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On the geometry of splitting models

S. Bijakowski, I. Zachos, Z. Zhao

Abstract

We consider Shimura varieties associated to a unitary group of signature $(n-s,s)$ where $n$ is even. For these varieties, by using the spin splitting models from Zachos-Zhao, we construct flat, Cohen-Macaulay, and normal $p$-adic integral models with reduced special fiber and with an explicit moduli-theoretic description over odd primes $p$ which ramify in the imaginary quadratic field with level subgroup at $p$ given by the stabilizer of a $π$-modular lattice in the hermitian space. We prove that the special fiber of the corresponding splitting model is stratified by an explicit poset with a combinatorial description, similar to Bijakowski-Hernandez, and we describe its irreducible components. Additionally, we prove the closure relations for this stratification.

On the geometry of splitting models

Abstract

We consider Shimura varieties associated to a unitary group of signature where is even. For these varieties, by using the spin splitting models from Zachos-Zhao, we construct flat, Cohen-Macaulay, and normal -adic integral models with reduced special fiber and with an explicit moduli-theoretic description over odd primes which ramify in the imaginary quadratic field with level subgroup at given by the stabilizer of a -modular lattice in the hermitian space. We prove that the special fiber of the corresponding splitting model is stratified by an explicit poset with a combinatorial description, similar to Bijakowski-Hernandez, and we describe its irreducible components. Additionally, we prove the closure relations for this stratification.
Paper Structure (15 sections, 24 theorems, 119 equations)

This paper contains 15 sections, 24 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Integral models of unitary Shimura varieties
  3. Unitary Shimura varieties
  4. Integral models of unitary Shimura varieties
  5. Splitting models of unitary Shimura varieties
  6. Local model diagrams
  7. Local Models and Variants
  8. Geometry of the Special fiber of ${\rm M}^{\rm spl}$
  9. Closure relations
  10. Flatness and Reducedness of ${\rm M}^{\rm spl}$
  11. The main result
  12. Local ring calculations
  13. Applications
  14. Relation of $X_{h,l}$ with Schubert cells
  15. Demazure Resolutions

Key Result

Theorem 1.1

For any signature $(n-s,s)$ and for every $K^p$ as above, there is a scheme $\mathcal{A}^{\rm spl}_{\mathbf{K}}$, flat over ${\rm Spec \, } (O_{K_1})$, with and which supports a local model diagram \begin{tikzcd} &\wti{\mathcal{A}}^{\rm spl}_{\mathbf{K}}(G, X)\arrow[dl, "\pi^{\rm reg}_K"']\arrow[dr, "q^{\rm reg}_K"] & \\ \mathcal{A}^{\rm spl}_{\mathbf{K}} && {\rm M}^{\rm spl} \end{tikzcd}such

Theorems & Definitions (49)

  • Theorem 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Definition 3.4
  • Theorem 3.5
  • ...and 39 more