[論文レビュー] On quasi-split orthogonal local models of PEL type D

J. H. Yang, I. Zachos+1 more|arXiv (Cornell University)|Feb 27, 2026

Geometry and complex manifolds被引用数 0

ひとこと要約

The paper proves flatness and related geometric properties of spin local models for quasi-split but non-split even orthogonal groups of PEL type D, and establishes topological flatness of naive local models; it also provides moduli interpretations and a resolution in maximal parahoric cases.

ABSTRACT

We study local models for the quasi-split but non-split even orthogonal similitude group over a complete discretely valued field of residue characteristic $p>2$. For arbitrary parahoric level, we prove that the Pappas-Rapoport spin local model is flat, normal, Cohen-Macaulay, with reduced special fiber. Equivalently, it agrees with the canonical local model, yielding an explicit moduli-theoretic description of the latter and confirming a conjecture of Pappas-Rapoport in the quasi-split orthogonal case. In the course of the proof we also show that the Rapoport-Zink local model is topologically flat, verifying a conjecture of Pappas-Rapoport-Smithling. Finally, for a maximal parahoric case we construct an explicit regular semi-stable model by blowing up the spin local model along the unique closed Schubert cell in its special fiber. As arithmetic applications, we deduce corresponding flatness and moduli-theoretic descriptions for integral PEL moduli spaces of type D and for the associated orthogonal Rapoport-Zink spaces.

研究の動機と目的

• Motivate and construct flat integral moduli schemes of PEL type D and corresponding flat orthogonal Rapoport–Zink spaces with parahoric level structure.
• Prove Pappas–Rapoport conjectures: spin local models are flat with reduced special fiber for even orthogonal similitude groups.
• Show topological flatness of naive local models in the quasi-split non-split setting and describe Schubert varieties in the maximal parahoric fiber.
• Provide moduli-theoretic interpretations and explicit resolutions in maximal parahoric cases.
• Outline arithmetic applications to Shimura varieties and Rapoport–Zink spaces.

提案手法

• Define and study naive local models Mnaive_L and spin local models M±_L for a quasi-split non-split even orthogonal similitude group.
• impose the spin condition to obtain the spin local model and compare with canonical local models (Pappas–Zhu framework).
• Prove flatness of M±_L over OF using a reduction to standard lattice chains and parahoric subgroups; establish reduced, Cohen–Macaulay fibers.
• Demonstrate topological flatness of Mnaive_L by affine-flag-geometry arguments and stratify reduced special fibers into Schubert cells.
• Provide explicit affine charts and a Schubert-cell stratification in maximal parahoric cases to describe reduced fibers and irreducibility.
• Construct a resolution via blowing up the spin local model along the unique closed Schubert cell in its special fiber in the maximal parahoric case.

実験結果

リサーチクエスチョン

• RQ1 Does the spin local model for quasi-split but non-split even orthogonal groups at arbitrary parahoric level yield a flat OF0-scheme with reduced special fiber?
• RQ2 Is the naive local model topologically flat in the quasi-split non-split setting?
• RQ3 What is the Schubert-cell structure of the special fiber in maximal parahoric cases, and is the special fiber irreducible?
• RQ4 Can one give a moduli-theoretic description matching canonical/local model constructions (Pappas–Zhu) in PEL type D?
• RQ5 Can explicit regular semi-stable models be obtained by blow-ups in maximal parahoric cases?

主な発見

• The spin local model M±_L is flat over OF0, normal, Cohen–Macaulay with reduced special fiber (Theorem 1.3(1)).
• The naive local model Mnaive_L is topologically flat (Theorem 1.3(2)).
• In maximal parahoric cases, the reduced special fiber admits a Schubert-cell stratification with irreducibility (Theorem 1.7).
• A moduli-theoretic description of the Schubert varieties in the special fiber is established (Proposition 1.8).
• The spin local model coincides with the canonical schematic local model and with the Pappas–Zhu local model; the flatness result confirms the Pappas–Rapoport conjecture in the quasi-split setting (Remark 1.4, Theorem 1.3).
• For a specific maximal parahoric level, a regular semi-stable model is obtained by blowing up the spin local model along the unique closed Schubert cell (Theorem 1.11).