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Topological flatness of orthogonal spin local models

Jie Yang

TL;DR

The paper studies integral structures attached to Shimura varieties of PEL type D for the split even orthogonal similitude group and addresses the non-flatness of naive local models by introducing spin refinements. It provides a detailed lattice-theoretic and Bruhat--Tits framework to classify parahoric subgroups, embeds the special fibers into affine flag varieties, and analyzes Schubert stratifications via admissible sets. The main achievement is proving topological flatness of the spin refined objects for arbitrary parahoric levels, with a reduction to pseudo-maximal cases and explicit descriptions of strata and components in singleton settings. These results advance the understanding of the geometry of local models in orthogonal similtude settings and connect to the broader theory of integral models of Shimura varieties and affine flag varieties.

Abstract

Let $p$ be an odd prime and $F$ be a complete discretely valued field with residue field of characteristic $p$. For any parahoric level structure of the split even orthogonal similitude group $\operatorname{GO}_{2n}$ over $F$, we prove a preliminary form of the Pappas-Rapoport flatness conjecture: the associated spin local model is topologically flat.

Topological flatness of orthogonal spin local models

TL;DR

The paper studies integral structures attached to Shimura varieties of PEL type D for the split even orthogonal similitude group and addresses the non-flatness of naive local models by introducing spin refinements. It provides a detailed lattice-theoretic and Bruhat--Tits framework to classify parahoric subgroups, embeds the special fibers into affine flag varieties, and analyzes Schubert stratifications via admissible sets. The main achievement is proving topological flatness of the spin refined objects for arbitrary parahoric levels, with a reduction to pseudo-maximal cases and explicit descriptions of strata and components in singleton settings. These results advance the understanding of the geometry of local models in orthogonal similtude settings and connect to the broader theory of integral models of Shimura varieties and affine flag varieties.

Abstract

Let be an odd prime and be a complete discretely valued field with residue field of characteristic . For any parahoric level structure of the split even orthogonal similitude group over , we prove a preliminary form of the Pappas-Rapoport flatness conjecture: the associated spin local model is topologically flat.

Paper Structure

This paper contains 14 sections, 31 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Organization
  4. Notation
  5. Parahoric subgroups of the orthogonal similitude groups
  6. Orthogonal similitude group
  7. Bruhat--Tits theory for $G$
  8. Parahoric subgroups
  9. Naive and spin local models
  10. Embedding ${\operatorname {M}}^\mathrm{naive}_I\otimes k$ in the affine flag variety
  11. Schubert cells in ${\operatorname {M}}^\mathrm{naive}_{I,k}$ and ${\operatorname {M}}^\pm_{I,k}$
  12. Topological flatness of spin local models
  13. Pseudo-maximal parahoric case
  14. General parahoric case

Key Result

Theorem 1.2

Let $I$ be a non-empty subset of $[0,n]$ (notation as in § subsec-notation-intro). Define Then the subgroup where $\kappa\colon G^\circ(F)\twoheadrightarrow \pi_1(G)$ denotes the Kottwitz map, is a parahoric subgroup of $G(F)$. Furthermore, any parahoric subgroup of $G(F)$ is $G^\circ(F)$-conjugate to a subgroup $P_I^\circ$ for some (not necessarily unique) $I\subset [0,n]$. The $G^\circ(F)$-con

Theorems & Definitions (85)

  • Conjecture 1.1: Pappas--Rapoport
  • Theorem 1.2: Proposition \ref{['prop-Jform']}
  • Theorem 1.3: Corollary \ref{['coro-mainresults']}
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Definition 1.7: Definition \ref{['defn-Mell']}
  • Definition 2.1
  • Proposition 2.2
  • proof
  • ...and 75 more