Division polynomials for arbitrary isogenies

2025/03/19 by Stange, Katherine E. · 1 citation
#Algebraic Geometry (math.AG) #Cryptography and Security (cs.CR) #FOS: Computer and information sciences #FOS: Mathematics #Number Theory (math.NT) #Primary: 11G05 11B37 14H52 Secondary: 11B39 11R37 14K02

paper · doi:10.48550/arxiv.2503.15428

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).

Cited by

• Explicit valuation of elliptic nets for elliptic curves with complex multiplication

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• Using the Charlap-Coley-Robbins polynomials for computing isogenies
• A note on factorisation of division polynomials
• Computing Isogenies at Singular Points of the Modular Polynomial
• Isogenies of Elliptic Curves: A Computational Approach