Reductions of Galois representations for slopes in $(1,2)$

TL;DR

This paper characterizes the semi-simplified mod p reductions of specific crystalline Galois representations with slopes between 1 and 2, utilizing the mod p Local Langlands Correspondence for GL_2(Q_p).

Contribution

It provides a detailed description of the mod p reductions for Galois representations with slopes in (1,2) and all weights, expanding understanding of their structure and submodules.

Findings

Semi-simplification of mod p reductions described for slopes in (1,2)

Complete characterization of submodules generated by second highest monomials

Application of mod p Local Langlands Correspondence in the analysis

Abstract

We describe the semi-simplification of the mod $p$ reduction of certain crystalline two dimensional local Galois representations of slopes in the interval $(1,2)$ and all weights. The proof uses the mod $p$ Local Langlands Correspondence for $GL_2(Q_p)$. We also give a complete description of the submodules generated by the second highest monomial in the mod $p$ symmetric power representations of $GL_2(F_p)$.