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arxiv: 2503.15428 · v3 · submitted 2025-03-19 · 🧮 math.NT · cs.CR· math.AG

Division polynomials for arbitrary isogenies

Katherine E. Stange This is my paper

Pith reviewed 2026-05-22 23:31 UTC · model grok-4.3

classification 🧮 math.NT cs.CRmath.AG
keywords division polynomialselliptic curvesisogeniesrecurrence relationselliptic netsMazur-TateSatoh
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The pith

Division polynomials extend to arbitrary isogenies of elliptic curves, including kernels that do not sum to the identity.

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

The paper generalizes the classical division polynomials, previously tied to multiplication-by-n maps, so that they apply to any isogeny between elliptic curves. It verifies that the new polynomials obey the same recurrence relations, connect to elliptic functions in the expected way, and satisfy a chain rule relating polynomials on the source and target curves. The construction also carries over to the higher-dimensional setting of elliptic nets. A sympathetic reader would care because isogenies appear throughout the arithmetic of elliptic curves, and a uniform polynomial description simplifies many calculations that previously required case-by-case handling.

Core claim

Following the constructions of Mazur-Tate and Satoh, division polynomials are defined for an arbitrary isogeny by applying the classical recurrence relations to the kernel subgroup without requiring that the kernel points sum to the identity. The resulting polynomials satisfy the expected recurrence relations, admit identities with Weierstrass sigma and zeta functions, obey a chain-rule relation under composition of isogenies, and extend directly to the theory of elliptic nets in higher dimension.

What carries the argument

The generalized division polynomials obtained by direct application of the Mazur-Tate–Satoh recurrence to the kernel of an arbitrary isogeny.

If this is right

  • Recurrence relations hold for the new division polynomials exactly as in the multiplication-by-n case.
  • Identities relating the polynomials to classical elliptic functions remain valid.
  • A chain rule relates the division polynomials on the source curve to those on the target curve under an isogeny.
  • The same construction produces elliptic nets for arbitrary isogenies in higher dimension.

Where Pith is reading between the lines

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

  • The uniform definition removes the need to treat multiplication maps separately from other isogenies in algorithmic settings.
  • The chain rule may allow recursive evaluation of composed isogenies without recomputing kernels at each step.
  • Generalization to elliptic nets suggests that similar polynomial machinery could apply to isogenies of abelian varieties of higher dimension.

Load-bearing premise

The classical recurrence relations and functional identities continue to hold when the kernel is replaced by an arbitrary finite subgroup of the elliptic curve.

What would settle it

An explicit computation, for a concrete isogeny whose kernel does not sum to the identity, showing that the candidate division polynomial fails to satisfy the claimed recurrence relation.

read the original abstract

Following work of Mazur-Tate and Satoh, we extend the definition of division polynomials to arbitrary isogenies of elliptic curves, including those whose kernels do not sum to the identity. In analogy to the classical case of division polynomials for multiplication-by-n, we demonstrate recurrence relations, identities relating to classical elliptic functions, the chain rule describing relationships between division polynomials on source and target curve, and generalizations to higher dimension (i.e., elliptic nets).

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 / 1 minor

Summary. The manuscript extends the Mazur-Tate and Satoh construction of division polynomials from the multiplication-by-n endomorphism to arbitrary isogenies of elliptic curves, including those whose kernel subgroups K satisfy sum_{P in K} P ≠ O. It asserts that the same recurrence relations, elliptic-function identities, chain rule relating source and target curves, and generalizations to elliptic nets continue to hold under this broader definition.

Significance. If the claimed extension and preservation of algebraic properties hold, the work supplies a uniform algebraic framework for division polynomials attached to any isogeny, which may streamline computations involving arbitrary kernels in elliptic curve theory and isogeny-based cryptography. The explicit retention of recurrence relations and the chain rule would constitute a non-trivial structural result.

minor comments (1)
  1. The abstract refers to 'generalizations to higher dimension (i.e., elliptic nets)' without indicating the precise dimension or the form of the net; a brief clarification in the introduction would help readers locate the relevant statements.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for noting its potential significance in providing a uniform framework for division polynomials. The recommendation of 'uncertain' appears to stem from whether the claimed extensions and preservations of properties hold for arbitrary isogenies (including kernels with nonzero sum). We address this below and confirm that the proofs are contained in the paper.

read point-by-point responses

  1. Referee: The manuscript extends the Mazur-Tate and Satoh construction of division polynomials from the multiplication-by-n endomorphism to arbitrary isogenies of elliptic curves, including those whose kernel subgroups K satisfy sum_{P in K} P ≠ O. It asserts that the same recurrence relations, elliptic-function identities, chain rule relating source and target curve, and generalizations to elliptic nets continue to hold under this broader definition.

    Authors: The manuscript defines division polynomials for arbitrary isogenies (Section 2), explicitly including the case sum_{P in K} P ≠ O. It then proves that the recurrence relations (Section 3), identities with elliptic functions (Section 4), the chain rule between source and target curves (Section 5), and the generalization to elliptic nets (Section 6) all continue to hold under this definition. These results are established algebraically without additional restrictions on the kernel. revision: no

Circularity Check

0 steps flagged

No significant circularity; explicit algebraic extension with supplied proofs

full rationale

The paper extends the Mazur-Tate/Satoh division polynomials to isogenies whose kernels need not sum to the identity. It supplies explicit definitions, recurrence relations, elliptic-function identities, chain-rule relations, and elliptic-net generalizations directly in the manuscript. No load-bearing step reduces by construction to a fitted parameter, self-citation, or renamed input; the cited prior results are external (different authors) and the new constructions are self-contained algebraic generalizations whose correctness is internal to the provided recurrences and proofs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5586 in / 908 out tokens · 40704 ms · 2026-05-22T23:31:08.762564+00:00 · methodology

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Reference graph

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