Towards a mod-p Lubin–Tate theory for GL2 over totally real fields

Abstract

In this paper, we show that the conjectural mod [Formula: see text] local Langlands correspondence can be realized in the mod [Formula: see text] cohomology of the Lubin–Tate towers. The proof utilizes a well-known conjecture of Buzzard–Diamond–Jarvis [8, Conjecture 4.9], a study of completed cohomology of the ordinary and supersingular locus of the Shimura curves for a totally real field [Formula: see text] and of mod [Formula: see text] local Langlands correspondence as given by Emerton–Helm [20]. In the case of modular curves, a similar theorem was obtained by Chojecki [13].