1.3. HALF-INTEGRAL WEIGHT MODULAR FORMS 11

important and fundamental properties require results on half-integral weight Hecke

operators in Shimura’s 1973 Annals of Mathematics paper

[Shi2]1.

This important

paper provided a general framework for studying half-integral weight modular forms

by introducing the so-called “Shimura correspondence”, a family of maps which

relate the Fourier expansions of half-integral weight modular forms to those of

integer weight forms. Here we briefly recall basic facts about half-integral weight

forms. For background information, one may consult [Kob2, SSt, Shi2].

To define these forms, we first define

(

c

d

)

and

d

. If d is an odd prime, then let

(

c

d

)

be the usual Legendre symbol. For positive odd d, define

(

c

d

)

by multiplicativity.

For negative odd d, we let

(1.13)

c

d

:=

(

c

|d|

)

if d 0 and c 0,

−

(

c

|d|

)

if d 0 and c 0.

Also let

(

0

±1

)

= 1. Define

d

, for odd d, by

(1.14)

d

:=

1 if d ≡ 1 mod 4,

i if d ≡ 3 mod 4.

Throughout, we let

√

z be the branch of the square root having argument in

(−π/2, π/2]. Hence,

√

z is a holomorphic function on the complex plane with

the negative real axis removed.

Definition 1.36. Suppose that λ is a nonnegative integer and that N is a

positive integer. Furthermore, suppose that χ is a Dirichlet character modulo 4N.

A meromorphic function g(z) on H is called a meromorphic half-integral weight

modular form with Nebentypus χ and weight λ +

1

2

if it is meromorphic at the cusps

of Γ, and if

g

az + b

cz + d

= χ(d)

c

d

2λ+1

−1−2λ(cz

d

+

d)λ+

1

2

g(z)

for all

a b

c d

∈ Γ0(4N). If g(z) is holomorphic on H and at the cusps of Γ0(4N),

then g(z) is referred to as a holomorphic half-integral weight modular form. If g(z)

is a holomorphic modular form which vanishes at the cusps of Γ0(4N), then g(z) is

known as a cusp form. If g(z) is a meromorphic form whose poles (if there are any)

are supported at the cusps of Γ0(4N), then g(z) is known as a weakly holomorphic

modular form.

Remark 1.37. The cusp conditions in Definition 1.36 are determined in natural

way which is analogous to the integer weight case (see Definition 1.8 and Remark

1.10).

Remark 1.38. As in the integer weight case, we refer to a holomorphic half-

integral weight modular form as a half-integral weight modular form, and we con-

tinue to use the terminology meromorphic (resp. weakly holomorphic) half-integral

weight modular form.

1In

1977 Shimura was awarded the Frank Nelson Cole Prize by the American Mathematical

Society for two of his research papers; one of these was [Shi2].