On torsion line bundles and torsion points on Abelian varieties
type: Item
instance of: scholarly article
title: On torsion line bundles and torsion points on Abelian varieties
zbMATH Open document ID: 1029.11020
DOI: 10.1215/S0012-7094-98-09110-4
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: In this paper, the author studies torsion line bundles on Abelian schemes and their restriction to the subschemes of torsion points. This is motivated by problems on the Galois module structure of rings of integers obtained by adjoining values of Abelian functions at torsion points. By definition, an Abelian scheme \(A@> f >>S\) of relative dimension \(g\) is a group scheme that is proper and smooth with geometrically connected fibers of dimension \(g\). Denote by \(0:S\rightarrow A\) the identity section. For any line bundle \({\mathcal L}\) on \(A\), a rigidification of \({\mathcal L}\) is an isomorphism \({\mathcal O}_S @>\sim>> 0^*{\mathcal L}\). For each nonzero integer \(m\), we can consider the (closed) subgroup scheme \(A_m\rightarrow S\) of \(m\)-torsion points of \(A\). The group scheme \(A_m\) is finite and locally free of rank \(m^{2g}\) over \(S\). The first aim of this paper is to show the following theorems. Theorem A. Assume \(g=1\) and \(\gcd(6,m)=1\), and let \({\mathcal L}\) be a torsion line bundle on \(A\) with a rigidification. Then the restriction of \({\mathcal L}\) to the subscheme of \(m\)-torsion points \(A_m\) is a trivial line bundle. Theorem B. Assume \(g=1\), and denote by \({\mathcal P}\) the Poincaré line bundle on \(A\times_S A\). Assume that one of the two nonzero integers \(n\) and \(m\) is coprime to 6. Then the restriction of \({\mathcal P}\) to the subscheme \(A_n\times_S A_m\) is a trivial line bundle. The proofs actually give trivializations for these restrictions. When \(g\geq 2\), the situation is dramatically different. He shows the following. Theorem C. Assume that \(g\geq 2\). For any two distinct prime numbers \(r\), \(l\), there is a smooth affine curve \(D\) over a finite field of characteristic \(r\), an Abelian scheme \(A\rightarrow D\) of relative dimension \(g\), and an \(l\)-torsion line bundle \({\mathcal L}\) on \(A\) with a rigidification, such that the restriction of \({\mathcal L}\) to \(A_l\) is not trivial…
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 14K15
zbMATH DE Number: 1425081
zbMATH Keywords: Abelian schemes
zbMATH Keywords: subschemes of torsion points
zbMATH Keywords: Galois module structure of rings of integers
author: Georgios Pappas
MaRDI profile type: Publication
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full work available at URL: s0012-7094-98-09110-4
OpenAlex ID: W1535308072
description: scientific article; zbMATH DE number 1425081