Connected fundamental domains for congruence subgroups

Abstract

We give explicit sets of right coset representatives for the congruence subgroups

,

Γ 1 ( N ) , $\Gamma _1(N),$ normal upper Gamma 1 left parenthesis upper N right parenthesis comma

and

Γ ( N ) $\Gamma (N)$ normal upper Gamma left parenthesis upper N right parenthesis

, 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

P 1 ( Z / N Z ) ${\mathbb P}^1({\mathbb Z}/N{\mathbb Z})$ double struck upper P Superscript 1 Baseline left parenthesis double struck upper Z divided by upper N double struck upper Z right parenthesis

. For every residue class

j ∈ Z / N Z $j\in {\mathbb Z}/N{\mathbb Z}$ j element of double struck upper Z divided by upper N double struck upper Z

, the number of representatives above j is governed by the simple function

W j = min { m ∈ Z > 0 ∣ m j − 1 ∈ ( Z / N Z ) ∗ } . $$\begin{align*}W_j=\min\{m\in{\mathbb{Z}}_{>0}\mid mj-1\in ({\mathbb Z}/N{\mathbb Z})^*\}.\end{align*}$$