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We show that the categories of smooth ${\rm SL}_2({\mathbbQ}_p)$-representations (resp. ${\rm GL}_2({\mathbb Q}_p)$-representations) oflevel $1$ on $p$-torsion modules are equivalent with certain explicitlydescribed equivariant coefficient systems on the Bruhat-Tits tree; thecoefficient 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 compactopen subgroup group $N_0$ of the unipotent radical of a Borel subgroup.

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