Pith. sign in

REVIEW 2 minor 13 references

Integrality of height-one formal groups

T0 review · 0 major / 2 minor · reviewed 2026-06-25 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A one-dimensional formal group law over a finite extension of the p-adics has integral coefficients if and only if its multiplication-by-n endomorphisms do, when the group has height one.

desk verdict The paper proves an if-and-only-if for integrality of height-one formal groups over p-adics, with the converse relying on p-adic Hodge theory applied to the associated character. read the letter →

arxiv 2606.25726 v1 pith:R4Y4JXRV submitted 2026-06-24 math.NT math.DS

Martin Debaisieux This is my paper · ORCID
classification math.NTmath.DS
keywords formalgrouplawsintegralityp-adicfieldsheightoneHodgetheoryendomorphismsWeierstrassdegree
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves an if-and-only-if statement for one-dimensional formal group laws over a finite extension K of Q_p: the law itself has coefficients in the ring of integers of K precisely when every multiplication-by-n endomorphism does, but only in the height-one case where the multiplication-by-p map has Weierstrass degree p. The argument translates between these integrality conditions by means of p-adic Hodge theory. A reader would care because the result supplies a practical test for integrality that bypasses direct inspection of the full power series defining the group law.

What carries the argument

The equivalence, for height-one groups, between integrality of the formal group law coefficients and integrality of all its [n]-endomorphisms, established via p-adic Hodge theory.

What would settle it

An explicit height-one formal group law over some finite extension of Q_p whose power-series coefficients lie outside the ring of integers while all its multiplication-by-n maps have integral coefficients, or the converse.

Watch

Extended reading notes

Core claim

Over a finite extension K of Q_p, a one-dimensional formal group law has integral coefficients if and only if its multiplication-by-n endomorphisms have integral coefficients for every integer n, provided the formal group has height one (that is, the multiplication-by-p map has Weierstrass degree p). The proof proceeds by applying p-adic Hodge theory to equate the two integrality statements.

Load-bearing premise

The formal group must have height one so that p-adic Hodge theory can be used to relate the integrality of the group law to the integrality of the endomorphisms.

Editorial extensions

If this is right

  • Integrality of a height-one formal group law can be verified by checking only the endomorphisms rather than the entire group law.
  • The criterion applies directly to formal groups attached to elliptic curves or Lubin-Tate extensions of height one over p-adic fields.
  • Questions about integral models of formal groups can be rephrased as questions about integral endomorphisms.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The equivalence might serve as a template for checking integrality in other p-adic settings where endomorphisms are easier to compute than the full law.
  • It raises the question whether an analogous statement holds for formal groups of height greater than one, possibly after replacing p-adic Hodge theory with a different tool.
  • Concrete examples such as the formal multiplicative group or the formal group of an ordinary elliptic curve could be used to test the sharpness of the height-one restriction.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that for a one-dimensional formal group law over a finite extension K of Q_p, the group law has integral coefficients if and only if all its multiplication-by-n endomorphisms have integral coefficients, but only in the height-one case (i.e., when the multiplication-by-p map has Weierstrass degree p). One direction is purely algebraic; the converse applies p-adic Hodge theory to the one-dimensional Galois representation on the torsion points, using the height-one hypothesis to ensure the representation is a character to which the filtered phi-module correspondence applies directly.

Significance. If the result holds, it supplies a verifiable criterion for integrality of formal groups in terms of endomorphisms, which may be more accessible in computations. The explicit use of the height-one condition to reduce to a character and invoke p-adic Hodge theory without extra ramification hypotheses is a clear strength, as is the separation of the algebraic direction from the Hodge-theoretic one.

minor comments (2)
  1. [Abstract] Abstract: the parenthetical explanation of height one could be expanded by one sentence to note that this condition is used only for the converse direction.
  2. [Proof of converse] The manuscript would benefit from an explicit citation or short recall of the precise p-adic Hodge theorem (filtered phi-module correspondence for characters) invoked in the converse, even if standard.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. Their summary correctly identifies the if-and-only-if criterion in the height-one case and the separation between the algebraic direction and the p-adic Hodge theoretic direction.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper proves an if-and-only-if equivalence between integrality of a one-dimensional formal group law and integrality of its [n] endomorphisms, restricted to the height-one case, via one algebraic direction and one direction that invokes p-adic Hodge theory on the associated Galois representation. No equations, fitted parameters, self-citations, or ansatzes are shown to reduce the claimed result to its own inputs by construction. The derivation is self-contained against external benchmarks (algebraic identities and standard p-adic Hodge correspondences).

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The claim rests on the height-one restriction and on the applicability of p-adic Hodge theory; no free parameters or invented entities are mentioned.

assumptions (1)
  • domain assumption p-adic Hodge theory applies to one-dimensional formal groups of height one over finite extensions of Q_p
    The abstract states that the proof uses some p-adic Hodge theory.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Integrality of height-one formal groups." pith.science (2026). https://pith.science/paper/R4Y4JXRV

@misc{pith2026260625726,
  author       = {Pith},
  title        = {Pith review of: Integrality of height-one formal groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R4Y4JXRV}},
  note         = {Machine review of arXiv:2606.25726}
}
abstract

Let $K$ be a finite extension of $\mathbb{Q}_p$. We prove that a one-dimensional formal group law over $K$ has integral coefficients if and only if its multiplication-by-$n$ endomorphisms have integral coefficients for all integers $n$, in the height-one case, i.e. when the multiplication by $p$ has Weierstrass degree $p$. The proof uses some $p$-adic Hodge theory.

Figures

Figures reproduced from arXiv: 2606.25726 by the authors.

