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).
@article{urn:eudml:doc:41333,
title = {On the p-rank of an abelian variety and its endomorphism algebra.},
journal = {Publicacions Matem\`atiques},
volume = {42},
year = {1998},
pages = {119-130},
mrnumber = {MR1628150},
zbl = {0941.14015},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41333}
}
González, Josep. On the p-rank of an abelian variety and its endomorphism algebra.. Publicacions Matemàtiques, Tome 42 (1998) pp. 119-130. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41333/