Reduction mod $p$ of semi-stable representations of some super-Breuil weights

TL;DR

This paper computes the mod p reductions of certain semi-stable Galois representations for specific weights and demonstrates the applicability of local Langlands techniques beyond previously known weight ranges.

Contribution

It extends the range of weights for which mod p reductions of semi-stable representations can be explicitly determined using local Langlands methods.

Findings

Mod p reductions are determined for weights outside the previously known range.

Techniques from [CG24] are effective for weights in [p+5, 2p] and [2p+6, 3p+1].

The bound on v_p(ℒ) can be improved for certain weights.

Abstract

We determine the mod $p$ reductions of the semi-stable representations $V_{k, \mathcal{L}}$ of weight $k \in [p + 5, 2p]\cup[2p + 6, 3p + 1]$ and $v_p(\mathcal{L}) < 1-k/2$ for primes $p \geq 5$. In particular, this shows that the techniques introduced in [CG24] involving the $p$-adic and mod $p$ local Langlands correspondences can be used to compute the reduction of $V_{k, \mathcal{L}}$ outside the range $k \in [3, p + 1]$. Moreover, this shows that the bound on $v_p(\mathcal{L})$ given by Bergdall-Levin-Liu [BLL23] can be improved, at least for weights $k \in [2p + 6, 3p + 1]$.