The semi-stable Local Langlands Correspondence

Abstract

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 [CG24]. Emphasis is placed on motivation and background rather than completeness.

45 pages