Supersingular loci of some Shimura varieties for CM unitary groups

TL;DR

This paper describes the structure of supersingular loci in certain Shimura varieties associated with CM unitary groups, focusing on cases where the Hermitian space dimension is at most 4 and specific prime splitting conditions hold.

Contribution

It provides a detailed description of the supersingular locus for Shimura varieties of low-dimensional CM unitary groups under particular prime splitting assumptions.

Findings

Explicit description of supersingular loci for dim_V ≤ 4

Conditions on prime splitting in CM extension

Application to understanding special fibers of Shimura varieties

Abstract

In this note, we study Shimura varieties for the groups $\mathrm{GU}(V)$, where $V$ is a Hermitian space relative to a CM extension $E/E^+$. We give a description of the supersingular locus of the fiber at a prime $\nu$ over $p$ of such a Shimura variety, under the assumptions that $\mathrm{dim}_E(V) \leq 4$ and that the prime $p$ splits completely in $E^+$ and is unramified in $E$.