pith. sign in

arxiv: 2308.11065 · v2 · submitted 2023-08-21 · 🧮 math.NT · math.AG

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

Sean Howe , Christian Klevdal This is my paper

Pith reviewed 2026-05-24 06:43 UTC · model grok-4.3

classification 🧮 math.NT math.AG
keywords p-adic Hodge structuresadmissible pairscomplex multiplicationtranscendencede Rham periodsTannakian categoryBreuil-Kisin-Fargues modules
0
0 comments X

The pith

p-adic Hodge structures over C are equivalent to basic admissible pairs, with CM ones characterized by transcendence of p-adic periods.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a Tannakian category of p-adic Hodge structures by equipping Q_p-vector spaces with B_dR^+-lattices that satisfy a transversality condition, mirroring classical Hodge theory in a p-adic setting. It establishes that this category is equivalent to a full subcategory of basic objects in the category of admissible pairs, which serve as a toy model for cohomological motives. Using this equivalence, the authors characterize basic admissible pairs with complex multiplication through the transcendence of their p-adic periods, providing a local analog to global results on CM motives.

Core claim

We introduce p-adic Hodge structures as Q_p-vector spaces with B_dR^+-lattices satisfying a natural transversality condition. These structures form a Tannakian category equivalent to a full subcategory of basic admissible pairs. Basic admissible pairs with complex multiplication are then characterized in terms of the transcendence of p-adic periods.

What carries the argument

B_dR^+-lattices with transversality condition on Q_p-vector spaces, which define p-adic Hodge structures and link them to admissible pairs.

If this is right

  • This equivalence allows transferring properties between p-adic Hodge structures and admissible pairs.
  • CM basic admissible pairs are determined by transcendence of their de Rham periods.
  • The result generalizes the characterization for one-dimensional formal groups.
  • It provides an unconditional local p-adic version of conditional global characterizations of CM motives.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The framework could extend to relate p-adic periods to other arithmetic invariants.
  • Similar transcendence criteria might apply to non-CM cases or higher-dimensional objects.
  • Connections to Breuil-Kisin-Fargues modules suggest applications in p-adic geometry.

Load-bearing premise

The transversality condition on B_dR^+-lattices produces a Tannakian category whose equivalence to admissible pairs holds.

What would settle it

Finding a basic admissible pair with complex multiplication where the p-adic periods are algebraic but the pair does not satisfy the expected transcendence condition, or vice versa.

read the original abstract

For an algebraically closed non-archimedean extension $C/\mathbb{Q}_p$, 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 $\mathbb{Q}$-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 $B^+_\mathrm{dR}$, and a pure $p$-adic Hodge structure is then a $\mathbb{Q}_p$-vector space equipped with a $B^+_\mathrm{dR}$-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 $\mathbb{C}$ 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).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript defines a Tannakian category of p-adic Hodge structures over an algebraically closed non-archimedean extension C/Q_p. Filtrations are enriched to B_dR^+-lattices with a transversality-type condition. It claims an equivalence between this category and a full subcategory of basic objects in the category of admissible pairs (equivalent to the isogeny category of rigidified Breuil-Kisin-Fargues modules). As an application, basic admissible pairs with complex multiplication are characterized by the transcendence of p-adic periods. This generalizes a result for one-dimensional formal groups and provides an unconditional local p-adic analog of global CM motive characterizations.

Significance. If substantiated, the work establishes a local p-adic counterpart to classical Hodge theory, connecting it to admissible pairs and Breuil-Kisin-Fargues modules in a Tannakian framework. The CM characterization via period transcendence is unconditional, offering a p-adic parallel to archimedean results that often rely on conjectures. This could advance understanding of p-adic motives and Hodge structures.

minor comments (1)
  1. The abstract introduces specialized terms such as 'admissible pairs' and 'Breuil-Kisin-Fargues modules' without brief contextual definitions, which may reduce accessibility for readers outside the narrow subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of the manuscript and for recognizing its potential contributions to p-adic Hodge theory and motives. The full text is available on arXiv as 2308.11065, and we welcome the opportunity to address any specific concerns. No major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract defines a new Tannakian category via enrichment of filtrations to B_dR^+-lattices with a transversality condition, then asserts an equivalence to a subcategory of admissible pairs (themselves related to existing Breuil-Kisin-Fargues modules and Fontaine theory) and an application to CM characterization via transcendence. No equations, fitted parameters, or self-citations are exhibited that would reduce the claimed equivalence or characterization to the inputs by construction. The generalization of an earlier result is mentioned but not shown to be load-bearing or self-referential in a way that collapses the derivation. With only the abstract available and no technical steps provided, the construction remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only; full definitions and axioms not visible. The work relies on standard background from p-adic Hodge theory and Fontaine rings but introduces new category definitions whose precise axioms cannot be audited.

axioms (2)
  • domain assumption The enriched lattice condition produces a Tannakian category
    Invoked when defining p-adic Hodge structures as a category equivalent to admissible pairs.
  • domain assumption Admissible pairs form a toy category of cohomological motives
    Used to relate the new structures to Breuil-Kisin-Fargues modules.

