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Uniform Bounds for Periods of Endomorphisms of Varieties

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Pith's one-line read Primitive periods of integral periodic points under endomorphisms of varieties are bounded explicitly by data of the special fiber

desk verdict The main theorem is false as stated at d'=0, but the proof method is sound and the statement is repairable with a small hypothesis fix. read the letter →

arxiv 1908.06231 v3 pith:IL7ZR7IK submitted 2019-08-17 math.NT math.AG

Keping Huang This is my paper · ORCID
classification math.NTmath.AG MSC 37P2014G20
keywords arithmeticdynamicsprimitiveperiodp-adicperiodicpointsuniformboundednessweakNeronmodelspecialfibercotangentspaceendomorphismsofvarieties
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 explicit uniform upper bound for the primitive period of a periodic point that is defined over the ring of integers of a p-adic field and lies on a variety with an endomorphism extending to an integral model. The bound depends only on reduction data: the number of points in the special fiber, the residue-field size, and the maximum cotangent-space dimension at special-fiber points. This matters because a uniform bound on rational preperiodic points is a known open problem in arithmetic dynamics, and bounding periods is a direct route toward it. The result works for morphisms without good reduction and for singular models, so it applies more broadly than earlier period bounds.

What carries the argument

The argument is carried by the reduced orbit subscheme $\operatorname{Spec} A$ attached to the orbit of $P$ under a suitable iterate $g$. The ring $A$ is local of finite rank over $R$, with maximal ideal $m$, and $g$ induces an $R$-endomorphism $\sigma$ of $A$. The action on the cotangent space $m/m^2$ (the maximal ideal modulo its square) controls the non-$p$ part of the period, while the decomposition $\sigma = \mathrm{id} + h$ with $h(m)\subseteq m^2$ controls the $p$-power part through a binomial expansion. A filtration of $m$ gives $\dim_k(m/m^2)\le d'+1$, with the span of $\pi$ invariant under $\sigma$, so the order of the linear action is at most $|k|^{d'}-1$; the binomial argument shows the remaining order is a $p$-power bounded by the valuation of $p$.

What would settle it

Compute the reduced orbit ring $A$ for an integral periodic point of a p-adic endomorphism on a singular model with bad reduction and check whether the uniformizer $\pi$ is a zero divisor; a single example with $\dim_k((\pi)/(m\pi,\pi^2)) \ne 1$ would invalidate the Step 2 bound on the cotangent order, and any example with primitive period larger than the stated bound would refute the theorem directly.

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Extended reading notes

Core claim

The central claim is Theorem 1.2. Let $K$ be a finite extension of $\mathbb{Q}_p$, $R$ its ring of integers, $k$ its residue field, and $\pi$ a uniformizer. If $P\in \mathcal{X}(R)$ is periodic under an endomorphism $f$ that extends to an integral model $\mathcal{X}$, and $d'$ is the maximum dimension of the cotangent spaces at points of the special fiber $\bar{\mathcal{X}}(k)$, then the primitive period $n$ of $P$ satisfies $n \le |\bar{\mathcal{X}}(k)|\, p^{v(p)-1}(|k|^{d'}-1)$ for $p>2$ and $n \le |\bar{\mathcal{X}}(k)|\, 2^{v(2)}(|k|^{d'}-1)$ for $p=2$. The proof splits the period as $n = n_0 r t$: $n_0$ is the period of the reduction of $P$ in the special fiber, $r$ is the order of the induced map on the cotangent space, and $t$ is a $p$-power. It bounds $n_0$ by $|\bar{\mathcal{X}}(k)|$, $r$ by $|k|^{d'}-1$, and $t$ by $v(p)-1$ (or $v(2)$ when $p=2$). This extends a known method to singular models and maps without good reduction.

Load-bearing premise

The proof relies on the orbit ring $A$ being free over $R$, so that the uniformizer $\pi$ is not a zero divisor; the paper states this property but does not prove it.

