$p$-adic Limit of Weakly Holomorphic Modular Forms of Half Integral
  Weight
p-ADIC LIMIT OF THE FOURIER COEFFICIENTS OF WEAKLY
HOLOMORPHIC MODULAR FORMS OF HALF INTEGRAL WEIGHT
D. CHOI AND Y. CHOIE
Abstract. Serre obtained the p-adic limit of the integral Fourier coefficients of modular
forms on SL2(Z) for p = 2, 3, 5, 7. In this paper, we extend the result of Serre to weakly
holomorphic modular forms of half integral weight on Γ0(4N) for N = 1, 2, 4. The proof
is based on linear relations among Fourier coefficients of modular forms of half integral
weight. As applications of our main result, we obtain congruences on various modular
objects, such as those for Borcherds exponents, for Fourier coefficients of quotients of
Eisentein series and for Fourier coefficients of Siegel modular forms on the Maass Space.
November 4, 2018
1. Introduction and Statement of Main Results
Serre obtained the p-adic limits of the integral Fourier coefficients of modular forms on
SL2(Z) for p = 2, 3, 5, 7 (see Théorème 7 and Lemma 8 in [20]). In this paper, we extend
the result of Serre to weakly holomorphic modular forms of half integral weight on Γ0(4N)
forN = 1, 2, 4. The proof is based on linear relations among Fourier coefficients of modular
forms of half integral weight. As applications of our main result, we obtain congruences
for various modular objects, such as those for Borcherds exponents, for Fourier coefficients
of quotients of Eisentein series and for Fourier coefficients of Siegel modular forms on the
Maass Space.
For odd d, let
:= γtΓ0(4N)tγ
where γt = (
c d ) ∈ Γ(1) and γt(t) = ∞. We denote the q-expansion of a modular form
f ∈Mλ+ 1
(Γ0(4N)) at each cusp t of Γ0(4N) by
(1.1) (f |λ+ 1
γt)(z) = (cz + d)
−λ− 1
az + b
cz + d
atf (n)q
t , qt := q
where
(1.2) r(t) ∈
2000 Mathematics Subject Classification. 11F11,11F33.
Key words and phrases. modular forms, p-adic limit, Borcherds exponents, Maass space .
This work was partially supported by KOSEF R01-2003-00011596-0 , ITRC and BRSI-POSTECH.
http://arxiv.org/abs/0704.0013v2
2 D. CHOI AND Y. CHOIE
When t ∼ ∞, we denote atf (n) by af (n). Note that the number r(t) is independent of the
choice of f ∈Mλ+ 1
(Γ0(4N)) and λ. We call t a regular cusp if r(t) = 0 (see Chapter IV.
§1. of [15] for a more general definition of a λ-regular cusp ).
Remark 1.1. Our definition of a regular cusp is different from the usual one.
Let U4N := {t1, · · · , tν(4N)} be the set of all inequivalent regular cusps of Γ0(4N). Note
that the genus of Γ0(4N) is zero if and only if 1 ≤ N ≤ 4. LetMλ+ 1
(Γ0(4N)) be the space
of weakly holomorphic modular forms of weight λ + 1
on Γ0(4N) and let M0λ+ 1
(Γ0(N))
denote the set of f(z) ∈ Mλ+ 1
(Γ0(N)) such that the constant term of its q-expansion at
each cusp is zero. Let Up be the operator defined by
(f |Up)(z) :=
af(pn)q
Let OL be the ring of integers of a number field L with a prime ideal p ⊂ OL. For
f(z) :=
af(n)q
n and g(z) :=
ag(n)q
n ∈ L[[q−1, q]] we write
f(z) ≡ g(z) (mod p)
if and only if af (n)− ag(n) ∈ p for every integer n.
With these notations we state the following theorem.
Theorem 1. For N = 1, 2, 4 consider
f(z) :=
af (n)q
n ∈ M0
(Γ0(4N)) ∩ L[[q−1, q]].
