A prismatic Riemann-Hilbert correspondence

Abstract

We study the category of constructible $p$-Kummer-\'{e}tale sheaves on the log adic generic fiber $(\mathrm{X},\mathcal{M}_{\mathrm{D}})_\eta$ for the stratification induced by a horizontal normal crossings divisor $\mathrm{D}$, showing that it is antiequivalent to a category of finitely generated unit Laurent crystals on the log prismatic site. We give a local description of finitely generated unit crystals on a framed log affine in terms of $(\varphi,\Gamma)$-modules, and show pushforward compatibility for strict log smooth proper morphisms.