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On properness of moduli stacks of $D^{\times}$-shtukas over ramified legs

Yong-Gyu Choi, Wansu Kim, Junyeong Park

TL;DR

The paper provides a concrete, explicit sufficient condition for the properness of moduli stacks of $D^{ imes}$-shtukas over ramified legs, refining Lau's degeneration analysis to allow legs to meet the ramification locus. The main result ties properness to a global inequality that compares ramified local invariants $[ ext{inv}_y(D) ]_{bQ}$ against bounds $oldsymbololdsymbol\lambda$, ensuring degenerations are ruled out. In addition, the authors establish non-emptiness results for Newton and Kottwitz–Rapoport strata for moduli stacks of $B^{ imes}$-shtukas, and discuss implications for parahoric unitary groups via Mazur-type inequalities. The work also develops a robust local–global framework using $(D,oldsymbol )$-spaces, Dieudonné $D_x$-modules, and BD-Schubert theory to control degenerations and to analyze the leg morphism, with potential applications to arithmetic intersection theory and Mantovan-type formalisms for inner forms of $ ext{GL}_d$. Overall, the results provide a versatile toolkit for understanding compactifications and degenerations in function-field shtuka moduli spaces.

Abstract

Given a maximal order $D$ of a central division algebra over a global function field $F$, we prove an explicit sufficient condition for moduli stacks of $D^\times$-shtukas to be proper over a finite field in terms of the local invariants of $D$ and bounds. Our proof is a refinement of E.~Lau's result (Duke Math. J. 140 (2007)), which showed the properness of the leg morphism (or characteristic morphism) away from the ramification locus of $D$. We also establish non-emptiness of Newton and Kottwitz--Rapoport strata for moduli stacks of $B^\times$-shtukas, where $B$ is a maximal order of a central simple algebra over $F$.

On properness of moduli stacks of $D^{\times}$-shtukas over ramified legs

TL;DR

The paper provides a concrete, explicit sufficient condition for the properness of moduli stacks of -shtukas over ramified legs, refining Lau's degeneration analysis to allow legs to meet the ramification locus. The main result ties properness to a global inequality that compares ramified local invariants against bounds , ensuring degenerations are ruled out. In addition, the authors establish non-emptiness results for Newton and Kottwitz–Rapoport strata for moduli stacks of -shtukas, and discuss implications for parahoric unitary groups via Mazur-type inequalities. The work also develops a robust local–global framework using -spaces, Dieudonné -modules, and BD-Schubert theory to control degenerations and to analyze the leg morphism, with potential applications to arithmetic intersection theory and Mantovan-type formalisms for inner forms of . Overall, the results provide a versatile toolkit for understanding compactifications and degenerations in function-field shtuka moduli spaces.

Abstract

Given a maximal order of a central division algebra over a global function field , we prove an explicit sufficient condition for moduli stacks of -shtukas to be proper over a finite field in terms of the local invariants of and bounds. Our proof is a refinement of E.~Lau's result (Duke Math. J. 140 (2007)), which showed the properness of the leg morphism (or characteristic morphism) away from the ramification locus of . We also establish non-emptiness of Newton and Kottwitz--Rapoport strata for moduli stacks of -shtukas, where is a maximal order of a central simple algebra over .

Paper Structure

This paper contains 12 sections, 23 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. General conventions
  4. Linear algebraic groups and loop groups
  5. $\mathcal{G}$-bundles
  6. Central simple algebras
  7. Moduli stacks of $\mathcal{G}$-shtukas
  8. Kottwitz--Rapoport and Newton stratifications
  9. "Special fibres" of Beilinson--Drinfeld Schubert varieties
  10. Review of $(D,\varphi)$-spaces and Dieudonné $D_x$-modules
  11. Isogeny classes and localisations for $\mathcal{G}$-shtukas
  12. Valuative criterion

Key Result

Theorem 1.1

Let $I$ be a non-empty finite set indexing legs, and choose a partition $I_\bullet$ of $I$. We fix $\boldsymbol\lambda = (\lambda_i)_{i\in I}$ where $\lambda_i = (\lambda_{i,1},\cdots,\lambda_{i,d})\in \mathbb{Z}^d$ such that $\lambda_{i,j}\geq\lambda_{i,j+1}$ for any $i,j$, and we have $\sum_{i\in for any integer $0<m<d$, where $[-]_\mathbb{Q}\colon\mathbb{Q}/\mathbb{Z}\to\mathbb{Q}\cap [0,1)$ i

Theorems & Definitions (65)

  • Theorem 1.1: Lau:Degeneration
  • Theorem 1.2: Theorem \ref{['thm:from-inequality-to-properness']}
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Definition 3.4: cf. Bieker:IntModels
  • Definition 3.5
  • Remark 3.6
  • Proposition 3.7: Arasteh Rad--Hartl ArastehRad-Hartl:Uniformizing
  • proof
  • ...and 55 more