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Large integers coprime to the characteristic appear as exact reduction orders of non-torsion points on elliptic curves over function fields, and the same holds fiberwise for sections of abelian schemes whose Betti map is generically submers

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2026-07-13 01:04 UTC pith:J2XBG6BO

load-bearing objection Solid, self-contained Silverman-type results for function fields and a clean Betti-map criterion; methodological rather than dramatic, but the proofs hold and the limitations are honest.

arxiv 2607.09002 v1 pith:J2XBG6BO submitted 2026-07-10 math.NT math.AG

Remarks on a theorem of Silverman

Khai-Hoan Nguyen-Dang , Quang-Khai Nguyen This is my paper
classification math.NT math.AG MSC 11G0514G0514G1711G5014K99
keywords elliptic curvereduction orderglobal function fieldSilverman theoremBetti mapabelian schemedivisibility sequenceunlikely intersections
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper asks when a non-torsion rational point on an abelian variety over a global field realizes every sufficiently large integer as the exact order of its reduction at some place. It answers the question affirmatively for elliptic curves over global function fields of characteristic greater than 3, but only for orders coprime to the characteristic. The same phenomenon is then established in a relative setting over the complex numbers: if a non-torsion section of an abelian scheme has generically submersive Betti map, every large enough integer appears as the exact order of that section on some complex fiber. The results give a function-field and a geometric counterpart to classical theorems of Bang, Zsigmondy, Schinzel and Silverman, and they show that the obstruction for p-power orders is forced by supersingular reduction. A reader interested in arithmetic dynamics or unlikely intersections therefore obtains a clean existence statement for reduction orders under hypotheses that can be checked via heights or Betti rank.

Core claim

For an elliptic curve over a global function field of characteristic p>3 and a non-torsion rational point P, every sufficiently large integer n coprime to p is realized as the exact order of the reduction of P at some place of good reduction. Independently, if a non-torsion section of an abelian scheme over C has generically submersive Betti map, every large enough integer appears as the exact order of the section on some complex fiber.

What carries the argument

Height decomposition together with the formal-group identity that local height is preserved under multiplication by n coprime to p at good places (so that a non-primitive divisor would force the Néron–Tate height of P to vanish); in the relative case, the real-analytic Betti map and the elementary fact that open sets in the real torus contain points of every large exact order.

Load-bearing premise

The formal-group calculation that, away from the characteristic, multiplying a point that reduces to the identity by an integer coprime to p does not change its local height; without this equality the height inequality that produces a contradiction fails.

What would settle it

Exhibit a single elliptic curve over a global function field of characteristic p>3 and a non-torsion rational point such that infinitely many integers n coprime to p never appear as the order of the reduction of the point at any place of good reduction.

Watch this falsifier — get emailed when new claim-graph text bears on it.

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 / 5 minor

Summary. The paper studies when a non-torsion point on an algebraic group over a global field realizes every sufficiently large integer n as the exact order of its reduction at some place. Theorem A (Theorem 2) proves this for elliptic curves over global function fields of characteristic p>3, for all large n coprime to p, via a Silverman–Cheon–Hahn style height comparison that uses the formal-group isomorphism at good places (Lemma 1), Northcott, and quadraticity of the Néron–Tate height. The authors also give explicit supersingular isotrivial examples (Propositions 2–3) showing that Siegel’s local-to-global height decay and the unrestricted “all but finitely many P” claim both fail in positive characteristic. Theorem B (Theorem 3) treats the relative problem over C: if the Betti map of a non-torsion section of an abelian scheme is generically submersive, then every large N appears as the exact order of the section on some complex fiber; this is deduced from real-analytic openness of a submersion together with density of exact-order torsion on the torus (Lemma 5), and is linked to the relative Manin–Mumford theorem of Gao–Habegger. An appendix proves the analogous Schinzel statement for one-dimensional tori over global function fields.

