On the \(p\)-rank of an abelian variety and its endomorphism algebra
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.
scientific article; zbMATH DE number 1247845
scientific article