From $p$-modular to $p$-adic Langlands correspondences for $\operatorname{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$: deformations in the non-supercuspidal case

TL;DR

This paper reviews the current understanding of $p$-adic and $p$-modular Langlands correspondences for a specific unitary group, emphasizing how deformation theory could connect these conjectural frameworks.

Contribution

It provides a survey of known results and explores the role of deformation theory in relating $p$-adic and $p$-modular Langlands correspondences in the non-supercuspidal case.

Findings

Highlights the current state of conjectural correspondences.

Emphasizes deformation theory's potential in linking these frameworks.

Focuses on the unramified quasi-split unitary group U(1,1).

Abstract

This paper surveys what is known about (conjectural) $p$-adic and $p$-modular semisimple Langlands correspondences in the non-supercuspidal setting for the unramified quasi-split unitary group $\operatorname{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$. It focuses in particular on the potential of deformation theory to relate these correspondences.