Significance. The function-field result supplies a clean, self-contained Silverman-type statement that covers supersingular curves without the constant-j or ordinary restrictions of earlier EDS work, while the counter-examples to Siegel’s theorem in characteristic p are concrete and useful. The relative theorem cleanly interfaces the reduction-order question with the Betti-map formalism and relative Manin–Mumford, giving a transparent geometric criterion that applies in all situations where generic submersivity is already known. Both proofs are short, standard, and free of circularity or fitted parameters; the limitations (coprimality to p, non-effectivity, restriction to generically submersive Betti maps) are stated honestly. The contribution is methodological and expository rather than a breakthrough on the open higher-dimensional number-field case, but it is solid and of clear interest to arithmetic geometry.

minor comments (5)
  1. In the introduction the authors write “a fortiori, they do not cover Theorem A”; the Latin phrase is slightly misused (the preceding sentence already states the stronger claim). Replace by “in particular” or “hence”.
  2. Lemma 1 and Remark 3: the local height is written both as -min{0,v(x),v(y)}deg(v) and as -(3/2)min{0,v(x)}deg(v). A single consistent formula (or an explicit cross-reference) would avoid momentary confusion.
  3. In the proof of Theorem 2 the elementary bound sum 1/r^{2} < 1/2 is correct, but the intermediate identity “1/4 + (π^{2}/8 - 1) = π^{2}/8 - 3/4” is unnecessary; a direct comparison with the geometric series already suffices and shortens the argument.
  4. Appendix A: the product-formula argument for Φ_n(x) is clean, yet the four cases would be easier to follow if each case ended with an explicit statement of the contribution to the sum rather than only the vanishing or non-vanishing of v(Φ_n(x)).
  5. Several bibliographic items (e.g., [18], [19], [13]) appear both as published and as arXiv preprints; a uniform citation style would improve readability.

Circularity Check

0 steps flagged

No circularity: height comparison and Betti-submersion arguments are self-contained against standard external tools.

full rationale

The paper's two main theorems are proved by direct arguments that do not reduce to their own conclusions. Theorem 2 assumes for contradiction that large n coprime to p never appear as reduction orders, obtains the local-height inequality (1) from the formal-group isomorphism of Lemma 1 (valid precisely when gcd(n,p)=1), bounds the finitely many bad places by Lemma 2, and reaches ^h(P)=0 via quadraticity and Northcott (Proposition 1), contradicting non-torsion. None of these steps is definitional of the claim; the coprimality restriction is independently justified by supersingular reductions (Remark 5) and by explicit counter-examples (Propositions 2–3). Theorem 3 likewise uses only the external geometric hypothesis of generic Betti submersivity, the real-analytic submersion theorem, and the elementary density of exact-order torsion on the torus (Lemma 5). Corollary 3 invokes the relative Manin–Mumford theorem of Gao–Habegger as an independent external input, not a self-citation. There are no fitted parameters, no self-definitional normalizations, and no load-bearing uniqueness theorems imported from the authors. The derivation chain is therefore free of circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The paper rests on classical arithmetic geometry (heights, formal groups, product formula, Néron models) and on one deep external theorem (relative Manin–Mumford of Gao–Habegger) used only for the corollary. No free parameters are fitted; the only ad-hoc restrictions are the characteristic and coprimality hypotheses forced by the formal-group argument.

axioms (4)
  • standard math Northcott property and quadraticity of the Néron–Tate height on elliptic curves over global function fields with finite constant field (Proposition 1).
    Invoked throughout §2 to obtain the contradiction ^h(P)=0.
  • standard math Formal-group isomorphism E1(Kv) ≅ Ê(mv) and the expansion [n]z = nz + O(z^{2}) for n coprime to p (Lemma 1).
    Load-bearing for the equality hv(nP)=hv(P) at good places.
  • domain assumption Relative Manin–Mumford conjecture as proved by Gao–Habegger (arXiv:2303.05045), used to equate Zariski density of torsion with maximal Betti rank.
    Invoked only for Corollary 3; the main Theorem 3 does not need it.
  • domain assumption Characteristic p>3 and n coprime to p so that the formal group behaves as in characteristic zero and supersingular reductions do not force p-power orders.
    Explicitly necessary (Remark 5); the paper supplies counter-examples when the hypothesis is dropped.

