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

Variations in the distribution of principally polarized abelian varieties among isogeny classes