Abelian varieties over finite fields and their groups of rational points

Over a finite field Fq, abelian varieties with commutative endomorphism rings can be described by using modules over orders in & eacute;tale algebras. By exploiting this connection, we produce four theorems regarding groups of rational points and self-duality, along with explicit examples. First, when End(A) is locally Gorenstein, we show that the group structure of A(Fq) is determined by End(A). In fact, the same conclusion is attained if End(A) has local Cohen-Macaulay type at most 2, under the additional assumption that A is ordinary or q is prime, although the conclusion is not true in general. Second, the description in the Gorenstein case is used to characterize cyclic isogeny classes in terms of conductor ideals. Third, going in the opposite direction, we characterize squarefree isogeny classes of abelian varieties with N rational points in which every abelian group of order N is realized as a group of rational points. Finally, we study when an abelian variety A over Fq and its dual A" satisfy or fail to satisfy several interrelated properties, namely A similar to= A", A(Fq) similar to= A"(Fq), and End(A) = End(A"). In the process, we exhibit a sufficient condition for A not congruent to A" involving the local Cohen-Macaulay type of End(A). In particular, such an abelian variety A is not a Jacobian, or even principally polarizable.