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

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