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

TL;DR

This paper establishes an equivalence between certain smooth p-torsion representations of ${ m SL}_2({f Q}_p)$ and ${ m GL}_2({f Q}_p)$ and explicitly described equivariant coefficient systems on the Bruhat-Tits tree, using group cohomology computations.

Contribution

It provides a new categorical equivalence linking p-torsion representations to geometric coefficient systems on the Bruhat-Tits tree.

Findings

Categories of smooth p-torsion representations are equivalent to equivariant coefficient systems.

Explicit description of the coefficient system assigns invariants under pro-p-Iwahori subgroups.

Proof involves computations of group cohomology of certain compact open subgroups.

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 $\tau$ 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.