Abelian subvarieties
type: Item
instance of: scholarly article
title: Abelian subvarieties
zbMATH Open document ID: 0861.14037
author: Frans Oort
author: Yu. G. Zarhin
author: H. W. jun. Lenstra
published in: Journal of Algebra
publication date: 1996-06-05T00:00:00Z
review text: It is shown that if \(X\) is an abelian variety defined over a field \(K\), the number of orbits of the set of abelian subvarieties of \(X\) under the natural action of the group of automorphisms of \(X\) is finite. It follows that there are, up to isomorphism, only finitely many abelian subvarieties of \(X\). The argument is brief, elegant, and elementary: One proves that given a finite-dimensional semi-simple \(\mathbb{Q}\)-algebra \(F\), a finitely generated right \(F\)-module \(M\) and a lattice \(L\) in \(M\), the number of orbits of the set of \(F\)-submodules of \(M\) under the action of the group \(G\) of automorphisms \(\sigma\) of \(M\) such that \(\sigma L=L\), is finite. This is applied to \(M=F= \text{End} X \otimes\mathbb{Q}\), \(L= \text{End} X\) and \(G= \Aut X\).
Mathematics Subject Classification ID: 14K05
Mathematics Subject Classification ID: 14E07
Mathematics Subject Classification ID: 14K10
zbMATH DE Number: 869790
zbMATH Keywords: finiteness of number of abelian subvarieties of abelian variety
zbMATH Keywords: group of automorphisms
MaRDI profile type: Publication
full work available at URL: jabr.1996.0079
OpenAlex ID: W4210456254
DOI: 10.1006/JABR.1996.0079
description: scientific article; zbMATH DE number 869790