Distribution of integral Fourier Coefficients of a Modular Form of Half
  Integral Weight Modulo Primes
DISTRIBUTION OF INTEGRAL FOURIER COEFFICIENTS OF A
MODULAR FORM OF HALF INTEGRAL WEIGHT MODULO
PRIMES
D. CHOI
Abstract. Recently, Bruinier and Ono classified cusp forms f(z) :=
af (n)q
Sλ+ 1
(Γ0(N), χ) ∩ Z[[q]] that does not satisfy a certain distribution property for modulo
odd primes p. In this paper, using Rankin-Cohen Bracket, we extend this result to
modular forms of half integral weight for primes p ≥ 5. As applications of our main
theorem we derive distribution properties, for modulo primes p ≥ 5, of traces of singular
moduli and Hurwitz class number. We also study an analogue of Newman’s conjecture
for overpartitions.
1. Introduction and Results
Let Mλ+ 1
(Γ0(N), χ) and Sλ+ 1
(Γ0(N), χ) be the spaces, respectively, of modular forms
and cusp forms of weight λ + 1
on Γ0(N) with a Dirichlet character χ whose conductor
divides N . If f(z) ∈Mλ+ 1
(Γ0(N), χ), then f(z) has the form
f(z) =
a(n)qn,
where q := e2πiz. It is well-known that the coefficients of f are related to interesting
objects in number theory such as the special values of L-function, class number, traces of
singular moduli and so on. In this paper, we study congruence properties of the Fourier
coefficients of f(z) ∈Mλ+ 1
(Γ0(N), χ) ∩ Z[[q]] and their applications.
Recently, Bruinier and Ono proved in [3] that g(z) ∈ Sλ+ 1
(Γ0(N), χ) ∩ Z[[q]] has a
special form (see (2.1)) by modulo p when p is an odd prime and the coefficients of f(z)
do not satisfy the following property for p:
Property A. IfM is a positive integer, we say that a sequence α(n) ∈ Z satisfies Property
A for M if for every integer r
♯{ 1 ≤ n ≤ X | α(n) ≡ r (mod M) and gcd(M,n) = 1}
if r 6≡ 0 (mod M),
X if r ≡ 0 (mod M).
2000 Mathematics Subject Classification. 11F11,11F33.
Key words and phrases. Modular forms, Congruences.
http://arxiv.org/abs/0704.0012v1
2 D. CHOI
θ(f(z)) :=
f(z) =
n · a(n)qn.
Using Rankin-Cohen Bracket (see (2.3)), we prove that there exists
f̃(z) ∈ Sλ+p+1+ 1
(Γ0(4N), χ) ∩ Z[[q]]
such that θ(f(z)) ≡ f̃(z) (mod p). We extend the results in [3] to modular forms of half
integral weight.
Theorem 1. Let λ be a non-negative integer. We assume that f(z) =
n=0 a(n)q
Mλ+ 1
(Γ0(4N), χ) ∩ Z[[q]], where χ is a real Dirichlet character. If p ≥ 5 is a prime and
there exists a positive integer n for which gcd(a(n), p) = 1 and gcd(n, p) = 1, then at least
one of the following is true:
(1) The coefficients of θp−1(f(z)) satisfies Property A for p.
(2) There are finitely many square-free integers n1, n2, · · · , nt for which
(1.1) θp−1(f(z)) ≡
a(nim
2)qnim
(mod p).
Moreover, if gcd(4N, p) = 1 and an odd prime ℓ divides some ni, then
p|(ℓ− 1)ℓ(ℓ+ 1)N or ℓ | N.
Remark 1.1. Note that for every odd prime p ≥ 5,
θp−1(f(z)) ≡
a(n)qn (mod p).
