On the supersingular locus of the Shimura variety for GU(2,2) over a ramified prime

Abstract

We study the structure of the supersingular locus of the Rapoport–Zink integral model of the Shimura variety for [Formula: see text] over a ramified odd prime with the special maximal parahoric level. We prove that the supersingular locus equals the disjoint union of two basic loci, one of which is contained in the flat locus, and the other is not. We also describe explicitly the structure of the basic loci. More precisely, the former one is purely [Formula: see text]-dimensional, and each irreducible component is birational to the Fermat surface. On the other hand, the latter one is purely [Formula: see text]-dimensional, and each irreducible component is birational to the projective line.