Prismatic crystals and $p$-adic 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.