The complexity of Ford domains of ${\Gamma }_0(N)$

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.

21 pages, to appear in Result in Mathematics