On the superspecial loci of orthogonal type Shimura varieties

Abstract

In this note, we study the superspecial loci of orthogonal type Shimura varieties of signature (n−2, 2) with n ≥ 3. We prove a conjecture of Gross on the parametrizations of the superspecial locus in the special fiber of an orthogonal type Shimura variety and its lift in the integral model by certain homogeneous spaces. As applications, we provide a mass formula for the superspecial locus. We also indicate how Gross's conjectures can be generalized to the Coxeter type Shimura varieties using the group theoretic method of Görtz and He.