  We show that for a large class of rings $R$, the number of principally
polarized abelian varieties over a finite field in a given simple ordinary
isogeny class and with endomorphism ring $R$ is equal either to 0, or to a
ratio of class numbers associated to $R$, up to some small computable factors.
This class of rings includes the maximal order of the CM field $K$ associated
to the isogeny class (for which the result was already known), as well as the
order $R$ generated over $\mathbf{Z}$ by Frobenius and Verschiebung.
  For this latter order, we can use results of Louboutin to estimate the
appropriate ratio of class numbers in terms of the size of the base field and
the Frobenius angles of the isogeny class. The error terms in our estimates are
quite large, but the trigonometric terms in the estimate are suggestive:
Combined with a result of Vladut on the distribution of Frobenius angles of
isogeny classes, they give a heuristic argument in support of the theorem of
Katz and Sarnak on the limiting distribution of the multiset of Frobenius
angles for principally polarized abelian varieties of a fixed dimension over
finite fields.
