# Counting abelian varieties over finite fields via Frobenius densities

## TL;DR

This paper develops a mass formula for counting isogeny classes of principally polarized abelian varieties over finite fields, using Frobenius densities and measure analysis without relying on class number calculations.

## Contribution

Introduces a new mass formula based on Frobenius densities for counting abelian varieties over finite fields, avoiding class number computations.

## Key findings

- The product of local factors computes the size of isogeny classes.
- The formula applies to ordinary abelian varieties and prime fields.
- Uses Kottwitz's formula and symplectic measure analysis.

## Abstract

Let $[X,\lambda]$ be a principally polarized abelian variety over a finite field with commutative endomorphism ring; further suppose that either $X$ is ordinary or the field is prime. Motivated by an equidistribution heuristic, we introduce a factor $\nu_v([X,\lambda])$ for each place $v$ of $\mathbb Q$, and show that the product of these factors essentially computes the size of the isogeny class of $[X,\lambda]$. The derivation of this mass formula depends on a formula of Kottwitz and on analysis of measures on the group of symplectic similitudes, and in particular does not rely on a calculation of class numbers.