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Integral filtered Sen theory and applications

Hui Gao, Tong Liu

TL;DR

This work develops an integral version of filtered Sen theory for integral semi-stable Galois representations by integrating Nygaard, Hodge, and conjugate filtrations on Breuil–Kisin modules and their reductions. It introduces an amplified Sen operator and proves 1-degree and, in mod $p$, p-degree shrinking on the conjugate filtration, with a robust link to prismatic F-crystals and Hodge–Tate crystals. The authors establish vanishing and torsion bounds for graded pieces of the integral Hodge filtration, provide new perspectives on Frobenius shapes, and reprove mod $p$ filtrations results originally due to Gee–Kisin, Bhatt–Gee–Kisin, and GLS14 through a more explicit, filtration-centric approach. A significant part of the paper connects these filtered Sen phenomena to prismatic interpretations, stabilized and truncated operators, and a flat mod $p$ subring that enables control over the interaction between filtrations and Frobenius. The framework yields a cohesive, explicit bridge between classical integral p-adic Hodge operators and modern prismatic machinery, with potential implications for Serre weight problems and future ramified generalizations.

Abstract

We study Nygaard-, conjugate-, and Hodge filtrations on the many variants of Breuil--Kisin modules associated to integral semi-stable Galois representations. This leads to an integral Sen operator satisfying certain ``$1$-degree shrinking" on the increasing conjugate filtration, and (in special cases) a mod $p$ Sen operator satisfying certain ``$p$-degree shrinking". These constructions are related with prismatic $F$-crystals, Hodge--Tate crystals and $F$-gauges, and have explicit relations with classical (non-prismatic) operators. As applications, we obtain vanishing and torsion bound results on graded of the integral Hodge filtration; our explicit methods also recover results of Gee--Kisin and Bhatt--Gee--Kisin concerning the mod $p$ Hodge filtrations and Frobenius structures.

Integral filtered Sen theory and applications

TL;DR

This work develops an integral version of filtered Sen theory for integral semi-stable Galois representations by integrating Nygaard, Hodge, and conjugate filtrations on Breuil–Kisin modules and their reductions. It introduces an amplified Sen operator and proves 1-degree and, in mod , p-degree shrinking on the conjugate filtration, with a robust link to prismatic F-crystals and Hodge–Tate crystals. The authors establish vanishing and torsion bounds for graded pieces of the integral Hodge filtration, provide new perspectives on Frobenius shapes, and reprove mod filtrations results originally due to Gee–Kisin, Bhatt–Gee–Kisin, and GLS14 through a more explicit, filtration-centric approach. A significant part of the paper connects these filtered Sen phenomena to prismatic interpretations, stabilized and truncated operators, and a flat mod subring that enables control over the interaction between filtrations and Frobenius. The framework yields a cohesive, explicit bridge between classical integral p-adic Hodge operators and modern prismatic machinery, with potential implications for Serre weight problems and future ramified generalizations.

Abstract

We study Nygaard-, conjugate-, and Hodge filtrations on the many variants of Breuil--Kisin modules associated to integral semi-stable Galois representations. This leads to an integral Sen operator satisfying certain ``-degree shrinking" on the increasing conjugate filtration, and (in special cases) a mod Sen operator satisfying certain ``-degree shrinking". These constructions are related with prismatic -crystals, Hodge--Tate crystals and -gauges, and have explicit relations with classical (non-prismatic) operators. As applications, we obtain vanishing and torsion bound results on graded of the integral Hodge filtration; our explicit methods also recover results of Gee--Kisin and Bhatt--Gee--Kisin concerning the mod Hodge filtrations and Frobenius structures.

Paper Structure

This paper contains 47 sections, 67 theorems, 269 equations.

Table of Contents

  1. Introduction
  2. Filtered Sen theory
  3. Applications: Hodge filtrations and Frobenius shapes
  4. Structure of the paper
  5. Notations and conventions
  6. Review of conjugate filtration
  7. Conjugate filtrations and gradeds
  8. Adapted bases of filtrations
  9. A sub-conjugate filtration
  10. Modules attached to semi-stable representations
  11. Breuil's $S_{K_{0}}$-modules and Kisin's $\mathcal{O}$-modules
  12. Integral semi-stable representations and Breuil--Kisin ${G_K}$-modules
  13. Relation of Nygaard filtrations
  14. Integral Sen theory for semi-stable representations
  15. Sen theory over the Kummer tower
  16. ...and 32 more sections

Key Result

Theorem 1.2

(Let $T$ be a semi-stable representation). There is a constant ${\mathfrak{c}} \in K$ (with explicit expression ${\mathfrak{c}}=(\theta_{\mathrm{Fon}}(u\lambda'))^{-1}$, cf § sec: fil sen), such that the scaled operator ---which we call the negative ${K_{\infty}}$-Sen operator (cf. Remark rem: negative Sen op)--- satisfies the following:

Theorems & Definitions (176)

  • Theorem 1.2: cf. Thm. \ref{['thm: rational sen shift']}
  • proof : Idea of proof for Theorem \ref{['thm: intro rational sen']}.
  • Remark 1.3
  • Remark 1.4: "1-degree shrinking" vs. "Griffiths transversality"
  • Theorem 1.5: cf. Theorems \ref{['thm: integral sen integral conj fil']} and \ref{['thm: mod p Sen op fil']}
  • proof : Idea of proof.
  • Remark 1.6
  • Theorem 1.8: Theorem \ref{['prop: p griffiths']}
  • proof : Idea of proof.
  • Proposition 1.9: cf. Proposition \ref{['prop-pGT-A-level']}
  • ...and 166 more