Editing Modular curve (section)

== Examples ==
The most common examples are the curves ''X''(''N''), ''X''<sub>0</sub>(''N''), and ''X''<sub>1</sub>(''N'') associated with the subgroups Γ(''N''), Γ<sub>0</sub>(''N''), and Γ<sub>1</sub>(''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 symmetry|icosahedral group]] on the Riemann sphere. This group is a simple group of order 60 isomorphic to ''A''<sub>5</sub> and PSL(2,&nbsp;5).

The modular curve ''X''(7) is the [[Klein quartic]] of genus 3 with 24 cusps. It can be interpreted as a surface with three handles tiled by 24 heptagons, with a cusp at the center of each face. These tilings can be understood via [[dessins d'enfants]] and [[Belyi function]]s – 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)&nbsp;→&nbsp;''X''(1) is a simple group of order 168 isomorphic to [[PSL(2,7)|PSL(2,&nbsp;7)]].

There is an explicit classical model for ''X''<sub>0</sub>(''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 [[Modular arithmetic|modulo]] ''N''. Then Γ<sub>0</sub>(''N'') is the larger subgroup of matrices which are upper triangular modulo ''N'':

:<math>\left \{ \begin{pmatrix} a & b \\ c & d\end{pmatrix}  : \ c\equiv 0 \mod N \right \},</math>

and Γ<sub>1</sub>(''N'') is the intermediate group defined by:

:<math>\left \{ \begin{pmatrix} a & b \\ c & d\end{pmatrix}  : \ a\equiv d\equiv 1\mod N, c\equiv 0 \mod N \right \}.</math>

These curves have a direct interpretation as [[moduli space]]s for [[elliptic curve]]s with ''[[level structure (algebraic geometry)|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 (algebra)|torsion]]. For ''X''<sub>0</sub>(''N'') and ''X''<sub>1</sub>(''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 ''X''<sub>0</sub>(''N'') can be defined over '''Q'''.

The equations defining modular curves are the best-known examples of [[modular equation]]s. The "best models" can be very different from those taken directly from [[elliptic function]] theory. [[Hecke operator]]s may be studied geometrically, as [[Correspondence (algebraic geometry)|correspondence]]s connecting pairs of modular curves.

Quotients of '''H''' that ''are'' compact do occur for [[Fuchsian group]]s Γ other than subgroups of the modular group; a class of them constructed from [[quaternion algebra]]s is also of interest in number theory.