Prismatic crystals over the de Rham period sheaf

Abstract: Let $\mathcal{O}K$ be a mixed characteristic complete discrete valuation ring with perfect residue field. We study $\mathbb{B}\mathrm{dR}^+$-crystals on the (log-) prismatic site of $\mathcal{O}K$, which are crystals defined over the de Rham period sheaf. We first classify these crystals using certain log connections. By constructing a Sen--Fontaine theory for $\mathbf{B}{\mathrm{dR}}^+$-representations over a Kummer tower, we further classify these crystals by (log-) nearly de Rham representations. In addition, we compare (log-) prismatic cohomology of these crystals with the corresponding Sen--Fontaine cohomology and Galois cohomology.