Hurwitz integrality of the power series expansion of the sigma function for a telescopic curve
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arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866917160882798592 |
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| author | Ayano, Takanori |
| author_facet | Ayano, Takanori |
| contents | A telescopic curve is a certain algebraic curve defined by $m-1$ equations in the affine space of dimension $m$, which can be a hyperelliptic curve and an $(n,s)$ curve as a special case. The sigma function $σ(u)$ associated with the telescopic curve of genus $g$ is a holomorphic function on $\mathbb{C}^g$. For a subring $R$ of $\mathbb{C}$ and variables $u={}^t(u_1,\dots, u_g)$, let \[R\langle\langle u \rangle\rangle=\left\{\sum_{i_1,\dots,i_g\ge0}κ_{i_1,\dots,i_g}\frac{u_1^{i_1}\cdots u_g^{i_g}}{i_1!\cdots i_g!}\;\middle|\;κ_{i_1,\dots,i_g}\in R\right\}.\] If the power series expansion of a holomorphic function $f(u)$ on $\mathbb{C}^g$ around the origin belongs to $R\langle\langle u \rangle\rangle$, then $f(u)$ is said to be Hurwitz integral over $R$. In this paper, we show that the sigma function $σ(u)$ associated with the telescopic curve is Hurwitz integral over the ring generated by the coefficients of the defining equations of the curve and $\frac{1}{2}$ over $\mathbb{Z}$. Further, we show that $σ(u)^2$ is Hurwitz integral over the ring generated by the coefficients of the defining equations of the curve over $\mathbb{Z}$. Our results are a generalization of the results of Y. Ônishi for $(n,s)$ curves to telescopic curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_10079 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Hurwitz integrality of the power series expansion of the sigma function for a telescopic curve Ayano, Takanori Algebraic Geometry Complex Variables 14H42(Primary) 14K25, 32A05(Secondary) A telescopic curve is a certain algebraic curve defined by $m-1$ equations in the affine space of dimension $m$, which can be a hyperelliptic curve and an $(n,s)$ curve as a special case. The sigma function $σ(u)$ associated with the telescopic curve of genus $g$ is a holomorphic function on $\mathbb{C}^g$. For a subring $R$ of $\mathbb{C}$ and variables $u={}^t(u_1,\dots, u_g)$, let \[R\langle\langle u \rangle\rangle=\left\{\sum_{i_1,\dots,i_g\ge0}κ_{i_1,\dots,i_g}\frac{u_1^{i_1}\cdots u_g^{i_g}}{i_1!\cdots i_g!}\;\middle|\;κ_{i_1,\dots,i_g}\in R\right\}.\] If the power series expansion of a holomorphic function $f(u)$ on $\mathbb{C}^g$ around the origin belongs to $R\langle\langle u \rangle\rangle$, then $f(u)$ is said to be Hurwitz integral over $R$. In this paper, we show that the sigma function $σ(u)$ associated with the telescopic curve is Hurwitz integral over the ring generated by the coefficients of the defining equations of the curve and $\frac{1}{2}$ over $\mathbb{Z}$. Further, we show that $σ(u)^2$ is Hurwitz integral over the ring generated by the coefficients of the defining equations of the curve over $\mathbb{Z}$. Our results are a generalization of the results of Y. Ônishi for $(n,s)$ curves to telescopic curves. |
| title | Hurwitz integrality of the power series expansion of the sigma function for a telescopic curve |
| topic | Algebraic Geometry Complex Variables 14H42(Primary) 14K25, 32A05(Secondary) |
| url | https://arxiv.org/abs/2208.10079 |