Difference of solutions for the inversion problem of ultra-elliptic integrals
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910367234392064 |
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| author | Ayano, Takanori |
| author_facet | Ayano, Takanori |
| contents | Let $V$ be a hyperelliptic curve of genus 2 defined by $Y^2=f(X)$, where $f(X)$ is a polynomial of degree 5. The sigma function associated with $V$ is a holomorphic function on $\mathbb{C}^2$. For a point $P$ on $V$, we consider the problem to express the $X$-coordinate of $P$ in terms of the image of $P$ under the Abel-Jacobi map. Two meromorphic functions $f_2$ and $g_2$ on $\mathbb{C}^2$ which give solutions of this problem are known. Since $f_2$ and $g_2$ coincide on the zero set of the sigma function, it is expected that $f_2-g_2$ can be divided by the sigma function. In this paper, we decompose $f_2-g_2$ into a product of the sigma function and a meromorphic function explicitly. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_09406 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Difference of solutions for the inversion problem of ultra-elliptic integrals Ayano, Takanori Complex Variables Algebraic Geometry 14H42(Primary) 14K25, 32A15(Secondary) Let $V$ be a hyperelliptic curve of genus 2 defined by $Y^2=f(X)$, where $f(X)$ is a polynomial of degree 5. The sigma function associated with $V$ is a holomorphic function on $\mathbb{C}^2$. For a point $P$ on $V$, we consider the problem to express the $X$-coordinate of $P$ in terms of the image of $P$ under the Abel-Jacobi map. Two meromorphic functions $f_2$ and $g_2$ on $\mathbb{C}^2$ which give solutions of this problem are known. Since $f_2$ and $g_2$ coincide on the zero set of the sigma function, it is expected that $f_2-g_2$ can be divided by the sigma function. In this paper, we decompose $f_2-g_2$ into a product of the sigma function and a meromorphic function explicitly. |
| title | Difference of solutions for the inversion problem of ultra-elliptic integrals |
| topic | Complex Variables Algebraic Geometry 14H42(Primary) 14K25, 32A15(Secondary) |
| url | https://arxiv.org/abs/2403.09406 |