On the supersingular locus of Shimura varieties for quaternionic unitary groups

Abstract

We study a Shimura variety attached to a unitary similitude group of a skew-Hermitian form over a totally indefinite quaternion algebra over a totally real number field. We give a necessary and sufficient condition for the existence of skew-Hermitian self-dual lattices. Under this condition we show that the superspecial locus in the fiber at $p$ of the associated Shimura variety is non-empty. We also give an explicit formula for the number of irreducible components of the supersingular locus when $p$ is odd and unramified in the quaternion algebra.