Editing Modular form (section)

===The Riemann surface ''G''\H<sup>&lowast;</sup>===
Let {{mvar|G}} be a subgroup of {{math|SL(2, '''Z''')}} that is of finite [[Index of a subgroup|index]]. Such a group {{mvar|G}} [[Group action (mathematics)|acts]] on '''H''' in the same way as {{math|SL(2, '''Z''')}}. The [[quotient topological space]] ''G''\'''H''' can be shown to be a [[Hausdorff space]]. Typically it is not compact, but can be [[compactification (mathematics)|compactified]] by adding a finite number of points called ''cusps''. These are points at the boundary of '''H''', i.e. in '''[[Rational numbers|Q]]'''∪{∞},<ref group="note">Here, a matrix <math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}</math> sends ∞ to ''a''/''c''.</ref> such that there is a parabolic element of {{mvar|G}} (a matrix with [[trace of a matrix|trace]] ±2) fixing the point. This yields a compact topological space ''G''\'''H'''<sup>∗</sup>. What is more, it can be endowed with the structure of a [[Riemann surface]], which allows one to speak of holo- and meromorphic functions.

Important examples are, for any positive integer ''N'', either one of the [[congruence subgroup]]s
:<math>\begin{align}
  \Gamma_0(N) &= \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \text{SL}(2, \mathbf{Z}): c \equiv 0 \pmod{N} \right\} \\
    \Gamma(N) &= \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \text{SL}(2, \mathbf{Z}) : c \equiv b \equiv 0, a \equiv d \equiv 1 \pmod{N} \right\}.
\end{align}</math>

For ''G'' = Γ<sub>0</sub>(''N'') or {{math|Γ(''N'')}}, the spaces ''G''\'''H''' and ''G''\'''H'''<sup>∗</sup> are denoted ''Y''<sub>0</sub>(''N'') and ''X''<sub>0</sub>(''N'') and ''Y''(''N''), ''X''(''N''), respectively.

The geometry of ''G''\'''H'''<sup>∗</sup> can be understood by studying [[fundamental domain]]s for ''G'', i.e. subsets ''D'' ⊂ '''H''' such that ''D'' intersects each orbit of the {{mvar|G}}-action on '''H''' exactly once and such that the closure of ''D'' meets all orbits. For example, the [[Genus (mathematics)|genus]] of ''G''\'''H'''<sup>∗</sup> can be computed.<ref>{{Citation | last1=Gunning | first1=Robert C. | title=Lectures on modular forms | publisher=[[Princeton University Press]] | series=Annals of Mathematics Studies | year=1962 | volume=48}}, p. 13</ref>