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

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 $U(1,1)({\mathbb{Q}}_{p^2}/{\mathbb{Q}}_p)$. It focuses in particular on the potential of deformation theory to relate these correspondences.

To appear in the Conference Proceedings of Women In Numbers Europe III