On l-th roots and division by l
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| Format: | Preprint |
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2024
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| _version_ | 1866916155912880128 |
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| author | Miret, Josep M. Pujolàs, Jordi Thériault, Nicolas |
| author_facet | Miret, Josep M. Pujolàs, Jordi Thériault, Nicolas |
| contents | We give a characterization of the codomain $[\ell]E(k)$ of the multiplication-by-$\ell$ map $[\ell]$ in the case of elliptic curves over a field $k$ of characteristic $\ne 2,3$ with $\ell$-torsion $E[\ell]=\langle W_1,W_2 \rangle$ fully defined over $k$, for primes $\ell$ different from the characteristic. We show that a point $Q\in E(k)$ lies in $[\ell]E(k)$ if and only if $h_{W_1}(-Q)$ and $h_{W_2}(-Q)$ are $\ell$-powers of $k$, where $h_{W_1}$ and $h_{W_2}$ are functions on $E$ with divisor ${\rm div}(h_{W_i})=\ell W_i- \ell P_{\infty}$. Our characterization leads to an effective procedure to find pre-images of $[\ell]$ by solving an order $\ell$ system of linear equations and computing a polynomial gcd. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_06619 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On l-th roots and division by l Miret, Josep M. Pujolàs, Jordi Thériault, Nicolas Number Theory 11G20 (Primary) 14H52, 14G50 (Secondary) We give a characterization of the codomain $[\ell]E(k)$ of the multiplication-by-$\ell$ map $[\ell]$ in the case of elliptic curves over a field $k$ of characteristic $\ne 2,3$ with $\ell$-torsion $E[\ell]=\langle W_1,W_2 \rangle$ fully defined over $k$, for primes $\ell$ different from the characteristic. We show that a point $Q\in E(k)$ lies in $[\ell]E(k)$ if and only if $h_{W_1}(-Q)$ and $h_{W_2}(-Q)$ are $\ell$-powers of $k$, where $h_{W_1}$ and $h_{W_2}$ are functions on $E$ with divisor ${\rm div}(h_{W_i})=\ell W_i- \ell P_{\infty}$. Our characterization leads to an effective procedure to find pre-images of $[\ell]$ by solving an order $\ell$ system of linear equations and computing a polynomial gcd. |
| title | On l-th roots and division by l |
| topic | Number Theory 11G20 (Primary) 14H52, 14G50 (Secondary) |
| url | https://arxiv.org/abs/2403.06619 |