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.

10 pages, 1 figure