1 Introduction

The motivation for this article came from the previous work [Reference Nie and ParentNP24], in which we produced canonical connected fundamental domains for the congruence subgroups. Let $N>1$ and let us concentrate on the heart of that work, the $\Gamma _0(N)$ case.

The connected fundamental domain produces some natural cusps with their own widths, specified by a function W (see (1.1)). The cusps produced this way are not necessarily inequivalent to each other. It was one motivation of this article to classify these cusps by their equivalence classes and to reconcile the corresponding widths. The particular case of $N=30$ was worked out in [Reference Nie and ParentNP24, Example 3.2], and here we aim to study the case for a general N and to prove the corresponding identities.

From the author’s point of view, the advantage of a connected fundamental domain is the convenience it provides in understanding and getting a feel for the modular curve $X_0(N)$ . As opposed to a disconnected fundamental domain consisting of ideal geodesic triangles in ${\mathbb H}$ , the common edges for a connected one are already identified, giving a clearer picture. The remaining task is to specify how the boundary arcs are identified by elements of $\Gamma _0(N)$ . Another goal of this article is to write out the list of boundary arcs and the gluing patterns. This, a priori, seems a hopeless task for a general N, but the final result turns out to be rather neat. A key role is played by the geometry of the projective line ${\mathbb P}^1({\mathbb Z}/N)$ (see (2.3)).

After the motivation, let us describe these two results, about the cusps and boundaries of the connected fundamental domains, in more detail.

A key tool in our construction [Reference Nie and ParentNP24] is a function

(1.1) $$ \begin{align} W:{\mathbb Z}/N\to {{\mathbb N}};\quad j\mapsto W_j = \min\{m\in {{\mathbb N}}\,|\, mj-1\in ({\mathbb Z}/N)^*\}, \end{align} $$

where ${{\mathbb N}}={\mathbb Z}_{>0}$ and $({\mathbb Z}/N)^*$ is the group of units. The related function

(1.2) $$ \begin{align} M: {\mathbb Z}/N\to {\mathbb Z}_{\geq 0};\quad j\mapsto M_j := W_j-1 \end{align} $$

is actually first studied and more fundamental in our work [Reference Nie and ParentNP24].

Let

$$ \begin{align*}S = \begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix}, \quad T = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \end{align*} $$

be the generators of $\Gamma (1)=SL_2({\mathbb Z})$ . Let A be a set in ${\mathbb Z}$ of consecutive residue class representatives for ${\mathbb Z}/N$ , such as $\{0,1,\dots , N-1\}$ , or

(1.3) $$ \begin{align} \{-N_1,\dots,N_2\}, \quad N_1= \left\lfloor {\frac{N-1}{2}} \right\rfloor ,\ N_2= \left\lfloor {\frac{N}{2}} \right\rfloor , \end{align} $$

where $ \left \lfloor {\cdot } \right \rfloor $ is the usual floor function. For $x\in {\mathbb Z} $ , we define $\widetilde {x}$ to be the unique integer such that $\widetilde {x}\equiv x \mod N$ and $-N_1\leq \widetilde {x}\leq N_2$ .

The main result of [Reference Nie and ParentNP24] is that

(1.4) $$ \begin{align} \Theta = \{ST^i\,|\, i\in A\} \cup \{ST^jST^m\,|\, j\in A,\ \gcd(j, N)> 1,\ 0\leq m\leq M_j\} \end{align} $$

is a set of right coset representatives for $\Gamma _0(N)\backslash \Gamma (1)$ , which gives a connected fundamental domain for $\Gamma _0(N)$ .

Figure 1 is the picture of the connected fundamental domain for $\Gamma _0(12)$ , where the labels are at the corresponding images of the standard fundamental domain

(1.5) $$ \begin{align} D = \Big\{z\in {\mathbb H}\,\Big|\, |z|>1, |\text{Re}z|<\frac{1}{2}\Big\} \end{align} $$

for $\Gamma (1)$ acting on the upper half plane ${\mathbb H}$ .

Figure 1

Our connected fundamental domain for $\Gamma _0(12)$ .

Figure 1 Long description

A mathematical plot on a Cartesian coordinate system where the vertical axis represents imaginary values and the horizontal axis represents real values. The vertical axis is labeled with I at the top and has tick marks from 0 to 1. The horizontal axis ranges from negative 0.5 to 0.5.

At the top center, a large region is bounded by a green arc and labeled with the letter S. Below this, the space is divided into symmetric curved regions by red, blue, and green arcs that converge toward the horizontal axis.

On the left side, regions are labeled from top to bottom as S T, S T super 2, S T super 3 S, and S T super 4 S. Smaller regions near the bottom are labeled with S T super 3 S T and S T super 4 S T.

On the right side, the regions are labeled symmetrically as S T super minus 1, S T super minus 2, S T super minus 3, and S T super minus 4. Further down, labels include S T super minus 2 S, S T super minus 3 S, and S T super minus 3 S T.