pith-pipeline@v0.9.0 · 5811 in / 1206 out tokens · 19304 ms · 2026-05-24T06:43:08.190936+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · 2 internal anchors

  1. [1]

    Une introduction aux motifs

    Yves Andr´ e. Une introduction aux motifs. Panoramas et syntheses , 17, 2004

    work page 2004

  2. [2]

    Breuil-Kisin-Fargues modules wit h complex multiplication

    Johannes Ansch¨ utz. Breuil-Kisin-Fargues modules wit h complex multiplication. J. Inst. Math. Jussieu , 20(6):1855–1904, 2021

    work page 1904

  3. [3]

    Un lemme de descente

    Arnaud Beauville and Yves Laszlo. Un lemme de descente. Comptes Rendus de l’Academie des Sciences-Serie I-Mathematique , 320(3):335–340, 1995

    work page 1995

  4. [4]

    Integ ral p-adic Hodge theory

    Bhargav Bhatt, Matthew Morrow, and Peter Scholze. Integ ral p-adic Hodge theory. Publ. Math. Inst. Hautes ´Etudes Sci. , 128:219–397, 2018

    work page 2018

  5. [5]

    Prisms and prismatic co homology

    Bhargav Bhatt and Peter Scholze. Prisms and prismatic co homology. Ann. of Math. (2) , 196(3):1135–1275, 2022

    work page 2022

  6. [6]

    Linear algebraic groups , volume 126

    Armand Borel. Linear algebraic groups , volume 126. Springer Science & Business Media, 2012

    work page 2012

  7. [7]

    Bruhat and J

    F. Bruhat and J. Tits. Groupes r´ eductifs sur un corps loc al. Inst. Hautes ´Etudes Sci. Publ. Math., (41):5–251, 1972

    work page 1972

  8. [8]

    On the generic part of the cohomology of compact unitary shimura varieties

    Ana Caraiani and Peter Scholze. On the generic part of the cohomology of compact unitary shimura varieties. Annals of Mathematics , 186(3):649–766, 2017

    work page 2017

  9. [9]

    On the generic part of the cohomology of non-compact unitary Shimura varieties

    Ana Caraiani and Peter Scholze. On the generic part of the cohomology of non-compact unitary Shimura varieties. Ann. of Math. (2) , 199(2):483–590, 2024

    work page 2024

  10. [10]

    On the locus of hodge classes

    Eduardo Cattani, Pierre Deligne, and Aroldo Kaplan. On the locus of hodge classes. Journal of the American Mathematical Society , 8(2):483–506, 1995

    work page 1995

  11. [11]

    Humbert surfaces and transcenden ce properties of automorphic func- tions

    Paula Beazley Cohen. Humbert surfaces and transcenden ce properties of automorphic func- tions. volume 26, pages 987–1001. 1996. Symposium on Diopha ntine Problems (Boulder, CO, 1994)

    work page 1996

  12. [12]

    Pseudo-re ductive groups

    Brian Conrad, Ofer Gabber, and Gopal Prasad. Pseudo-re ductive groups. Preprint, 2008

    work page 2008

  13. [13]

    Vari´ et´ es de Shimura: interpr´ etation modulaire, et techniques de construc- tion de mod` eles canoniques

    Pierre Deligne. Vari´ et´ es de Shimura: interpr´ etation modulaire, et techniques de construc- tion de mod` eles canoniques. In Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 19 77), Part 2 , volume XXXIII of Proc. Sympos. Pure Math. , pages 247–289. Amer. Math. Soc., Providence, RI, 1979

    work page 1979

  14. [14]

    Pierre Deligne and James S. Milne. Tannakian categorie s. In Hodge cycles, motives, and Shimura varieties , volume 900 of Lecture Notes in Mathematics , pages 101–228. Springer- Verlag, Berlin-New York, 1982

    work page 1982

  15. [15]

    A p-adic variational hodge conjectur e and modular forms with complex multiplication

    Matthew Emerton. A p-adic variational hodge conjectur e and modular forms with complex multiplication. https: // math. uchicago. edu/~ emerton/ pdffiles/ cm. pdf. 9Here we use that the image of ρ is a Zariski dense set of Qp-points of G, and that this implies the induced set of L-points in GL for any L/Qp is also Zariski dense — to see this second point, ...

    work page

  16. [16]

    L’isomorphisme entre les tours de Lub in-Tate et de Drinfeld et applications cohomologiques

    Laurent Fargues. L’isomorphisme entre les tours de Lub in-Tate et de Drinfeld et applications cohomologiques. In L’isomorphisme entre les tours de Lubin-Tate et de Drinfeld , volume 262 of Progr. Math., pages 1–325. Birkh¨ auser, Basel, 2008

    work page 2008

  17. [17]

