On the supersingular locus of the Shimura variety for $\mathrm{GU}(2,2)$ over a ramified prime

TL;DR

This paper investigates the structure of the supersingular locus in a specific Shimura variety for GU(2,2) over a ramified prime, revealing its decomposition into two basic loci with explicit geometric descriptions.

Contribution

It explicitly describes the supersingular locus as a union of two basic loci, detailing their dimensions and geometric structures, which was previously unknown.

Findings

Supersingular locus decomposes into two disjoint basic loci.

One locus is 2-dimensional with components birational to Fermat surfaces.

The other locus is 1-dimensional with components birational to projective lines.

Abstract

We study the structure of the supersingular locus of the Rapoport--Zink integral model of the Shimura variety for $\mathrm{GU}(2,2)$ 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 structures of basic loci. More precisely, the former one is purely $2$-dimensional, and each irreducible component is birational to the Fermat surface. On the other hand, the latter one is purely $1$-dimensional, and each irreducible component is birational to the projective line.