Division polynomials for arbitrary isogenies

• URL: http://arxiv.org/abs/2503.15428v2
• Date: Thu, 17 Apr 2025 19:23:39 GMT
• Title: Division polynomials for arbitrary isogenies
• Authors: Katherine E. Stange,
• Abstract summary: We extend the definition of divisions to arbitrary isogenies of elliptic curves, including those whose kernels do not sum to the identity.<n>In analogy to the case of divisions for multiplication-by-n, we demonstrate relations, identities relating to classical elliptic functions.
• Score: 0.24864093375172572
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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).

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