On the flatness of spin local models for split even orthogonal groups
Jie Yang
TL;DR
<3-5 sentence high-level summary>The paper addresses the problem of constructing well-behaved integral models for Shimura varieties of PEL type with split orthogonal similitude groups, focusing on spin local models ${\operatorname{M}}^{\pm}_{{\mathcal{L}}}$ and their flatness over ${\mathcal{O}}$ at parahoric levels. It develops a precise local-coordinate framework near the worst point, reduces the flatness question to showing the reducedness of a class of matrix-defined rings ${\mathcal{R}}^{\pm}_N$ and ${\mathcal{R}}_N$, and proves reducedness (hence flatness) under either large residue characteristic or small $N$ (notably $N\in\{0,2,4\}$) and for large primes. The core technical advance is a representation-theoretic straightening argument for $O(N)$-standard bideterminants that yields a basis and reducedness, enabling a global flatness result for the spin local models in the pseudo-maximal and general parahoric cases. As a consequence, moduli spaces of type D gain flat, normal, Cohen–Macaulay integral models, aligning the local and global geometric structures with the expectations of Pappas–Rapoport and providing explicit moduli interpretations via the local-model diagram.
Abstract
Let $F$ be a complete discretely valued field with ring of integers $\mathcal{O}$ and residue field of characteristic $p>2$. Let $G=\operatorname{GO}_{2n}$ denote the split orthogonal similitude group over $F$. For any parahoric level structure, we prove that the associated spin local model for $G$ is a flat $\mathcal{O}$-scheme with reduced special fiber, provided either $p$ is sufficiently large or $n\leq 4$. This partially confirms a conjecture of Pappas and Rapoport. As a corollary, we obtain a flat (integral) moduli space of PEL-type D under the same assumptions.