The shape of the Ford domains for Γ₀(𝑁)

This is a second paper on the Ford domains for the Hecke congruence subgroups Γ 0 ( N ) = { ( a a m p ; a m p ; b c a m p ; a m p ; d ) ∈ S L ( 2 , Z ) ∣ c ≡ 0 mod N } . \begin{equation*} \Gamma _0(N) = \left \{ \begin {pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm {SL}(2, \mathbb {Z}) \mid c\equiv 0 \mod N \right \}. \end{equation*} The author establishes techniques to calculate the number of sides of these domains; in the process the shape of such polygons becomes apparent in many cases. Explicit formulas are given for numbers which have no more than four prime factors. The main result ( Theorem 1 ) exhibits the existence of a universal symmetric polynomial which evaluated at p 1 , p 2 , … , p r p_1,p_2,\dots ,p_r yields the number of finite vertices of the Ford polygon for Γ 0 ( N ) \Gamma _0(N) , for all numbers N = p 1 p 2 ⋯ p r N=p_1\,p_2\dotsb p_r whose prime factors are larger than a constant which depends only on r r . In all cases the formulas are in terms of symmetric polynomials which generalize the Euler ϕ \phi function. The techniques developed to count the number of visible isometric circles show that the study of these circles might also be a useful tool to simplify or solve problems in number theory.