Prismatic crystals and $p$-adic Riemann--Hilbert correspondence

TL;DR

This paper develops a comprehensive classification of prismatic crystals on formal schemes using explicit stratification computations, connecting them to nilpotent and enhanced connections, and relates these to $p$-adic Riemann--Hilbert correspondence and local systems.

Contribution

It introduces a new classification framework for relative and absolute prismatic crystals via explicit stratifications and connections, extending the $p$-adic Riemann--Hilbert correspondence.

Findings

Classified local relative crystals by nilpotent connections.

Classified local absolute crystals by enhanced connections.

Connected prismatic crystals to $p$-adic local systems and Riemann--Hilbert correspondence.

Abstract

We systematically study relative and absolute ${\Delta}_{\mathrm{dR}}^+$-crystals on the (log-) prismatic site of a smooth (resp.~ semi-stable) formal scheme. Using explicit computation of stratifications, we classify (local) relative crystals by certain nilpotent connections, and classify (local) absolute crystals by certain enhanced connections. By using a $p$-adic Riemann--Hilbert functor and an infinite dimensional Sen theory over the Kummer tower, we globalize the results on absolute crystals and further classify them by certain small (global) $\mathbb{B}_{\mathrm{dR}}^+$-local systems.