On the Bruhat-Tits stratification for GU(2,2) type Rapoport-Zink space: unramified case

TL;DR

This paper investigates the structure of the supersingular locus in a specific GU(2,2) Shimura variety at an unramified prime, providing a direct description of its Bruhat-Tits stratification without relying on exceptional isomorphisms.

Contribution

It offers a new, more direct method to describe the Bruhat-Tits stratification of the supersingular locus in the unramified case, avoiding the use of exceptional isomorphisms.

Findings

Explicit description of the supersingular locus's stratification

Connection between the locus and the basic Rapoport-Zink space

Methodology that bypasses exceptional isomorphisms

Abstract

In this note we study the supersingular locus of the GU(2,2) Shimura variety modulo a prime which is unramified in the imaginary quadratic extension. The supersingular locus of this Shimura variety can be related to the basic Rapoport-Zink space whose special fibre is described by the Bruhat-Tits stratification. The description for this supersingular locus in the case where the prime is inert in imaginary quadratic field is already known to Howard and Pappas by exploiting the exceptional isomorphism. Our method is more direct without using the exceptional isomorphism.