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, End 0 (A). As is well known, End 0 (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 End 0 (A) are related. For example, if the center of End 0 (A) is an abelian extension of Q, then A is ordinary if and only if End 0 (A) is a commutative field. Nevertheless, we

more »

... ive an example in dimension 3 which shows that the algebra End 0 (A) does not determine the value r(A).