Principal polarization on products of abelian varieties over finite fields

We refine and generalize the results of K. E. Lauter and E. W. Howe on principal polarizations on products of abelian varieties over finite fields. Firstly, we study the reasons for the absence of an irreducible principal polarization in the isogeny class of the product of an ordinary and a supersingular abelian variety. Secondly, we provide a necessary condition for the existence of a principal polarization on an abelian variety in the isogeny class of the product of a geometrically simple abelian surface and an elliptic curve. As an application, we prove that this abelian threefold or its quadratic twist is a Jacobian.