The structure of the group of rational points of an abelian variety over a finite field

Let A be a simple abelian variety of dimension g defined over a finite field F-q with Frobenius endomorphism pi. This paper describes the structure of the group of rational points A(F-qn), for all n >= 1, as a module over the ring R of endomorphisms which are defined over F-q, under certain technical conditions. If [Q(pi) : = 2g and R is a Gorenstein ring, then A(F-qn) congruent to R/R(pi(n) - 1). This includes the case when A is ordinary and has maximal real multiplication. Otherwise, if Z is the center of R and (r(n)- 1)Z is the product of invertible prime ideals in Z, then A(F-qn)(d) congruent to R/R(pi(n)-1) where d = 2g/[Q(pi) : Q]. Finally, we deduce the structure of A((F) over bar (q)) as a module over R under similar conditions. These results generalize results of Lenstra for elliptic curves.