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

Author:
Antonio Lascurain Orive

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

Published electronically:
February 9, 1999

MathSciNet review:
1668275

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Abstract | References | Similar Articles | Additional Information

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

Article copyright:
© Copyright 1999
American Mathematical Society