Local heuristics and an exact formula for abelian varieties of odd prime dimension over finite fields

TL;DR

This paper explores local heuristics and an exact formula for counting abelian varieties of odd prime dimension over finite fields, linking local densities to class number ratios and proposing a conjecture for genus three cases.

Contribution

It introduces a new approach to relate local matrix densities to class numbers for abelian varieties of odd prime dimension, including a conjecture for genus three.

Findings

Defined a product of local densities for matrices with given characteristic polynomial.

Computed this product under an equidistribution assumption.

Conjectured the product's relation to the size of isogeny classes for genus three.

Abstract

Consider a $q$-Weil polynomial $f$ of degree $2g$. Using an equidistribution assumption that is too strong to be true, we define and compute a product of local relative densities of matrices in $\rm{GSp}_{2g}(\mathbb{F}_\ell)$ with characteristic polynomial $f\mod\ell$ when $g$ is an odd prime. This infinite product is closely related to a ratio of class numbers. When $g=3$ we conjecture that the product gives the size of an isogeny class of principally polarized abelian threefolds.