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On the moduli description of ramified unitary local models of signature $(n-1,1)$

Yu Luo

TL;DR

This work resolves the moduli-theoretic description of ramified unitary local models with signature $(n-1,1)$ by showing the local model is represented by the strengthened-spin model for all nonempty parahoric sets $I$, under the assumption $p eq 2$. It provides explicit defining equations for the special fiber at a worst-point affine chart and proves the model is flat, normal, and Cohen–Macaulay (with Gorenstein cases when $n=4\, ho+2$). The analysis reduces to strongly non-special parahoric subgroups, where two irreducible components are described by explicit moduli data and their intersection by a normal Cohen–Macaulay cone; these descriptions extend to the ramified Pappas–Zhu model and yield modular interpretations of irreducible components and Shimura-variety integral models. The results give sharp moduli descriptions, connect local models to affine flag varieties, and establish flatness in the $p eq 3$ regime and in characteristic zero, with broad applications to Shimura varieties and arithmetic geometry. Overall, the paper provides a concrete, moduli-theoretic realization of ramified unitary local models, advancing both geometric and group-theoretic understandings of these objects. The explicit equations and constructions also offer a platform for further investigations into singularities and potential semi-stable reductions in broader ramified settings.

Abstract

We provide a moduli description of the ramified unitary local model of signature $(n-1,1)$ with arbitrary parahoric level structure, assuming the residue field has characteristic not equal to $2$, thereby confirming a conjecture of Smithling. Our approach involves writing down explicit equations for the special fiber and proving that they define a normal, Cohen-Macaulay scheme, which is also of independent interest. As applications, we obtain moduli descriptions for: (1) ramified unitary Pappas-Zhu local models with arbitrary parahoric level; (2) the irreducible components of their special fiber in the maximal parahoric case; (3) integral models of ramified unitary Shimura varieties with arbitrary (quasi-)parahoric level.

On the moduli description of ramified unitary local models of signature $(n-1,1)$

TL;DR

This work resolves the moduli-theoretic description of ramified unitary local models with signature by showing the local model is represented by the strengthened-spin model for all nonempty parahoric sets , under the assumption . It provides explicit defining equations for the special fiber at a worst-point affine chart and proves the model is flat, normal, and Cohen–Macaulay (with Gorenstein cases when ). The analysis reduces to strongly non-special parahoric subgroups, where two irreducible components are described by explicit moduli data and their intersection by a normal Cohen–Macaulay cone; these descriptions extend to the ramified Pappas–Zhu model and yield modular interpretations of irreducible components and Shimura-variety integral models. The results give sharp moduli descriptions, connect local models to affine flag varieties, and establish flatness in the regime and in characteristic zero, with broad applications to Shimura varieties and arithmetic geometry. Overall, the paper provides a concrete, moduli-theoretic realization of ramified unitary local models, advancing both geometric and group-theoretic understandings of these objects. The explicit equations and constructions also offer a platform for further investigations into singularities and potential semi-stable reductions in broader ramified settings.

Abstract

We provide a moduli description of the ramified unitary local model of signature with arbitrary parahoric level structure, assuming the residue field has characteristic not equal to , thereby confirming a conjecture of Smithling. Our approach involves writing down explicit equations for the special fiber and proving that they define a normal, Cohen-Macaulay scheme, which is also of independent interest. As applications, we obtain moduli descriptions for: (1) ramified unitary Pappas-Zhu local models with arbitrary parahoric level; (2) the irreducible components of their special fiber in the maximal parahoric case; (3) integral models of ramified unitary Shimura varieties with arbitrary (quasi-)parahoric level.
Paper Structure (52 sections, 55 theorems, 214 equations)

This paper contains 52 sections, 55 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Ramified unitary local model
  4. Ramified unitary local models of strongly non-special parahoric subgroups
  5. Moduli description of the local model of the unitary group
  6. The structure of the paper
  7. Acknowledgments
  8. Conflict of Interest Statement
  9. The moduli problems
  10. Basic setup
  11. Notation
  12. Linear algebra setup of the split hermitian form
  13. Naive model
  14. Wedge and spin conditions
  15. Wedge condition
  16. ...and 37 more sections

Key Result

Theorem 1.2.1

For any nonempty subset $I\subset\{0,1,\cdots,m\}$ satisfying intro_back:local-dynkin, the inclusion $M_I^\mathrm{loc}\xspace(n-1,1)\subset M_I(n-1,1)$ is an equality. In other words, for signature $(n-1,1)$, the quasi-split ramified unitary local model is represented by the strengthened spin model.

Theorems & Definitions (115)

  • Theorem 1.2.1
  • Remark 1.2.2
  • Theorem 1.2.3
  • Remark 1.2.4
  • Definition 1.3.1
  • Theorem 1.3.2
  • Theorem 1.3.4
  • Remark 1.3.5
  • Remark 1.4.1
  • Remark 2.2.1
  • ...and 105 more