Let p be a prime number, K a finite unramified extension of \(\mathbb {Q}_{p}\) and \(\mathbb {F}\) a finite extension of \(\mathbb {F}_{p}\) . Using perfectoid spaces we associate to any finite-dimensional continuous representation \(\overline{\rho }\) of \(\textrm{Gal}({\overline{K}}/K)\) over \(\mathbb {F}\) an étale \((\varphi ,\mathcal {O}_K^\times )\) -module \(D_A^\otimes (\overline{\rho })\) over a completed localization A of \(\mathbb {F}\llbracket \mathcal {O}_K\rrbracket \) . We conjecture that one can also associate an étale \((\varphi ,\mathcal {O}_K^\times )\) -module \(D_A(\pi )\) to any smooth representation \(\pi \) of \({{\,\textrm{GL}\,}}_2(K)\) occurring in some Hecke eigenspace of the mod p cohomology of a Shimura curve, and that moreover \(D_A(\pi )\) is isomorphic (up to twist) to \(D_A^\otimes (\overline{\rho })\) , where \(\overline{\rho }\) is the underlying 2-dimensional representation of \(\textrm{Gal}({\overline{K}}/K)\) . Using previous work of the same authors, we prove this conjecture when \(\overline{\rho }\) is semi-simple and sufficiently generic.
