On $p$-adic integral moduli schemes and local models for PEL type D

TL;DR

This paper constructs flat integral moduli schemes of PEL type D and related Rapoport--Zink spaces over p-adic rings, proving conjectures on flatness, normality, and Cohen--Macaulay properties, and providing explicit models.

Contribution

It proves the flatness and normality of spin local models for orthogonal groups, confirming conjectures and constructing explicit semi-stable models.

Findings

Spin local models are flat, normal, and Cohen--Macaulay.

Naive Rapoport--Zink local models are topologically flat in certain cases.

Explicit semi-stable models are constructed via blow-ups.

Abstract

We construct flat integral moduli schemes of PEL type D and the corresponding flat orthogonal Rapoport--Zink spaces with parahoric level structure over a $p$-adic integer ring. The construction relies on proving a conjecture of Pappas--Rapoport: for an even orthogonal similitude group over a complete discretely valued field of residue characteristic $p>2$, and for arbitrary parahoric level, the associated spin local model is flat, normal, Cohen--Macaulay, with reduced special fiber. In the course of the proof, we also show that in the quasi-split but non-split case, the Rapoport--Zink (naive) local model is topologically flat, verifying a conjecture of Pappas--Rapoport--Smithling. In the maximal parahoric case, we also describe the Schubert varieties in the special fiber in moduli-theoretic terms. 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.