Combinatoire des sous-groupes de congruence du groupe modulaire

TL;DR

This paper explores the combinatorial structure of congruence subgroups of the modular group, introducing irreducible solutions to systematically construct all solutions, including explicit classifications for small levels.

Contribution

It generalizes combinatorial results from the non-modular case to the modular group and defines a new notion of irreducible solutions for congruence subgroups.

Findings

Provided a universal irreducible solution for any level N

Listed all irreducible solutions for N ≤ 6

Established a method to generate all solutions from irreducibles

Abstract

In this paper, we study the combinatorics of congruence subgroups of the modular group by generalizing results obtained in the non-modular case. For this, we define a notion of irreducible solutions from which we can build all the solutions. In particular, we give a particular solution, irreducible for any $N$, and the list of irreducible solutions for $N \leq 6$.