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arxiv: 2605.16092 · v1 · pith:UKTPAHZVnew · submitted 2026-05-15 · 🧮 math.NT · math.AG

On Drinfeld's representability theorem

Arnaud Vanhaecke This is my paper

Pith reviewed 2026-05-19 18:42 UTC · model grok-4.3

classification 🧮 math.NT math.AG
keywords Drinfeld representability theoremp-adic symmetric spacemoduli spacep-divisible groupsquasi-isogeniesp-adic Hodge theorysemi-stable modelformal model
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The pith

Drinfeld's representability theorem for moduli of p-divisible groups holds via a new transparent proof.

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

The paper establishes Drinfeld's representability theorem through a new and more transparent proof while supplying a detailed account of the associated moduli space and formal model. The theorem asserts that a moduli problem classifying deformations by quasi-isogenies of certain p-divisible groups with extra actions is represented by an explicit semi-stable model of the p-adic symmetric space. A reader would care because the result forms a cornerstone for geometric methods in p-adic Hodge theory, and greater transparency in the proof can make these constructions more usable in arithmetic geometry. The notes also clarify the structure of the moduli space itself.

Core claim

Drinfeld's representability theorem holds, and the notes prove it by a new transparent method. The moduli problem is defined by deformations by quasi-isogenies of p-divisible groups with extra actions, and this problem is represented by the explicit semi-stable model of the p-adic symmetric space. The notes further present Drinfeld's moduli space and the formal model in detail.

What carries the argument

Drinfeld's moduli problem of deformations by quasi-isogenies of p-divisible groups with extra actions, represented by the semi-stable formal model of the p-adic symmetric space.

If this is right

  • The explicit model permits direct geometric constructions within p-adic Hodge theory.
  • The formal model of the p-adic symmetric space can be studied through the moduli interpretation.
  • Deformations of the p-divisible groups become more accessible for explicit calculations.

Where Pith is reading between the lines

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

  • The transparent proof technique could extend to analogous representability questions for other group actions on p-divisible groups.
  • The detailed moduli presentation may help relate the p-adic symmetric space to nearby objects such as Shimura varieties.
  • One could test the construction computationally for low-dimensional cases to verify the semi-stable model.

Load-bearing premise

The moduli problem must be defined exactly as the original deformation problem by quasi-isogenies of certain p-divisible groups with extra actions considered by Drinfeld.

What would settle it

A mismatch between the semi-stable reduction or the formal completion constructed in the new proof and the corresponding objects in Drinfeld's original construction would show the representability claim fails.

Figures

Figures reproduced from arXiv: 2605.16092 by Arnaud Vanhaecke.

Figure 1
Figure 1. Figure 1: The formal scheme H∆ Z3 (upper left) for d = 2 and E = Q3 of the Drinfeld symmetric space lying above a maximal simplex ∆ = {η0, η1} ⊂ BT 2 (lower right), its special fiber H∆ F3 (upper right) and its rigid generic fiber H∆ Q3 (lower left) [PITH_FULL_IMAGE:figures/full_fig_p029_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: In red, the Newton polygon of G, in blue the Hodge polygon of G. Hence, there exists an integer r ∈ J0, d − 1K such that the slopes of D are 0 (with multiplicity r) and 1 d−r (with multiplicity d − r), which proves the proposition. Remark 4.36. — Note that the condition on the polygons in [PITH_FULL_IMAGE:figures/full_fig_p040_2.png] view at source ↗
read the original abstract

In the seventies, V. G. Drinfeld proved that a moduli problem of deformations by quasi-isogenies of certain $p$-divisible groups with extra actions is representable by an explicit semi-stable model of the $p$-adic symmetric space. This theorem, known as \emph{Drinfeld's representability theorem}, has been one of the cornerstones of geometric aspects in $p$-adic Hodge theory. The purpose of these notes is twofold. On the one hand we give a new and more transparent proof of Drinfeld's representability theorem; on the other hand, we give a detailed presentation of Drinfeld's moduli space and the formal model of the $p$-adic symmetric space.

