Reduction modulo $p$ of certain semi-stable representations

TL;DR

This paper constructs Galois stable lattices in certain 3-dimensional semi-stable representations of the Galois group of Q_p, computes their mod p reductions, and analyzes irreducibility properties.

Contribution

It provides explicit constructions of strongly divisible modules for semi-stable Galois representations with specific weights, advancing understanding of their mod p reductions.

Findings

Identified Galois stable lattices in semi-stable representations

Computed Breuil modules for mod p reductions

Determined conditions for irreducibility of reductions

Abstract

Let $p>3$ be a prime number and let $G_{\mathbb{Q}_p}$ be the absolute Galois group of $\mathbb{Q}_p$. In this paper, we find Galois stable lattices in the irreducible $3$-dimensional semi-stable and non-crystalline representations of $G_{\mathbb{Q}_p}$ with Hodge--Tate weights $(0,1,2)$ by constructing their strongly divisible modules. We also compute the Breuil modules corresponding to the mod $p$ reductions of the strongly divisible modules, and determine which of the semi-stable representations has an absolutely irreducible mod $p$ reduction.