Figure 1
Figure 1. Tree associated to SD in the case p = 3. Indexed from bottom to top, the vertices at level n ⩾ 0 are the elements of Λn(D). For the sake of readability, we have omitted the arrowheads in the figure. There is an edge from a vertex v2 to a vertex v1 if and only if sp(v2) = v1. Moreover, recall that Z × p ≃ F × p × (1 + pZp) as topological groups. The horizontal connected components (by a dotted line) at level n ⩾ 1 ar… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 10 canonical work pages

  1. [1]

    Laurent Berger, Lubin's conjecture for full p -adic dynamical systems (English, with English and French summaries), Publ. Math. Besançon Algèbre Théorie Nr. (2016), 19--24. DOI 10.5802/pmb.o-2 https://doi.org/10.5802/pmb.o-2. MR: 3645058 https://mathscinet.ams.org/mathscinet/relay-station?mr=3645058

    work page doi:10.5802/pmb.o-2 2016

  2. [2]

    Laurent Berger, Nonarchimedean dynamical systems and formal groups, Proc. Amer. Math. Soc. 147 (2019), no. 4, 1413--1419. DOI: 10.1090/proc/14401 https://doi.org/10.1090/proc/14401. MR: 3910408 https://mathscinet.ams.org/mathscinet/relay-station?mr=3910408

    work page doi:10.1090/proc/14401 2019

  3. [3]

    Christophe Breuil, Groupes p -divisibles, groupes finis et modules filtrés, Ann. of Math. 152, (2000), no. 2, 489--549. DOI: 10.2307/2661391 https://doi.org/10.2307/2661391. MR: 1804530 https://mathscinet.ams.org/mathscinet/relay-station?mr=1804530

    work page doi:10.2307/2661391 2000

  4. [4]

    Lubin's conjecture for height-one $p$-adic dynamical systems over cyclo-tame extensions

    Martin Debaisieux, Lubin's conjecture for height-one p -adic dynamical systems over cyclo-tame extensions, arXiv (2026). arXiv: 2603.03873v2 https://arxiv.org/abs/2603.03873v2

    work page Pith review arXiv 2026

  5. [5]

    MR: 1293971 https://mathscinet.ams.org/mathscinet/relay-station?mr=1293971

    Jean-Marc Fontaine, Le corps des périodes p -adiques (French), with an appendix by Pierre Colmez, Périodes p -adiques (Bures-sur-Yvette, 1988), Astérisque 223 (1994), 59--111. MR: 1293971 https://mathscinet.ams.org/mathscinet/relay-station?mr=1293971

    1988

  6. [6]

    Michiel Hazewinkel, Formal Groups and Applications, AMS Chelsea Publishing, Amer. Math. Soc., 2012, 573 pp. ISBN: 978-0-8218-5349-8 https://isbnsearch.org/isbn/978-0-8218-5349-8

    2012

  7. [7]

    Number Theory 168 (2016), 135--153

    Liang-Chung Hsia, Hua-Chieh Li, Ramification filtrations of certain abelian Lie extensions of local fields , J. Number Theory 168 (2016), 135--153. DOI: 10.1016/j.jnt.2016.04.008 https://doi.org/10.1016/j.jnt.2016.04.008. MR: 3515812 https://mathscinet.ams.org/mathscinet/relay-station?mr=3515812

    work page doi:10.1016/j.jnt.2016.04.008 2016

  8. [8]

    Mark Kisin, Crystalline representations and F -crystals, Algebraic Geometry and Number Theory, Progr. Math. 253 (2006), 459--496. DOI: 10.1007/978-0-8176-4532-8\_7 https://doi.org/10.1007/978-0-8176-4532-8_7. MR: 2263197 https://mathscinet.ams.org/mathscinet/relay-station?mr=2263197

    work page doi:10.1007/978-0-8176-4532-8 2006

Show all 13 references
  1. [9]

    Jonathan Lubin, Nonarchimedean dynamical systems, Compos. Math. 94 (1994), no. 3, 321--346. MR: 1310863 https://mathscinet.ams.org/mathscinet/relay-station?mr=1310863

    1994

  2. [10]

    Algebra 293 (2005), no

    Ghassan Sarkis, On lifting commutative dynamical systems, J. Algebra 293 (2005), no. 1, 130--154. DOI: 10.1016/j.jalgebra.2005.08.007 https://doi.org/10.1016/j.jalgebra.2005.08.007. MR: 2173969 https://mathscinet.ams.org/mathscinet/relay-station?mr=2173969

    2005 doi

  3. [11]

    Ghassan Sarkis, Height-one commuting power series over Z _p , Bull. Lond. Math. Soc. 42 (2010), no. 3, 381--387. DOI: 10.1112/blms/bdp130 https://doi.org/10.1112/blms/bdp130. MR: 2651931 https://mathscinet.ams.org/mathscinet/relay-station?mr=2651931

    2010 doi

  4. [12]

    Joel Specter, The crystalline period of a height one p -adic dynamical system, Trans. Amer. Math. Soc. 370 (2018), no. 5, 3591--3608. DOI: 10.1090/tran/7057 https://doi.org/10.1090/tran/7057. MR: 3766859 https://mathscinet.ams.org/mathscinet/relay-station?mr=3766859

    2018 doi

  5. [13]

    DOI: 10.1007/978-3-642-87942-5\_12 https://doi.org/10.1007/978-3-642-87942-5_12

    John Tate, p -divisible groups, Proceedings of a Conference on Local Fields (Driebergen, 1966), Springer, (1967), 158--183. DOI: 10.1007/978-3-642-87942-5\_12 https://doi.org/10.1007/978-3-642-87942-5_12. MR: 231827 https://mathscinet.ams.org/mathscinet/relay-station?mr=231827

    1966 doi

Pith tools

Reviewed June 25, 2026 · model on record in the stance chip above.