pith-pipeline@v1.1.0-grok45 · 24314 in / 2738 out tokens · 25888 ms · 2026-07-13T01:04:48.789167+00:00 · methodology

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Motivated by a theorem of Silverman, we consider the following problem. Let $A$ be an abelian variety over a global field $K$. Given a non-torsion point $P \in A(K)$, for a sufficiently large positive integer $n$, whether there exists a place $v$ of $K$ such that the order of the reduction of $P$ modulo $v$ is $n$? In this article, we first show that this holds for an elliptic curve over a global function field of positive characteristic $p>3$ and for sufficiently large positive integers $n$ coprime to $p$. In the second part of the paper, we consider its relative version over $\mathbb C$. More precisely, let $\pi: \mathcal{A} \rightarrow S$ be an abelian scheme over some variety $S$ over $\mathbb C$, and let $P$ be a non-torsion section of $\pi$. If the Betti map associated to $P$ is generically submersive, then for every sufficiently large $n$, there is a point $s$ in $S(\mathbb C)$ such that $P(s)$ is a point of order $n$ in the corresponding fiber.

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Works this paper leans on

35 extracted references · 3 linked inside Pith

  1. [1]

    The Betti map associated to a section of an abelian scheme.Inventiones Mathematicae, 222(1):161–202, 2020

    Yves André, Pietro Corvaja, and Umberto Zannier. The Betti map associated to a section of an abelian scheme.Inventiones Mathematicae, 222(1):161–202, 2020

    2020

  2. [2]

    Andreas S. Bang. Taltheoretiske undersøgelser (fortsat, se s. 80).Tidsskrift for Mathematik, 5:70–80, 130– 137, 1886. In Danish

  3. [3]

    Divisibility sequences related to abelian varieties isogenous to a power of an elliptic curve.Journal of Number Theory, 279:170–183, 2025

    Stefan Bara ´ nczuk, Bartosz Naskr˛ ecki, and Matteo Verzobio. Divisibility sequences related to abelian varieties isogenous to a power of an elliptic curve.Journal of Number Theory, 279:170–183, 2025. Preprint version: arXiv:2309.09699

    Pith/arXiv arXiv 2025

  4. [4]

    Greatest common divisor results on semia- belian varieties and a conjecture of Silverman.Res

    Fabrizio Barroero, Laura Capuano, and Amos Turchet. Greatest common divisor results on semia- belian varieties and a conjecture of Silverman.Res. Number Theory, 10(1):16, 2024. Id/No 17

    2024

  5. [5]

    Betti maps, Pell equations in polynomials and almost-Belyi maps.Forum of Mathematics, Sigma, 10:e84, 2022

    Fabrizio Barroero, Laura Capuano, and Umberto Zannier. Betti maps, Pell equations in polynomials and almost-Belyi maps.Forum of Mathematics, Sigma, 10:e84, 2022. 19

    2022

  6. [6]

    Fabrizio Barroero and Gabriel A. Dill. On the Zilber–Pink conjecture for complex abelian varieties. Annales Scientifiques de l’École Normale Supérieure (4), 55(1):261–282, 2022

    2022

  7. [7]

    Baum and Melvin M

    Leonard E. Baum and Melvin M. Sweet. Continued fractions of algebraic power series in characteristic 2.Ann. Math. (2), 103:593–610, 1976

    1976

  8. [8]

    The orders of the reductions of a point in the Mordell–Weil group of an elliptic curve.Acta Arithmetica, 88(3):219–222, 1999

    Jung Cheon and Seong Hahn. The orders of the reductions of a point in the Mordell–Weil group of an elliptic curve.Acta Arithmetica, 88(3):219–222, 1999

    1999

  9. [9]

    On the torsion values for sections of an elliptic scheme.Journal für die reine und angewandte Mathematik (Crelles Journal), 782:1–41, 2022

    Pietro Corvaja, Julian Demeio, David Masser, and Umberto Zannier. On the torsion values for sections of an elliptic scheme.Journal für die reine und angewandte Mathematik (Crelles Journal), 782:1–41, 2022

    2022

  10. [10]

    Number 62 in Mémoires de la Société Mathématique de France (N.S.)

