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
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
. If d is an odd prime, then let
be the usual Legendre symbol. For positive odd d, define
For negative odd d, we let
if d 0 and c 0,
if d 0 and c 0.
= 1. Define
, for odd d, by
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 +
if it is meromorphic at the cusps
of Γ, and if
az + b
cz + 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
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
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.
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].