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The Bruhat-Tits stratification for basic unramified $GU(1,n-1)$ Rapoport-Zink spaces at arbitrary parahoric level

Joseph Muller

TL;DR

The paper develops a comprehensive BT stratification for the basic unramified unitary Rapoport-Zink space of signature $$(1,n-1)$$ at arbitrary parahoric level. It introduces Bruhat-Tits indices to index locally closed strata, proves that closed strata are isomorphic to closures of fine Deligne-Lusztig varieties for products of unitary and GL groups, and uses He’s results to decompose these closures into DL pieces. The strata are shown to be smooth, projective, and irreducible, and their incidence is controlled by a BT-combinatorial framework; the action of the auto-quasi-isogeny group $J(E)$ organizes irreducible components into a finite number of orbits, with explicit counts (e.g., $n-m+1$ total orbits) and dimension formulas. By passing to $p$-adic uniformization, these local results inform the basic locus of unitary Shimura varieties and have arithmetic applications such as Kudla-Rapoport and Tate-type conjectures. The work generalizes known hyperspecial and maximal parahoric results to arbitrary parahoric level, providing a unified stratification framework with explicit DL-models and combinatorics.

Abstract

In this paper, we describe a stratification on the reduced special fiber of the basic unramified unitary Rapoport-Zink space of signature $(1,n-1)$ and at arbitrary parahoric level. We prove the smoothness, irreducibility and compute the dimensions of the closed strata, which are isomorphic to the closure of certain fine Deligne-Lusztig varieties for a product of unitary and general linear groups. We also describe the incidence relations of the stratification by using Bruhat-Tits indices, which are related to the Bruhat-Tits building of an underlying $p$-adic unitary group.

The Bruhat-Tits stratification for basic unramified $GU(1,n-1)$ Rapoport-Zink spaces at arbitrary parahoric level

TL;DR

The paper develops a comprehensive BT stratification for the basic unramified unitary Rapoport-Zink space of signature at arbitrary parahoric level. It introduces Bruhat-Tits indices to index locally closed strata, proves that closed strata are isomorphic to closures of fine Deligne-Lusztig varieties for products of unitary and GL groups, and uses He’s results to decompose these closures into DL pieces. The strata are shown to be smooth, projective, and irreducible, and their incidence is controlled by a BT-combinatorial framework; the action of the auto-quasi-isogeny group organizes irreducible components into a finite number of orbits, with explicit counts (e.g., total orbits) and dimension formulas. By passing to -adic uniformization, these local results inform the basic locus of unitary Shimura varieties and have arithmetic applications such as Kudla-Rapoport and Tate-type conjectures. The work generalizes known hyperspecial and maximal parahoric results to arbitrary parahoric level, providing a unified stratification framework with explicit DL-models and combinatorics.

Abstract

In this paper, we describe a stratification on the reduced special fiber of the basic unramified unitary Rapoport-Zink space of signature and at arbitrary parahoric level. We prove the smoothness, irreducibility and compute the dimensions of the closed strata, which are isomorphic to the closure of certain fine Deligne-Lusztig varieties for a product of unitary and general linear groups. We also describe the incidence relations of the stratification by using Bruhat-Tits indices, which are related to the Bruhat-Tits building of an underlying -adic unitary group.

Paper Structure

This paper contains 22 sections, 61 theorems, 200 equations, 3 tables.

Table of Contents

  1. Introduction
  2. The moduli space of formal $\mathcal{O}_F$-modules of signature $(1,n-1)$
  3. Maximal parahoric level
  4. Some results on vertex lattices
  5. The moduli space $\mathcal{N}_{E/F}^{\mathbbm h}$ at arbitrary parahoric level
  6. Subsets $\mathcal{N}_{I,\mathbf{\Lambda}}^{\mathbbm h}(k)$ of $\mathcal{N}_{E/F}^{\mathbbm h}(k)$
  7. Subschemes $\mathcal{N}_{I,\mathbf{\Lambda}}^{\mathbbm h}$ of $\mathcal{N}_{E/F,\mathrm{red}}^{\mathbbm h}$ and Deligne-Lusztig varieties
  8. Subschemes $\mathcal{N}_{I,\mathbf{\Lambda}}^{\mathbbm h}$ attached to Bruhat-Tits indices $(I,\mathbf{\Lambda})$
  9. Deligne-Lusztig varieties
  10. Coarse, parabolic and fine Deligne-Lusztig varieties
  11. Example: the general linear group
  12. Example: the fake unitary case
  13. Example: the unitary group
  14. Some combinatorial lemmas
  15. The varieties $X_{I,\mathbf{\Lambda}}^{\mathbbm h}$
  16. ...and 7 more sections

Key Result

Lemma 1

The Bruhat-Tits type $I$ of any point $(A_m \subset \ldots \subset B_m) \in \mathcal{N}_{E/F}^{\mathbbm h}(k)$ is not empty.

Theorems & Definitions (135)

  • Lemma
  • Lemma
  • Corollary
  • Corollary
  • Theorem
  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Lemma 2.4
  • Theorem 2.5
  • ...and 125 more