Optimal Farey sequence for the Congruence subgroup $Γ_0(2^{n})$
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917183086395392 |
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| author | Doan, Nhat Minh Kim, Sang-hyun Lang, Mong Lung Tan, Ser Peow |
| author_facet | Doan, Nhat Minh Kim, Sang-hyun Lang, Mong Lung Tan, Ser Peow |
| contents | We prove that $Γ_0(2^n)$ ($n\ge2$) has a Farey sequence $\{e_i\}$ such that $e_i \le 2^{n-1}$ for all $e_i$. The above upper bound is optimal, and there exists a unique $j$ such that $e_j= 2^{n-1} $. For each $e_i$, there exists a unique $a_i$ such that $\{ a_i/e_i\}\cup \{\infty\}$ is the set of ideal vertices of a fundamental domain of $Γ_0(2^n)$ whose side-pairings give a set of independent generators of $Γ_0(2^n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01324 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimal Farey sequence for the Congruence subgroup $Γ_0(2^{n})$ Doan, Nhat Minh Kim, Sang-hyun Lang, Mong Lung Tan, Ser Peow Number Theory Group Theory Geometric Topology 20D08 We prove that $Γ_0(2^n)$ ($n\ge2$) has a Farey sequence $\{e_i\}$ such that $e_i \le 2^{n-1}$ for all $e_i$. The above upper bound is optimal, and there exists a unique $j$ such that $e_j= 2^{n-1} $. For each $e_i$, there exists a unique $a_i$ such that $\{ a_i/e_i\}\cup \{\infty\}$ is the set of ideal vertices of a fundamental domain of $Γ_0(2^n)$ whose side-pairings give a set of independent generators of $Γ_0(2^n)$. |
| title | Optimal Farey sequence for the Congruence subgroup $Γ_0(2^{n})$ |
| topic | Number Theory Group Theory Geometric Topology 20D08 |
| url | https://arxiv.org/abs/2601.01324 |