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

TL;DR

This paper investigates the distribution of principally polarized abelian varieties within isogeny classes over finite fields, establishing a link between their counts and class numbers of associated rings, and providing heuristic support for Katz and Sarnak's distribution conjecture.

Contribution

It extends known results to a broader class of rings, including orders generated by Frobenius and Verschiebung, and offers estimates relating abelian variety counts to class numbers.

Findings

Number of such abelian varieties is 0 or a ratio of class numbers.

Provides estimates of class number ratios using Frobenius angles.

Supports the Katz-Sarnak distribution conjecture heuristically.

Abstract

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.