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

Howe, Sean; Klevdal, Christian

doi:10.2140/ant.2026.20.971

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

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

Sean Howe and Christian Klevdal

Vol. 20 (2026), No. 5, 971–1028

DOI: 10.2140/ant.2026.20.971

Abstract

For an algebraically closed nonarchimedean extension $C ∕ ℚ_{p}$ , 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 $ℚ$ -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 $ℚ_{p}$ -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 $ℂ$ 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).

Keywords

$p$-adic transcendence, $p$-adic Hodge theory, complex multiplication, $p$-divisible group, Mumford–Tate group

Mathematical Subject Classification

Primary: 11G09, 11G18, 11J81

Milestones

Received: 25 January 2024

Revised: 28 February 2025

Accepted: 2 May 2025

Published: 6 May 2026

Authors

Sean Howe
Department of Mathematics University of Utah Salt Lake City, UT United States

Christian Klevdal
Department of Mathematics Johns Hopkins University Baltimore, MD United States

Open Access made possible by participating institutions via Subscribe to Open.

Vol. 20, No. 5, 2026