Suppose that p ⊂ OL is any prime ideal such that p|p, p prime, and that af(n) is p-integral
for every integer n ≥ n0.
(1) If p = 2 and af (0) = 0, then there exists a positive integer b such that
(f |(Up)b)(z) ≡ 0 (mod pj) for each j ∈ N.
(2) If p ≥ 3 and f(z) ∈ M0
(Γ0(4N)) with λ ≡ 2 or 2+
(mod p−1
), then there
exists a positive integer b such that
(f |(Up)b)(z) ≡ 0 (mod pj) for each j ∈ N.
Remark 1.2. The p-adic limit of a sum of Fourier coefficients of f ∈ M 3
(Γ0(4N)) was
studied in [13].
Our method only allows to prove a weaker result if f(z) 6∈ M0
(Γ0(4N)).
THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 3
Theorem 2. For N = 1, 2 or 4, let
f(z) :=
af (n)q
n ∈ Mλ+ 1
(Γ0(4N)) ∩ L[[q−1, q]].
Suppose that p ⊂ OL is any prime ideal with p|p, p prime, p ≥ 5, and that af (n) is
p-integral for every integer n ≥ n0. If λ ≡ 2 or 2 +
(mod p−1
), then there exists a
positive integer b0 such that
p2b−m(p:λ)
t∈U4N
∆4N,3−α(p:λ)(z)
R4N (z)
e·ω(4N)
(0)atf (0) (mod p)
for every positive integer b > b0 (see Section 3 for detailed notation ).
Example 1.3. Recall that the generating function of the overpartition P̄ (n) of n(see [11])
P̄ (n)qn =
η(2z)
η(z)2
is in M− 1
(Γ0(16)), where η(z) := q
n=1(1− qn). Therefore, theorem 2 implies that
P̄ (52b) ≡ 1 (mod 5), ∀b ∈ N.
2. Applications: More Congruences
In this section, we study congruences for various modular objects such as those for
Borcherds exponents and for quotients of Eisenstein series.
2.1. p-adic Limits of Borcherds Exponents. Let MH denote the set of meromorphic
modular forms of integral weight on SL2(Z) with Heegner divisor, integer coefficients and
leading coefficient 1. Let
(Γ0(4)) := {f(z) =
af(n)q
n ∈ M 1
(Γ0(4)) | a(n) = 0 for n ≡ 2, 3 (mod 4)}.
If f(z) =
af(n)q
n ∈ M+1
(Γ0(4)), then define Ψ(f(z)) by
Ψ(f(z)) := q−h
(1− qn)af (n2),
where h = − 1
af(0) +
1<n≡0,1 (mod 4) af (−n)H(−n). Here H(−n) denotes the usual
Hurwitz class number of discriminant −n. The following was proved by Borcherds.
Theorem 2.1 ([4]). The map Ψ is an isomorphism from M+1
(Γ0(4)) to MH , and the
weight of Ψ(f(z)) is af (0).
4 D. CHOI AND Y. CHOIE
Let j(z) be the usual j-invariant function with the product expansion
j(z) = q−1
(1− qn)A(n).
Let F (z) := q−h
n=1(1 − qn)c(n) be a meromorphic modular form of weight k in MH .
The p-adic limit of
d|n d · c(d) was studied in [5] for p = 2, 3, 5, 7. Here we obtain the
p-adic limit of c(d) for p = 2, 3, 5, 7.
Theorem 3. Let F (z) := q−h
n=1(1− qn)c(n) be a meromorphic modular form of weight
k in MH .
(1) If p = 2, then for each j ∈ N there exists a positive integer b such that
c(mpb) ≡ 2k (mod pj)
for every positive integer m.
(2) If p ∈ {3, 5, 7}, then, for each j ∈ N there exists a positive integer b such that
5c(mpb)−̟(F )A(mpb) ≡ 10k (mod pj)
for every positive integer m. Here, ̟(F ) is a constant determined by the constant
term of the q-expansion of Ψ−1(F ) at 0.
