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

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 $\tilde{\mathfrak{X}}$ over $\Ainf$. Let $X$ be the generic fiber of $\mathfrak{X}$ and $\tilde{X}$ be its lifting over $\BdRp$ induced by $\tilde{\mathfrak{X}}$. 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 $X_v$.

submitted version. Compared with version 1(where we are in the good reduction case), we generalize all results to the semi-stable reduction case. Comments are welcome!