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For $N\ge 2$, the effective principal congruence subgroup is torsion-free. Its modular curve cover of $X(1)$ has analytic ramification indices $2,3,N$ above the two elliptic orbits and the modular cusp. If $\mu$ is its projective index, the numbers of points above these three values are $\mu /2,\mu /3,\mu /N$. Applying the Riemann-Hurwitz formula gives $2g-2=\mu (1/6-1/N)$. Here $\mu =6$ for $N=2$ and $\mu =N^3{\prod }_{p\mid N}(1-p^{-2})/2$ for $N\ge 3$; level one is the identity cover.

Let ${\mu }_N=[PS{L}_2(\mathbb{Z}):\overline{\Gamma (N)}]$, where the bar denotes the effective projective image. This is the degree of the modular curve projection, rather than the full $S{L}_2$ index when $-I$ is not in $\Gamma (N)$.

Reduction modulo $N$ is onto $S{L}_2(\mathbb{Z}/N\mathbb{Z})$. One proof uses the Chinese remainder theorem and elementary matrices: over $\mathbb{Z}/p^r\mathbb{Z}$, a unimodular first column has a unit entry, so elementary row operations reduce a determinant-one matrix to the identity. Each elementary matrix lifts integrally. Over ${\mathbb{F}}_p$, the first column has $p^2-1$ choices and the second has $p$ choices with prescribed determinant, giving $p(p^2-1)$. Each subsequent prime-power lift has $p^3$ choices, because the determinant-one linearized condition is one trace equation on four entries. HenceFor $N=1,2$, the subgroup contains $-I$; for $N\ge 3$ it does not. ConsequentlyFor $N\ge 2$ there are no effective elliptic stabilizer subgroups in the subgroup. Indeed, write $\gamma =I+NB\in \Gamma (N)$. The determinant condition gives $\mathrm{tr}\gamma =2-N^2\det B$. A noncentral elliptic element in $S{L}_2(\mathbb{Z})$ has trace $-1,0$ or $1$, none congruent to $2$ modulo $N^2$ for $N\ge 2$. Thus the projective action is torsion-free.

All cusp widths are $N$. At infinity, $T^h$ lies in the effective subgroup exactly when $h$ is divisible by $N$: the alternative $T^h\equiv -I(\mathrm{mod}N)$ can only occur for $N=1,2$, and for $N=2$ it imposes the same condition. Every rational modular cusp is a modular translate of infinity, and the subgroup is normal, so the same width holds there. The sum of modular cusp analytic ramification indices equals ${\mu }_N$, giving ${\mu }_N/N$ modular cusps.

The base $X(1)$ is a sphere. For instance $j=E_4^3/\Delta$ is a meromorphic function on it with a single simple pole at the modular cusp and no other poles, using the nonvanishing of $\Delta$ proved above. It therefore defines a degree-one map to the Riemann sphere, making the base genus zero.

The only branch values of $X(N)\to X(1)$ are the two elliptic orbits and the modular cusp. For $N\ge 2$, the numbers of points above them and their analytic ramification indices are respectively ${\mu }_N/2$ with index two, ${\mu }_N/3$ with index three, and ${\mu }_N/N$ with index $N$. Applying the Riemann-Hurwitz formula directly,This is the elliptic and cusp ramification for principal level calculation. Level one is the identity projection. The complete result isFor example the genera for $N=3,4,5,6,7$ are $0,0,0,1,3$. Keeping the exceptional projective index at level two is essential; applying the factor one-half there would give a nonintegral answer. This agrees with the principal congruence modular curve genus formula obtained from these branch counts.

The modular group is $S{L}_2(\mathbb{Z})$ acting on the complex upper half-plane byIts central element $-I$ acts trivially on points, so the effective group is $PS{L}_2(\mathbb{Z})$. It is generated by $S:\tau \mapsto -1/\tau$ and $T:\tau \mapsto \tau +1$: the Euclidean algorithm reduces the bottom row of a matrix using these operations until it is a translation. The relations $S^2=(ST{)}^3=1$ hold in the projective group.