As an applications of Theorem 1, we study the distribution of traces of singular moduli
modulo primes p ≥ 5. Let j(z) be the usual j-invariant function. We denote by Fd the
set of positive definite binary quadratic forms
F (x, y) = ax2 + bxy + cy2 = [a, b, c]
with discriminant −d = b2−4ac. For each F (x, y), let αF be the unique complex number
in the complex upper half plane, which is a root of F (x, 1). We define ωF ∈ {1, 2, 3} as
ωF :=
2 if F ∼Γ [a, 0, a],
3 if F ∼Γ [a, a, a],
1 otherwise,
where Γ := SL2(Z). Here, F ∼Γ [a, b, c] denotes that F (x, y) is equivalent to [a, b, c].
From these notations, we define the Hecke trace of singular moduli.
DISTRIBUTION OF INTEGRAL FOURIER COEFFICIENTS MODULO PRIMES 3
Definition 1.2. If m ≥ 1, then we define the mth Hecke trace of the singular moduli of
discriminant −d as
tm(d) :=
F∈Fd/Γ
jm(αF )
where Fd/Γ denotes a set of Γ−equivalence classes of Fd and
jm(z) := j(z)|T0(m) =
az + b
Here, T0(m) denotes the normalized mth weight zero Hecke operator.
Note that t1(d) = t(d), where
t(d) :=
F∈Fd/Γ
j(αF )− 744
is the usual trace of singular moduli. Let
h(z) :=
η(z)2
η(2z)
· E4(4z)
η(4z)6
and Bm(1, d) denote the coefficient of q
d in h(z)|T (m2, 1, χ0), where
E4(z) := 1 + 240
d3qn, η(z) := q
(1− qn) ,
and χ0 is a trivial character. Here, T (m
2, λ, χ) denotes the mth Hecke operator of weight
λ + 1
with a Dirichlet chracter χ (see VI. §3. in [5] or (2.5)). Zagier proved in [11] that
for all m and d
(1.2) tm(d) = −Bm(1, d).
Using these generating functions, Ahlgren and Ono studied the divisibility properties
of traces and Hecke traces of singular moduli in terms of the factorization of primes in
imaginary quadratic fields (see [2]). For example, they proved that a positive proportion
of the primes ℓ has the property that tm(ℓ
3n) ≡ 0 (mod ps) for every positive integer n
coprime to ℓ such that p is inert or ramified in Q
. Here, p is an odd prime, and
s and m are integers with p ∤ m. In the following theorem, we give the distribution of
traces and Hecke traces of singular moduli modulo primes p.
4 D. CHOI
Theorem 2. Suppose that p ≥ 5 is a prime such that p ≡ 2 (mod 3).
(1) Then, for every integer r, p ∤ r,
♯{ 1 ≤ n ≤ X | t1(n) ≡ r (mod p)} ≫r,p
if r 6≡ 0 (mod p)
X if r ≡ 0 (mod p).
(2) Then, a positive proportion of the primes ℓ has the property that
♯{ 1 ≤ n ≤ X | tℓ(n) ≡ r (mod p)} ≫r,p
if r 6≡ 0 (mod p)
X if r ≡ 0 (mod p).
for every integer r, p ∤ r.
As another application we study the distribution of Hurwitz class number modulo
primes p ≥ 5. The Hurwitz class number H(−N) is defined as follows: the class number
of quadratic forms of the discriminant −N where each class C is counted with multiplicity
Aut(C)
. The following theorem gives the distribution of Hurwitz class number modulo
primes p ≥ 5.
Theorem 3. Suppose that p ≥ 5 is a prime. Then, for every integer r
♯{ 1 ≤ n ≤ X | H(n) ≡ r (mod p)} ≫r,p
if r 6≡ 0 (mod p),
X if r ≡ 0 (mod p).
We also use the main theorem to study an analogue of Newman’s conjecture for overpar-
titions. Newman’s conjecture concerns the distribution of the ordinary partition function
modulo primes p.
Newman’s Conjecture. Let P (n) be an ordinary partition function. If M is a positive
integer, then for every integer r there are infinitely many nonnegative integer n for which
P (n) ≡ r (mod M).