2.2. Sums of n-Squares. For u ∈ Z>0, let
rn(u) := ♯{(s1, · · · , sn) ∈ Zn : s21 + · · ·+ s2n = u}.
Theorem 4. Suppose that p ≥ 5 is a prime. If λ ≡ 2 or 3 (mod p−1
), then there exists
a positive integer C0 such that
r2λ+1
p2b−m(p:λ)
≡ − (14− 4α (p : λ)) + 16
)[ λp−1 ]+α(p:λ)m(p:λ)
(mod p),
for every b > C0.
Remark 2.2. As for an example, if λ ≡ 2 (mod p− 1) and p is an odd prime, then there
exists a positive integer C0 such that
r2λ+1
≡ 10 (mod p), ∀b > C0
2.3. Quotients of Eisenstein Series. Congruences for the coefficients of quotients of
elliptic Eisenstein series have been studied in [3]. Let us consider the Cohen Eisenstein
series Hr+ 1
(z) :=
N=0H(r,N)q
n of weight r+ 1
, r ≥ 2 (see [7]). We derive congruences
for the coefficients of quotients of Hr+ 1
(z) and Eisenstein series.
THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 5
Theorem 5. Let
F (z) :=
E4(z)
aF (n)q
G(z) :=
E6(z)
aG(n)q
W (z) :=
E6(z)
aW (n)q
Then there exists a positive integer C0 such that
aF (11
2b+1) ≡ 1 (mod 11),
aG(11
2b+1) ≡ 6 (mod 11),
aW (11
2b+1) ≡ 2 (mod 11)
for every integer b > C0.
2.4. The Maass Space. Next we deal with congruences for the Fourier coefficients of
a Siegel modular form in the Maass space. To define the Maass space, let us introduce
notations given in [17]: let T ∈ M2g(Q) be a rational, half-integral, symmetric, non-
degenerate matrix of size 2g with discriminant
DT := (−1)g det(2T ).
Let DT = DT,0f
T , where DT,0 is the corresponding fundamental discriminant. Further-
more, let
G8 :=

2 0 −1 0 0 0 0 0
0 2 0 −1 0 0 0 0
−1 0 2 −1 0 0 0 0
0 −1 −1 2 −1 0 0 0
0 0 0 −1 2 −1 0 0
0 0 0 0 −1 2 −1 0
0 0 0 0 0 −1 2 −1
0 0 0 0 0 0 −1 2

and G7 be the upper (7, 7)-submatrix of G8. Define
Sg :=
(g−1)/8
2, if g ≡ 1 (mod 8),
(g−7)/8
G7, if g ≡ −1 (mod 8).
6 D. CHOI AND Y. CHOIE
For each m ∈ N such that (−1)gm ≡ 0, 1 (mod 4), define a rational, half-integral, sym-
metric, positive definite matrix Tm of size 2g by
Tm :=


0 m/4
, if m ≡ 0 (mod 4),
e2g−1
e′2g−1 [m+ 2 + (−1)n]/4
, if m ≡ (−1)g (mod 4)
Here e2g−1 ∈ Z(2n−1,1) is the standard column vector and e′2g−1 is its transpose.
Definition 2.3. (The Maass Space) Take g, k ∈ N such that g ≡ 0, 1 (mod 4) and
g ≡ k (mod 2). Let
SMaassk+g (Γ2g)
F (Z) =
A(T )qtr(TZ) ∈ Sk+g(Γ2g)
∣∣∣∣∣∣
A(T ) =
ak−1φ(a;T )A(T|DT |/a2)
(see (6.2) for details). This space is called the Maass space of genus 2g and weight g + k.
In [17] it was proved that the Maass space is the same as the image of the Ikeda lifting
when g ≡ 0, 1 (mod 4). Using this fact together with Theorem 1, we derive the following
congruences for the Fourier coefficients of F (Z) in SMaassk+g (Γ2g).
