Connected fundamental domains for congruence subgroups

Abstract

We give explicit sets of right coset representatives for the congruence subgroups $\Gamma_0(N)$, $\Gamma_1(N)$ and $\Gamma(N)$, and prove that the corresponding unions of standard modular triangles are connected fundamental domains. The construction is based on a study of the projective line ${\mathbb P}^1({\mathbb Z}/N{\mathbb Z})$. For every residue class $j\in{\mathbb Z}/N{\mathbb Z}$, the number of representatives above $j$ is governed by the simple function \[W_j=\min\{m\in{\mathbb Z}_{>0}\mid mj-1\in ({\mathbb Z}/N{\mathbb Z})^*\}.\] We also include examples illustrating how the connected domains make cusp and boundary data visible.