Strong local-global compatibility in the p-adic Langlands program for U(2)

TL;DR

This paper establishes a strong local-global compatibility result in the p-adic Langlands program for U(2), connecting Galois representations, completed cohomology, and local Langlands correspondences in a unified framework.

Contribution

It proves a new compatibility result linking Galois-isotypic parts of completed cohomology with tensor products of local p-adic and classical Langlands representations.

Findings

Galois-isotypic parts of completed cohomology decompose as tensor products.

Established compatibility between p-adic and classical local Langlands correspondences.

Advances the understanding of the p-adic Langlands program for unitary groups.

Abstract

We prove that certain Galois-isotypic parts of the completed cohomology group for U(2) can be written as a completed tensor product of a representation coming from the p-adic Langlands correspondence for $GL_2(\mathbb{Q}_p)$ and a representation arising via the local Langlands correspondence in famillies of Emerton and Helm.