# Generalized division polynomials

@article{Satoh2004GeneralizedDP,
title={Generalized division polynomials},
author={Takakazu Satoh},
journal={Mathematica Scandinavica},
year={2004},
volume={94},
pages={161-184}
}
Let $E$ be an elliptic curve with complex multiplication by the ring $O_{F}$ of integers of an imaginary quadratic field $F$. We give an explicit condition on $\alpha\in O_{F}$ so that there exists a rational function $\psi_{\alpha}$ satisfying $\div\psi_{\alpha}=\sum_{P\in\mathrm{Ker}[\alpha]}[P] - N_{F /Q}(\alpha)[{\mathcal{O}}]$ where $[\alpha ]$ is the multiplication by $\alpha$ map. We give an algorithm to compute $\psi_{\alpha}$ based on recurrence formulas among these functions. We prove… Expand
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