Une \'etude des repr\'esentations modulo $p$ de SL(2,F)

TL;DR

This paper investigates mod p representations of SL(2,F), connecting them to GL(2,F) representations and providing explicit descriptions of supersingular representations for F = Q_p, revealing their packet structures.

Contribution

It extends the study of mod p representations from GL(2,F) to SL(2,F), offering explicit descriptions of supersingular representations in the case F = Q_p.

Findings

Linked SL(2,F) mod p representations to GL(2,F) representations

Explicitly described supersingular representations for F = Q_p

Identified packet sizes of supersingular representations as at most 2

Abstract

Following what Barthel-Livn\'e and Breuil made for GL(2,F), we study mod $p$ representations of SL(2,F) for F a complete non-archimedean local field of residual characteristic p and with finite residue field. In particular, we link these representations to the mod p representations of GL(2,F) and, when F = Q_p, we give an explicit description of the so-called supersingular representations, that do appear by packets of size at most 2.