    Sinnou David.Minorations de formes linéaires de logarithmes elliptiques. Number 62 in Mémoires de la Société Mathématique de France (N.S.). Société Mathématique de France, 1995

    1995

  11. [11]

    A polynomial Zsigmondy theorem.Journal of Algebra, 343(1):138– 142, 2011

    Anthony Flatters and Thomas Ward. A polynomial Zsigmondy theorem.Journal of Algebra, 343(1):138– 142, 2011

    2011

  12. [12]

    Generic rank of Betti map and unlikely intersections.Compositio Mathematica, 156(12):2469–2509, 2020

    Ziyang Gao. Generic rank of Betti map and unlikely intersections.Compositio Mathematica, 156(12):2469–2509, 2020

    2020

  13. [13]

    The Relative Manin-Mumford Conjecture

    Ziyang Gao and Philipp Habegger. The Relative Manin-Mumford Conjecture. Preprint, arXiv:2303.05045 [math.NT] (2023), 2023

    Pith/arXiv arXiv 2023

  14. [14]

    Springer, New York, 1 edition, 1977

    Robin Hartshorne.Algebraic Geometry, volume 52 ofGraduate Texts in Mathematics. Springer, New York, 1 edition, 1977

    1977

  15. [15]

    Silverman, Katherine E

    Patrick Ingram, Valéry Mahé, Joseph H. Silverman, Katherine E. Stange, and Marco Streng. Algebraic divisibility sequences over function fields.Journal of the Australian Mathematical Society, 92(1):99–126, 2012

    2012

  16. [16]

    Springer, New York, 1 edition, 1983

    Serge Lang.Fundamentals of Diophantine Geometry. Springer, New York, 1 edition, 1983. Originally pub- lished by Wiley-Interscience

    1983

  17. [17]

    On a theorem of liouville in fields of positive characteristic.Canadian Journal of Mathemat- ics, 1:397–400, 1949

    Kurt Mahler. On a theorem of liouville in fields of positive characteristic.Canadian Journal of Mathemat- ics, 1:397–400, 1949

    1949

  18. [18]

    Divisibility sequences of polynomials and heights estimates.New York Journal of Mathematics, 22:989–1020, 2016

    Bartosz Naskr˛ ecki. Divisibility sequences of polynomials and heights estimates.New York Journal of Mathematics, 22:989–1020, 2016. Also available as arXiv:1609.04750

    Pith/arXiv arXiv 2016

  19. [19]

    Primitive divisors of elliptic divisibility sequences over function fields with constantj-invariant.Journal of Number Theory, 213:152–186, 2020

    Bartosz Naskr˛ ecki and Marco Streng. Primitive divisors of elliptic divisibility sequences over function fields with constantj-invariant.Journal of Number Theory, 213:152–186, 2020

    2020

  20. [20]

    Common valuations of division polynomials.Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 155(5):1646–1660, 2025

    Bartosz Naskr˛ ecki and Matteo Verzobio. Common valuations of division polynomials.Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 155(5):1646–1660, 2025

    2025

  21. [21]

    Charles F. Osgood. Effective bounds on the ’diophantine approximation’ of algebraic functions over fields of arbitrary characteristic and applications to differential equations.Nederl. Akad. Wet., Proc., Ser. A, 78:105–119, 1975

    1975

  22. [22]

