Editing Modular form (section)

==Modular forms for more general groups==
The functional equation, i.e., the behavior of ''f'' with respect to <math>z \mapsto \frac{az+b}{cz+d} </math> can be relaxed by requiring it only for matrices in smaller groups.

===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>

===Definition===
A modular form for {{mvar|G}} of weight ''k'' is a function on '''H''' satisfying the above functional equation for all matrices in {{mvar|G}}, that is holomorphic on '''H''' and at all cusps of {{mvar|G}}. Again, modular forms that vanish at all cusps are called cusp forms for {{mvar|G}}. The '''C'''-vector spaces of modular and cusp forms of weight ''k'' are denoted {{math|''M<sub>k</sub>''(''G'')}} and {{math|''S<sub>k</sub>''(''G'')}}, respectively. Similarly, a meromorphic function on ''G''\'''H'''<sup>∗</sup> is called a modular function for {{mvar|G}}. In case ''G'' = Γ<sub>0</sub>(''N''), they are also referred to as modular/cusp forms and functions of ''level'' ''N''. For {{math|''G'' {{=}} Γ(1) {{=}} SL(2, '''Z''')}}, this gives back the afore-mentioned definitions.

===Consequences===
The theory of Riemann surfaces can be applied to ''G''\'''H'''<sup>∗</sup> to obtain further information about modular forms and functions. For example, the spaces {{math|''M<sub>k</sub>''(''G'')}} and {{math|''S<sub>k</sub>''(''G'')}} are finite-dimensional, and their dimensions can be computed thanks to the [[Riemann–Roch theorem]] in terms of the geometry of the {{mvar|G}}-action on '''H'''.<ref>{{Citation | last1=Shimura | first1=Goro | title=Introduction to the arithmetic theory of automorphic functions | publisher=Iwanami Shoten | location=Tokyo | series=Publications of the Mathematical Society of Japan | year=1971 | volume=11}}, Theorem 2.33, Proposition 2.26</ref> For example,

:<math>\dim_\mathbf{C} M_k\left(\text{SL}(2, \mathbf{Z})\right) = \begin{cases}
  \left\lfloor k/12 \right\rfloor     & k \equiv 2 \pmod{12} \\
  \left\lfloor k/12 \right\rfloor + 1 & \text{otherwise}
\end{cases}</math>

where <math>\lfloor \cdot \rfloor</math> denotes the [[floor function]] and <math>k</math> is even.

The modular functions constitute the [[function field of an algebraic variety|field of functions]] of the Riemann surface, and hence form a field of [[transcendence degree]] one (over '''C'''). If a modular function ''f'' is not identically 0, then it can be shown that the number of zeroes of ''f'' is equal to the number of [[pole (complex analysis)|pole]]s of ''f'' in the [[closure (mathematics)|closure]] of the [[fundamental region]] ''R''<sub>Γ</sub>.It can be shown that the field of modular function of level ''N'' (''N'' ≥ 1) is generated by the functions ''j''(''z'') and ''j''(''Nz'').<ref>{{Citation |last=Milne |first=James |title=Modular Functions and Modular Forms |url=https://www.jmilne.org/math/CourseNotes/MF.pdf#page=88 |year=2010 |page=88 }}, Theorem 6.1.</ref>

===Line bundles===
The situation can be profitably compared to that which arises in the search for functions on the [[projective space]] P(''V''): in that setting, one would ideally like functions ''F'' on the vector space ''V'' which are polynomial in the coordinates of ''v''&nbsp;≠&nbsp;0 in ''V'' and satisfy the equation ''F''(''cv'')&nbsp;=&nbsp;''F''(''v'') for all non-zero ''c''. Unfortunately, the only such functions are constants. If we allow denominators (rational functions instead of polynomials), we can let ''F'' be the ratio of two [[homogeneous function|homogeneous]] polynomials of the same degree. Alternatively, we can stick with polynomials and loosen the dependence on ''c'', letting ''F''(''cv'')&nbsp;=&nbsp;''c''<sup>''k''</sup>''F''(''v''). The solutions are then the homogeneous polynomials of degree {{mvar|k}}. On the one hand, these form a finite dimensional vector space for each&nbsp;''k'', and on the other, if we let ''k'' vary, we can find the numerators and denominators for constructing all the rational functions which are really functions on the underlying projective space&nbsp;P(''V'').

One might ask, since the homogeneous polynomials are not really functions on P(''V''), what are they, geometrically speaking? The [[algebraic geometry|algebro-geometric]] answer is that they are ''sections'' of a [[sheaf (mathematics)|sheaf]] (one could also say a [[vector bundle|line bundle]] in this case). The situation with modular forms is precisely analogous.

Modular forms can also be profitably approached from this geometric direction, as sections of line bundles on the moduli space of elliptic curves.