The functional equation, i.e., the behavior of
f with respect to z \mapsto \frac{az+b}{cz+d} can be relaxed by requiring it only for matrices in smaller groups.
The Riemann surface G\H∗ Let be a subgroup of that is of finite
index. Such a group
acts on
H in the same way as . The
quotient topological space G\
H can be shown to be a
Hausdorff space. Typically it is not compact, but can be
compactified by adding a finite number of points called
cusps. These are points at the boundary of
H, i.e. in
Q∪{∞}, such that there is a parabolic element of (a matrix with
trace ±2) fixing the point. This yields a compact topological space
G\
H∗. 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 subgroups :\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} For
G = Γ0(
N) or , the spaces
G\
H and
G\
H∗ are denoted
Y0(
N) and
X0(
N) and
Y(
N),
X(
N), respectively. The geometry of
G\
H∗ can be understood by studying
fundamental domains for
G, i.e. subsets
D ⊂
H such that
D intersects each orbit of the -action on
H exactly once and such that the closure of
D meets all orbits. For example, the
genus of
G\
H∗ can be computed.
Definition A modular form for of weight
k is a function on
H satisfying the above functional equation for all matrices in , that is holomorphic on
H and at all cusps of . Again, modular forms that vanish at all cusps are called cusp forms for . The
C-vector spaces of modular and cusp forms of weight
k are denoted and , respectively. Similarly, a meromorphic function on
G\
H∗ is called a modular function for . In case
G = Γ0(
N), they are also referred to as modular/cusp forms and functions of
level N. For , this gives back the afore-mentioned definitions.
Consequences The theory of Riemann surfaces can be applied to
G\
H∗ to obtain further information about modular forms and functions. For example, the spaces and are finite-dimensional, and their dimensions can be computed thanks to the
Riemann–Roch theorem in terms of the geometry of the -action on
H. For example, :\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} where \lfloor \cdot \rfloor denotes the
floor function and k is even. The modular functions constitute the
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
poles of
f in the
closure of the
fundamental region RΓ. It can be shown that the field of modular functions of level
N (
N ≥ 1) is generated by the functions
j(
z) and
j(
Nz).
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 ≠ 0 in
V and satisfy the equation
F(
cv) =
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 polynomials of the same degree. Alternatively, we can stick with polynomials and loosen the dependence on
c, letting
F(
cv) = ''c'
k'F
(v
). The solutions are then the homogeneous polynomials of degree . On the one hand, these form a finite dimensional vector space for each 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 P(V''). One might ask, since the homogeneous polynomials are not really functions on P(
V), what are they, geometrically speaking? The
algebro-geometric answer is that they are
sections of a
sheaf (one could also say a
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. ==Rings of modular forms==