This conjecture was already studied by many mathematicians (see Chapter 5. in [8]).
The overpartition of a natural number n is a partition of n in which the first occurrence
of a number may be overlined. Let P̄ (n) be the number of the overpartition of an integer
n. As an analogue of Newman’s conjecture, the following theorem gives a distribution
property of P̄ (n) modulo odd primes p.
Theorem 4. Suppose that p ≥ 5 is a prime such that p ≡ 2 (mod 3). Then, for every
integer r,
♯{ 1 ≤ n ≤ X | P̄ (n) ≡ r (mod p)} ≫r,p
if r 6≡ 0 (mod p),
X if r ≡ 0 (mod p).
Remark 1.3. When r ≡ 0 (mod p), Theorem 2, 3 and 4 were proved in [2] and [10].
DISTRIBUTION OF INTEGRAL FOURIER COEFFICIENTS MODULO PRIMES 5
Next sections are detailed proofs of theorems: Section 2 gives a proof of Theorem 1. In
Section 3, we give the proofs of Theorem 2, 3, and 4.
2. Proof of Theorem 1
We begin by stating the following theorem proved in [3].
Theorem 2.1 ([3]). Let λ be a non-negative integer. Suppose that g(z) =
n=0 ag(n)q
Sλ+ 1
(Γ0(4N), χ) ∩ Z[[q]], where χ is a real Dirichlet character. If p is an odd prime and
a positive integer n exists for which gcd(ag(n), p) = 1, then at least one of the following
is true:
(1) If 0 ≤ r < p, then
♯{ 1 ≤ n ≤ X | ag(n) ≡ r (mod p)} ≫r,M
if r 6≡ 0 (mod p),
X if r ≡ 0 (mod p).
(2) There are finitely many square-free integers n1, n2, · · · , nt for which
(2.1) g(z) ≡
ag(nim
2)qnim
(mod p).
Moreover, if gcd(p, 4N) = 1, ǫ ∈ {±1}, and ℓ ∤ 4Np is a prime with
∈ {0, ǫ}
for 1 ≤ i ≤ t, then (ℓ−1)g(z) is an eigenform modulo p of the half-integral weight
Hecke operator T (ℓ2, λ, χ). In particular, we have
(2.2) (ℓ− 1)g(z)|T (ℓ2, λ, χ) ≡ ǫχ(p)
(−1)λ
ℓλ + ℓλ−1
(ℓ− 1)g(z) (mod p).
Recall that f(z) =
a(n)qn ∈ Mλ+ 1
(Γ0(4N), χ) ∩ Z[[q]]. Thus, to apply Theorem
2.1, we show that there exists a cusp form f̃(z) such that f̃(z) ≡ θp−1(f(z)) (mod p) for
a prime p ≥ 5.
Lemma 2.2. Suppose that p ≥ 5 is a prime and
f(z) =
a(n)qn ∈Mλ+ 1
(Γ0(N), χ) ∩ Z[[q]].
Then, there exists a cusp form f̃(z) ∈ Sλ+(p+1)(p−1)+ 1
(Γ0(N), χ) ∩ Z[[q]] such that
f̃(z) ≡ θp−1(f(z)) (mod p).
Proof of Lemma 2.2. For F (z) ∈Mk1
(Γ0(N), χ1) and G(z) ∈Mk2
(Γ0(N), χ2), let
(2.3) [F (z), G(z)]1 :=
θ(F (z)) ·G(z)−
F (z) · θ(G(z)).
