Algorithms for Modular Parametrizations of Elliptic Curves over $\mathbb{Q}$
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| Format: | Preprint |
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2025
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| _version_ | 1866912591686664192 |
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| author | Wang, SanMin |
| author_facet | Wang, SanMin |
| contents | Let \( E \) be a complex elliptic curve with conductor \( N \) and modular invariant \( j(E) \in \mathbb{Q} \). We construct a class of modular polynomials $F_N(x,j)$ that relate the modular function $x$ on $X_0(N)$ to the $j$-invariant $j$, where $x$ is obtained by composing the first coordinate function of $E$ with the modular parametrization $φ: X_0(N) \rightarrow E$. Using $F_N(x,j)$, we can precisely determine the poles of $φ$, compute exact values of $φ$ at cusps, and develop an algorithm for calculating ramification points of $φ$. Moreover, $F_N(x,j)$ yields an efficient algorithm for computing the fibres of $φ$ over arbitrary points on $E$. In some sense, $F_N(x,j)$ also provides a ``total" formula for computing the minimal polynomial of the images of Heegner points on $X_0(N)$ under $φ$. Especially, we compute the semi-trace of the image $φ([\frac{{ - 1 + \sqrt { - 3} }}{2}])$ of the CM-point $[\frac{-1 + \sqrt{-3}}{2}]$ on $X_{0}(389)$, under the action of a 65-element subgroup of the 260-element Galois group of $\mathbb{Q}(\sqrt{-3}, j(389 \cdot \frac{-1 + \sqrt{-3}}{2}))$. Finally, we associate a point of infinite order in~\( E(\mathbb{Q}) \) with an infinite sequence~$\{ (j(τ_n), j(Nτ_n)) \}_{n \in \mathbb{Z}^+} $ of algebraic numbers whose degrees are bounded by the degree of~$φ$. This provides one seemingly practicable approach to addressing the BSD conjecture. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_14747 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algorithms for Modular Parametrizations of Elliptic Curves over $\mathbb{Q}$ Wang, SanMin Number Theory Primary 11G05, 11F03, Secondary 11Y16, 14H05 Let \( E \) be a complex elliptic curve with conductor \( N \) and modular invariant \( j(E) \in \mathbb{Q} \). We construct a class of modular polynomials $F_N(x,j)$ that relate the modular function $x$ on $X_0(N)$ to the $j$-invariant $j$, where $x$ is obtained by composing the first coordinate function of $E$ with the modular parametrization $φ: X_0(N) \rightarrow E$. Using $F_N(x,j)$, we can precisely determine the poles of $φ$, compute exact values of $φ$ at cusps, and develop an algorithm for calculating ramification points of $φ$. Moreover, $F_N(x,j)$ yields an efficient algorithm for computing the fibres of $φ$ over arbitrary points on $E$. In some sense, $F_N(x,j)$ also provides a ``total" formula for computing the minimal polynomial of the images of Heegner points on $X_0(N)$ under $φ$. Especially, we compute the semi-trace of the image $φ([\frac{{ - 1 + \sqrt { - 3} }}{2}])$ of the CM-point $[\frac{-1 + \sqrt{-3}}{2}]$ on $X_{0}(389)$, under the action of a 65-element subgroup of the 260-element Galois group of $\mathbb{Q}(\sqrt{-3}, j(389 \cdot \frac{-1 + \sqrt{-3}}{2}))$. Finally, we associate a point of infinite order in~\( E(\mathbb{Q}) \) with an infinite sequence~$\{ (j(τ_n), j(Nτ_n)) \}_{n \in \mathbb{Z}^+} $ of algebraic numbers whose degrees are bounded by the degree of~$φ$. This provides one seemingly practicable approach to addressing the BSD conjecture. |
| title | Algorithms for Modular Parametrizations of Elliptic Curves over $\mathbb{Q}$ |
| topic | Number Theory Primary 11G05, 11F03, Secondary 11Y16, 14H05 |
| url | https://arxiv.org/abs/2509.14747 |