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.