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 / 3 minor

Summary. The manuscript provides a new and more transparent proof of Drinfeld's representability theorem: the moduli functor of deformations by quasi-isogenies of certain p-divisible groups equipped with extra actions is represented by an explicit semi-stable formal model of the p-adic symmetric space. It also supplies a detailed presentation of Drinfeld's moduli space and the formal model.

Significance. If the central claim holds, the work is significant for p-adic Hodge theory: it supplies an alternative proof of a foundational representability result together with a detailed exposition of the moduli problem and formal model. These elements can serve as a reference for further geometric constructions in the area.

minor comments (3)
  1. [§2.1] §2.1: the definition of the moduli functor via quasi-isogenies matches the classical setup, but an explicit side-by-side comparison table with Drinfeld's original formulation would improve readability.
  2. [§4] §4, after Eq. (4.3): the verification that the formal model is semi-stable would be clearer if the reduction steps were summarized in a short diagram or flowchart.
  3. [Abstract] The manuscript refers to itself as 'notes' in the abstract but is formatted as a self-contained paper; a brief statement of intended audience or publication venue would help.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript, the recognition of its significance for p-adic Hodge theory, and the recommendation of minor revision. The paper offers a new transparent proof of Drinfeld's representability theorem together with a detailed exposition of the moduli problem and the formal model.

Circularity Check

0 steps flagged

New proof of Drinfeld representability is self-contained with no circular reduction

full rationale

The paper redefines the moduli functor exactly as Drinfeld's original deformation problem by quasi-isogenies of p-divisible groups with extra actions, then verifies representability criteria directly through deformation theory and an explicit semi-stable formal model whose generic fiber matches the p-adic symmetric space. No step reduces a claimed result to a fitted parameter, self-citation chain, or ansatz imported from the author's prior work; the argument is presented as an independent verification matching the classical setup without internal redefinition or load-bearing reliance on the original Drinfeld construction beyond the shared definition of the moduli problem itself.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone.

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On optimal $p$-adic uniformization of unitary Shimura curves

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    Extends p-adic uniformization results to RSZ and unitary group variants of Shimura curves via maximal levels and explicit integral local Shimura varieties.

Reference graph

Works this paper leans on

66 extracted references · 66 canonical work pages · cited by 1 Pith paper · 2 internal anchors

  1. [1]

    Abramenko and K

    P. Abramenko and K. S. Brown.Buildings: Theory and applications. Grad. Texts Math. 248, Springer (2008)

    work page 2008

  2. [2]

    Ahsendorf, C

    T. Ahsendorf, C. Cheng, and T. Zink.O-displays andπ-divisible formalO-modules.J. Algebra, 457:129–193 (2016)

    work page 2016

  3. [3]

    Bartling,The universal special formalO D-module ford= 2, arXiv:2206.13195

    S. Bartling. The universal special formalO D-module ford= 2. Preprint, arXiv:2206.13195 (2022)

    work page arXiv 2022

  4. [4]

    Bartling and M

    S. Bartling and M. Hoff. Moduli spaces of nilpotent displays.Int. Math. Res. Not., 2025(3):rnaf005 (2025)

    work page 2025

  5. [5]

    Bass.AlgebraicK-theory

    H. Bass.AlgebraicK-theory. Math. Lect. Note Ser., Benjamin/Cummings, Reading, MA (1968)

    work page 1968

  6. [6]

    V. G. Berkovich.Spectral theory and analytic geometry over non-Archimedean fields. Math. Surveys Monogr. 33, Amer. Math. Soc., Providence, RI (1990)

    work page 1990

  7. [7]

    Berthelot, L

    P. Berthelot, L. Breen, and W. Messing.Th´ eorie de Dieudonn´ e cristalline. II. Lecture Notes in Math. 930, Springer (1982)

    work page 1982

  8. [8]

    Bertolini and H

    M. Bertolini and H. Darmon. Heegner points,p-adicL-functions, and the Cherednik–Drinfeld uni- formization.Invent. Math., 131(3):453–491 (1998)

    work page 1998

  9. [9]

