Reduction modulo $p$ of certain semi-stable representations

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.

34 pages, Contains minor correction from the previous version, Comments welcome