Supersingular loci of some Shimura varieties for CM unitary groups

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 ${\dim }_E(V)\le 4$ and that the prime $p$ splits completely in $E^{+}$ and is unramified in $E$.