On the reductions of certain two-dimensional crystalline representations

TL;DR

This paper investigates the reductions modulo p of two-dimensional crystalline p-adic Galois representations, providing partial proofs for a conjecture relating irreducibility to weight and slope conditions.

Contribution

It proves specific cases of a conjecture on the irreducibility of reductions for certain slopes up to (p-1)/2.

Findings

Confirmed irreducibility for slopes up to (p-1)/2 in some cases

Extended understanding of the structure of crystalline Galois representations

Provided evidence supporting the conjecture by Breuil, Buzzard, and Emerton.

Abstract

The question of computing the reductions modulo $p$ of two-dimensional crystalline $p$-adic Galois representations has been studied extensively, and partial progress has been made for representations that have small weights, very small slopes, or very large slopes. It was conjectured by Breuil, Buzzard, and Emerton that these reductions are irreducible if they have even weight and non-integer slope. We prove some instances of this conjecture for slopes up to $\frac{p-1}{2}$.