Theorem 6. For g ≡ 0, 1 (mod 4), let
F (Z) :=
A(T )qtr(TZ) ∈ SMaassk+g (Γ2g)
with integral coefficients A(T ), T > 0. If k ≡ 2 or 3 (mod p−1
) for some prime p, then,
for each j ∈ N, there exists a positive integer b for which
A(T ) ≡ 0 (mod pj)
for every T > 0, det(2T ) ≡ 0 (mod pb).
This paper is organized as follows. Section 3 gives a linear relation among Fourier
coefficients of modular forms of half integral weight. The remaining sections contain
detailed proofs of the main theorems.
3. Linear Relation among Fourier Coefficients of modular forms of Half
Integral Weight
Let V (N ; k, n) be the subspace of Cn generated by the first n coefficients of the q-
expansion of f at ∞ for f ∈ Sk(Γ0(N)), where Sk(Γ0(N)) denotes the space of cusp forms
of weight k ∈ Z on Γ0(N). Let L(N ; k, n) be the orthogonal complement of V (N ; k, n)
THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 7
in Cn with the usual inner product of Cn. The vector space L(1; k, d(k) + 1), d(k) =
dim(Sk(Γ(1))), was studied by Siegel to evaluate the value of the Dedekind zeta function
at a certain point. The vector space L(1; k, n) is explicitly described in terms of the
principal part of negative weight modular forms in [9]. These results were extended in [8]
to the groups Γ0(N) of genus zero. For 1 ≤ N ≤ 4, let
4N, λ+
at1f (0), · · · , a
tν(4N)
f (0), af(1), · · · , af(n)
∈ Cn+ν(4n)
∣∣∣ f ∈Mλ+ 1
(Γ0(4N))
where U4N := {t1, · · · , tν(4N)} is the set of all inequivalent regular cusps of Γ0(4N). We
define EL(4N, λ+ 1
;n) to be the orthogonal complement of EV (4N, λ+ 1
;n) in Cn+ν(4N).
Let ∆4N,λ := q
δλ(4N)+O(qδλ(4N)+1) be inMλ+ 1
(Γ0(4N) with the maximum order at ∞,
that is, its order at ∞ is bigger than that of any other modular form of the same level
and weight. Furthermore, let
R4(z) :=
η(4z)8
η(2z)4
, R8(z) :=
η(8z)8
η(4z)4
R12(z) :=
η(12z)12η(2z)2
η(6z)6η(4z)4
and R16(z) :=
η(16z)8
η(8z)4
For ℓ, n ∈ N, define
m(ℓ : n) :=
≡ 0 (mod 2)
≡ 1 (mod 2)
α(ℓ : n) := n− ℓ− 1
Let ω(4N) be the order of zero of R4N (z) at ∞. Note that R4N (z) ∈ M2(Γ0(4N)) has
its only zero at ∞. So, using the definition of η(z) = q 124
n=1(1− qn), we find that
(3.1) ω(4) = 1, ω(8) = 2, ω(12) = 4, ω(16) = 4.
For each g ∈Mr+ 1
(Γ0(4N)) and e ∈ N, let
(3.2)
R4N (z)e
e·ω(4N)∑
b(4N, e, g; ν)q−ν +O(1) at ∞.
With these notations we state the following theorem:
Theorem 3.1. Suppose that λ ≥ 0 is an integer and 1 ≤ N ≤ 4. For each e ∈ N
such that e ≥ λ
− 1, take r = 2e − λ + 1. The linear map Φr,e(4N) : Mr+ 1
(Γ0(4N)) →
8 D. CHOI AND Y. CHOIE
EL(4N, λ+ 1
; e · ω(4N)), defined by
Φr,e(4N)(g)
R4N (z)
(0), · · · , htν(4N)a
tν(4N)
R4N (z)
(0), b(4N, e, g; 1), · · · , b(4N, e, g; e · ω(4N))
is an isomorphism.
