On non-abelian Lubin-Tate theory and analytic cohomology

Abstract

We prove that the p p -adic local Langlands correspondence for GL 2 ⁡ ( Q p ) \operatorname {GL}_2(\mathbb {Q}_p) appears in the étale cohomology of the Lubin-Tate tower at infinity. We use global methods using recent results of Emerton on the local-global compatibility, and hence our proof applies to local Galois representations which come via a restriction from global pro-modular Galois representations.