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

TL;DR

This paper advances the understanding of two-dimensional crystalline Galois representations by classifying their slopes over non-subtle weight space components, confirming a conjecture about integrality of slopes.

Contribution

It provides a complete classification of certain reducible p-adic Galois representations over non-subtle components, extending previous partial results to all slopes.

Findings

Confirmed the integrality of slopes in the specified setting.

Classified reducible crystalline representations over non-subtle components.

Extended previous work to include non-integer slopes.

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 completely classify the aforementioned representations over the non-subtle components of weight space, both for integer and non-integer slopes.