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

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