Division polynomials for arbitrary isogenies
For mathematicians studying elliptic curves and isogenies, this provides a theoretical generalization of division polynomials, but the contribution is incremental as it builds on existing work by Mazur-Tate and Satoh.
The paper extends division polynomials to arbitrary isogenies of elliptic curves, including those with kernels not summing to identity, and demonstrates recurrence relations, identities, chain rules, and generalizations to higher dimensions.
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).