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Relative representability and parahoric level structures

Yuta Takaya

TL;DR

The article develops a representability criterion for $v$-sheaf modifications of formal schemes by embedding thick prekimberlites into a framework of good/maximal good covers and perfectoid colimits. Central to the method is the dilatation theory for $v$-sheaves and the analysis of geometric quotients by profinite groups, which enables gluing from affine/local data to global formal models. These formal models yield integral models for local and global Shimura varieties at hyperspecial levels, and provide a $v$-sheaf theoretic avenue toward local model diagrams and parahoric level structures for local shtukas. The work thus bridges $v$-sheaf moduli problems with classical integral-model constructions, offering tools for depth-zero and level-structure generalizations in the Shimura-variety landscape.

Abstract

We establish a representability criterion of $v$-sheaf theoretic modifications of formal schemes and apply this criterion to moduli spaces of parahoric level structures on local shtukas. In the proof, we introduce nice classes of equivariant profinite perfectoid covers and study geometric quotients of perfectoid formal schemes by profinite groups. As a corollary, we obtain a construction of (part of) integral models of local and global Shimura varieties under hyperspecial levels from those at hyperspecial levels.

Relative representability and parahoric level structures

TL;DR

The article develops a representability criterion for -sheaf modifications of formal schemes by embedding thick prekimberlites into a framework of good/maximal good covers and perfectoid colimits. Central to the method is the dilatation theory for -sheaves and the analysis of geometric quotients by profinite groups, which enables gluing from affine/local data to global formal models. These formal models yield integral models for local and global Shimura varieties at hyperspecial levels, and provide a -sheaf theoretic avenue toward local model diagrams and parahoric level structures for local shtukas. The work thus bridges -sheaf moduli problems with classical integral-model constructions, offering tools for depth-zero and level-structure generalizations in the Shimura-variety landscape.

Abstract

We establish a representability criterion of -sheaf theoretic modifications of formal schemes and apply this criterion to moduli spaces of parahoric level structures on local shtukas. In the proof, we introduce nice classes of equivariant profinite perfectoid covers and study geometric quotients of perfectoid formal schemes by profinite groups. As a corollary, we obtain a construction of (part of) integral models of local and global Shimura varieties under hyperspecial levels from those at hyperspecial levels.
Paper Structure (31 sections, 110 theorems, 4 equations)

This paper contains 31 sections, 110 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. The structure of the paper
  3. Preliminaries on adic spaces
  4. General properties
  5. Adic spectra of reduced excellent complete adic rings
  6. Finiteness criterion
  7. Formal modifications
  8. The theory of $v$-sheaves and kimberlites
  9. Basic facts on $v$-sheaves
  10. Kimberlites and thick closures
  11. Geometric quotients
  12. Perfectoid formal schemes
  13. $F$-standard ideals
  14. Adic perfectoid rings
  15. Continuous arc-topology
  16. ...and 16 more sections

Key Result

Theorem 1

(cor:vshfmodif) Let $R$ be a reduced excellent complete adic ring admitting a good cover $(R_\bullet,\Gamma_\bullet)$. Let $Y$ be a thick prekimberlite formally adic over ${\operatorname{Spd}}(R)$ with $Y^{\operatorname{an}} \cong {\operatorname{Spd}}(R)^{\operatorname{an}}$. Suppose that $Y^{\opera

Theorems & Definitions (237)

  • Theorem 1
  • Theorem 2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 227 more