    G-torseurs en th´ eorie de Hodge p-adique

    Laurent Fargues. G-torseurs en th´ eorie de Hodge p-adique. Compos. Math. , 156(10):2076– 2110, 2020

    work page 2076

  18. [18]

    Courbes et fibr ´ es vectoriels en th´ eorie de Hodge p-adique

    Laurent Fargues and Jean-Marc Fontaine. Courbes et fibr ´ es vectoriels en th´ eorie de Hodge p-adique. Ast´ erisque, (406):xiii+382, 2018. With a preface by Pierre Colmez

    work page 2018

  19. [19]

    Transcendence of the Hodge-Tate filtration

    Sean Howe. Transcendence of the Hodge-Tate filtration. J. Th´ eor. Nombres Bordeaux , 30(2):671–680, 2018

    work page 2018

  20. [20]

    J. Huisman. The exponential sequence in real algebraic geometry and Harnack’s inequality for proper reduced real schemes. Comm. Algebra, 30(10):4711–4730, 2002

    work page 2002

  21. [21]

    Nicholas M. Katz. Slope filtration of F -crystals. In Journ´ ees de G´ eom´ etrie Alg´ ebrique de Rennes (Rennes, 1978), Vol. I , volume 63 of Ast´ erisque, pages 113–163. Soc. Math. France, Paris, 1979

    work page 1978

  22. [22]

    B(G) for all local and global fields

    Robert Kottwitz. B(G) for all local and global fields. arXiv:1401.5728, 2014

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  23. [23]

    Isocrystals with additional struct ure

    Robert E Kottwitz. Isocrystals with additional struct ure. Compositio Mathematica , 56(2):201–220, 1985

    work page 1985

  24. [24]

    Isocrystals with additional struct ure

    Robert E Kottwitz. Isocrystals with additional struct ure. ii. Compositio Mathematica , 109(3):255–339, 1997

    work page 1997

  25. [25]

    G. D. Mostow. Fully reducible subgroups of algebraic gr oups. Amer. J. Math. , 78:200–221, 1956

    work page 1956

  26. [26]

    Rapoport and Th

    M. Rapoport and Th. Zink. Period spaces for p-divisible groups , volume 141 of Annals of Mathematics Studies . Princeton University Press, Princeton, NJ, 1996

    work page 1996

  27. [27]

    Towards a theory of l ocal Shimura varieties

    Michael Rapoport and Eva Viehmann. Towards a theory of l ocal Shimura varieties. M¨ unster J. Math. , 7(1):273–326, 2014

    work page 2014

  28. [28]

    Cat´ egories tannakiennes, volume 265

    N Saavedra Rivano. Cat´ egories tannakiennes, volume 265. Springer, 2006

    work page 2006

  29. [29]

    Arithmetische Untersuchungen ell iptischer Integrale

    Theodor Schneider. Arithmetische Untersuchungen ell iptischer Integrale. Math. Ann. , 113(1):1–13, 1937

    work page 1937

  30. [30]

    On the p-adic cohomology of the Lubin-Tate tower

    Peter Scholze. On the p-adic cohomology of the Lubin-Tate tower. Ann. Sci. ´Ec. Norm. Sup´ er. (4), 51(4):811–863, 2018. With an appendix by Michael Rapoport

    work page 2018

  31. [31]

    p-adic geometry

    Peter Scholze. p-adic geometry. Proceedings of the ICM 2018 , 2018

    work page 2018

  32. [32]

    Berkeley Lectures on p-adic Geometry: (AMS-207)

    Peter Scholze and Jared W einstein. Berkeley Lectures on p-adic Geometry: (AMS-207) . Princeton University Press, 2020

    work page 2020

  33. [33]

    Groupes alg´ ebriques associ´ es aux modules de Hodge-Tate

    Jean-Pierre Serre. Groupes alg´ ebriques associ´ es aux modules de Hodge-Tate. In Journ´ ees de G´ eom´ etrie Alg´ ebrique de Rennes (Rennes, 1978), Vol. III, volume 65 of Ast´ erisque, pages 155–188. Soc. Math. France, Paris, 1979

    work page 1978

  34. [34]

    Criteria for compl ex multiplication and transcendence properties of automorphic functions

    Hironori Shiga and J¨ urgen W olfart. Criteria for compl ex multiplication and transcendence properties of automorphic functions. J. Reine Angew. Math. , 463:1–25, 1995

    work page 1995

  35. [35]

    Mixed twistor structures

    Carlos Simpson. Mixed twistor structures. arXiv:alg-geom/9705006, 1997

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  36. [36]

    Stacks Project

    The Stacks Project Authors. Stacks Project. https://stacks.math.columbia.edu, 2018

    work page 2018

  37. [37]

    Groupes alg´ ebriques associ´ es ` a certaines representationsp-adiques

    Jean-Pierre Wintenberger. Groupes alg´ ebriques associ´ es ` a certaines representationsp-adiques. Amer. J. Math. , 108(6):1425–1466, 1986. Email address : sean.howe@utah.edu Email address : cklevdal@ucsd.edu

    work page 1986