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

Abstract

In this paper, we show that the conjectural mod p local Langlands correspondence can be realized in the mod p cohomology of the Lubin-Tate towers. The proof utilizes a wellknown 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 F and of mod l(? p) local Langlands correspondence as given by Emerton- Helm [20]. In the case of modular curves, a similar theorem was obtained by Chojecki [13].