Connected fundamental domains for congruence subgroups

Abstract

We produce canonical sets of right coset representatives for the congruence subgroups $Γ_0(N)$, $Γ_1(N)$ and $Γ(N)$, 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.

Figures (4)

• Figure 1: Our Fundamental Domain for $\Gamma_0(6)$ Labeled
• Figure 2: Our Fundamental Domain for $\Gamma_1(8)$ Partially Labeled
• Figure 3: Our Fundamental Domain for $\Gamma_1(8)$ Zoomed in for Further Labeling
• Figure 4: Our Fundamental Domain for $\Gamma_0(30)$

Theorems & Definitions (17)

• Definition 1.1
• Theorem 1.7
• Definition 1.10
• Theorem 1.12
• Lemma 2.1
• proof
• Definition 2.6
• Remark 2.7
• Lemma 2.8
• proof