On the Rapoport-Zink space for $\mathrm{GU}(2, 4)$ over a ramified prime

TL;DR

This paper investigates the supersingular locus of a Shimura variety linked to the unitary group GU(2,4) at a ramified prime, providing explicit descriptions of the associated Rapoport-Zink space and its irreducible components.

Contribution

It offers a detailed analysis of the Rapoport-Zink space for GU(2,4) over a ramified prime, including flatness and explicit component descriptions.

Findings

Rapoport-Zink space is flat.

Irreducible components are homeomorphic to Deligne-Lusztig varieties.

Provides explicit geometric descriptions of the basic locus.

Abstract

In this work, we study the supersingular locus of the Shimura variety associated to the unitary group $\mathrm{GU}(2,4)$ over a ramified prime. We show that the associated Rapoport-Zink space is flat, and we give an explicit description of the irreducible components of the reduction modulo $p$ of the basic locus. In particular, we show that these are universally homeomorphic to either a generalized Deligne-Lusztig variety for a symplectic group or to the closure of a vector bundle over a classical Deligne-Lusztig variety for an orthogonal group. Our results are confirmed in the group-theoretical setting by the reduction method \`a la Deligne and Lusztig and the study of the admissible set.