The semi-stable Local Langlands Correspondence

We start with background that goes into an Iwahori-theoretic reformulation of the mod $p$ Local Langlands Correspondence (\S 2). We then explain some classical $p$-adic functional analytic results (\S 3) that go into defining the $p$-adic Banach space (\S 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 $3 \leq k \leq p+1$ and all ${\mathcal L}$ for $p \geq 5$ (\S 5). These notes are a summary of our joint work with Anand Chitrao [CG24]. Emphasis is placed on motivation and background rather than completeness.