Proof of Theorem 3.1. Suppose that G(z) is a meromorphic modular form of weight 2 on
Γ0(4N). For τ ∈ H∪C4N , let Dτ be the image of τ under the canonical map from H∪C4N
to a compact Riemann surface X0(4N). Here H is the usual complex upper half plane,
and C4N denotes the set of all inequivalent cusps of Γ0(4N). The residue ResDτGdz of
G(z) at Dτ ∈ X0(4N) is well-defined since we have a canonical correspondence between a
meromorphic modular form of weight 2 on Γ0(4N) and a meromorphic 1-form of X0(4N).
If ResτG denotes the residue of G at τ on H, then
ResDτGdz =
ResτG.
Here lτ is the order of the isotropy group at τ . The residue of G at each cusp t ∈ C4N is
(3.3) ResDtGdz = ht ·
atG(0)
Now we give a proof of Theorem 3.1.
To prove Theorem 3.1, take
G(z) =
R4N (z)e
f(z),
where g ∈Mr+ 1
(Γ0(4N)) and f(z) =
n=1 af(n)q
n ∈Mλ+ 1
(Γ0(4N)). Note that G(z) is
holomorphic on H. Since g(z), R4N (z) and f(z) are holomorphic and R4N (z) has no zero
on H, it is enough to compute the residues of G(z) only at all inequivalent cusps to apply
the Residue Theorem. The q-expansion of
R4N (z)
ef(z) at ∞ is
R4N(z)e
f(z) =
e·ω(4N)∑
b(4N, e, g; ν)q−ν + a g(z)
R4N (z)
(0) +O(q)
af(n)q
Since R4N (z) has no zero at t ≁ ∞, we have
R4N (z)e
γt = a
R4N (z)
(0)af(0) +O(qt).
Further note that, for an irregular cusp t,
at g(z)
R4N (z)
(0)af(0) = 0.
THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 9
So the Residue Theorem and (3.3) imply that
(3.4)
t∈U4N
e·ω(4N)
(0)atf(0) +
e·ω(4N)∑
b(4N, e, g; ν)af(ν) = 0.
This shows that Φr,e(4N) is well-defined. The linearity of the map Φr,e(4N) is clear.
It remains to check that Φr,e(4N) is an isomorphism. Since there exists no holomorphic
modular form of negative weight except the zero function, we obtain the injectivity of
Φr,e(4N). Note that for e ≥ λ−12 ,
4N ;λ+
, e · ω(4N)
= e · ω(4N) + ν(4N)− dimC
Mλ+ 1
(Γ0(4N))
However, the set C4N , 1 ≤ N ≤ 4, of all inequivalent cusps of Γ0(4N) are
∞, 0, 1
∞, 0, 1
C12 =
∞, 0, 1
C16 =
∞, 0, 1
and it can be checked that
(3.5) ν(4) = 2, ν(8) = 3, ν(12) = 4, ν(16) = 6
(see §1 of Chapter 4. in [15] for details). The dimension formula of Mλ+ 1
(Γ0(4N)) (see
Table 1) together with the results in (3.1) and (3.5), implies that
4N, λ+
; e · ω(N)
= dimC(Mr+ 1
(Γ0(4N)))
since r = 2e− λ+ 1.
Table 1. Dimension Formula for Mk(Γ0(4N))
N k = 2n + 1
k = 2n+ 3
k = 2n
N = 1 n + 1 n + 1 n + 1
N = 2 2n+ 1 2n+ 2 2n+ 1
N = 3 4n+ 1 4n+ 3 4n+ 1
N = 4 4n+ 2 4n+ 4 4n+ 1
So Φr,e(4N) is surjective since the map Φr,e(4N) is injective. This completes our claim.
10 D. CHOI AND Y. CHOIE
4. Proofs of Theorem 1 and 2
4.1. Proof of Theorem 1. First, we obtain linear relations among Fourier coefficients
of modular forms of half integral weight modulo p. Let
Op := {α ∈ L | α is p-integral}.
