Integral filtered Sen theory and applications

TL;DR

This paper develops an integral Sen theory framework for semi-stable Galois representations, connecting prismatic crystals and classical operators, leading to new results on Hodge filtrations and Frobenius structures.

Contribution

It introduces an integral Sen operator with shrinking properties and relates prismatic F-crystals to classical operators, advancing the understanding of Galois representations.

Findings

Vanishing and torsion bounds on graded Hodge filtrations

Explicit relations between prismatic and classical operators

Recovery of known results on mod p Hodge filtrations

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.