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An exact geometric mass formula. (English) Zbl 1213.11129
Let $$B$$ be a totally indefinite quaternion algebra over a totally real number field $$F$$. Let $$p$$ be a rational prime which is unramified in $$B$$. Let $$O_B$$ be a maximal order of $$B$$ stable under a positive involution *. For a positive integer $$g$$, let $$\Lambda_g^B$$ denote the set of isomorphism classes of $$g$$-dimensional superspecial principally polarized abelian $$O_B$$-varieties over an algebraically closed field of characteristic $$p$$. The main theorem of this paper is a formula for the mass of $$\Lambda_g^B$$. As a consequence the author obtains a formula for the number of superspecial points in the moduli space $$\mathcal{M}$$ of $$g$$-dimensional principally polarized abelian $$O_B$$-varieties over $$\overline{\mathbb{F}}_p$$ with prescribed level-structure.

##### MSC:
 11G18 Arithmetic aspects of modular and Shimura varieties 14G35 Modular and Shimura varieties
##### Keywords:
mass formula; Shimura variety; superspecial abelian variety
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