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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2503.12352 |
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| _version_ | 1866911483217051648 |
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| author | Nie, Zhaohu |
| author_facet | Nie, Zhaohu |
| contents | For $N>1$, we constructed a canonical connected fundamental domain for $Γ_0(N)$ in [Nie, Parent], utilizing an interesting function $W: {\mathbb Z}/N\to {\mathbb N}$. In this paper, we further study the function $W$, prove some identities, and use it to match the cusps, with widths, produced by our connected fundamental domain with the known cusp classes of $Γ_0(N)$. Furthermore, we list the boundary arcs and the gluing patterns of our connected fundamental domain, a key step in understanding the modular curve $X_0(N)$ by this approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_12352 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cusps and boundaries of connected fundamental domains for $Γ_0(N)$ Nie, Zhaohu Number Theory Combinatorics Rings and Algebras 11F06, 20H05 For $N>1$, we constructed a canonical connected fundamental domain for $Γ_0(N)$ in [Nie, Parent], utilizing an interesting function $W: {\mathbb Z}/N\to {\mathbb N}$. In this paper, we further study the function $W$, prove some identities, and use it to match the cusps, with widths, produced by our connected fundamental domain with the known cusp classes of $Γ_0(N)$. Furthermore, we list the boundary arcs and the gluing patterns of our connected fundamental domain, a key step in understanding the modular curve $X_0(N)$ by this approach. |
| title | Cusps and boundaries of connected fundamental domains for $Γ_0(N)$ |
| topic | Number Theory Combinatorics Rings and Algebras 11F06, 20H05 |
| url | https://arxiv.org/abs/2503.12352 |