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

TL;DR

This paper studies the complexity of Ford domains for the subgroup (N), classifies when this complexity is zero, and explores how it grows with the prime factors of N.

Contribution

It provides a complete classification of integers N with zero complexity and analyzes the growth of complexity relative to prime factors.

Findings

Classified all N with c(N)=0.

Proved c(N) tends to infinity as prime factors increase.

Linked complexity to shape properties of Ford domains.

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.