This operator is referred to as a Rankin-Cohen 1-bracket, and it was proved in [4] that
[F (z), G(z)]1 ∈ S k1+k2
(Γ0(N), χ1χ2χ
6 D. CHOI
where χ′ = 1 if k1
and k2
∈ Z, χ′(d) =
2 if ki
∈ Z and k3−i
+ Z, and
χ′(d) =
) k1+k2
2 if k1
and k2
For even k ≥ 4, let
Ek(z) := 1−
dk−1qn
be the usual normalized Eisenstein series of weight k. Here, the number Bk denotes the
kth Bernoulli number. The function Ek(z) is a modular form of weight k on SL2(Z), and
(2.4) Ep−1(z) ≡ 1 (mod p)
(see [6]). From (2.3) and (2.4), we have
[Ep−1(z), f(z)]1 ≡ θ(f(z)) (mod p)
and [Ep−1(z), f(z)]1 ∈ Sλ+p+1+ 1
(Γ0(N), χ). Repeating this method p− 1 times, we com-
plete the proof. �
Using the following lemma, we can deal with the divisibility of ag(n) for positive integers
n, p ∤ n, where g(z) =
n=1 ag(n)q
n ∈ Sλ+ 1
(Γ0(N), χ) ∩ Z[[q]].
Lemma 2.3 (see Chapter 3 in [8]). Suppose that g(z) =
n=1 ag(n)q
n ∈ Sλ+ 1
(Γ0(N), χ)
has coefficients in OK , the algebraic integers of some number field K. Furthermore,
suppose that λ ≥ 1 and that m ⊂ OK is an ideal norm M .
(1) Then, a positive proportion of the primes Q ≡ −1 (mod 4MN) has the property
g(z)|T (Q2, λ, χ) ≡ 0 (mod m).
(2) Then a positive proportion of the primes Q ≡ 1 (mod 4MN) has the property that
g(z)|T (Q2, λ, χ) ≡ 2g(z) (mod m).
We can now prove Theorem 1.
Proof of Theorem 1. From Lemma 2.2, there exists a cusp form
f̃(z) ∈ Sλ+(p+1)(p−1)+ 1
(Γ0(N), χ) ∩ Z[[q]]
such that
f̃(z) ≡ θp−1(f(z)) (mod p).
Note that, for F (z) =
n=0 aF (n)q
n ∈ Mk+ 1
(Γ0(N), χ) and each prime Q ∤ N , the
half-integral weight Hecke operator T (Q2, λ, χ) is defined as
(2.5)
F (z)|T (Q2, k, χ)
aF (Q
2n) + χ∗(Q)
Qk−1aF (n) + χ
∗(Q2)Q2k−1aF (n/Q
DISTRIBUTION OF INTEGRAL FOURIER COEFFICIENTS MODULO PRIMES 7
where χ∗(n) := χ∗(n)
(−1)k
and aF (n/Q
2) = 0 if Q2 ∤ n. If F (z)|T (Q2, k, χ) ≡ 0
(mod p) for a prime Q ∤ N , then we have
aF (Q
2 ·Qn) + χ∗(Q)
Qk−1aF (Qn) + χ
∗(Q2)Q2k−1aF
Qn/Q2
≡ aF (Q3n) ≡ 0 (mod p)
for every positive integer n such that gcd(Q, n) = 1. Thus, we have the following by
Lemma 2.3-(1):
♯{ 1 ≤ n ≤ X | a(n) ≡ 0 (mod p) and gcd(p, n) = 1} ≫ X.
We apply Theorem 2.1 with f̃(z). Then the purpose of the remaining part of the proof
is to show the following: if gcd(p, 4N) = 1, an odd prime ℓ divides some ni, and
(2.6) θp−1(f(z)) ≡
a(nim
2)qnim
(mod p),
then p|(ℓ− 1)ℓ(ℓ+ 1)N or ℓ | N . We assume that there exists a prime ℓ1 such that ℓ1|n1,
p ∤ (ℓ1 − 1)ℓ1(ℓ1 + 1)N and ℓ | N . We also assume that nt = 1 and that ni ∤ n1 for every
i, 2 ≤ i ≤ t − 1. Then, we can take a prime ℓi for each i, 2 ≤ i ≤ t − 1, such that ℓi|ni
and ℓi ∤ n1. For convention, we define
(−1)(n−1)2/8 if n is odd,
0 otherwise,
and χQ(d) :=
for a prime Q. Let ψ(d) :=
i=2 χℓi(d). We take a prime β such that
ψ(n1)χβ(n1) = −1. If we denote the ψ-twist of f̃(z) by f̃ψ(z) and the ψχβ-twist of f̃(z)
by f̃ψχβ(z), then
f̃ψχ2
(z)− f̃ψχβ(z) ≡ 2
gcd(m,β
ℓj)=1
a(n1m
2)qn1m
(mod p)
and f̃ψχβ(z) ∈ Sλ+(p+1)(p−1)+ 1
(Γ0(Nα
2β2), χ) ∩ Z[[q]] (see Chapter 3 in [8]). Note that
gcd(Nα2β2, p) = gcd(Nα2β2, ℓ1) = 1.
