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arXiv · 2508.18511

The complexity of Ford domains of $\Gamma_0(N)$

Pengcheng Zhang
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Abstract

We investigate a particular choice of the Ford fundamental domain of the congruence subgroup $\Gamma_0(N)$ 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 $\Gamma_0(N)$. 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.

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Pengcheng Zhang. 2025-08-25. The complexity of Ford domains of $\Gamma_0(N)$. https://arxiv.org/abs/2508.18511

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