    Bhatt and P

    B. Bhatt and P. Scholze. Projectivity of the Witt vector affine Grassmannian.Invent. Math., 209(2):329–423 (2017)

    work page 2017

  10. [10]

    Bosch, U

    S. Bosch, U. G¨ untzer, and R. Remmert.Non-Archimedean analysis: A systematic approach to rigid analytic geometry. Grundlehren Math. Wiss. 261, Springer (1984)

    work page 1984

  11. [11]

    Boucksom and D

    S. Boucksom and D. Eriksson. Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry.Adv. Math., 378:107501 (2021)

    work page 2021

  12. [12]

    P. Boyer. Bad reduction of Drinfeld varieties and local Langlands correspondence.Invent. Math., 138(3):573–629 (1999)

    work page 1999

  13. [13]

    Bourbaki

    N. Bourbaki. ´El´ ements de math´ ematique. Alg` ebre. Chapitres 4 ` a 7. Springer, Berlin (2007)

    work page 2007

  14. [14]

    Bourbaki

    N. Bourbaki. ´El´ ements de math´ ematique. Alg` ebre commutative. Chapitres 1 ` a 4. Springer, Berlin (2006)

    work page 2006

  15. [15]

    Boutot and H

    J.-F. Boutot and H. Carayol.p-adic uniformization of Shimura curves: the theorems of Cherednik and Drinfeld.Ast´ erisque, 196–197:45–158 (1991). 58ARNAUD VANHAECKE

    work page 1991

  16. [16]

    Barthel, T

    T. Barthel, T. M. Schlank, N. Stapleton, and J. Weinstein. On the rationalization of theK(n)-local sphere. Preprint, arXiv:2402.00960 (2025)

    work page arXiv 2025

  17. [17]

    H. Carayol. Sur les repr´ esentations galoisiennes moduloℓattach´ ees aux formes modulaires.Duke Math. J., 59(3):785–801 (1989)

    work page 1989

  18. [18]

    H. Carayol. Non-abelian Lubin–Tate theory. InAutomorphic Forms, Shimura Varieties, andL- Functions, Vol. II, pages 15–39 (1990)

    work page 1990

  19. [19]

    M. Chen, L. Fargues, and X. Shen. On the structure of somep-adic period domains.Camb. J. Math., 9(1):213–267 (2021)

    work page 2021

  20. [20]

    I. V. Cherednik. Algebraische Kurven, die durch diskrete arithmetische Untergruppen von P GL2(kw) uniformisierbar sind.Usp. Mat. Nauk, 30(3):181–182 (1975)

    work page 1975

  21. [21]

    P. Colmez. Espaces de Banach de dimension finie.J. Inst. Math. Jussieu, 1(3):331–439 (2002)

    work page 2002

  22. [22]

    P. Colmez. Espaces vectoriels de dimension finie et repr´ esentations de de Rham. InRepr´ esentations p-adiques de groupesp-adiques I, Ast´ erisque 319, pages 117–186 (2008)

    work page 2008

  23. [23]

    Colmez, G

    P. Colmez, G. Dospinescu, and W. Nizio l. Cohomology ofp-adic Stein spaces.Invent. Math., 219(3):873–985 (2020)

    work page 2020

  24. [24]

    Colmez, G

    P. Colmez, G. Dospinescu, and W. Nizio l. On the geometrization of thep-adic local Langlands correspondence. InProceedings of the International Conference of Basic Science 2024, pages 92–

    work page 2024

  25. [25]

    International Press (2025)

    work page 2025

  26. [26]

    J.-F. Dat. Nonabelian Lubin–Tate theory and elliptic representations.Invent. Math., 169(1):75–152 (2007)

    work page 2007

  27. [27]

    A. J. de Jong. Crystalline Dieudonn´ e module theory via formal and rigid geometry.Publ. Math. IH ´ES, 82:5–96 (1995)

    work page 1995

  28. [28]

    Deligne and G

    P. Deligne and G. Lusztig. Representations of reductive groups over finite fields.Ann. of Math. (2), 103:103–161 (1976)

    work page 1976

  29. [29]