Thus, (f̃ψ(z)− f̃ψχβ(z))|T (ℓ21, λ+ (p+ 1)(p− 1), χ) satisfies the formula (2.2) of Theorem
2.1 for both of ǫ = 1 and ǫ = −1. This results in a contradiction since
(f̃ψ(z)− f̃ψχβ(z))|T (ℓ
1, λ+ (p+ 1)(p− 1), χ) 6≡ 0 (mod p)
and p ≥ 5. Thus, we complete the proof. �
8 D. CHOI
3. Proofs of Theorem 2, 3, and 4
3.1. Proof of Theorem 2. Note that h(z) =
η(z)2
η(2z)
·E4(4z)
η(4z)6
is a meromorphic modular form.
In [2] it was obtained a holomorphic modular form on Γ0(4p
2) whose Fourier coefficients
generate traces of singular moduli modulo p (see the formula (3.1) and (3.2)). Since the
level of this modular form is not relatively prime to p, we need the following proposition.
Proposition 3.1 ([1]). Suppose that p ≥ 5 is a prime. Also, suppose that p ∤ N , j ≥ 1 is
an integer, and
g(z) =
a(n)qn ∈ Sλ+ 1
(Γ0(Np
j)) ∩ Z[[q]].
Then, there exists a cusp form G(z) ∈ Sλ′+ 1
(Γ0(N)) ∩ Z[[q]] such that
G(z) ≡ g(z) (mod p),
where λ′ + 1
= (λ+ 1
)pj + pe(p− 1) for a sufficiently large e ∈ N.
Using Theorem 1 and Proposition 3.1, we give the proof of Theorem 2.
Proof of Theorem 2. Let
(3.1) h1,p(z) := h(z)−
hχp(z),
where hχp(z) is the χp-twist of h(z). From (1.2), we have
h1,p(z) := −2 −
0<d≡0,3 (mod 4)
t1(d)q
d − 2
0<d≡0,3 (mod 4)
(−dp )=−1
t1(d)q
hm,p(z) := h1,p(z)|T (m2, 1, χ0)
= −2 −
0<d≡0,3 (mod 4)
tm(d)q
d − 2
0<d≡0,3 (mod 4)
(−dp )=−1
tm(d)q
for every positive integer m. Let
Fp(z) :=
η(4z)p
η(4pz)
It was proved in [2] that if α is a sufficiently large positive integer, then h1,p(z)Fp(z)
(Γ0(4p
2)) and
(3.2) h1,p(z)Fp(z)
α ≡ h1,p(z) (mod p),
DISTRIBUTION OF INTEGRAL FOURIER COEFFICIENTS MODULO PRIMES 9
where k0 = α · p
. Lemma 2.2 and Proposition 3.1 imply that there exists f1,p(z) ∈
Sλ′+ 1
(Γ0(4)) ∩ Z[[q]] such that
f1,p(z) ≡ −2
0<d≡0,3 (mod 4)
(−dp )=−1
tm(d)q
d (mod p),
where λ′ = (k0 + 1 + (p− 1)(p+ 1) + 12)p
2 + pe(p− 1) for a sufficiently large e ∈ N.
