On the \(p\)-rank of an abelian variety and its endomorphism algebra
type: Item
instance of: scholarly article
title: On the \(p\)-rank of an abelian variety and its endomorphism algebra
zbMATH Open document ID: 0941.14015
DOI: 10.5565/PUBLMAT_42198_05
author: Josep González
published in: Publicacions Matemàtiques
publication date: 2000-07-23T00:00:00Z
full work available at URL: https://eudml.org/doc/41333
review text: Let \(E\) be an elliptic curve defined over an algebraically closed field \(k\) of characteristic \(p>0\). It is well known that \(E\) is an ordinary curve if, and only if, \(\text{End}^0(E):=\text{End}(E)\otimes_{\mathbb Z}{\mathbb Q}\) is a commutative field and that \(E\) is supersingular if, and only if, \(\text{End}^0(E)\) is a quaternion algebra over \({\mathbb Q}\). This paper summarizes some relations between the \(p\)-rank, \(r(A)\), of the abelian variety \(A\), defined over a finite field \({\mathbb F}_q\), and the structure of its algebra of endomorphisms \(\text{End}^0(A)\). In particular, the author proves (cf. theorem 3.7) that for an absolutely simple abelian variety of dimension \(g=2\), the algebra \(\text{End}^0(A)\) determines \(r(A)\) and actually it determines \(A\), up to isogeny, if \(p\) does not split completely in \(\text{End}^0(A)\). The last section of the paper is devoted to give an example of two absolutely simple abelian varieties of dimension \(3\) which have isomorphic endomorphism algebras but different \(p\)-ranks.
Mathematics Subject Classification ID: 14K15
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 14G15
zbMATH DE Number: 1247845
zbMATH Keywords: abelian varieties
zbMATH Keywords: endomorphism algebra
zbMATH Keywords: characteristic \(p\)
reviewed by: Maurizio Candilera
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1247845