Potentially crystalline deformation rings in the ordinary case

TL;DR

This paper investigates the structure of potentially crystalline deformation rings for a specific class of Galois representations, showing smoothness and ordinarity under certain conditions, advancing understanding in p-adic Hodge theory.

Contribution

It demonstrates that the potentially crystalline deformation space is formally smooth and all lifts are ordinary in the considered setting, with a detailed analysis of Breuil modules and monodromy conditions.

Findings

Deformation space is formally smooth over 

All lifts are ordinary in this setting

The mod p locus of Breuil modules with descent datum is smooth

Abstract

We study potentially crystalline deformation rings for a residual, ordinary Galois representation $\overline{\rho}: G_{\mathbb{Q}_p}\rightarrow \mathrm{GL}_3(\mathbb{F}_p)$. We consider deformations with Hodge-Tate weights $(0,1,2)$ and inertial type chosen to contain exactly one Fontaine-Laffaille modular weight for $\overline{\rho}$. We show that, in this setting, the potentially crystalline deformation space is formally smooth over $\mathbb{Z}_p$ and any potentially crystalline lift is ordinary. The proof requires an understanding of the condition imposed by the monodromy operator on Breuil modules with descent datum, in particular, that this locus mod p is formally smooth.