On the superspecial loci of orthogonal type Shimura varieties

TL;DR

This paper proves Gross's conjecture on the parametrization of superspecial loci in orthogonal type Shimura varieties and provides a mass formula for these loci, advancing understanding of their structure.

Contribution

It confirms Gross's conjecture on the parametrization of superspecial loci and derives a mass formula, offering new insights into the structure of these loci.

Findings

Confirmed Gross's conjecture on parametrization

Derived a mass formula for superspecial loci

Enhanced understanding of orthogonal 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.