A standard fundamental domain of the modular group isTo see that each orbit meets it, use $\mathrm{Im}(\gamma \tau )=\mathrm{Im}\tau /|c\tau +d{|}^2$. Choose a coprime integer pair $(c,d)$ minimizing the nonzero lattice modulus $|c\tau +d|$; a modular matrix with that bottom row maximizes the imaginary part. Translate to the central strip. If the result had modulus below one, inversion would further increase its imaginary part, a contradiction. Interior representatives are unique: for a point in the interior, $|c\tau +d|>1$ when $c\ne 0$. For $|c|\ge 2$ use $\mathrm{Im}\tau >\sqrt{3}/2$; for $|c|=1$ use the strip and the strict unit-circle bound. Thus such a matrix lowers the imaginary part, which would contradict the same fact applied to its inverse if both representatives were in the interior. Matrices with $c=0$ are translations, and cannot move one interior strip point to another.

The vertical sides are identified by $T$, and the circular halves by $S$. The exceptional points with elliptic stabilizers of the modular group are represented by $i$ and $\rho =e^{2\pi i/3}$, of projective stabilizer orders two and three. The rational boundary points together with infinity form one modular cusp orbit. In the hyperbolic metric the region has area ${\int }_{-1/2}^{1/2}(1-x^2{)}^{-1/2}dx=\pi /3$.

A useful subgroup is the level-two principal congruence subgroup,Reduction modulo two is onto $S{L}_2({\mathbb{F}}_2)$, which has six elements and permutes the three nonzero vectors. Hence its kernel has index six, and the quotient is $S_3$. In the projective action, $\Gamma (2)$ is torsion-free. Indeed finite-order nonidentity modular elements have trace zero or $\pm 1$; matrices in $\Gamma (2)$ have even trace, and trace zero is impossible because $a,d$ are odd while $b,c$ are even: $ad-bc=1$ with $d=-a$ would imply $bc=-a^2-1\equiv 2(\mathrm{mod}4)$. The subgroup has three modular cusp classes, represented by $\infty ,0,1$, each of width two. Parity of a primitive pair distinguishes these classes, and elementary level-two row operations reduce each class to its representative. Six translates of $\mathcal{F}$ supply a fundamental region. Its compactified level-two modular curve is obtained by adjoining these three modular cusp points.

Invariant holomorphic functions arise naturally from theta constants. In the radian convention of Question 4, put ${\Theta }_j(\tau )={\theta }_j(0,\tau )$, and defineHere ${\Theta }_2={\sum }_{n\in \mathbb{Z}}{q}^{(n+1/2{)}^2}$. The theta-constant inversion and translation laws, obtained by Gaussian Poisson summation and by reindexing, interchange ${\Theta }_2,{\Theta }_4$ under $S$ with the common factor $\sqrt{-i\tau }$, leave ${\Theta }_3$ fixed up to that factor, and under $T$ interchange ${\Theta }_3,{\Theta }_4$ while multiplying ${\Theta }_2$ by $e^{\pi i/4}$. Together with the Jacobi abstruse identity ${\Theta }_3^4={\Theta }_2^4+{\Theta }_4^4$, this givesThese transformations generate the six permutations of $0,1,\infty$. In particular $T^2$ and $S{T}^2{S}^{-1}$ fix $\lambda$; with $-I$, these generate $\Gamma (2)$. Thus the modular lambda function is invariant under this subgroup, but not under the whole modular group. Its even theta constants do not vanish on $\mathbb{H}$, so it is holomorphic there and avoids $0,1,\infty$. Near infinity, $\lambda =16q+O(q^2)$, with $q=e^{\pi i\tau }$, which is the width-two modular cusp coordinate. The two transformation laws give modular cusp values one at zero and infinity at one, with the same simple local orders. Thus $\lambda$ has exactly one simple pole on the compactified level-two quotient and no pole in the half-plane. A meromorphic map with one simple pole has degree one, so it identifies that quotient with the sphere and the open quotient with $\mathbb{C}\setminus \{0,1\}$. This proves the coordinate assertion and describes all three modular cusps.

