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arXiv · 2512.09601

Explicit valuation of elliptic nets for elliptic curves with complex multiplication

Edison H L Au-Yeung
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Abstract

Division polynomials associated to an elliptic curve $E/K$ are polynomials $\phi_n, \psi_n^2$ that arise from the sequence of points $\{nP\}_{n \in \N}$ on this curve. If one wishes to study $\Z$--linear combination of points on $E(K)$, we can use net polynomials $\Phi_{\bm{v}}, \Psi_{\bm{v}}^2$ which are higher--dimensional \textcolor{red}{analogues} of division polynomials. It turns out they are also elliptic nets, an $n$--dimensional array with values in $K$ satisfying the same nonlinear recurrence relation that division polynomials do as well. Now further assume the elliptic curve $E/K$ has complex multiplication by an order of a quadratic imaginary field $F \subseteq K$, we will prove a formula for the common valuation of $\Phi_{\bm{v}}$ and $\Psi_{\bm{v}}^2$ associated to multiples of points by elements of an order in $F$. As an application, we will use the formula to show that elliptic divisibility sequences associated to multiples of points indexed by elements of an order also satisfy a recurrence relation when indexed by elements of an order, subject to certain conditions on the indices.

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Edison H L Au-Yeung. 2025-12-10. Explicit valuation of elliptic nets for elliptic curves with complex multiplication. https://doi.org/10.4153/s2976859426100058

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