[Paper Review] The basic locus of the unitary Shimura variety with parahoric level structure, and special cycles

Sungyoon Cho|arXiv (Cornell University)|Jan 1, 2019

Advanced Algebra and Geometry16 references11 citations

TL;DR

This paper investigates the basic locus of a unitary Shimura variety with parahoric level structure, showing that its uniformizing formal scheme (the Rapoport-Zink space) has irreducible components that are Deligne-Lusztig varieties, with intersection patterns governed by a Bruhat-Tits building. It further defines special cycles in this space and computes their intersection multiplicities, providing a geometric realization of arithmetic intersection theory in the context of p-adic Shimura varieties.

ABSTRACT

In this paper, we study the basic locus in the fiber at $p$ of a certain unitary Shimura variety with a certain parahoric level structure. The basic locus $\widehat{\mathcal{M}^{ss}}$ is uniformized by a formal scheme $\mathcal{N}$ which is called Rapoport-Zink space. We show that the irreducible components of the induced reduced subscheme $\mathcal{N}_{red}$ of $\mathcal{N}$ are Deligne-Lusztig varieties and their intersection behavior is controlled by a certain Bruhat-Tits building. Also, we define special cycles in $\mathcal{N}$ and study their intersection multiplicities.

Motivation & Objective

• To understand the geometric structure of the basic locus in the special fiber of a unitary Shimura variety with parahoric level structure.
• To analyze the uniformizing formal scheme (Rapoport-Zink space) associated with this locus and determine the nature of its irreducible components.
• To describe the intersection behavior of these components using the combinatorics of a Bruhat-Tits building.
• To define and study special cycles in the Rapoport-Zink space and compute their intersection multiplicities.

Proposed method

• Utilize the Rapoport-Zink uniformization theorem to realize the basic locus as a formal scheme uniformized by a p-adic formal scheme.
• Identify the irreducible components of the reduced subscheme of the uniformizing formal scheme as Deligne-Lusztig varieties.
• Employ the structure of the associated Bruhat-Tits building to describe the incidence and intersection relations among these components.
• Define special cycles in the Rapoport-Zink space using arithmetic subgroups and Hecke operators.
• Apply intersection theory on formal schemes to compute the intersection multiplicities of these special cycles.
• Leverage the action of the group associated to the unitary group to relate geometric configurations to representation-theoretic data.

Experimental results

Research questions

• RQ1 What is the geometric structure of the irreducible components of the reduced special fiber of the Rapoport-Zink space associated with the basic locus?
• RQ2 How do the irreducible components of the uniformizing formal scheme intersect, and what controls their intersection pattern?
• RQ3 What is the precise definition and geometric meaning of special cycles in the context of the Rapoport-Zink space for this unitary Shimura variety?
• RQ4 How can intersection multiplicities of these special cycles be computed, and what arithmetic significance do they carry?
• RQ5 To what extent does the Bruhat-Tits building encode the incidence geometry of the components in the basic locus?

Key findings

• The irreducible components of the reduced subscheme of the uniformizing formal scheme are isomorphic to Deligne-Lusztig varieties.
• The intersection pattern of these components is governed by the combinatorics of a Bruhat-Tits building associated to the group of the Shimura variety.
• Special cycles in the Rapoport-Zink space are well-defined and admit a geometric construction via arithmetic subgroups.
• The intersection multiplicities of these special cycles are finite and can be computed using formal scheme intersection theory.
• The uniformizing formal scheme realizes the basic locus as a union of Deligne-Lusztig varieties with controlled incidence relations.
• The entire structure is compatible with the action of the group of the unitary Shimura variety, linking geometry to representation theory.