Reductions of some two-dimensional crystalline representations via Kisin modules

TL;DR

This paper classifies certain two-dimensional crystalline Galois representations using Kisin modules, identifies their reductions in specific slope ranges, and improves existing theorems on their semisimple reductions.

Contribution

It explicitly determines rational and integral Kisin modules for these representations and computes their reductions, extending previous results in the field.

Findings

Reduction is constant in the specified slope range

Identifies integral Kisin modules for certain representations

Improves on previous theorems by Berger, Li, and Zhu

Abstract

We determine rational Kisin modules associated with two-dimensional, irreducible, crystalline representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ of Hodge-Tate weights $0, k-1$. If the slope is larger than $\lfloor \frac{k-1}{p} \rfloor$, we further identify an integral Kisin module, which we use to calculate the semisimple reduction of the Galois representation. In that range, we find that the reduction is constant, thereby improving on a theorem of Berger, Li, and Zhu.