Modular curves $X_1(n)$ as moduli spaces of point arrangements and applications
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916196095361024 |
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| author | Borisov, Lev Roulleau, Xavier |
| author_facet | Borisov, Lev Roulleau, Xavier |
| contents | For a complex elliptic curve $E$ and a point $p$ of order $n$ on it, the images of the points $p_k=kp$ under the Weierstrass embedding of $E$ into $\mathbb{C}\mathbb{P}^2$ are collinear if and only if the sum of indices is divisible by $n$. Thus, it provides a realization of a certain matroid. We study this matroid in detail and prove that its realization space is isomorphic (over $\mathbb{C}$) to the modular curve $X_1(n)$, provided $n\geq 10$, which also provides an integral model of $X_1(n)$. In the process, we find a connection to the classical Ceva and Böröczky examples of special point and line configurations. We also discuss the situation for smaller values of $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_04364 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Modular curves $X_1(n)$ as moduli spaces of point arrangements and applications Borisov, Lev Roulleau, Xavier Algebraic Geometry Number Theory 14N20, 14G35, 11G05, 14K10, 11Fxx For a complex elliptic curve $E$ and a point $p$ of order $n$ on it, the images of the points $p_k=kp$ under the Weierstrass embedding of $E$ into $\mathbb{C}\mathbb{P}^2$ are collinear if and only if the sum of indices is divisible by $n$. Thus, it provides a realization of a certain matroid. We study this matroid in detail and prove that its realization space is isomorphic (over $\mathbb{C}$) to the modular curve $X_1(n)$, provided $n\geq 10$, which also provides an integral model of $X_1(n)$. In the process, we find a connection to the classical Ceva and Böröczky examples of special point and line configurations. We also discuss the situation for smaller values of $n$. |
| title | Modular curves $X_1(n)$ as moduli spaces of point arrangements and applications |
| topic | Algebraic Geometry Number Theory 14N20, 14G35, 11G05, 14K10, 11Fxx |
| url | https://arxiv.org/abs/2404.04364 |