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 [Formula: see text] with [Formula: see text]. We prove a conjecture of Gross that the superspecial loci and their lift in the integral models can be parametrized by certain homogeneous spaces. As applications, we provide mass formulas for the superspecial loci. We also indicate how Gross’s conjectures can be proved in the case of Coxeter type Shimura varieties using the group theoretic methods of Görtz and He.