Mass formulas and the basic locus of unitary Shimura varieties

TL;DR

This paper computes mass formulas for unimodular lattices in Hermitian spaces over CM fields and analyzes the geometry and arithmetic of the basic locus in certain Shimura varieties, providing explicit component counts and EO stratum data.

Contribution

It introduces explicit mass formulas and detailed geometric and arithmetic descriptions of the basic locus in GU(r,s)-Shimura varieties over CM fields.

Findings

Mass formulas for unimodular lattices over CM fields

Explicit counts of irreducible and connected components of the basic locus

Descriptions of EO strata components in specific signatures

Abstract

In this article we compute the mass associated to any unimodular lattice in a Hermitian space over an arbitrary CM field under a condition at 2. We study the geometry and arithmetic of the basic locus of the GU(r,s)-Shimura variety associated to an imaginary quadratic field modulo a good prime p>2. We give explicit formulas for the numbers of irreducible and connected components of the basic locus, and of points of the zero-dimensional Ekedahl-Oort (EO) stratum, as well as of the irreducible components of basic EO strata when the signature is either (1, n-1) or (2,2).