbMATH Keywords: abelian variety Néron model
zbMATH Keywords: Tate-Shafarevich group
MaRDI profile type: Publication
arXiv ID: 0909.4803
cites work: Products in sheaf-cohomology
cites work: Algebraic Groups Over Finite Fields
cites work: Duality theorems for Néron models
OpenAlex ID: W2963072832
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published in: Journal für die Reine und Angewandte Mathematik
review text: Let \(F\) be a global field, \(S\) a nonempty finite set of primes (containing all the infinite primes in the number field case) and \({\mathcal O}_{F,S}\) the corresponding ring of \(S\)-integers. Let \(A\) be an abelian variety over \(F\), and \(\mathcal A\) the Néron model of \(A\) over \(U=\text{Spec}\, {\mathcal O}_{F,S}\) with connected component \({\mathcal A}^\circ\). Write \(D^1(U, {\mathcal A})\) for the subgroup of elements of the étale cohomology group \(H^1(U, {\mathcal A})\) that are mapped to 0 by the restriction maps to completions \(H^1(U, {\mathcal A})\to H^1(F_v, {\mathcal A})\) for all \(v\in S\), and similarly for \({\mathcal A}^\circ\). Finally, denote by \(C_{A, F, S}\) (resp. \(C^1_{A, F, S}\)) the kernel (resp. cokernel) of the natural map \(D^1(U, {\mathcal A}^\circ)\to D^1(U, {\mathcal A})\).NEWLINENEWLINEThe main result of the paper constructs, under assumption of the finiteness of the Tate-Shafarevich group of \(A\), a perfect pairing of finite groups \(C_{A, F, S}\times C^1_{B, F, S}\to{\mathbb Q}/{\mathbb Z}\), where \(B\) is the dual abelian variety of \(A\). This is achieved by first constructing a perfect pairing \(D^1(U, {\mathcal A}^\circ)\times D^1(U, {\mathcal B})\to{\mathbb Q}/{\mathbb Z}\) following the method of [\textit{J. S. Milne}, Arithmetic duality theorems. 2nd ed. Charleston, SC: BookSurge, LLC (2006; Zbl 1127.14001)], section II.5 and [\textit{D. Harari} and \textit{T. Szamuely}, J. Reine Angew. Math. 578, 93--128 (2005; Zbl 1088.14012)], and then combining it with the usual Cassels-Tate duality for \(A\).NEWLINENEWLINEThe main interest of this result comes from the interpretation of the groups \(C_{A, F, S}\) and \(C^1_{B, F, S}\). The first group can be identified with the cokernel of the natural reduction map \(A(F)\to\bigoplus_{v\in U}\Phi_v(A)(k(v))\), where \(\Phi_v(A)\) is the group scheme of connected components of the special fibre of the Néron model of \(A_{F_v}\). Grothendieck constructed a non-degenerate p…
reviewed by: Tamás Szamuely
description: scientific article; zbMATH DE number 6022594