Characterization and enumeration on Lamé equations with finite monodromy
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909120145129472 |
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| author | Chou, You-Cheng Wang, Chin-Lung Wu, Po-Sheng |
| author_facet | Chou, You-Cheng Wang, Chin-Lung Wu, Po-Sheng |
| contents | We give a complete characterization of the classical Lamé equations $y'' = (n(n + 1)\wp(z) + B)y$, $n \in \Bbb R$, $B \in \Bbb C$ on flat tori $E_τ= \Bbb C/(\Bbb Z + \Bbb Z\,τ)$ with finite monodromy groups $M$.
Beuker--Waall had shown that such $n$ must lie in a finite number of arithmetic progressions $n_i + \Bbb N \subset \Bbb Q$ and they determined all corresponding $M$. By combining the theory of dessin d'enfants with the geometry of spherical tori, we prove the existence of $(B, τ)$ for each such $n$ and provide a description of all such $(n, B, τ, M)$.
In particular, for a given $(n, M)$ with $n \not\in \tfrac{1}{2} + \Bbb Z$, we prove the finiteness of $(B, τ)$ and derive an explicit counting formula of them. (The case $n \in \tfrac{1}{2} + \Bbb Z$ is a classical result due to Brioschi--Halphen--Crawford.)
The main ingredients in this work are (1) the definition and classification of basic spherical triangles with finite monodromy and (2) the process of attaching cells corresponding to $n \mapsto n + 1$ which reduces the problem to the basic case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16286 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characterization and enumeration on Lamé equations with finite monodromy Chou, You-Cheng Wang, Chin-Lung Wu, Po-Sheng Differential Geometry We give a complete characterization of the classical Lamé equations $y'' = (n(n + 1)\wp(z) + B)y$, $n \in \Bbb R$, $B \in \Bbb C$ on flat tori $E_τ= \Bbb C/(\Bbb Z + \Bbb Z\,τ)$ with finite monodromy groups $M$. Beuker--Waall had shown that such $n$ must lie in a finite number of arithmetic progressions $n_i + \Bbb N \subset \Bbb Q$ and they determined all corresponding $M$. By combining the theory of dessin d'enfants with the geometry of spherical tori, we prove the existence of $(B, τ)$ for each such $n$ and provide a description of all such $(n, B, τ, M)$. In particular, for a given $(n, M)$ with $n \not\in \tfrac{1}{2} + \Bbb Z$, we prove the finiteness of $(B, τ)$ and derive an explicit counting formula of them. (The case $n \in \tfrac{1}{2} + \Bbb Z$ is a classical result due to Brioschi--Halphen--Crawford.) The main ingredients in this work are (1) the definition and classification of basic spherical triangles with finite monodromy and (2) the process of attaching cells corresponding to $n \mapsto n + 1$ which reduces the problem to the basic case. |
| title | Characterization and enumeration on Lamé equations with finite monodromy |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2402.16286 |