Modular curve

For a congruence subgroup $\Gamma$, the compact modular curve isAt an ordinary point it inherits a coordinate from $\mathfrak{h}$; at an elliptic point of stabilizer order $e$ a coordinate is the $e$th power of a local disk coordinate; and at a cusp of width $h$ a coordinate is $e^{2\pi i\tau /h}$.

An elliptic point of a modular curve is the image of a point of the complex upper half-plane with a nontrivial stabilizer subgroup in the effective modular subgroup. For the full modular group the stabilizer subgroup orders are two and three, represented by $i$ and $e^{2\pi i/3}$. A local disk coordinate $u$ upstairs descends as $u^e$ when the stabilizer subgroup order is $e$. The central matrices $\pm I$ are factored out before counting this order.

A modular symbol is the relative homology class of a path between two modular cusps. Oriented paths satisfy $\{\alpha ,\beta \}+\{\beta ,\gamma \}=\{\alpha ,\gamma \}$. Continued fractions subdivide rational-endpoint paths into unimodular edges, giving a finite presentation through Manin symbols. The Hecke operators act by finite sums of transformed paths, so their matrices can be computed by rational linear algebra.

For a finite-index modular subgroup, a Manin symbol records a coset and its associated oriented unimodular edge. Over the rationals the presentation has the relations $[g]+[gS]=0$ and $[g]+[gST]+[g(ST{)}^2]=0$, for inversion $S$ and translation $T$. They express reversing an edge and the boundary of a triangle. Integral computation also removes torsion caused by elliptic stabilizers. The boundary map sends the symbol to its terminal cusp minus its initial cusp; its kernel is absolute homology.

The compact quotient for the Gamma 0 congruence subgroup at level two has projective index three, two modular cusps of cusp widths one and two, one order-two elliptic orbit and no order-three elliptic orbit. It has genus zero. It differs from the level-two principal-congruence curve for $\Gamma (2)$, which has three modular cusps.

The ratio of modular discriminants has weight zero and level ${\Gamma }_0(2)$. It has one simple pole at infinity and one simple zero at the zero modular cusp, with no zeros or poles in the half-plane. It is a spherical coordinate by the single-pole criterion for a spherical coordinate. The Klein j-invariant is $(t+256{)}^3/t^2$, with $t(e^{2\pi i/3})=-256$.

For a finite-index subgroup of the modular group, the quotient of the complex upper half-plane becomes a compact Riemann surface after adjoining its cusp classes. At an elliptic fixed point use the quotient coordinate for the finite stabilizer; at a cusp use its exponential local parameter. Weight-zero modular forms descend to holomorphic functions on this compact surface, and hence are constant by the maximum modulus principle.

The quotient $\mathbb{H}/\Gamma (2)$ is a genus-zero Riemann surface with three cusps and no elliptic Möbius transformations, hence is biholomorphic to $\mathbb{C}\setminus \{0,1\}$. The integer level-two principal congruence subgroup has projective index six; reduction modulo two gives three cusp classes.

Let $\mu =[PS{L}_2(\mathbb{Z}):\overline{\Gamma }]$, let $e_2,e_3$ count elliptic orbits of orders two and three, and let $c$ be the number of cusps. Then

For $N\ge 3$, the principal congruence subgroup has projective index $\mu =\frac{1}{2}{N}^3{\prod }_{p\mid N}(1-p^{-2})$, no effective elliptic stabilizers, and all cusp widths equal to $N$. Hence the number of cusps is $\mu /N$, and the genus formula for a modular curve gives the displayed formula. The exceptional levels one and two both have genus zero; the full genus-zero list is $N=1,2,3,4,5$. Level six has genus one and every larger level has positive genus.

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.

The modular curve $X_0(3)$ has index four, two cusps, no elliptic orbit of order two, and one elliptic orbit of order three. Its genus is therefore zero.