We assume that the coefficients of f1,p(z) do not satisfy Property A for an odd prime
p ≡ 2 (mod 3). Note that
= −1 and that p ∤ (3−1)3(3+1). So, Theorem 1 implies
2t1(3) ≡ 0 (mod p).
This results in a contradiction since 2t1(3) = 2
4 ·31. Thus, we obtain a proof when m = 1.
For every odd prime ℓ, we have
f1,p(z)|T (ℓ2, λ′, χ0) ≡ θp−1(h1,p(z))|T (ℓ2, λ′, χ0)
≡ θp−1(h1,p(z)|T (ℓ2, 1, χ0)) ≡ θp−1(hℓ,p(z)) (mod p).
Moreover, Lemma 2.3 implies that a positive proportion of the primes ℓ satisfies the
property
f1,p(z)|T (ℓ2, λ′, χ0) ≡ 2f1,p (mod p).
This completes the proof. �
3.2. Proofs of Theorem 3. The following theorem gives the formula for the Hurwitz
class number in terms of the Fourier coefficients of a modular form of half integral weight.
Theorem 3.2. Let T (z) := 1 + 2
n=1 q
n2. If integers r3(n) are defined as
r3(n)q
n := T (z)3,
r(n) =


12H(−4n) if n ≡ 1, 2 (mod 4),
24H(−n) if n ≡ 3 (mod 8),
r(n/4) if n ≡ 0 (mod 4),
0 if n ≡ 7 (mod 8).
Note that T (z) is a half integral weight modular form of weight 1
on Γ0(4). Combining
Theorem 1 and Theorem 3.2, we derive the proof of Theorem 3.
Proof of Theorem 3. Let G(z) be the
-twist of T (z)3. Then, from Theorem 3.2, we
G(z) = 1 +
n≡1 (mod 4)
12H(−4n)qn +
n≡3 (mod 8)
24H(−n)qn
10 D. CHOI
and G(z) ∈ M 3
(Γ0(16)). Note that 24H(−3) = 8. This gives the complete proof by
Theorem 1. �
3.3. Proofs of Theorem 4. In the following, we prove Theorem 4.
Proof of Theorem 4. Let
W (z) :=
η(2z)
η(z)2
It is known that
W (z) =
P̄ (n)qn
and that W (z) is a weakly holomorphic modular form on Γ0(16). Let
G(z) :=
W (z)−
Wχp(z)
Fp(z)
where Fp(z) =
η(4z)p
η(4p2z)
and β are positive integers. Then we have
G(z) ≡ 2
(−np )=−1
P̄ (n)qn +
P̄ (n)qn (mod p).
We claim that there exists a positive integer β such that G(z) is a holomorphic modular
form of half integral weight on Γ0(16p
2). To prove our claim, we follow the arguments
of Ahlgren and Ono ([1], Lemma 4.2). Note that, by a well-known criterion, Fp(z) is a
holomorphic modular form on Γ0(4p
2) that vanishes at each cusp a
∈ Q for which p2 ∤ c
(see [7]). This implies that G(z) is a weakly holomorphic modular form on Γ0(16p
2). If β
is sufficiently large, then G(z) is holomorphic except at each cusp a
for which p2|c′.
Thus, we prove that G(z) is holomorphic at 1
for 0 ≤ m ≤ 3. Let, for odd d,
ǫd :=
1 if d ≡ 1 (mod 4),
i if d ≡ 3 (mod 4).
If f(z) is a function on the complex upper half plane, λ ∈ Z, and γ = ( a bc d ) ∈ Γ0(4), then
we define the usual slash operator by
f(z) |λ+ 1
)2λ+1
ǫ−1−2λd (cz + d)
−λ− 1
az + b
cz + d
Let g :=
e2πiv/p be the usual Gauss sum. Note that
Wχp(z) =
W (z)|− 1
1 −v/p
Choose an integer kv satisfying
16kv ≡ 15v (mod p).
