Type numbers of quaternion hermitian forms and supersingular abelian varieties

Abstract

The word type number of an algebra means classically the number of isomorphism classes of maximal orders in the algebra, but here we consider quaternion hermitian lattices in a fixed genus and their right orders. Instead of inner isomorphism classes of right orders, we consider isomorphism classes realized by similitudes of the quaternion hermitian forms.The number T of such isomorphism classes are called type number or G-type number , where G is the group of quaternion hermitian similitudes. We express T in terms of traces of some special Hecke operators. This is a generalization of the result announced in [5] (I) from the principal genus to general lattices. We also apply our result to the number of isomorphism classes of any polarized superspecial abelian varieties which have a model over F p such that the polarizations are in a ”fixed genus of lattices”. This is a generalization of [8] and has an application to the number of components in the supersingular locus which are defined over F p .