Explicit valuation of elliptic nets for elliptic curves with complex multiplication
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915666290802688 |
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| author | Au-Yeung, Edison H L |
| author_facet | Au-Yeung, Edison H L |
| contents | Division polynomials associated to an elliptic curve $E/K$ are polynomials $ϕ_n, ψ_n^2$ 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 $Φ_{v}, Ψ_{v}^2$ 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 $Φ_{v}$ and $Ψ_{v}^2$ 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_09601 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Explicit valuation of elliptic nets for elliptic curves with complex multiplication Au-Yeung, Edison H L Number Theory 11G05, 11B37, 11B39, 14H52 Division polynomials associated to an elliptic curve $E/K$ are polynomials $ϕ_n, ψ_n^2$ 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 $Φ_{v}, Ψ_{v}^2$ 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 $Φ_{v}$ and $Ψ_{v}^2$ 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). |
| title | Explicit valuation of elliptic nets for elliptic curves with complex multiplication |
| topic | Number Theory 11G05, 11B37, 11B39, 14H52 |
| url | https://arxiv.org/abs/2512.09601 |