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

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.