On the p-rank of an abelian variety and its endomorphism algebra.

• Volume: 42, Issue: 1, page 119-130
• ISSN: 0214-1493

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Abstract

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Let A be an abelian variety defined over a finite field. In this paper, we discuss the relationship between the p-rank of A, r(A), and its endomorphism algebra, End0(A). As is well known, End0(A) determines r(A) when A is an elliptic curve. We show that, under some conditions, the value of r(A) and the structure of End0(A) are related. For example, if the center of End0(A) is an abelian extension of Q, then A is ordinary if and only if End0(A) is a commutative field. Nevertheless, we give an example in dimension 3 which shows that the algebra End0(A) does not determine the value r(A).

How to cite

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González, Josep. "On the p-rank of an abelian variety and its endomorphism algebra.." Publicacions Matemàtiques 42.1 (1998): 119-130. <http://eudml.org/doc/41333>.

@article{González1998,
abstract = {Let A be an abelian variety defined over a finite field. In this paper, we discuss the relationship between the p-rank of A, r(A), and its endomorphism algebra, End0(A). As is well known, End0(A) determines r(A) when A is an elliptic curve. We show that, under some conditions, the value of r(A) and the structure of End0(A) are related. For example, if the center of End0(A) is an abelian extension of Q, then A is ordinary if and only if End0(A) is a commutative field. Nevertheless, we give an example in dimension 3 which shows that the algebra End0(A) does not determine the value r(A).},
author = {González, Josep},
journal = {Publicacions Matemàtiques},
keywords = {Endomorfismos; Variedades abelianas; Rango; Funciones elípticas; abelian varieties; endomorphism algebra; characteristic },
language = {eng},
number = {1},
pages = {119-130},
title = {On the p-rank of an abelian variety and its endomorphism algebra.},
url = {http://eudml.org/doc/41333},
volume = {42},
year = {1998},
}

TY - JOUR
AU - González, Josep
TI - On the p-rank of an abelian variety and its endomorphism algebra.
JO - Publicacions Matemàtiques
PY - 1998
VL - 42
IS - 1
SP - 119
EP - 130
AB - Let A be an abelian variety defined over a finite field. In this paper, we discuss the relationship between the p-rank of A, r(A), and its endomorphism algebra, End0(A). As is well known, End0(A) determines r(A) when A is an elliptic curve. We show that, under some conditions, the value of r(A) and the structure of End0(A) are related. For example, if the center of End0(A) is an abelian extension of Q, then A is ordinary if and only if End0(A) is a commutative field. Nevertheless, we give an example in dimension 3 which shows that the algebra End0(A) does not determine the value r(A).
LA - eng
KW - Endomorfismos; Variedades abelianas; Rango; Funciones elípticas; abelian varieties; endomorphism algebra; characteristic
UR - http://eudml.org/doc/41333
ER -

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