Connected fundamental domains for congruence subgroups

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.

💡 Research Summary

The paper addresses a classical problem in the theory of modular forms: constructing explicit, canonical sets of right coset representatives for the three standard congruence subgroups of the modular group, namely Γ₀(N), Γ₁(N) and Γ(N), and proving that the associated fundamental domains are connected. The authors begin by recalling the usual fundamental domain D for the full modular group Γ(1)=SL₂(ℤ), defined by |z|>1 and |Re z|<½ in the upper half‑plane ℍ. They prove a general lemma (Lemma 2.1) stating that if a finite collection of right coset representatives {γ₁,…,γ_n} is given, then the union E=⋃_{i=1}^n γ_i D is a fundamental domain for the subgroup generated by those cosets provided that E is connected. Thus the whole problem reduces to producing a representative list whose associated union of copies of D is connected.

The key insight is to translate the coset problem for Γ₀(N) into a combinatorial problem on the projective line over the finite ring ℤ/Nℤ, denoted ℙ¹(ℤ/Nℤ). There is a well‑known bijection between right cosets in Γ₀(N)\Γ(1) and the points of ℙ¹(ℤ/Nℤ). The authors split ℙ¹(ℤ/Nℤ) into two subsets: the “affine part” A₁={ (1:b) } and the “hyperbolic part” H={ (a:b) | gcd(a,N)>1, gcd(a,b,N)=1 }. For each point (a:b) they define a multiplicity function

M(a:b)=min{ m≥0 | gcd(m a−b, N)=1 }.

This measures how many applications of the generator T are needed before the resulting matrix has a lower‑left entry coprime to N. While M is well defined, its direct computation can be cumbersome. Therefore the authors introduce a more tractable function

W(j)=min{ m∈ℕ | m j−1∈(ℤ/Nℤ)^{×} },

which simply asks for the smallest m such that m j−1 is a unit modulo N. They prove (Theorem 1.12) that for each residue class j, W(j)=M_j+1, where M_j is the maximal value of M over all points whose first coordinate equals j. This relation allows one to compute the needed multiplicities efficiently.

With these tools, the authors construct an explicit list of representatives for Γ₀(N). Let N₁=⌊(N−1)/2⌋ and N₂=⌊N/2⌋, and for each residue class x∈ℤ/Nℤ define a symmetric integer e_x∈