On families of strongly divisible modules of rank 2

Let $p$ be an odd prime, and $\mathbf{Q}_{p^f}$ the unramified extension of $\mathbf{Q}_p$ of degree $f$. In this paper, we reduce the problem of constructing strongly divisible modules for $2$-dimensional semi-stable non-crystalline representations of $\mathrm{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_{p^f})$ with Hodge--Tate weights in the Fontaine--Laffaille range to solving systems of linear equations and inequalities. We also determine the Breuil modules corresponding to the mod-$p$ reduction of the strongly divisible modules. We expect our method to produce at least one Galois-stable lattice in each such representation for general $f$. Moreover, when the mod-$p$ reduction is an extension of distinct characters, we further expect our method to provide the two non-homothetic lattices. As applications, we show that our approach recovers previously known results for $f=1$ and determine the mod-$p$ reduction of the semi-stable representations with some small Hodge--Tate weights when $f=2$.