On level 2 Modular differential equations
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929617842995200 |
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| author | Besrour, Khalil Sebbar, Abdellah |
| author_facet | Besrour, Khalil Sebbar, Abdellah |
| contents | In this paper, we explore the modular differential equation $\displaystyle y'' + F(z)y = 0$ on the upper half-plane $\mathbb{H}$, where $F$ is a weight 4 modular form for $Γ_0(2)$. Our approach centers on solving the associated Schwarzian equation $\displaystyle \{h, z\} = 2F(z)$, where $\{h, z\}$ represents the Schwarzian derivative of a meromorphic function $h$ on $\mathbb{H}$. We derive conditions under which the solutions to this equation are modular functions for subgroups of the modular group and provide explicit expressions for these solutions in terms of classical modular functions. Key tools in our analysis include the theory of equivariant functions on the upper half-plane and the representation theory of level 2 subgroups of the modular group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_14006 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On level 2 Modular differential equations Besrour, Khalil Sebbar, Abdellah Number Theory Classical Analysis and ODEs 11F03, 11F11, 34M05 In this paper, we explore the modular differential equation $\displaystyle y'' + F(z)y = 0$ on the upper half-plane $\mathbb{H}$, where $F$ is a weight 4 modular form for $Γ_0(2)$. Our approach centers on solving the associated Schwarzian equation $\displaystyle \{h, z\} = 2F(z)$, where $\{h, z\}$ represents the Schwarzian derivative of a meromorphic function $h$ on $\mathbb{H}$. We derive conditions under which the solutions to this equation are modular functions for subgroups of the modular group and provide explicit expressions for these solutions in terms of classical modular functions. Key tools in our analysis include the theory of equivariant functions on the upper half-plane and the representation theory of level 2 subgroups of the modular group. |
| title | On level 2 Modular differential equations |
| topic | Number Theory Classical Analysis and ODEs 11F03, 11F11, 34M05 |
| url | https://arxiv.org/abs/2410.14006 |