    A Jacquet-Langlands functor for $p$-adic locally analytic representations

    G. Dospinescu and J. E. Rodr´ ıguez Camargo. A Jacquet–Langlands functor forp-adic locally ana- lytic representations. Preprint, arXiv:2411.17082 (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  30. [30]

    V. G. Drinfeld. Coverings ofp-adic symmetric regions.Funct. Anal. Appl., 10:107–115 (1976)

    work page 1976

  31. [31]

    Faltings

    G. Faltings. A relation between two moduli spaces studied by Drinfeld. InAlgebraic Number Theory and Algebraic Geometry, pages 115–129. Amer. Math. Soc. (2002)

    work page 2002

  32. [32]

    L. Fargues. The curve. InProc. ICM 2018, Vol. II, pages 291–319 (2018)

    work page 2018

  33. [33]

    Fargues and J.-M

    L. Fargues and J.-M. Fontaine.Courbes et fibr´ es vectoriels en th´ eorie de Hodgep-adique. Ast´ erisque 406 (2018)

    work page 2018

  34. [34]

    Fargues, A

    L. Fargues, A. Genestier, and V. Lafforgue.L’isomorphisme entre les tours de Lubin–Tate et de Drinfeld. Progr. Math. 262, Birkh¨ auser, Basel (2008)

    work page 2008

  35. [35]

    [GI23] Ian Gleason and Alexander B

    L. Fargues and P. Scholze. Geometrization of the local Langlands correspondence. Preprint, arXiv:2102.13459 (2021)

    work page arXiv 2021

  36. [36]

    Fontaine.Groupesp-divisibles sur les corps locaux

    J.-M. Fontaine.Groupesp-divisibles sur les corps locaux. Ast´ erisque 47–48, Soc. Math. France (1977)

    work page 1977

  37. [37]

    Fujiwara and F

    K. Fujiwara and F. Kato.Foundations of rigid geometry. I. EMS Monogr. Math. (2018)

    work page 2018

  38. [38]

    Genestier.Espaces sym´ etriques de Drinfeld

    A. Genestier.Espaces sym´ etriques de Drinfeld. Ast´ erisque 234 (1996)

    work page 1996

  39. [39]

    Grothendieck

    A. Grothendieck. ´El´ ements de g´ eom´ etrie alg´ ebrique. II:´Etude globale ´ el´ ementaire de quelques classes de morphismes.Publ. Math. IH ´ES, 4:1–222 (1961)

    work page 1961

  40. [40]

    Grothendieck

    A. Grothendieck. ´El´ ements de g´ eom´ etrie alg´ ebrique. IV:´Etude locale des sch´ emas et des morphismes de sch´ emas (deuxi` eme partie).Publ. Math. IH´ES, 24:1–231 (1965)

    work page 1965

  41. [41]

    Grothendieck

    A. Grothendieck. ´El´ ements de g´ eom´ etrie alg´ ebrique. IV:´Etude locale des sch´ emas et des morphismes de sch´ emas (troisi` eme partie).Publ. Math. IH´ES, 28:5–255 (1966)

    work page 1966

  42. [42]

    Grothendieck

    A. Grothendieck. ´El´ ements de g´ eom´ etrie alg´ ebrique. IV:´Etude locale des sch´ emas et des morphismes de sch´ emas (quatri` eme partie).Publ. Math. IH´ES, 32:5–361 (1967)

    work page 1967

  43. [43]

    Grothendieck and J

    A. Grothendieck and J. A. Dieudonn´ e. ´El´ ements de g´ eom´ etrie alg´ ebrique. I. Grundlehren Math. Wiss. 166, Springer (1971)

    work page 1971

  44. [44]

    Fulton.Intersection theory

    W. Fulton.Intersection theory. Ergeb. Math. Grenzgeb. (3) 2, Springer (1984)

    work page 1984

  45. [45]

    M. Harris. Supercuspidal representations in the cohomology of Drinfeld upper half spaces.Invent. Math., 129(1):75–119 (1997)

    work page 1997

  46. [46]

