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

TL;DR

This paper explicitly determines the mod p reductions of all two-dimensional semi-stable Galois representations with specific weights and invariants, using the Iwahori mod p Local Langlands Correspondence and Breuil's Banach spaces.

Contribution

It provides a complete description of mod p reductions for a class of semi-stable representations, including constants in unramified characters, advancing understanding of the mod p Local Langlands Correspondence.

Findings

Explicit mod p reductions for all representations in the specified range.

Complete description of constants in unramified characters.

Application of Iwahori theoretic methods to the mod p Local Langlands Correspondence.

Abstract

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 $3 \leq k \leq p+1$ and $\mathcal{L}$-invariants $\mathcal{L}$ for primes $p \geq 5$. In particular, we describe the constants appearing in the unramified characters completely. The proof involves computing the reduction of Breuil's $\mathrm{GL}_2(\mathbb{Q}_p)$-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.