Abelian varieties over finite fields isogenous to a power

Let A be an ordinary square-free abelian variety over a finite field. In this talk we will describe the category of abelian varieties isogenous to A^r in terms of R-modules, where R is an order in a certain étale algebra.

We will describe also polarizations and groups of automophisms. Under certain mild assumption on the order R, we will be able to effectively compute the abelian varieties up to isomorphism and in the case r=1 we can also list all polarizations (up to polarized isomorphisms) with automorphism groups and period matrix of the canonical lift.