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

TL;DR

This work develops and extends $p$-adic Simpson and Riemann--Hilbert correspondences to the imperfect residue field setting, building on Liu--Zhu’s framework to represent $G_K$-modules via Higgs and filtered-connection data. The authors establish base-change compatibilities across embeddings $K \hookrightarrow L$ with finite $p$-basis and prove Hodge--Tate and de Rham rigidity theorems, showing that HT/de Rham properties descend along suitable field extensions. The approach synthesizes Fontaine rings, Sen theory (including Brinon and Andreatta--Brinon variants), Tate--Sen decompletion, and a decompleted RH theory to produce a coherent, functorial picture that generalizes Morita’s rigidity and supports integral $p$-adic Hodge theory in the imperfect residue field context. By framing the constructions through base-change and decomposed structures, the paper provides a robust platform for relative, integral $p$-adic Hodge theory in non-perfect residue fields, with potential applications to relative $p$-adic monodromy and beyond.

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.

Theorems & Definitions (97)

• Theorem 1.1.2
• Remark 1.1.3
• Theorem 1.1.4
• Remark 1.1.5
• Theorem 1.1.6
• Remark 1.1.7
• Remark 2.0.2
• Definition 2.1.4
• Remark 2.1.5
• Definition 2.1.6