All arcs meet at the origin 0 on the horizontal axis, creating a dense, fan-like structure of increasingly smaller geometric tiles.

The author finds the function W in (1.1) interesting and first establishes some identities for it.

Proposition 1.6 We have

(1.7) $$ \begin{align} \sum_{j\in {\mathbb Z}/N} W_j &= \psi(N):=N\prod_{p|N} \left( 1+\frac{1}{p}\right), \end{align} $$

(1.8) $$ \begin{align} \sum_{j\in ({\mathbb Z}/N)^*} W_j &= N, \end{align} $$

(1.9) $$ \begin{align} \sum_{\gcd(j, N)>1} W_j &= \psi(N) - N. \end{align} $$

$\Gamma _0(N)$ acts on ${\mathbb P}^1({\mathbb Q})={\mathbb Q}\cup \infty $ by Möbius transformations, and the quotient set is the set $C_0(N)$ of cusp classes of $\Gamma _0(N)$ . The cusp class of $s\in {\mathbb P}^1({\mathbb Q})$ is denoted by

$$ \begin{align*}[s] =\Gamma_0(N)\cdot s\in C_0(N). \end{align*} $$

An element $s=a/c\in {\mathbb P}^1({\mathbb Q})$ with $a, c\in {\mathbb Z}$ and $\gcd (a, c)=1$ is called reduced. By convention, $\infty =\pm 1/0$ . In this article, we will only work with reduced elements.

We use $d|N$ to denote that d is a positive integer divisor of N. Then

(1.10) $$ \begin{align} \text{for }d|N,\quad d' := N/d,\quad d":=\gcd(d, d'),\quad \tilde d := d'/d". \end{align} $$

Proposition 1.11 Let $N>1$ . The number of cusp classes for $\Gamma _0(N)$ is

$$ \begin{align*}\epsilon_\infty(\Gamma_0(N)) = \sum_{d|N} \phi(\gcd(d, N/d)) = \sum_{d|N} \phi(d"),\end{align*} $$

where $\phi $ is the Euler totient function. Furthermore, with

$$ \begin{align*}S_0(N):=\bigcup_{d|N}({\mathbb Z}/d")^*, \end{align*} $$

we have a bijection

(1.12) $$ \begin{align} \chi: C_0(N)\to S_0(N);\quad [a/c] \mapsto (d=\gcd(c, N); \pi_{d"}(a\cdot c/d)), \end{align} $$

where $\pi _{d"}:{\mathbb Z}\to {\mathbb Z}/d"$ is the natural homomorphism. The width is

$$ \begin{align*}\mathrm{wd}([a/c]) = \tilde d = d'/d". \end{align*} $$

This proposition can be verified using standard resources, such as [Reference Diamond and ShurmanDS05, Section 3.8], [Reference ShimuraShi71, Section 1.6], [Reference CremonaCre97, Section 2.2], and [Reference SteinSte07, Section 1.4]. We will not provide a detailed proof.

The representatives $ST^i$ in (1.4) produce the cusp

$$ \begin{align*}ST^i(\infty) = S(\infty) = 0. \end{align*} $$

This in reduced form is $0/1$ and corresponds to $d=1$ in (1.12). The width is ${\tilde d = d'/d" = N}$ , and this corresponds to the fact that i runs through the representatives for ${\mathbb Z}/N$ in (1.4). This is the trivial part of our identification goal for cusps.

On the other hand, the representatives $ST^jST^m$ in (1.4) produce the cusps

(1.13) $$ \begin{align} ST^jST^m(\infty) = \begin{pmatrix} -1 & -m \\ j & mj-1 \end{pmatrix}(\infty) = -\frac{1}{j}. \end{align} $$

Here, $\gcd (j, N)>1$ , and $0\leq m\leq M_j$ . So the cusp $-1/j$ has a natural width $W_j$ , in view of (1.2).

If we let j run through the residue class representatives $A=\{0,-1,\dots ,-(N-1)\}$ of ${\mathbb Z}/N$ , then

$$ \begin{align*}\{1/j \text{ with width }W_j\,|\, 0\leq j< N,\ \gcd(j, N)>1\} \end{align*} $$

are natural cusps of $\Gamma _0(N)$ , produced by the work [Reference Nie and ParentNP24].

Now we achieve the goal of identifying these with the cusp classes in Proposition 1.11 and obtain the identity for widths.

By (1.12), we have

(1.14) $$ \begin{align} \chi([1/j])= (d; \pi_{d"}(j/d)),\quad \text{where }d=\gcd(j, N)>1. \end{align} $$

Going the other way, with $d>1$ , $d|N$ , and $b\in ({\mathbb Z}/d")^*$ , for $\chi ([1/j])= (d; b)$ , we need

$$ \begin{align*}d|j,\ j/d\in ({\mathbb Z}/d')^*,\ \pi^{*d'}_{d"} (j/d)=b, \end{align*} $$

where $\pi ^{*d'}_{d"}: ({\mathbb Z}/d')^*\to ({\mathbb Z}/d")^*$ is the natural homomorphism. Therefore, we have

(1.15) $$ \begin{align} j=dk,\quad k\in K_b := (\pi^{*d'}_{d"})^{-1}(b), \end{align} $$

where the notation k is abused to also denote its integer lift for $\pi _{d'}: {\mathbb Z}\to {\mathbb Z}/d'$ between 1 and $d'$ .

