Reductions of semi-stable representations using the Iwahori mod $p$ Local Langlands Correspondence

We determine the mod $p$ reductions of all two-dimensional semi-stable representations $V_{k,\mathcal{L}}$ of the Galois group of ${\mathbb{Q}}_p$ of weights $3\le k\le p+1$ and $\mathcal{L}$-invariants $\mathcal{L}$ for primes $p\ge 5$. In particular, we describe the constants appearing in the unramified characters completely. The proof involves computing the reduction of Breuil’s ${\mathrm{G}\mathrm{L}}_2({\mathbb{Q}}_p)$-Banach space $\tilde{B}(k,\mathcal{L})$, by studying certain logarithmic functions using background material developed by Colmez, and then applying an Iwahori theoretic version of the mod $p$ Local Langlands Correspondence.