Revistes Catalanes amb Accés Obert (RACO)

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

J. González


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, $\operatorname{End}^0(A)$. As is well known, $\operatorname{End}^0(A)$ determines $r(A)$ when $A$ is an elliptic curve. We show that, under some conditions, the value of $r(A)$ and the structure of $\operatorname{End}^0(A)$ are related. For example, if the center of $\operatorname{End}^0(A)$ is an abelian extension of $\Bbb Q$, then $A$ is ordinary if and only if $\operatorname{End}^0(A)$ is a commutative field. Nevertheless, we give an example in dimension 3 which shows that the algebra $\operatorname{End}^0(A)$ does not determine the value $r(A)$.

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