Fields of definition of abelian subvarieties
type: Item
instance of: scholarly article
title: Fields of definition of abelian subvarieties
author: Séverin Philip
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2022-10-25T00:00:00Z
full work available at URL: https://arxiv.org/abs/2010.12845
review text: Let \(A\) be an abelian variety over a field \(K\) of characteristic zero. In this paper, the author shows that, if no isotypic component of \(A_{\overline{K}}\) is simple, then there are infinitely many abelian subvarieties of \(A_{\overline{K}}\) whose field of definition is the one of the geometric endomorphisms of \(A\). The idea of the proof is to relate the set of abelian subvarieties to the rational points of a Grassmannian \(Gr(D)\) over a skew field \(D\). The Galois action of \(\mathrm{Gal}(\overline{K}/K)\) on subvarieties factors through the finite group \(G=\mathrm{Gal}(K_A/K)\), where \(K_A\) denotes the field of definition of the endomorphisms of \(A_{\overline{K}}\). The fixed points of a nontrivial element of \(G\) are given by the rational points of a proper closed subvariety of \(Gr(D)\). Since the finite union of these subvarieties cannot cover the whole \(Gr(D)\), one finds infinitely many subvarieties by density of the rational points of \(Gr(D)\).
reviewed by: Fumio Hazama
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 14K15
Mathematics Subject Classification ID: 14G99
zbMATH DE Number: 7607141
zbMATH Keywords: abelian varieties
zbMATH Keywords: fields of definition
MaRDI profile type: Publication
OpenAlex ID: W3094312093
cites work: Logarithmes des points rationnels des variétés abéliennes
cites work: Q5602064
cites work: Abelian varieties and maximal orders
cites work: Degree of definition of endomorphisms of an abelian variety
cites work: Q4256372
cites work: Q3708857
DOI: 10.5802/JTNB.1214
description: scientific article; zbMATH DE number 7607141