2 1. Modular Forms
Since the determinant of γ is 1, we have
d
dz
γ (z) =
1
(cz + d)2
.
Definition 1.1 (Modular Group). The modular group is the group of all
matrices
(
a b
c d
)
with a, b, c, d Z and ad bc = 1.
For example, the matrices
(1.1.2) S =
0 −1
1 0
and T =
1 1
0 1
are both elements of SL2(Z); the matrix S induces the function z −1/z
on h, and T induces the function z z + 1.
Theorem 1.2. The group SL2(Z) is generated by S and T .
Proof. See e.g. [Ser73, §VII.1].
In SAGE we compute the group SL2(Z) and its generators as follows:
sage: G = SL(2,ZZ); G
Modular Group SL(2,Z)
sage: S, T = G.gens()
sage: S
[ 0 -1]
[ 1 0]
sage: T
[1 1]
[0 1]
Definition 1.3 (Holomorphic and Meromorphic). Let R be an open subset
of C. A function f : R C is holomorphic if f is complex differentiable at
every point z R, i.e., for each z R the limit
f (z) = lim
h→0
f(z + h) f(z)
h
exists, where h may approach 0 along any path. A function f : R C∪{∞}
is meromorphic if it is holomorphic except (possibly) at a discrete set S of
points in R, and at each α S there is a positive integer n such that
(z
α)nf(z)
is holomorphic at α.
The function f(z) =
ez
is a holomorphic function on C; in contrast,
1/(z i) is meromorphic on C but not holomorphic since it has a pole at i.
The function
e−1/z
is not even meromorphic on C.
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