2023/08/21 by Howe, Sean, Klevdal, Christian · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2308.11065
For an algebraically closed non-archimedean extension C/ℚp, we define a Tannakian category of p-adic Hodge structures over C that is a local, p-adic analog of the global, archimedean category of ℚ-Hodge 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 ℚp-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 ℂ conditional on the standard conjectures, the Hodge conjecture, and the Grothendieck period conjecture (known unconditionally for abelian varieties by work Cohen and Shiga and Wolfart).