On quasi-split orthogonal local models of PEL type D

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.