Editorial extensions

If this is right

  • For $K=\mathbb{Q}_p$ the bound becomes $n \le |\bar{\mathcal{X}}(\mathbb{F}_p)|(p^{d'}-1)$ for odd $p$ and $n \le |\bar{\mathcal{X}}(\mathbb{F}_p)|\,2(2^{d'}-1)$ for $p=2$.
  • Any $K$-rational periodic point on a variety admitting a weak Neron model satisfies the same bound, because such a model is bijective on $K$-points.
  • For cubic polynomial maps with no $K$-rational repelling fixed point, the worked example yields $n \le (|k|+1)p^{v(p)-1}(|k|-1)$.
  • Because the model is not required to be nonsingular and the morphism only needs to extend to the model, the bound covers singular and non-projective examples beyond earlier constructions.

Reading between the lines

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

  • A direct computational test of the proof's core assumption is to form the reduced orbit ring $A$ for a periodic point on a singular model with bad reduction and check whether $\pi$ is a zero divisor; if any example gives $\dim_k((\pi)/(m\pi,\pi^2))>1$, the Step 2 bound on $r$ would be unsupported.
  • The same three-factor decomposition might extend from periodic points to periodic subvarieties, replacing the cotangent space of a point by infinitesimal data along the orbit; the paper does not pursue this.
  • The bound addresses periods, not the full uniform-boundedness conjecture: bounding the primitive period of a single point does not by itself bound the number of preperiodic points, since the special fiber may support many distinct orbits. Combining this period control with a count of periodic points of bounded period on the reduction would be the natural next step.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 3 minor

Summary. The paper proves an explicit uniform bound for the primitive period of an R-point that is periodic under an endomorphism f of a projective K-variety X, assuming f extends to a model X over the ring of integers R of a p-adic field K. The bound is expressed in terms of |\bar X(k)|, the maximum cotangent-space dimension d' on the special fiber, and p-adic data. The proof follows Fakhruddin's three-step strategy: bound the period of the reduction, bound the order of the induced action on the cotangent space, and show the remaining factor is a p-power with controlled valuation. The paper also gives a weak Neron model corollary and a cubic polynomial example.

Significance. If the technical gaps are repaired, the result is a useful contribution: it gives a fully explicit, reduction-only bound for primitive periods of integral periodic points, covering singular models and cases of bad reduction that earlier good-reduction results exclude. The strategy is transparent and the reliance on external results (Fakhruddin, Darafsheh, Poonen) is clearly stated; there is no circularity. The examples illustrate the scope of the method. However, the current manuscript contains several load-bearing gaps and one false statement in the main theorem, so the result is not yet established as written.

major comments (4)
  1. [§1, Theorem 1.2, display (1.1)] The theorem is false as stated when d'=0. Take X=Spec K, X=Spec R, f=id. Then P has primitive period n=1, |\bar X(k)|=1, d'=0, and the right-hand side of (1.1) is zero, so the asserted inequality 1≤0 fails. The same issue affects Theorem 1.3. The statement needs either an explicit hypothesis d'≥1 or the replacement of |k|^{d'}-1 by max(1, |k|^{d'}-1).
  2. [§3, Step 3] The proof writes σ=id+h and asserts h(m)⊂m^2 for the original σ. This is false; in Example 3.1, with A=Z_3[x]/(x(x-3)), m=(x,3), and σ(x)=3-x, one obtains h(x)=3-2x, which is not in m^2=(3x,9). The argument only works after replacing σ by σ^r, where r is the order of the induced map on m/m^2, so that the linear part is trivial. This replacement is missing, and the subsequent claim that the order s of σ is a p-power is false for the original σ (in Example 3.1 the order is 2 when p=3).
  3. [§3, paragraph 'Recall that A is local of finite rank over R...'] The assertions that A is finite flat over R and that π is not a zero divisor in A are used in Step 2 (for example, to identify dim_k(π)/(mπ,π^2) with dim_k A/(m,π)=1) but are not proved. The orbit subscheme should be defined as the scheme-theoretic image of the finite union of the orbit sections; since R is a DVR, the resulting quotient A embeds into R^t, is torsion-free, and hence is finite flat over R. The paper should supply this argument, as the current text merely states the needed properties.
  4. [§3, Step 3, p-power order proof] The contradiction argument that s is a p-power omits the choice of a nonzero h(a). One must choose a∈m with h(a)≠0 (which exists when s≠1) before applying the binomial expansion; then ν(s h(a))=ν(h(a)) while all remaining terms have strictly larger valuation, giving the contradiction. As written, if h(a)=0 the equation gives no contradiction, so the conclusion '0 ∉ m^{ν(h(a))+1}' is not justified.
minor comments (3)
  1. [§3, Step 2] The sentence 'σ is the identity precisely when \barσ is' is not true in general; an R-algebra automorphism can be nontrivial while acting trivially on m/m^2, a p-order effect that Step 3 is meant to control. Please rephrase to describe the exact sequence whose kernel is a p-group.
  2. [§3, Step 2] Please spell out how Corollary 2 of [Dar05] yields r ≤ |k|^{d'}-1. Since the induced linear map fixes the line spanned by π, the relevant group is an affine group of dimension d', not simply GL_{d'}(k), so the bound is not immediate from the usual general-linear-group order bound.
  3. [Abstract] There is a typographical spacing error in 'ove r' in the abstract.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1.2's bound is derived from reduction data via external results, with no fitted inputs or self-citation chains.

