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

Abstract

Let K be a mixed characteristic complete discrete valuation field with residue field admitting a finite p-basis, and let G K be the Galois group. Inspired by Liu and Zhu's construction of p-adic Simpson and Riemann-Hilbert correspondences over rigid analytic varieties, we construct such correspondences for representations of G K . As an application, we prove a Hodge-Tate (resp. de Rham) "rigidity" theorem for p-adic representations of G K , generalizing a result of Morita. Contents 1. Introduction 1 2. Fontaine rings and Fontaine modules 6 3. Change of base fields 11 4. Sen theory and Tate-Sen decompletion 13 5. p-adic Simpson correspondence and Hodge-Tate rigidity 18 6. Sen-Fontaine theory 22 7. p-adic Riemann-Hilbert correspondence and de Rham rigidity 25 References 27