Division polynomials for arbitrary isogenies
Number Theory
2026-04-20 v3 Cryptography and Security
Algebraic Geometry
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).
Cite
@article{arxiv.2503.15428,
title = {Division polynomials for arbitrary isogenies},
author = {Katherine E. Stange},
journal= {arXiv preprint arXiv:2503.15428},
year = {2026}
}
Comments
19 pages, v2: some additional exposition and corrections; v3: corrections and many new examples