Prismatic Cohomology and the Derived Geometry of Period Morphisms over p-adic Fields

This paper develops a novel framework for studying p-adic period morphisms through the lens of derived algebraic geometry and the recently developed theory of prismatic cohomology. We construct a derived enhancement of the classical period domain, termed the "derived prismatic period stack," which parameterizes filtered (φ, N, G)-modules equipped with a prismatic structure. For a smooth proper family of algebraic varieties X → S over a p-adic formal scheme S, we define a "derived prismatic period morphism" Π∆ : S → Pder which maps a point in the base to the derived data of the relative prismatic cohomology of the corresponding fiber. Our main result establishes that, under certain mild conditions on the family (e.g., abelian type), this morphism is a formal étale map of derived stacks. This can be interpreted as a derived and p-adic analogue of the infinitesimal Torelli theorem. We leverage the categorical framework of ∞-topoi and the cotangent complex formalism of derived algebraic geometry to analyze the infinitesimal properties of Π∆. The construction provides a unified perspective on several aspects of p-adic Hodge theory, including the Kodaira-Spencer map, comparison isomorphisms, and the variation of Galois representations, packaging them into a single geometric object. We discuss potential applications to the study of arithmetic moduli spaces and the p-adic Langlands program.