De Rham prismatic crystals over $\mathcal{O}_K$

TL;DR

This paper investigates de Rham prismatic crystals over the ring of integers of a p-adic field, establishing their structure via matrices and connecting them to nearly de Rham Galois representations through a decompletion theorem.

Contribution

It introduces a matrix-based description of de Rham prismatic crystals and proves a fully faithful functor to nearly de Rham representations, advancing the understanding of their structure.

Findings

De Rham crystals are controlled by a sequence of matrices with a nilpotent initial matrix.

The functor from de Rham crystals to nearly de Rham representations is fully faithful.

A Sen style decompletion theorem for de Rham representations is established.

Abstract

We study de Rham prismatic crystals on $(\mathcal{O}_K)_{\bbold{\Delta}}$. We show that a de Rham crystal is controlled by a sequence of matrices $\{A_{m,1}\}_{m \geq 0}$ with $A_{0,1}$ "nilpotent". Using this, we prove that the natural functor from de Rham crystals over $(\mathcal{O}_K)_{\bbold{\Delta}}$ to the category of nearly de Rham representations is fully faithful. The key ingredient is a Sen style decompletion theorem for de Rham representations of $G_K$.