On the reductions of certain two-dimensional crystalline representations, III

TL;DR

This paper advances the understanding of the slopes of reducible p-adic Galois representations, confirming a conjecture for subtle components with slopes below a certain threshold, thus extending previous results.

Contribution

It proves the Breuil-Buzzard-Emerton conjecture for subtle components with slopes less than (p-1)/2, expanding the scope of earlier work on non-subtle components.

Findings

Confirmed the conjecture for subtle components with slopes < (p-1)/2

Extended previous results to a broader class of components

Provided new insights into the structure of crystalline representations

Abstract

A conjecture of Breuil, Buzzard, and Emerton says that the slopes of certain reducible $p$-adic Galois representations must be integers. In previous work we showed this conjecture for representations that lie over certain non-subtle components of weight space. This article is a continuation of that work in which we also show the conjecture for the subtle components for slopes less than $\frac{p-1}{2}$.