DISTRIBUTION OF INTEGRAL FOURIER COEFFICIENTS MODULO PRIMES 11
Then, we have
(3.3)
2mp2 1
= γv,m
2mp2 1
1 −16v
+ 16kv
where
γv,m =
1− 2m+4p(v + kv + 2mv2p− 2mvkvp) 1p(15v − 16kv − 2
m+4(v2p+ vkvp))
22mp2(−16vp+ 16kvp) 2m+4vp− 2m+4kvp+ 1
Note that W (z) has its only pole at z ∼ 0 up to Γ0(16). Since γv,m ∈ Γ0(16), the
formula (3.3) implies that Wχp(z) is holomorphic at 2
mp2 for 1 ≤ m ≤ 3. Thus, G(z) is
holomorphic at 2mp2 for 1 ≤ m ≤ 3.
If m = 0, then we have
W (z)|− 1
γv,0 =
−16vp3 + 16kvp3
16vp− 16kvp+ 1
W (z) =
p2(−vp+ kvp)
16vp− 16kvp + 1
W (z) = W (z).
Note that
(3.4) W (z)|− 1
= α · q−
16 +O(1),
where α is a nonzero complex number. The q-expansion of Wχp(z) at
is given by
(3.5) Wχp(z)|− 1
Using (3.3) and (3.4), the only term in (3.5) with a negative exponent on q is the term
(v−kv).
If N is defined by 16N ≡ 1 (mod p), then we have
(v−kv) =
Thus, we have that
(W (z)−Wχp(z))|− 1
= O(1).
This implies that G(z) is a holomorphic modular form of half integral weight on Γ0(16p
Noting that
P̄ (3) = 8,
the remaining part of the proof is similar to that in Theorem 3. Thus, it is omitted. �
12 D. CHOI
References
[1] S. Ahlgren and M. Boylan, Central Critical Values of Modular L-functions and Coeffients of Half
Integral Weight Modular Forms Modulo ℓ, to appear in Amer. J. Math.
[2] S. Ahlgren and K. Ono, Arithmetic of singular moduli and class polynomials, Compos. Math. 141
(2005), no. 2, 293–312.
[3] J. H. Bruinier and K. Ono, Coefficients of half-integral weight modular forms, J. Number Theory 99
(2003), no. 1, 164–179.
[4] H. Cohen, Sums involving the values at negative integers of L-functions of quadratic characters,
Math. Ann. 217 (1975), no. 3, 271–285.
[5] N. Koblitz, Introduction to elliptic curves and modular forms, Springer-Verlag New York, GTM 97,
1993.
[6] S. Lang, Introduction to Modular Forms, Grundl. d. Math. Wiss. no. 222, Springer: Berlin Heidelberg
New York, 1976 Berlin, 1995.
[7] B. Gordon and K. Hughes, Multiplicative properties of eta-product, Cont. Math. 143 (1993), 415-430.
[8] K. Ono, The web of modularity: arithmetic of the coefficients of modular forms and q-series, Amer.
Math. Soc., CBMS Regional Conf. Series in Math., vol. 102, 2004.
[9] J.-P. Serre, Divisibilite de certaines fonctions arithmetiques, Enseignement Math. (2) 22 (1976), no.
3-4, 227–260.
[10] S. Treneer, Congruences for the Coefficients of Weakly Holomorphic Modular Forms, to appear in
the Proceedings of the London Mathematical Society.
[11] D. Zagier, Traces of singular moduli, Motives, polylogarithms and Hodge theory, Part I, Int. Press
Lect. Ser., 3, I, Int. Press, Somerville, MA, 2002, pp.211-244.
School of Mathematics, KIAS, 207-43 Cheongnyangni 2-dong 130-722, Korea
E-mail address : choija@postech.ac.kr
	1. Introduction and Results
	2. Proof of Theorem ??
	3. Proofs of Theorem ??, ??, and ??
	3.1. Proof of Theorem ??
	3.2. Proofs of Theorem ??
	3.3. Proofs of Theorem ??
	References