    Hazewinkel.Formal groups and applications

    M. Hazewinkel.Formal groups and applications. Pure Appl. Math. 78, Academic Press, New York (1978). ON DRINFELD’S REPRESENTABILITY THEOREM59

    work page 1978

  47. [47]

    S. M. H. Hedayatzadeh. Explicit isomorphism between Cartier and Dieudonn´ e modules.J. Algebra, 570:611–635 (2021)

    work page 2021

  48. [48]

    R. Huber. ´Etale cohomology of rigid analytic varieties and adic spaces. Aspects Math. E30, Vieweg, Wiesbaden (1996)

    work page 1996

  49. [49]

    R. E. Kottwitz. Isocrystals with additional structure.Compos. Math., 56:201–220 (1985)

    work page 1985

  50. [50]

    S. S. Kudla, M. Rapoport, and T. Yang.Modular forms and special cycles on Shimura curves. Ann. Math. Stud. 161, Princeton Univ. Press, Princeton, NJ (2006)

    work page 2006

  51. [51]

    Lazard.Commutative formal groups

    M. Lazard.Commutative formal groups. Lecture Notes in Math. 443, Springer (1975)

    work page 1975

  52. [52]

    A.-C. Le Bras. Espaces de Banach–Colmez et faisceaux coh´ erents sur la courbe de Fargues–Fontaine. Duke Math. J., 167(18):3455–3532 (2018)

    work page 2018

  53. [53]

    J. N. P. Louren¸ co. The Riemannian Hebbarkeitss¨ atze for pseudorigid spaces. Preprint, arXiv:1711.06903 (2017)

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  54. [54]

    Messing.The crystals associated to Barsotti–Tate groups

    W. Messing.The crystals associated to Barsotti–Tate groups. Lecture Notes in Math. 264, Springer (1972)

    work page 1972

  55. [55]

    G. A. Mustafin. Nonarchimedean uniformization.Math. USSR Sb., 34:187–214 (1978)

    work page 1978

  56. [56]

    Rapoport and E

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

    work page 2014

  57. [57]

    Rapoport and T

    M. Rapoport and T. Zink.Period Spaces forp-divisible Groups. Ann. Math. Stud. 141, Princeton Univ. Press (1996)

    work page 1996

  58. [58]

    Rapoport and T

    M. Rapoport and T. Zink. On the Drinfeld moduli problem ofp-divisible groups.Camb. J. Math., 5(2):229–279 (2017)

    work page 2017

  59. [59]

    P. Scholze. Perfectoid spaces: a survey. InCurrent Developments in Mathematics 2012, pages 193–

    work page 2012

  60. [60]

    International Press (2013)

    work page 2013

  61. [61]

    P. Scholze. On thep-adic cohomology of the Lubin–Tate tower.Ann. Sci. ´Ec. Norm. Sup´ er. (4), 51(4):811–863 (2018)

    work page 2018

  62. [62]

    Scholze and J

    P. Scholze and J. Weinstein. Moduli ofp-divisible groups.Camb. J. Math., 1(2):145–237 (2013)

    work page 2013

  63. [63]

    Scholze and J

    P. Scholze and J. Weinstein.Berkeley lectures onp-adic geometry. Ann. Math. Stud. 207, Princeton Univ. Press, Princeton, NJ (2020)

    work page 2020

  64. [64]

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

    work page

  65. [65]

    Zink.Cartier-Theorie kommutativer formaler Gruppen

    T. Zink.Cartier-Theorie kommutativer formaler Gruppen. Teubner-Texte zur Mathematik 68, Teub- ner, Leipzig (1984)

    work page 1984

  66. [66]

    T. Zink. The display of a formalp-divisible group. InCohomologiesp-adiques et applications arithm´ etiques (I), pages 127–248. Soc. Math. France (2002). May 18, 2026 Arnaud V anhaecke,Morningside Center of Mathematics, No. 55, Zhongguancun East Road, Beijing, 100190, China. Url :https://arnaudvanhaeckemaths.github.io/ •E-mail :arnaud@amss.ac.cn

    work page 2002