    La Sapienza

    Antonella Perucca.On the order of the reductions of points on abelian varieties and tori. PhD thesis, Univer- sità di Roma “La Sapienza”, 2008

    2008

  23. [23]

    Primitive divisors of the expressiona n −b n in algebraic number fields.Journal für die reine und angewandte Mathematik, 268/269:27–33, 1974

    Andrzej Schinzel. Primitive divisors of the expressiona n −b n in algebraic number fields.Journal für die reine und angewandte Mathematik, 268/269:27–33, 1974

    1974

  24. [24]

    Silverman

    Joseph H. Silverman. Arithmetic distance functions and height functions in diophantine geometry. Math. Ann., 279(1-2):193–216, 1987

    1987

  25. [25]

    Silverman

    Joseph H. Silverman. Wieferich’s criterion and the abc-conjecture.Journal of Number Theory, 30(2):226– 237, 1988

    1988

  26. [26]

    Silverman.Advanced Topics in the Arithmetic of Elliptic Curves, volume 151 ofGraduate Texts in Mathematics

    Joseph H. Silverman.Advanced Topics in the Arithmetic of Elliptic Curves, volume 151 ofGraduate Texts in Mathematics. Springer, New York, 1994

    1994

  27. [27]

    Silverman.The Arithmetic of Elliptic Curves, volume 106 ofGraduate Texts in Mathematics

    Joseph H. Silverman.The Arithmetic of Elliptic Curves, volume 106 ofGraduate Texts in Mathematics. Springer, New York, 2 edition, 2009

    2009

  28. [28]

    The Stacks project.https://stacks.math.columbia.edu, 2026

    The Stacks project authors. The Stacks project.https://stacks.math.columbia.edu, 2026

    2026

  29. [29]

    Katherine E. Stange. Integral points on elliptic curves and explicit valuations of division polynomials. Canadian Journal of Mathematics, 68(5):1120–1158, 2016

    2016

  30. [30]

    The rank of elliptic curves.Doklady Akademii Nauk SSSR, 175(4):770– 773, 1967

    John Tate and Igor Shafarevich. The rank of elliptic curves.Doklady Akademii Nauk SSSR, 175(4):770– 773, 1967

    1967

  31. [31]

    Bounding tangencies of sections on elliptic surfaces.International Mathematics Research Notices

    Douglas Ulmer and Giancarlo Urzúa. Bounding tangencies of sections on elliptic surfaces.International Mathematics Research Notices. IMRN, 2021(6):4768–4802, 2021. 20

    2021

  32. [32]

    Some effectivity results for primitive divisors of elliptic divisibility sequences.Pacific Journal of Mathematics, 325(2):331–351, 2023

    Matteo Verzobio. Some effectivity results for primitive divisors of elliptic divisibility sequences.Pacific Journal of Mathematics, 325(2):331–351, 2023

    2023

  33. [33]

    Multiplier ideal sheaves, Nevanlinna theory, and Diophantine approximation

    Paul Vojta. Multiplier ideal sheaves, Nevanlinna theory, and Diophantine approximation. InNumber theory, analysis and geometry. In memory of Serge Lang, pages 647–658. Berlin: Springer, 2012

    2012

  34. [34]

    Horst G. Zimmer. On the difference of the Weil height and the Néron–Tate height.Mathematische Zeitschrift, 147:35–51, 1976

    1976

  35. [35]

    Zur Theorie der Potenzreste.Monatshefte für Mathematik und Physik, 3(1):265–284, 1892

    Karl Zsigmondy. Zur Theorie der Potenzreste.Monatshefte für Mathematik und Physik, 3(1):265–284, 1892. Khai-Hoan Nguyen-Dang Khaihoann@gmail.com Morningside Center of Mathematics, Chinese Academy of Sciences, Beijing, China Quang-Khai Nguyen nguyen@math.univ-lyon1.fr Camille Jordan Institute, Claude Bernard University Lyon 1 21 avenue Claude Bernard, 6910...