We produce canonical sets of right coset representatives for the congruence subgroups $\Gamma_0(N)$, $\Gamma_1(N)$ and $\Gamma(N)$, and prove that the corresponding fundamental domains are connected. Key to our construction is a study of the projective line $P^1({ \mathbb Z}/N{ \mathbb Z})$ using a function $M: { \mathbb Z}/N{ \mathbb Z}\to { \mathbb Z}_{ \geq 0}$, representing multiplicities. We further study this function and show that it is simply one less than another much more computable function $W:{ \mathbb Z}/N{ \mathbb Z}\to { \mathbb N}$, of possible independent interest. We present some examples and pictures at the end.
Categories: math.NT math.GR