A stacky $p$-adic Riemann--Hilbert correspondence on Hitchin-small locus

TL;DR

This paper establishes a new equivalence between Hitchin-small integrable connections and local systems on a semi-stable formal scheme over a perfectoid field using a novel period sheaf with connection.

Contribution

It introduces a new period sheaf with connection to prove a $p$-adic Riemann--Hilbert correspondence for Hitchin-small loci.

Findings

Established an equivalence between Hitchin-small integrable connections and local systems.

Introduced a new period sheaf with connection $( ext{O}B_{ ext{dR}, ext{pd}}^+, d)$.

Extended the $p$-adic Riemann--Hilbert correspondence to semi-stable formal schemes.

Abstract

Let $C$ be an algebraically closed perfectoid field over $\mathbb{Q}_p$ with the ring of integer $\mathcal{O}_C$ and the infinitesimal thickening $\Ainf$. Let $\mathfrak X$ be a semi-stable formal scheme over $\mathcal{O}_C$ with a fixed flat lifting $\widetilde{\mathfrak X}$ over $\Ainf$. Let $X$ be the generic fiber of $\mathfrak{X}$ and $\widetilde X$ be its lifting over $\BdRp$ induced by $\widetilde{\mathfrak X}$. Let $\MIC_r(\widetilde X)^{{\rm H}\text{-small}}$ and $\rL\rS_r(X,\BBdRp)^{{\rm H}\text{-small}}$ be the $v$-stacks of rank-$r$ Hitchin-small integrable connections on $X_{\et}$ and $\BBdRp$-local systems on $X_{v}$, respectively. In this paper, we establish an equivalence between these two stacks by introducing a new period sheaf with connection $(\calO\bB_{\dR,\pd}^+,\rd)$ on $X_{v}$.