full rationale

The derivation of Theorem 1.2 is self-contained against external bounds rather than circular. Step 1 bounds the primitive period n0 of the reduced point by the size |Xbar(k)| of the special fiber; Step 2 bounds the order r of the induced cotangent-space map by |k|^{d'}-1 using the dimension estimate for m/m2 and Corollary 2 of Darafsheh [Dar05]; Step 3 bounds the remaining p-power part by v(p)-1 (or v(p) when p=2) via Proposition 3 of Fakhruddin [Fak01] and the argument following Theorem 1 of Poonen [Poo14]. No parameter is fitted to the desired inequality, no claim is defined in terms of the target bound, and the cited results are external works invoked under stated hypotheses, not self-citations. The paper does contain a genuine correctness concern: the assertion that π is not a zero divisor in the orbit algebra A is used to compute dim((π)/(mπ,π^2)) = 1, but flatness/torsion-freeness of A over R is not proved. Moreover, the theorem as stated fails when d'=0 because |k|^{d'}-1=0 makes the right-hand side of (1.1) zero (e.g., a zero-dimensional variety with a periodic point of period 1). These are mathematical gaps, however, not instances of the derivation reducing to its own inputs, so they do not affect the circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted constants and no new entities. The proof pulls in several external results, and the flatness of the orbit algebra is an unproved structural premise. The bound itself contains no free parameters beyond the input data of the special fiber and the cotangent dimension.

assumptions (6)
  • domain assumption The orbit subscheme A of P under the iterate g is a finite flat local R-algebra and pi is not a zero divisor in A.
    Asserted in Section 3 ('A is local of finite rank over R' and 'as pi is not a zero divisor in A') without proof; needed for the dimension count and valuation argument.
  • standard math Proposition 1 of [Fak01]: n <= n0 * r * p^t, where n0 is the reduction period, r the order on the cotangent space, and t the p-part exponent.
    Used as the framework at the top of Section 3; not reproved in the paper.
  • standard math Proposition 3 of [Fak01]: after trivializing the cotangent action, the remaining p-part t is at most v(p)-1 for p > 2 and at most v(p) for p = 2.
    Final step of Step 3; the paper cites this instead of proving it.
  • standard math Corollary 2 of [Dar05]: the order of any element of GL_{d'}(k) is at most |k|^{d'} - 1.
    Used in Step 2 to bound r; the paper does not reproduce the argument.
  • standard math Poonen's binomial argument: if sigma is an R-algebra automorphism with sigma = id + h and h(m) subset m^2, then a nontrivial finite order must be divisible by p.
    Step 3 follows [Poo14]; the manuscript's exposition skips the minimal-order step.
  • domain assumption Definition 2.1 of weak Neron model after [Hsi96], with X(K) isomorphic to X(R).
    Corollary 2.2 relies on it to transfer K-periodic points to R-points.

