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arxiv: 2412.10284 · v2 · submitted 2024-12-13 · 🧮 math.AG

Division polynomials in Mumford coordinates

Julia Bernatska This is my paper

Pith reviewed 2026-05-23 07:26 UTC · model grok-4.3

classification 🧮 math.AG
keywords division polynomialsMumford coordinatesgenus two curvestorsion divisorsJacobian varietyhyperelliptic curvesalgebraic geometry
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The pith

Division polynomials for 3- and 4-torsion divisors on genus two curves are obtained explicitly in Mumford coordinates.

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

The paper presents an effective method of computing division polynomials expressed directly in Mumford coordinates. Explicit formulas are derived for the cases of 3-torsion and 4-torsion divisors on a genus two curve, along with the x- and y-coordinates of points in their support. If correct, these formulas allow n-torsion divisors to be found by solving polynomial equations or by applying the Jacobi inversion problem at Jacobian points of order n. A sympathetic reader would care because the approach avoids intermediate coordinate transformations when locating torsion on the Jacobian.

Core claim

Division polynomials can be computed in terms of Mumford coordinates, with explicit expressions obtained for 3-torsion and 4-torsion divisors on a genus two curve; these expressions also give the x- and y-coordinates of the support points, so that n-torsion divisors on a given curve are obtained directly from the polynomials or by solving the Jacobi inversion problem at points of order n on the Jacobian.

What carries the argument

Division polynomials expressed in Mumford coordinates, which encode the conditions for a divisor to have order dividing n on the Jacobian of the genus two curve.

If this is right

  • n-torsion divisors on a given curve can be computed directly from the division polynomials.
  • The x- and y-coordinates of the support of the torsion divisors are obtained as part of the same computation.
  • The same divisors can alternatively be recovered by solving the Jacobi inversion problem at Jacobian points of order n.

Where Pith is reading between the lines

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

  • The method supplies a template that could be applied to obtain division polynomials for other small torsion orders on the same curves.
  • Explicit Mumford-form expressions might reduce the algebraic degree of equations that must be solved when searching for torsion points.
  • Verification on a specific curve with independently known torsion points would confirm that the derived polynomials vanish precisely on those points.

Load-bearing premise

The explicit expressions for the division polynomials in Mumford coordinates can be derived and verified for general genus-two curves without additional hidden assumptions on the curve coefficients or the base field.

What would settle it

Take a concrete genus two curve over a small finite field, apply the method to produce the 3-torsion division polynomial in Mumford form, and check whether its roots correspond exactly to the known 3-torsion divisors on that curve.

read the original abstract

An effective method of computing division polynomials in terms of Mumford coordinates is presented. As an example, division polynomials for $3$- and $4$-torsion divisors on a genus two curve are obtained explicitly in terms of Mumford coordinates, and $x$-, $y$-coordinates of the support of torsion divisors. As a result, $n$-torsion divisors on a given curve can be computed directly from the division polynomials. Alternatively, these divisors are obtained by solving the Jacobi inversion problem at points of the Jacobian variety of order $n$.

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

1 major / 0 minor

Summary. The paper claims to present an effective method for computing division polynomials in Mumford coordinates on genus-two curves. Explicit formulas are derived for the 3-torsion and 4-torsion cases, including the division polynomials themselves and the x- and y-coordinates of the support points of the corresponding divisors; these are then used to compute n-torsion divisors directly or via the Jacobi inversion problem on the Jacobian.

Significance. If the claimed method is effective and the explicit expressions are correct and general, the work supplies a practical computational tool for locating torsion points on Jacobians of genus-two curves. Such tools are relevant to arithmetic geometry and to cryptographic constructions based on hyperelliptic curves. The concrete 3- and 4-torsion examples constitute a verifiable contribution that could be reproduced or extended by other researchers.

major comments (1)
  1. [Abstract] Abstract: the central claim of an 'effective method' together with explicit 3- and 4-torsion formulas is asserted without any derivation steps, verification procedure, or error analysis. This absence is load-bearing for the soundness of the stated results and prevents assessment of whether the mathematics supports the claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim of an 'effective method' together with explicit 3- and 4-torsion formulas is asserted without any derivation steps, verification procedure, or error analysis. This absence is load-bearing for the soundness of the stated results and prevents assessment of whether the mathematics supports the claim.

    Authors: The abstract is a concise summary of the paper's results. The effective method for computing division polynomials in Mumford coordinates is developed in Section 2. Explicit derivations of the 3-torsion and 4-torsion division polynomials, together with the x- and y-coordinates of the support points, appear in Sections 3 and 4; each step follows from the standard relations between Mumford coordinates and the hyperelliptic curve equation. Verification consists of direct algebraic substitution confirming that the resulting divisors satisfy the n-torsion condition on the Jacobian. Because the identities are exact over the base field, a separate error analysis is not required. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper claims an effective computational method for obtaining division polynomials in Mumford coordinates, with explicit examples for 3- and 4-torsion divisors on genus-2 curves. No load-bearing steps reduce by construction to inputs, self-citations, fitted parameters renamed as predictions, or ansatzes smuggled via prior work. The output consists of concrete algebraic expressions derived from standard Jacobian arithmetic and Jacobi inversion, which are independently verifiable on general curves without hidden self-referential assumptions. The derivation chain is self-contained as a direct algorithmic construction rather than a closed loop.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information is available from the abstract to identify free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5600 in / 1146 out tokens · 27322 ms · 2026-05-23T07:26:02.144233+00:00 · methodology

