Modular Curve - Examples

Examples

The most common examples are the curves X(N), X₀(N), and X₁(N) associated with the subgroups Γ(N), Γ₀(N), and Γ₁(N).

The modular curve X(5) has genus 0: it is the Riemann sphere with 12 cusps located at the vertices of a regular icosahedron. The covering X(5) → X(1) is realized by the action of the icosahedral group on the Riemann sphere. This group is a simple group of order 60 isomorphic to A₅ and PSL(2,5).

The modular curve X(7) is the Klein quartic of genus 3 with 24 cusps. It can be interpreted as the Riemann sphere tiled by 24 heptagons, with a cusp at the center of each face. These tilings can be understood via dessins d'enfants and Belyi functions – the cusps are the points lying over ∞ (red dots), while the vertices and centers of the edges (black and white dots) are the points lying over 0 and 1. The Galois group of the covering X(7) → X(1) is a simple group of order 168 isomorphic to PSL(2,7).

There is an explicit classical model for X₀(N), the classical modular curve; this is sometimes called the modular curve. The definition of Γ(N) can be restated as follows: it is the subgroup of the modular group which is the kernel of the reduction modulo N. Then Γ₀(N) is the larger subgroup consisting of matrices with that are upper triangular modulo N and Γ₁(N) is the intermediate group represented by matrices with

These curves have a direct interpretation as moduli spaces for elliptic curves with level structure and for this reason they play an important role in arithmetic geometry. The level N modular curve X(N) is the moduli space for elliptic curves with a basis for the N-torsion. For X₀(N) and X₁(N), the level structure is, respectively, a cyclic subgroup of order N and a point of order N. These curves have been studied in great detail, and in particular, it is known that X₀(N) can be defined over Q.

The equations defining modular curves are the best-known examples of modular equations. The "best models" can be very different from those taken directly from elliptic function theory. Hecke operators may be studied geometrically, as correspondences connecting pairs of modular curves.

Remark: quotients of H that are compact do occur for Fuchsian groups Γ other than subgroups of the modular group; a class of them constructed from quaternion algebras is also of interest in number theory.