The $p$-arithmetic homology of mod $p$ representations of $\mathrm{GL}_2(\mathbb{Q}_p)$

We compute the non-Eisenstein systems of Hecke eigenvalues contributing to the $p$-arithmetic homology of irreducible smooth mod $p$ representations $\pi$ of $\mathrm{GL}_2(\mathbb{Q}_p)$ and to the cohomology of their duals. We show that in most cases they are associated to odd irreducible 2-dimensional Galois representations whose local component at $p$ corresponds under the mod $p$ local Langlands correspondence to a smooth representation that contains $\pi$ as a subrepresentation.