18 1. BASIC FACTS
where
χ12(n) :=
12
n
=

1







if n 1, 11 (mod 12),
−1 if n 5, 7 (mod 12),
0 otherwise.
Definition 1.63. Any function f(z) of the form
f(z) =
δ|N
η(δz)rδ
,
where N 1 and each is an integer, is known as an eta-quotient. If each 0,
then f(z) is known as an eta-product.
Using Theorem 1.61, it is not difficult to deduce the modular transformation
properties of eta-quotients. The following general result of Gordon, Hughes, and
Newman [GoH, N1, N3] comes in handy when working with eta-quotients and
eta-products.
Theorem 1.64. If f(z) =
δ|N
η(δz)rδ
is an eta-quotient with k =
1
2

δ|N

Z, with the additional properties that
δ|N
δrδ 0 (mod 24)
and
δ|N
N
δ
0 (mod 24),
then f(z) satisfies
f
az + b
cz + d
= χ(d)(cz +
d)kf(z)
for every
a b
c d
Γ0(N). Here the character χ is defined by χ(d) :=
((−1)ks)
d
,
where s :=
δ|N
δrδ
.
Suppose that k is a positive integer and that f(z) is an eta-quotient satisfying
the conditions of Theorem 1.64. If f(z) is holomorphic (resp. vanishes) at all of
the cusps of Γ0(N), then f(z) Mk(Γ0(N), χ) (resp. Sk(Γ0(N), χ)). Since η(z)
is analytic and never vanishes on H, it suffices to check that the orders at the
cusps are nonnegative (resp. positive). The following theorem (see, for example
[Bi, Lig, Ma]) is the necessary criterion for determining orders of an eta-quotient
at cusps in terms of the usual local variables.
Theorem 1.65. Let c, d and N be positive integers with d | N and gcd(c, d) = 1.
If f(z) is an eta-quotient satisfying the conditions of Theorem 1.64 for N, then the
order of vanishing of f(z) at the cusp
c
d
is
N
24
δ|N
gcd(d,
δ)2rδ
gcd(d,
N
d
)dδ
.
Previous Page Next Page