Editing Modular curve (section)

== Analytic definition ==
The modular group SL(2,&nbsp;'''Z''') acts on the upper half-plane by [[fractional linear transformation]]s. The analytic definition of a modular curve involves a choice of a congruence subgroup Γ of SL(2,&nbsp;'''Z'''), i.e. a subgroup containing the [[principal congruence subgroup|principal congruence subgroup of level ''N'']] for some positive integer ''N'', which is defined to be

:<math>\Gamma(N)=\left\{
\begin{pmatrix}
a & b\\
c & d\\
\end{pmatrix}  : \ a \equiv d \equiv 1 \mod N \text{ and } b, c \equiv0 \mod N \right\}.</math>

The minimal such ''N'' is called the '''level of Γ'''. A [[Complex manifold|complex structure]] can be put on the quotient Γ\'''H''' to obtain a [[noncompact]] Riemann surface called a '''modular curve''', and commonly denoted ''Y''(Γ).

=== Compactified modular curves ===
A common compactification of ''Y''(Γ) is obtained by adding finitely  many points called the cusps of Γ. Specifically, this is done by considering the action of Γ on the '''extended complex upper-half plane''' '''H'''*&nbsp;=&nbsp;{{nowrap|'''H''' ∪ '''Q''' ∪ {∞}}}. We introduce a topology on '''H'''* by taking as a basis:
* any open subset of '''H''',
* for all ''r'' > 0, the set <math>\{\infty\}\cup\{\tau\in \mathbf{H} \mid\text{Im}(\tau)>r\}</math>
* for all [[coprime integers]] ''a'', ''c'' and all ''r'' > 0, the image of <math>\{\infty\}\cup\{\tau\in \mathbf{H} \mid\text{Im}(\tau)>r\}</math> under the action of 
::<math>\begin{pmatrix}a & -m\\c & n\end{pmatrix}</math> 
:where ''m'', ''n'' are integers such that ''an'' + ''cm'' = 1.

This turns '''H'''* into a topological space which is a subset of the [[Riemann sphere]] '''P'''<sup>1</sup>('''C'''). The group Γ acts on the subset {{nowrap|'''Q''' ∪ {∞}}}, breaking it up into finitely many [[Orbit (group theory)|orbits]] called the '''cusps of Γ'''. If Γ acts transitively on {{nowrap|'''Q''' ∪ {∞}}}, the space Γ\'''H'''* becomes the [[Alexandroff compactification]] of Γ\'''H'''. Once again, a complex structure can be put on the quotient Γ\'''H'''* turning it into a Riemann surface denoted ''X''(Γ) which is now [[Compact space|compact]]. This space is a compactification of ''Y''(Γ).<ref>{{citation|last=Serre|first= Jean-Pierre|title=Cours d'arithmétique|edition=2nd|series= Le Mathématicien|volume= 2|publisher= Presses Universitaires de France|year= 1977}}</ref>