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Log Prismatic Dieudonné theory and its application to Shimura varieties

Kentaro Inoue

TL;DR

The paper develops a comprehensive log-prismatic Dieudonné theory and embeds it into the study of Shimura varieties of Hodge type with toroidal boundary. It introduces Kummer-flat descent and a robust kfl framework for log-prismatic crystals, tying log BT and log p-divisible structures to crystals and crystals with Frobenius. A Tannakian formalism for log-prismatic objects is established, enabling G-structure descriptions and torsor twists, which are then applied to toroidal compactifications to construct log prismatic realizations and crystalline-étale comparison isomorphisms. Consequently, Lovering’s conjecture on p-adic comparison isomorphisms is proven in the setting of Shimura varieties with boundary, broadening the scope of $p$-adic Hodge theory to degenerating boundary objects.

Abstract

We study the log version of the prismatic Dieudonné theory established by Anschütz-Le Bras. By applying this result to the integral toroidal compactification of a Shimura variety of Hodge type, we extend the prismatic realization, originally constructed by Imai-Kato-Youcis, to the compactification. This extension enables us to prove Lovering's conjecture on $p$-adic comparison isomorphisms for Shimura varieties.

Log Prismatic Dieudonné theory and its application to Shimura varieties

TL;DR

The paper develops a comprehensive log-prismatic Dieudonné theory and embeds it into the study of Shimura varieties of Hodge type with toroidal boundary. It introduces Kummer-flat descent and a robust kfl framework for log-prismatic crystals, tying log BT and log p-divisible structures to crystals and crystals with Frobenius. A Tannakian formalism for log-prismatic objects is established, enabling G-structure descriptions and torsor twists, which are then applied to toroidal compactifications to construct log prismatic realizations and crystalline-étale comparison isomorphisms. Consequently, Lovering’s conjecture on p-adic comparison isomorphisms is proven in the setting of Shimura varieties with boundary, broadening the scope of -adic Hodge theory to degenerating boundary objects.

Abstract

We study the log version of the prismatic Dieudonné theory established by Anschütz-Le Bras. By applying this result to the integral toroidal compactification of a Shimura variety of Hodge type, we extend the prismatic realization, originally constructed by Imai-Kato-Youcis, to the compactification. This extension enables us to prove Lovering's conjecture on -adic comparison isomorphisms for Shimura varieties.

Paper Structure

This paper contains 24 sections, 79 theorems, 217 equations.

Table of Contents

  1. Introduction
  2. Kfl topology on fs log formal schemes
  3. Kummer morphisms of log formal schemes
  4. Kfl sites of fs log formal schemes
  5. Kfl vector bundles on fs log formal schemes
  6. Log $p$-divisible groups on fs formal log schemes
  7. Log finite group schemes
  8. Log $p$-divisible groups
  9. Kfl log prismatic crystals
  10. Log prismatic crystals
  11. Kfl prismatic --crystals
  12. Kfl crystalline crystals
  13. Log prismatic Dieudonné theory
  14. Review on prismatic Dieudonné theory
  15. Definition of the log prismatic Dieudonné functor
  16. ...and 9 more sections

Key Result

Theorem A

There exists a unique $\mathbb{Z}_{p}$-linear exact tensor functor with $T_{\text{\'{e}t}}\circ \omega_{{\mathlarger{\mathbbl{\Delta}}},\mathrm{log}}\cong \omega_{\mathrm{k\text{\'{e}t}}}$ and $\omega_{{\mathlarger{\mathbbl{\Delta}}},\mathrm{log}}|_{\widehat{\mathscr{S}}_{K}(G,X)}\cong \omega_{{\mathlarger{\mathbbl{\Delta}}}}$, where $T_{\text{\'{e}t}}$ is the étal

Theorems & Definitions (199)

  • Theorem A: see Theorem \ref{['log pris realization']}
  • Theorem B
  • Theorem C: see Theorem \ref{['log pris dieudonne equiv']}
  • Remark 1.1
  • Definition 2.1: Kummer morphisms of log formal schemes
  • Lemma 2.2: kat21
  • proof
  • Lemma 2.3: cf. kat21
  • proof
  • Definition 2.4: Log flat morphisms of log schemes, kat21
  • ...and 189 more