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

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- by Antonio Lascurain Orive PDF
- Conform. Geom. Dyn.
**3**(1999), 1-23 Request permission

## 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.

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## Additional Information

**Antonio Lascurain Orive**- Affiliation: Havre 101, Colonia Villa Verdun, Mexico D. F. 01810 Mexico
- Email: lasc@hardy.fciencias.unam.mx
- Received by editor(s): February 1, 1998
- Received by editor(s) in revised form: November 23, 1998
- Published electronically: February 9, 1999
- © Copyright 1999 American Mathematical Society
- Journal: Conform. Geom. Dyn.
**3**(1999), 1-23 - MSC (1991): Primary 11F06, 20H10, 22E40, 30F35, 51M10
- DOI: https://doi.org/10.1090/S1088-4173-99-00030-2
- MathSciNet review: 1668275