Explicit valuation of elliptic nets for elliptic curves with complex multiplication

TL;DR

This paper derives explicit valuation formulas for elliptic nets related to elliptic curves with complex multiplication, enabling new insights into divisibility sequences and potential applications in computing integral points.

Contribution

It provides a new valuation formula for elliptic nets on CM elliptic curves, extending understanding of divisibility sequences and recurrence relations.

Findings

Derived valuation formula for elliptic nets with CM

Showed recurrence relations for elliptic divisibility sequences

Potential application in computing integral points

Abstract

Division polynomials associated to an elliptic curve $E/K$ are polynomials $\phi_n, \psi_n^2$ that arise from the sequence of points $\{nP\}_{n \in \mathbb{N}}$ on this curve. If one wishes to study $\mathbb{Z}$--linear combination of points on $E(K)$, we can use net polynomials $\Phi_{v}, \Psi_{v}^2$ which are higher--dimensional analogue 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 $\Phi_{v}$ and $\Psi_{v}^2$ 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. Additionally, we also expect that the formula may also be used in computing $\mathcal{O}_K$--integral points of an elliptic curve of rank $2$ with complex multiplication (this is future work).