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Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · 1 internal anchor

  1. [1]

    F., Abel’s theorem and the allied theory of theta functions , Cambridge Univ

    Baker, H. F., Abel’s theorem and the allied theory of theta functions , Cambridge Univ. Press, Cambridge, 1897

    work page

  2. [2]

    F., Multiply periodic functions , Cambridge Univ

    Baker, H. F., Multiply periodic functions , Cambridge Univ. Press, Cambridge, 1907

    work page 1907

  3. [3]

    Glasgow Mathematical Journal, 61:1 (2019) 169–193

    Bernatska, J.; Leykin, D., On degenerate sigma-functio n in genus 2. Glasgow Mathematical Journal, 61:1 (2019) 169–193

    work page 2019

  4. [4]

    Solution of the Jacobi inversion problem on non-hyperelliptic curves, Lett

    Bernatska J., Leykin D. Solution of the Jacobi inversion problem on non-hyperelliptic curves, Lett. Math. Phys. , 113 (2023) 110

    work page 2023

  5. [5]

    Bernatska, J., Computation of ℘-functions on plane algebraic curves, 2024, arXiv:2407.05 632

    work page 2024

  6. [6]

    J., Lange, T., Hyper-and-elliptic-curve cryptography

    Bernstein, D. J., Lange, T., Hyper-and-elliptic-curve cryptography. LMS Journal of Compu- tation and Mathematics , 17, Special Issue A: Algorithmic Number Theory Symposium XI, (2014) 181–202

    work page 2014

  7. [7]

    W., Costello, C, Hisil, H, Lauter, K, Fast Cryptog raphy in Genus 2, In: Johansson, T., Nguyen, P.Q

    Bos, J. W., Costello, C, Hisil, H, Lauter, K, Fast Cryptog raphy in Genus 2, In: Johansson, T., Nguyen, P.Q. (eds) Advances in Cryptology. EUROCRYPT 2013. Lecture Notes in Co mputer Science, 7881:194–210, Springer, Berlin, Heidelberg, 2013

    work page 2013

  8. [8]

    M., Leikin, D

    Buchstaber, V. M., Leikin, D. V., Hyperelliptic Additio n Law, JNMP, 12:1 (2005), 106–123

    work page 2005

  9. [9]

    M., Enolskii, V

    Buchstaber, V. M., Enolskii, V. Z., Leykin, D. V., Hyperelliptic Kleinian Functions and Ap- plications, preprint ESI 380, Vienna, 1996

    work page 1996

  10. [10]

    Multi-Dimensional Sigma-Functions

    Buchstaber, V. M., Enolskii, V. Z., Leikin, D. V., Multi-dimensional sigma-function , 2012, arXiv 1208.0990

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  11. [11]

    M., Enolskii, V

    Buchstaber, V. M., Enolskii, V. Z., Leikin, D. V., Ratio nal analogs of abelian functions, Functional Analysis and Its Applications , 33:2 (1999) 83–94

    work page 1999

  12. [12]

    M., Leikin, D

    Buchstaber, V. M., Leikin, D. V., Solution of the proble m of differentiation of Abelian func- tions over parameters for families of ( n, s)-curves, Functional Analysis and Its Applications , 42:4 (2008) 268–278

    work page 2008

  13. [13]

    Cantor, Computing in the Jacobian of a Hyperelliptic curve, Mathematics of Computation , 48:177 (1987), pp

    D. Cantor, Computing in the Jacobian of a Hyperelliptic curve, Mathematics of Computation , 48:177 (1987), pp. 95–101

    work page 1987

  14. [14]

    Cantor, D. G. On the analogue of the division polynomial s for hyperelliptic curves. J. reine angew. Math. 447 (1994) 91–145

    work page 1994

  15. [15]

    Cassels, J. W. S., Flynn, E. V., Prolegomena to a middlebrow arithmetics of curves of genus 2, CUP, Cambridge, 1996

    work page 1996

  16. [16]

    Group Law Computations on Jaco bians of Hyperelliptic Curves

    Costello, C., Lauter, K. Group Law Computations on Jaco bians of Hyperelliptic Curves. In: Miri, A., Vaudenay, S. (eds) Selected Areas in Cryptography. SAC 2011. Lecture Notes in Computer Science , 7118:92–117, Springer, Berlin, Heidelberg, 2012

    work page 2011

  17. [17]

    Kanayama, N., Division polynomials and multiplicatio n formulae of Jacobian varieties of dimension 2. Math. Proc. Camb. Philos. Soc. 139 (2005) 399–409

    work page 2005

  18. [18]

    Mathematics for Industry , 29, Springer, Singapore, 2018 18 J BERNATSKA

    Mathematical modelling for next-generation cryptography , CREST Crypto-Math Project. Mathematics for Industry , 29, Springer, Singapore, 2018 18 J BERNATSKA

    work page 2018

  19. [19]

    ˆOnishi,Y., Determinant expressions for hyperelliptic fun ctions, Proc. Edinb. Math. Soc. 48 (2005) 705–742

    work page 2005

  20. [20]

    , 134 (2011) 273–308

    Uchida, Y., Division polynomials and canonical local h eights on hyperelliptic Jacobians, Manuscripta Math. , 134 (2011) 273–308

    work page 2011