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Pith. "Pith review of Uniform Bounds for Periods of Endomorphisms of Varieties." pith.science (2026). https://pith.science/paper/IL7ZR7IK

@misc{pith2026190806231,
  author       = {Pith},
  title        = {Pith review of: Uniform Bounds for Periods of Endomorphisms of Varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IL7ZR7IK}},
  note         = {Machine review of arXiv:1908.06231}
}
abstract

Suppose $X$ is a projective variety defined over a finite extension $K$ of $\mathbb{Q}_p$ and suppose $X$ admits a model $\mathcal{X}$ defined over the ring of integers $R$ of $K$. Let $f:{X}\rightarrow {X}$ be an endomorphism of $X$ defined over $K$ that can be extended to an endomorphism of $\mathcal{X}$ defined over $R$. We apply a method of Fakhruddin to prove an explicit upper bound for the primitive period of periodic points defined over $R$.

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Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [1]

    Benedetto, Reduction, dynamics, and J ulia sets of rational functions , J

    Robert L. Benedetto, Reduction, dynamics, and J ulia sets of rational functions , J. Number Theory 86 (2001), no. 2, 175--195. 1813109

    work page 2001

  2. [2]

    Benedetto, Dragos Ghioca, Benjamin Hutz, P\" a r Kurlberg, Thomas Scanlon, and Thomas J

    Robert L. Benedetto, Dragos Ghioca, Benjamin Hutz, P\" a r Kurlberg, Thomas Scanlon, and Thomas J. Tucker, Periods of rational maps modulo primes, Math. Ann. 355 (2013), no. 2, 637--660. 3010142

    work page 2013

  3. [3]

    11, 3576--3597

    Jason Pierre Bell, Dragos Ghioca, and Thomas John Tucker, Applications of p-adic analysis for bounding periods of subvarieties under \'e tale maps , International Mathematics Research Notices 2015 (2014), no. 11, 3576--3597

    work page 2014

  4. [4]

    153 (2012), no

    Jean-Yves Briend and Liang-Chung Hsia, Weak N \' e ron models for cubic polynomial maps over a non- A rchimedean field , Acta Arith. 153 (2012), no. 4, 415--428. 2925380

    work page 2012

  5. [5]

    M. R. Darafsheh, Order of elements in the groups related to the general linear group, Finite Fields Appl. 11 (2005), no. 4, 738--747. 2181417

    work page 2005

  6. [6]

    i, Publications Math \'e matiques de l'Institut des Hautes \'E tudes Scientifiques 43 (1974), no

    Pierre Deligne, La conjecture de weil. i, Publications Math \'e matiques de l'Institut des Hautes \'E tudes Scientifiques 43 (1974), no. 1, 273--307

    work page 1974

  7. [7]

    Indian Acad

    Najmuddin Fakhruddin, Boundedness results for periodic points on algebraic varieties, Proc. Indian Acad. Sci. Math. Sci. 111 (2001), no. 2, 173--178. 1836365

    work page 2001

  8. [8]

    Ramanujan Math

    , Questions on self maps of algebraic varieties, J. Ramanujan Math. Soc. 18 (2003), no. 2, 109--122. 1995861

    work page 2003

Show all 29 references
  1. [9]

    3, 277--304

    Liang-Chung Hsia, A weak n \'e ron model with applications to p -adic dynamical systems , Compositio Mathematica 100 (1996), no. 3, 277--304

    1996

  2. [10]

    Benjamin Hutz, Good reduction of periodic points on projective varieties, Illinois J. Math. 53 (2009), no. 4, 1109--1126. 2741181

    2009

  3. [11]

    , Good reduction and canonical heights of subvarieties, Math. Res. Lett. 25 (2018), no. 6, 1837--1863. 3934847

    2018

  4. [12]

