$p$-torsion coefficient systems for ${\mathrm{SL}}_2({\mathbf{Q}}_p)$ and ${\mathrm{GL}}_2({\mathbf{Q}}_p)$,

Abstract

We show that the categories of smooth ${\mathrm{SL}}_2({\mathbb{Q}}_p)$-representations (resp. ${\mathrm{GL}}_2({\mathbb{Q}}_p)$-representations) of level $1$ on $p$-torsion modules are equivalent with certain explicitly described equivariant coefficient systems on the Bruhat-Tits tree; the coefficient system assigned to a representation $V$ assigns to an edge $\tau$ the invariants in $V$ under the pro-$p$-Iwahori subgroup corresponding to $\tau$. The proof relies on computations of the group cohomology of a compact open subgroup group $N_0$ of the unipotent radical of a Borel subgroup.