Towards a mod-$p$ Lubin-Tate theory for $\GL_2$ over totally real fields

TL;DR

This paper advances the understanding of the mod p local Langlands correspondence by connecting it with the cohomology of Lubin-Tate towers, leveraging conjectures and prior results in the context of totally real fields.

Contribution

It provides a realization of the conjectural mod p local Langlands correspondence within the cohomology of Lubin-Tate towers for totally real fields, building on existing conjectures and cohomological studies.

Findings

Realization of mod p local Langlands in Lubin-Tate cohomology

Connection between completed cohomology and Lubin-Tate towers

Extension of known results from modular curves to totally real fields

Abstract

We show that the conjectural mod $p$ local Langlands correspondence can be realised in the mod $p$ cohomology of the Lubin-Tate towers. The proof utilizes a well known conjecture of Buzzard-Diamond-Jarvis \cite[Conj. 4.9]{BDJ10}, a study of completed cohomology of the ordinary and supersingular locus of the Shimura curves for a totally real field $F$ and of mod $l(\neq p)$ local Langlands correspondence as given by Emerton-Helm \cite{EmertonHelm14}. %And then we connect the completed cohomlgy with the cohomology of Lubin-Tate towers. In the case of modular curves a similar theorem was obtained by Chojecki \cite{Cho15}.