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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.

11G18 Arithmetic aspects of modular and Shimura varieties
14G35 Modular and Shimura varieties
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