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

TL;DR

This paper extends $p$-adic Simpson and Riemann-Hilbert correspondences to mixed characteristic fields with imperfect residue fields, enabling new rigidity results for $p$-adic Galois representations.

Contribution

It constructs $p$-adic Simpson and Riemann-Hilbert correspondences in the imperfect residue field case, generalizing prior work to a broader class of fields.

Findings

Established $p$-adic Simpson and Riemann-Hilbert correspondences for $G_K$ representations.

Proved a Hodge-Tate and de Rham rigidity theorem for $p$-adic Galois representations.

Generalized Morita's results to fields with imperfect residue fields.

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.