# Conformal Geometry and Dynamics

Published by the American Mathematical Society since 1997, the purpose of this electronic-only journal is to provide a forum for mathematical work in related fields broadly described as conformal geometry and dynamics. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.49.

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## The shape of the Ford domains for $\Gamma _0(N)$HTML articles powered by AMS MathViewer

by Antonio Lascurain Orive
Conform. Geom. Dyn. 3 (1999), 1-23
DOI: https://doi.org/10.1090/S1088-4173-99-00030-2
Published electronically: February 9, 1999

## Abstract:

This is a second paper on the Ford domains for the Hecke congruence subgroups \begin{equation*} \Gamma _0(N) = \left \{ \begin {pmatrix} a &amp; b \\ c &amp; 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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Bibliographic 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