On non-abelian Lubin–Tate theory for

Abstract

We analyse the$\text{mod}~p$étale cohomology of the Lubin–Tate tower both with compact support and without support. We prove that there are no supersingular representations in the$H_{c}^{1}$of the Lubin–Tate tower. On the other hand, we show that in$H^{1}$of the Lubin–Tate tower appears the$\text{mod}~p$local Langlands correspondence and the$\text{mod}~p$local Jacquet–Langlands correspondence, which we define in the text. We discuss the local-global compatibility part of the Buzzard–Diamond–Jarvis conjecture which appears naturally in this context.