The shape of the Ford domains for $\Gamma _0(N)$

Abstract:

This is a second paper on the Ford domains for the Hecke congruence subgroups \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 $p_1,p_2,\dots ,p_r$ yields the number of finite vertices of the Ford polygon for $\Gamma _0(N)$, for all numbers $N=p_1\,p_2\dotsb p_r$ whose prime factors are larger than a constant which depends only on $r$. In all cases the formulas are in terms of symmetric polynomials which generalize the Euler $\phi$ 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.