Type numbers of quaternion Hermitian forms and supersingular abelian varieties (Q1750372)

The result of \textit{K. Hashimoto} and the author [J. Fac. Sci., Univ. Tokyo, Sect. I A 27, 549--601 (1980; Zbl 0452.10029)] is generalized from the principal genus to general lattices. This is furthermore applied to the number of isomorphism classes of any polarized superspecial abelian varieties which have a model over \(\mathbb{F}_p\) such that the polarizations are in a fixed genus of lattices. This generalizes the result of the author and \textit{T. Katsura} [Compos. Math. 91, No. 1, 37--46 (1994; Zbl 0817.14024)] and has an application to the number of components in the supersimple locus which are defined over \(\mathbb{F}_p\).