Symmetrizing this subgroup invariant produces a full modular-invariant function:Substitution shows that this rational expression is unchanged by both $\lambda \mapsto 1-\lambda$ and $\lambda \mapsto \lambda /(\lambda -1)$, hence by $S$ and $T$. It is the Klein j-invariant. Its denominator does not vanish on $\mathbb{H}$, so it is holomorphic there; at the modular cusp it has $j\sim q^{-2}$, a simple pole in $Q=e^{2\pi i\tau }$. It therefore remains nonconstant without being a bounded holomorphic function on the compactified quotient. In fact a holomorphic invariant extending holomorphically to the compactified modular curve would be constant by the maximum principle.

This also connects invariance to elliptic curves: changing an oriented basis of the lattice $\mathbb{Z}+\tau \mathbb{Z}$ gives precisely the modular action after a rescaling. On the compact full modular quotient, $j$ has just the one simple modular cusp pole, so its map to the sphere has degree one. It therefore classifies the resulting homothety of complex lattices classes, while the modular lambda function retains a level-two labeling of the branch points in the Legendre model $y^2=x(x-1)(x-\lambda )$. Finally, a modular form of nonzero weight obeys a transformation law with a factor $(c\tau +d{)}^k$, not plain invariance; ratios of equal-weight forms instead give weight-zero modular functions. The distinction between forms, invariant holomorphic functions on the open half-plane, and their modular cusp behavior is essential to this construction.

Suppose ${\tau }_1=(a{\tau }_2+b)/(c{\tau }_2+d)$, with $ad-bc=1$, and put $\alpha =(c{\tau }_2+d{)}^{-1}$. The identitiesshow that multiplication by $\alpha$ takes one period lattice isomorphically to the other: its basis matrix has determinant $ad-bc=1$. If $N\mid c$, the image of $1/N$ is congruent to $a/N$ modulo the target period lattice. Moreover $ad\equiv 1(\mathrm{mod}N)$, so $a$ is a unit modulo $N$. The image therefore generates the same cyclic subgroup.

Conversely, a holomorphic group isomorphism of complex tori lifts to $z\mapsto \alpha z$, with $\alpha {\Lambda }_{{\tau }_2}={\Lambda }_{{\tau }_1}$. Indeed its lift fixing zero is additive: its failure to be additive is a continuous lattice-valued function, hence zero. An additive holomorphic map is complex linear. WriteThe corresponding integral basis matrix has determinant $1$: complex multiplication preserves the real orientation, and both bases $(1,{\tau }_j)$ are positively oriented. Solving gives ${\tau }_1=(a{\tau }_2+b)/(c{\tau }_2+d)$. Preservation of the cyclic subgroup means $\alpha /N\equiv u/N(\mathrm{mod}{\Lambda }_{{\tau }_1})$ for some unit $u$ modulo $N$. Comparing the two real basis coefficients gives $N\mid c$. Thus the matrix lies in the Gamma 0 congruence subgroup, and the classification isThis is the noncompact modular curve classification; no cusp points have been added.

The modular curve isGive it the quotient topology on $\Gamma \backslash \mathfrak{h}$ and adjoin one end for each cusp of a modular group. At a point with trivial effective stabilizer, the quotient map identifies a sufficiently small disk with a chart. At an elliptic point with cyclic stabilizer of order $e$, choose a disk coordinate $z$ centered there; then $z^e$ descends to a quotient coordinate. At a cusp represented by $\sigma \in S{L}_2(\mathbb{Z})$ and having width $h$, a punctured neighborhood is described byand adjoining $q_h=0$ fills in the cusp. These charts make $X(\Gamma )$ a compact Riemann surface.