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.
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