12 1. BASIC FACTS

Remark 1.39. Since

−1 0

0 −1

∈ Γ0(4N), it follows that there are no nonzero

meromorphic half-integral weight modular forms with odd Nebentypus character χ

(i.e. with χ(−1) = −1).

As in the integer weight case, these forms constitute C-vector spaces. We denote

the C-vector space of weight λ +

1

2

modular (resp. cusp) forms on Γ0(4N) with

Nebentypus χ by

(1.15) Mλ+

1

2

(Γ0(4N), χ) (resp. Sλ+

1

2

(Γ0(4N), χ)).

If χ = χ0 is the trivial character modulo 4N, then we use the notation

(1.16) Mλ+

1

2

(Γ0(4N)) (resp. Sλ+

1

2

(Γ0(4N))).

1.3.1. Theta-functions. Theta-functions provide the first examples of half-

integral weight modular forms. We begin by defining the prototypical form.

Definition 1.40. The theta-function θ0(z) is given by the Fourier series

θ0(z) :=

∞

n=−∞

qn2

= 1 + 2q +

2q4

+

2q9

+ ··· .

Proposition 1.41. We have that

θ0(z) ∈ M

1

2

(Γ0(4)).

More generally, we have the following two families of theta-functions.

Definition 1.42. Suppose that ψ is a Dirichlet character.

(1) If ψ is even, then define θ(ψ, 0, z) by

θ(ψ, 0, z) :=

∞

n=−∞

ψ(n)qn2

.

(2) If ψ is odd, then define θ(ψ, 1, z) by

θ(ψ, 1, z) :=

∞

n=1

ψ(n)nqn2

.

By convention, we agree that

θ(χ0, 0, z) := θ0(z).

Remark 1.43. We shall refer to these theta-functions as single variable theta-

functions.

As modular forms, we have the following elegant fact.

Theorem 1.44. Suppose that ψ is a primitive Dirichlet character with conduc-

tor r(ψ).

(1) If ψ is even, then θ(ψ, 0, z) ∈ M

1

2

(Γ0(4 ·

r(ψ)2),

ψ).

(2) If ψ is odd, then θ(ψ, 1, z) ∈ S

3

2

(Γ0(4 ·

r(ψ)2),

ψχ−4), where χ−4 is the

nontrivial Dirichlet character modulo 4.

Serre and Stark [SSt] proved that every modular form of weight 1/2 is a linear

combination of theta-functions. In particular, they obtained the following complete

description of the spaces of weight 1/2 modular forms.