Admissible pairs and p-adic Hodge structures, I : Transcendence of the de Rham lattice

For an algebraically closed nonarchimedean extension C/‫ޑ‬ p , we define a Tannakian category of p-adic Hodge structures over C that is a local, p-adic structural analog of the global, archimedean category of ‫-ޑ‬Hodge structures in complex geometry.In this setting the filtrations of classical Hodge theory must be enriched to lattices over a complete discrete valuation ring, Fontaine's integral de Rham period ring B + dR , and a pure p-adic Hodge structure is then a ‫ޑ‬ p -vector space equipped with a B + dR -lattice satisfying a natural condition analogous to the transversality of the complex Hodge filtration with its conjugate.We show p-adic Hodge structures are equivalent to a full subcategory of basic objects in the category of admissible pairs, a toy category of cohomological motives over C that is equivalent to the isogeny category of rigidified Breuil-Kisin-Fargues modules and closely related to Fontaine's p-adic Hodge theory over p-adic subfields.As an application, we characterize basic admissible pairs with complex multiplication in terms of the transcendence of p-adic periods.This generalizes an earlier result for one-dimensional formal groups and is an unconditional, local, p-adic analog of a global, archimedean characterization of CM motives over ‫ރ‬ conditional on the standard conjectures, the Hodge conjecture, and the Grothendieck period conjecture (known unconditionally for abelian varieties by work of Cohen, Shiga and Wolfart).