In this paper I succeed in removing the ‘ordinary’ hypothesis from the major results in my thesis. In particular, I show that every simple odd-dimensional abelian variety over a finite field is isogenous to a principally polarized variety. To do this I need to investigate the Grothendieck group of the category of kernels of isogenies between elements of a given isogeny class. I also have to prove results on reduced norms in central simple algebras over number fields, results that can be viewed as extensions of the Hasse-Schilling-Maass theorem.