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

TL;DR

This paper computes the $p$-arithmetic homology for mod $p$ representations of $ ext{GL}_2(Q_p)$, linking Hecke eigenvalues to Galois representations via the mod $p$ local Langlands correspondence.

Contribution

It explicitly determines the non-Eisenstein Hecke eigenvalues associated with irreducible smooth mod $p$ representations of $ ext{GL}_2(Q_p)$ and their relation to Galois representations.

Findings

Hecke eigenvalues correspond to odd irreducible 2-dimensional Galois representations.

Most cases show a correspondence between local components and smooth representations containing $ ho$.

The work connects $p$-arithmetic homology with the mod $p$ local Langlands program.

Abstract

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.