On 𝑝-adic Simpson and Riemann-Hilbert correspondences in the imperfect residue field case

Let K K be a mixed characteristic complete discrete valuation field with residue field admitting a finite p p -basis, and let G K G_K be the Galois group. Inspired by Liu and Zhu’s construction of p p -adic Simpson and Riemann–Hilbert correspondences over rigid analytic varieties, we construct such correspondences for representations of G K G_K . As an application, we prove a Hodge–Tate (resp. de Rham) “rigidity” theorem for p p -adic representations of G K G_K , generalizing a result of Morita.