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Moduli stacks of crystals and isocrystals

Gyujin Oh, Koji Shimizu

Abstract

Given a liftable smooth proper variety over $\mathbb{F}_p$, we construct the moduli stacks of crystals and isocrystals on it. We show that the former is a formal algebraic stack over $\mathbb{Z}_p$ and the latter is an adic stack -- Artin stack in rigid geometry -- over $\mathbb{Q}_p$. Both stacks come equipped with the Verschiebung endomorphism $V$ corresponding to the Frobenius pullback of (iso)crystals. We study the geometry of the $V$-fixed points over the open substack of irreducible isocrystals, which we use to geometrically count the rank one $F$-isocrystals. Along the way, we carefully develop the theory of adic stacks.

Moduli stacks of crystals and isocrystals

Abstract

Given a liftable smooth proper variety over , we construct the moduli stacks of crystals and isocrystals on it. We show that the former is a formal algebraic stack over and the latter is an adic stack -- Artin stack in rigid geometry -- over . Both stacks come equipped with the Verschiebung endomorphism corresponding to the Frobenius pullback of (iso)crystals. We study the geometry of the -fixed points over the open substack of irreducible isocrystals, which we use to geometrically count the rank one -isocrystals. Along the way, we carefully develop the theory of adic stacks.

Paper Structure

This paper contains 40 sections, 148 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Formal algebraic spaces and stacks
  3. Classical formal schemes
  4. Formal algebraic spaces
  5. Small étale site and a.q.c. sheaves
  6. Formal algebraic stacks
  7. Étale spaces and adic stacks
  8. Generalities on adic spaces
  9. The category VK
  10. Étale spaces
  11. Adic stacks
  12. Formal deformation theory
  13. Analytification and generic fiber
  14. Analytification
  15. Generic fiber
  16. ...and 25 more sections

Key Result

Theorem 1.2

The substack $\mathcal{M}_{\mathrm{isoc},\mathrm{irr}}(Z)$ of absolutely irreducible isocrystals on $Z$ admits a coarse moduli space $M_{\mathrm{isoc},\mathrm{irr}}(Z)$. Furthermore, the space $M_{\mathrm{isoc},\mathrm{irr}}^{V=\mathrm{id}}(Z)$ of the Verschiebung fixed points is discrete and of the

Theorems & Definitions (430)

  • Theorem 1.2: Rough form of Theorems \ref{['thm:Misocirr is Gm-gerbe']} and \ref{['thm:irreducible F-isocrystals discrete']}
  • Theorem 1.3: See Theorems \ref{['thm:counting rank 1 F-isocrystals']}, \ref{['thm:counting rank 1 F-isocrystals Fqm']}
  • Definition 1.4: Definitions \ref{['def:etale spaces']}, \ref{['def:adic stacks']}
  • Example 1.5
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3: stacks-project
  • Definition 2.4: Fujiwara-Kato
  • Definition 2.5: Fujiwara-Kato
  • Definition 2.6: Fujiwara-Kato, stacks-project
  • ...and 420 more