We start with background that goes into an Iwahori-theoretic reformulation of the mod p Local Langlands Correspondence (§2). We then explain some classical p-adic functional analytic results (§3) that go into defining the p-adic Banach space (§4) attached to a two-dimensional semi-stable representation \(V_{k,\mathcal {L}}\) of the Galois group of \(\mathbb {Q}_p\) of weight k and \(\mathcal {L}\) -invariant \(\mathcal {L}\) under the p-adic Local Langlands correspondence. We then sketch how to compute the reduction of a lattice in this Banach space, which along with the Iwahori mod p LLC, allows one to completely determine the mod p reduction of \(V_{k,\mathcal {L}}\) for all weights \(3 \le k \le p+1\) and all \(\mathcal {L}\) for \(p \ge 5\) (§5). These notes are a summary of our joint work with Anand Chitrao [21]. Emphasis is placed on motivation and background rather than completeness.
