Hodge-Tate prismatic crystals and Sen theory

TL;DR

This paper develops a new approach to Sen theory using Kummer towers instead of cyclotomic towers, and proves that Hodge-Tate prismatic crystals are characterized by the Sen operator, linking them to nearly Hodge-Tate representations.

Contribution

It reconstructs Sen theory via Kummer towers and proves a conjecture relating Hodge-Tate prismatic crystals to the Sen operator, expanding the understanding of these structures.

Findings

Sen theory can be reconstructed using Kummer towers.

Hodge-Tate prismatic crystals are determined by the Sen operator.

Category of Hodge-Tate prismatic crystals is equivalent to nearly Hodge-Tate representations.

Abstract

Let $K$ be a mixed characteristic complete discrete valuation field with perfect residue field, and let $K_\infty/K$ be a Kummer tower extension by adjoining a compatible system of $p$-power roots of a chosen uniformizer. We use this Kummer tower to reconstruct Sen theory which classically is obtained using the cyclotomic tower. Using this Sen theory over the Kummer tower, we prove a conjecture of Min-Wang which predicts that Hodge-Tate prismatic crystals are determined by the Sen operator; this implies that the category of (rational) Hodge-Tate prismatic crystals is equivalent to the category of nearly Hodge-Tate representations.