    Kamienny, Torsion points on elliptic curves and q -coefficients of modular forms , Invent

    S. Kamienny, Torsion points on elliptic curves and q -coefficients of modular forms , Invent. Math. 109 (1992), no. 2, 221--229. 1172689

    1992

  5. [13]

    Kamienny and B

    S. Kamienny and B. Mazur, Rational torsion of prime order in elliptic curves over number fields, Ast\' e risque (1995), no. 228, 3, 81--100, With an appendix by A. Granville, Columbia University Number Theory Seminar (New York, 1992). 1330929

    1995

  6. [14]

    Eric Katz, Joseph Rabinoff, and David Zureick-Brown, Uniform bounds for the number of rational points on curves of small M ordell- W eil rank , Duke Math. J. 165 (2016), no. 16, 3189--3240. 3566201

    2016

  7. [15]

    100 (1996), no

    Hua-Chieh Li, Counting periodic points of p -adic power series , Compositio Math. 100 (1996), no. 3, 351--364. 1387670

    1996

  8. [16]

    Tucker, Thue equations and the method of C habauty- C oleman , Invent

    Dino Lorenzini and Thomas J. Tucker, Thue equations and the method of C habauty- C oleman , Invent. Math. 148 (2002), no. 1, 47--77. 1892843

    2002

  9. [17]

    4, 819--827

    Serge Lang and Andr \'e Weil, Number of points of varieties in finite fields, American Journal of Mathematics 76 (1954), no. 4, 819--827

    1954

  10. [18]

    Mazur, Modular curves and the E isenstein ideal , Inst

    B. Mazur, Modular curves and the E isenstein ideal , Inst. Hautes \' E tudes Sci. Publ. Math. (1977), no. 47, 33--186 (1978). 488287

    1977

  11. [19]

    Lo\" c Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124 (1996), no. 1-3, 437--449. 1369424

    1996

  12. [20]

    Silverman, Rational periodic points of rational functions, Internat

    Patrick Morton and Joseph H. Silverman, Rational periodic points of rational functions, Internat. Math. Res. Notices (1994), no. 2, 97--110. 1264933

    1994

  13. [21]

    Narkiewicz, Polynomial cycles in algebraic number fields, Colloq

    W. Narkiewicz, Polynomial cycles in algebraic number fields, Colloq. Math. 58 (1989), no. 1, 151--155. 1028168

    1989

  14. [22]

    Narkiewicz and T

    W. Narkiewicz and T. Pezda, Finite polynomial orbits in finitely generated domains, Monatsh. Math. 124 (1997), no. 4, 309--316. 1480362

    1997

  15. [23]

    Pezda, Polynomial cycles in certain local domains, Acta Arith

    T. Pezda, Polynomial cycles in certain local domains, Acta Arith. 66 (1994), no. 1, 11--22. 1262650

    1994

  16. [24]

    Bjorn Poonen, The classification of rational preperiodic points of quadratic polynomials over Q : a refined conjecture , Math. Z. 228 (1998), no. 1, 11--29. 1617987

    1998

  17. [25]

    3, 525--527

    , p-adic interpolation of iterates, Bulletin of the London Mathematical Society 46 (2014), no. 3, 525--527

    2014

  18. [26]

    Silverman, The field of definition for dynamical systems on P^1 , Compositio Math

    Joseph H. Silverman, The field of definition for dynamical systems on P^1 , Compositio Math. 98 (1995), no. 3, 269--304. 1351830

    1995

  19. [27]

    Michael Stoll, Uniform bounds for the number of rational points on hyperelliptic curves of small M ordell- W eil rank , J. Eur. Math. Soc. (JEMS) 21 (2019), no. 3, 923--956. 3908770

    2019

  20. [28]

    Andr \'e Weil, Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc 55 (1949), no. 5, 497--508

    1949

  21. [29]

    Michael Ernest Zieve, Cycles of polynomial mappings, ProQuest LLC, Ann Arbor, MI, 1996, Thesis (Ph.D.)--University of California, Berkeley. 2694837

    1996

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