The complexity of Ford domains of $Γ_0(N)$
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910175077597184 |
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| author | Zhang, Pengcheng |
| author_facet | Zhang, Pengcheng |
| contents | We investigate a particular choice of the Ford fundamental domain of the congruence subgroup $Γ_0(N)$ and define a notion of complexity $c(N)$ accordingly, which is a nonnegative integer and carries some information on the shape of the Ford domain. The property that $c(N)=0$ first appeared as a technical assumption in a paper by Pohl, which is closely related to a conjecture of Zagier on the "reduction theory" of $Γ_0(N)$. In this paper, we give a complete classification of positive integers $N$ with $c(N)=0$, and we also show that $c(N)$ goes to infinity if both the number of distinct prime factors of $N$ and the smallest prime factor of $N$ go to infinity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18511 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The complexity of Ford domains of $Γ_0(N)$ Zhang, Pengcheng Number Theory 11F06 We investigate a particular choice of the Ford fundamental domain of the congruence subgroup $Γ_0(N)$ and define a notion of complexity $c(N)$ accordingly, which is a nonnegative integer and carries some information on the shape of the Ford domain. The property that $c(N)=0$ first appeared as a technical assumption in a paper by Pohl, which is closely related to a conjecture of Zagier on the "reduction theory" of $Γ_0(N)$. In this paper, we give a complete classification of positive integers $N$ with $c(N)=0$, and we also show that $c(N)$ goes to infinity if both the number of distinct prime factors of $N$ and the smallest prime factor of $N$ go to infinity. |
| title | The complexity of Ford domains of $Γ_0(N)$ |
| topic | Number Theory 11F06 |
| url | https://arxiv.org/abs/2508.18511 |