p-torsion coefficient systems for SL2(Qp) and GL2(Qp)

We show that the categories of smooth SL2(Qp)-representations (resp. GL2(Qp)-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 N0 of the unipotent radical of a Borel subgroup.