Connected fundamental domains for congruence subgroups
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866918367936380928 |
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| author | Nie, Zhaohu Parent, C. Xavier |
| author_facet | Nie, Zhaohu Parent, C. Xavier |
| contents | We produce canonical sets of right coset representatives for the congruence subgroups $Γ_0(N)$, $Γ_1(N)$ and $Γ(N)$, and prove that the corresponding fundamental domains are connected. Key to our construction is a study of the projective line $P^1({\mathbb Z}/N{\mathbb Z})$ using a function $M: {\mathbb Z}/N{\mathbb Z}\to {\mathbb Z}_{\geq 0}$, representing multiplicities. We further study this function and show that it is simply one less than another much more computable function $W:{\mathbb Z}/N{\mathbb Z}\to {\mathbb N}$, of possible independent interest. We present some examples and pictures at the end. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17119 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Connected fundamental domains for congruence subgroups Nie, Zhaohu Parent, C. Xavier Number Theory Group Theory 11F06, 20H05 We produce canonical sets of right coset representatives for the congruence subgroups $Γ_0(N)$, $Γ_1(N)$ and $Γ(N)$, and prove that the corresponding fundamental domains are connected. Key to our construction is a study of the projective line $P^1({\mathbb Z}/N{\mathbb Z})$ using a function $M: {\mathbb Z}/N{\mathbb Z}\to {\mathbb Z}_{\geq 0}$, representing multiplicities. We further study this function and show that it is simply one less than another much more computable function $W:{\mathbb Z}/N{\mathbb Z}\to {\mathbb N}$, of possible independent interest. We present some examples and pictures at the end. |
| title | Connected fundamental domains for congruence subgroups |
| topic | Number Theory Group Theory 11F06, 20H05 |
| url | https://arxiv.org/abs/2411.17119 |