\'El\'ements de comptage sur les g\'en\'erateurs du groupe modulaire et les $\lambda$-quiddit\'es

TL;DR

This paper investigates counting specific integer solutions related to the modular group generators, utilizing $ ext{lambda}$-quiddities linked to Coxeter's friezes, to understand their structure and enumeration.

Contribution

It introduces a method to count $n$-tuples of positive integers solving matrix equations involving modular group generators, connecting to $ ext{lambda}$-quiddities and Coxeter's friezes.

Findings

Derived formulas for counting solutions for specific matrices

Connected $ ext{lambda}$-quiddities to modular group elements

Provided enumeration results for solutions with fixed last component

Abstract

The aim of this article is to count the $n$-tuples of positive integers $(a_{1},\ldots,a_{n})$ solutions of the equation $\begin{pmatrix} a_{n} & -1 \\[4pt] 1 & 0 \end{pmatrix} \begin{pmatrix} a_{n-1} & -1 \\[4pt] 1 & 0 \end{pmatrix} \cdots \begin{pmatrix} a_{1} & -1 \\[4pt] 1 & 0 \end{pmatrix}=\pm M$ when $M$ is equal to the generators of the modular group $S=\begin{pmatrix} 0 & -1 \\[4pt] 1 & 0 \end{pmatrix}$ and $T=\begin{pmatrix} 1 & 1 \\[4pt] 0 & 1 \end{pmatrix}$. To count these elements, we will study the $\lambda$-quiddities, which are the solutions of the equation in the case $M=Id$ (related to Coxeter's friezes), whose last component is fixed.