We have the following result to relate the widths.

Theorem 1.16 Let $N>1$ , $d>1,$ and $d|N$ . Then the width of the cusp class for $\Gamma _0(N)$ represented by $(d; b\in ({\mathbb Z}/d")^*)$ is the sum of the widths of all the $1/j$ such that $\chi ([1/j])=(d;b)$ , that is,

(1.17) $$ \begin{align} \tilde d=\frac{d'}{d"}=\sum_{k\in K_b} W_{dk}. \end{align} $$

Now to describe the boundary arcs of our fundamental domains, we consider the ideal geodesic triangle $\overline D$ , the closure of D in (1.5) in the upper half plane ${\mathbb H}$ . We introduce the notation for its edges:

$$ \begin{align*} L &= \{z\in {\mathbb H}\,|\, \text{Re}x=1/2,\ |z|\geq 1\},\\ R &= \{z\in {\mathbb H}\,|\, \text{Re}x=-1/2,\ |z|\geq 1\},\\ B &= \{z\in {\mathbb H}\,|\, |z|=1, |\text{Re}z|\leq {1}/{2}\}, \end{align*} $$

called the left, the right (from the viewpoint of the cusp $\infty $ ), and the base. Note that

(1.18) $$ \begin{align} TR=L,\quad SB=B. \end{align} $$

Also, $L, R,$ and their images under elements of $\Gamma (1)$ are all connected to cusps, while B and its images are not.

In this part, we use the symmetric choice (1.3) of residue classes, in agreement with [Reference Nie and ParentNP24].

We use the natural terminology that two boundary arcs $C_1$ and $C_2$ are equivalent, written as

(1.20) $$ \begin{align} C_1\sim C_2 \text{ if }\exists g\in \Gamma_0(N) \text{ such that }C_1 = gC_2. \end{align} $$

Theorem 1.21 We have the following gluing patterns for the boundary arcs of the connected fundamental domain for $\Gamma _0(N)$ :

1. (1) $ST^{N_2}L \sim ST^{-N_1}R$ .

2. (2) For $\gcd (i, N)=1$ , $ST^i B \sim ST^{\widetilde {-i^{-1}}} B$ .

3. (3) For $\gcd (j, N)>1$ ,

   1. (a) $ST^jST^{M_j}L\sim ST^{\widetilde {(1-jW_j)^{-1} j}} SR$ , and

   2. (b) for $1\leq m\leq M_j$ , $ST^jST^{m}B\sim ST^{j'}ST^{m'}B,$ where $(j',m')$ with $\gcd (j',N)>1$ , $1\le m'\le M_{j'}$ , is the unique such pair satisfying

(1.22) $$ \begin{align} jj' + (jm-1)(j'm'-1)\equiv 0\mod N. \end{align} $$

The boundary arcs of the fundamental domain for $\Gamma _0(12)$ in Figure 1, besides the two vertical rays $\text {Re}z=\pm 1/2$ which are naturally identified by $T\in \Gamma _0(12)$ , are listed below with equivalent pairs stacked, as an example of our Proposition 1.19 and Theorem 1.21.

Table 1

Table 1 Long description

The table is organized into four rows and six columns.

Row 1 (Header Row):

- Column 1: Arc

- Column 2: S T super 3 S T L

- Column 3: S T super 3 S T B

- Column 4: S T super 3 S T L

- Column 5: S T super 4 S T L

- Column 6: S T super 4 S T B

Row 2:

- Column 1: Pair

- Column 2: S T super minus 2 S T L

- Column 3: S T super minus 2 S T B

- Column 4: S T super minus 3 S T B

- Column 5: S T super minus 3 S T L

- Column 6: S T super minus 3 S T B

Row 3:

- Column 1: Arc

- Column 2: S T super 4 S T L

- Column 3: S T super 4 S T B

- Column 4: S T super 5 B

- Column 5: S T super 6 S T L

- Column 6: S T super 6 B

Row 4:

- Column 1: Pair

- Column 2: S T super minus 4 S T B

- Column 3: S T super minus 4 S T L

- Column 4: S T super minus 5 B

- Column 5: S T super 5 S T L

- Column 6: S T super minus 5 S B

The gluing pattern in this case turns out to be particularly simple, and this means that the corresponding modular curve $X_0(12)$ has genus 0 by standard topological considerations. This is consistent with the general genus formula [Reference Diamond and ShurmanDS05, Theorem 3.1.1].

The article is organized as follows. In Section 2, we further study the function W and prove Proposition 1.6. In Section 3, we prove Theorem 1.16 about the cusps. These two proofs are combinatorial and very concrete. In Section 4, we prove Theorem 1.21 about the gluing patterns of the boundary arcs.