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

Abstract

We show that the categories of smooth ${\rm SL}_2({\mathbb Q}_p)$-representations (resp. ${\rm GL}_2({\mathbb Q}_p)$-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 $τ$ the invariants in $V$ under the pro-$p$-Iwahori subgroup corresponding to $τ$. 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.