Division polynomials for arbitrary isogenies

TL;DR

This work generalizes division polynomials from the classical multiplication-by-$n$ case to arbitrary isogenies by defining $\Psi_\phi$ with divisor $D_\phi = \phi^*(\mathcal{O}') - \deg\phi(\mathcal{O}) + (P_\phi) - (\mathcal{O})$ and by introducing kernel-function normalizations to handle biased kernels. It establishes analogues of the key properties (recurrence, chain rule, and relation to the $x$-coordinate) for $\Psi_\phi$ and $\widehat{\Psi}_\phi$, including both unbiased and biased isogenies, plus generalized recurrences and a two-torsion analysis; it also extends the construction to higher dimensions and connects specialization to elliptic nets. The framework yields a coherent approach to compute and manipulate isogeny-related polynomials, with explicit examples and a path to applications in kernel polynomials, complex multiplication, and computational isogeny theory. Overall, the paper unifies and extends Mazur–Tate and Satoh’s perspectives, providing new tools for isogeny-based computations and a bridge to elliptic nets and higher-dimensional analogues.

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).

Theorems & Definitions (28)

• Lemma 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4: First chain rule
• proof
• Lemma 4.1
• proof
• Lemma 4.2