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 GSp_2g(F_l) with characteristic polynomial f mod l 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.

We thank Everett Howe for sharing with us his work related to isogeny classes of principally polarized abelian varieties, and Jeff Achter for very helpful conversations. We also thank the referee for useful suggestions and insightful questions.