M̃λ+ 1
, p(Γ0(4N)) := {H(z) =
aH(n)q
n ∈ Op/pOp[[q−1, q]] |
H ≡ h (mod p) for some h ∈ Op[[q−1, q]] ∩Mλ+ 1
(Γ0(4N))}.
S̃λ+ 1
, p(Γ0(4N)) := {H(z) =
aH(n)q
n ∈ Op/pOp[[q−1, q]] |
H ≡ h (mod p) for some h ∈ Op[[q−1, q]] ∩ Sλ+ 1
(Γ0(4N))}.
The following lemma gives the dimension of M̃λ+ 1
, p(Γ0(4N)).
Lemma 4.1. Take λ ∈ N, 1 ≤ N ≤ 4 and a prime p such that
p ≥ 3 if N = 1, 2, 4,
p ≥ 5 if N = 3.
Now take any prime ideal p ⊂ OL, p|p. Then
dim M̃λ+ 1
, p(Γ0(4N)) = dimMλ+ 1
(Γ0(4N))
dim S̃λ+ 1
, p(Γ0(4N)) = dimSλ+ 1
(Γ0(4N)).
Proof. Let
j4N (z) = q
−1 +O(q)
be a meromorphic modular function with a pole only at ∞. Explicitly, these functions
j4(z) =
η(z)8
η(4z)8
+ 8, j8(z) =
η(4z)12
η(2z)4η(8z)8
j12(z) =
η(4z)4η(6z)2
η(2z)2η(12z)4
, j16(z) =
η2(z)η(8z)
η(2z)η2(16z)
Since the Fourier coefficients of η(z) and 1
are integral, the q-expansion of j4N (z) has
integral coefficients.
Recall that ∆4N,λ = q
δλ(4N) + O(qδλ(4N)+1) is the modular form of weight λ + 1
Γ0(4N) such that the order of its zero at ∞ is higher than that of any other modular form
THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 11
of the same level and weight. Denote the order of zero of ∆4N,λ at ∞ by δλ(4N). Then
the basis of Mλ+ 1
(Γ0(4N)) can be chosen as
(4.1) {∆4N,λ(z)j4N (z)e | 0 ≤ e ≤ δλ(4N)} .
If ∆4N,λ(z) is p-integral, then {∆4N,λ(z)j4N (z)e | 0 ≤ e ≤ δλ(4N)} also forms a basis of
M̃λ+ 1
,p(Γ0(4N)). Note that δλ(4N) = dimMλ+ 1
(Γ0(4N))− 1. So from Table 1 we have
(4.2) ∆4N,λ(z) = ∆4N,j(z)R4N (z)
where λ ≡ j (mod 2), j ∈ {0, 1}. More precisely, one can choose ∆4N,j(z) as followings:
∆4,0(z) = θ(z), ∆4,1(z) = θ(z)
∆8,0(z) = θ(z), ∆8,1(z) =
(θ(z)3 − θ(z)θ(2z)2) ,
∆12,0(z) = θ(z), ∆12,1(z) =
x,y,z∈Z q
3x2+2(y2+z2+yz) −
x,y,z∈Z q
3x2+4y2+4z2+4yz
∆16,0(z) =
(θ(z)− θ(4z)) , ∆16,1(z) = 18 (θ(z)
3 − 3θ(z)2θ(4z) + 3θ(z)θ(4z)2 − θ(4z)3) .
Since θ(z) = 1+ 2
n=1 q
n, the coefficients of the q-expansion of ∆4N,j(z), j ∈ {0, 1}, are
p-integral. This completes the proof. �
Remark 4.2. The proof of Lemma 4.1 implies that the spaces of Mλ+ 1
(Γ0(4N)) for
N = 1, 2, 4 are generated by eta-quotients since θ(z) =
η(2z)5
η(z)2η(4z)2
For 1 ≤ N ≤ 4 set
4N, λ+
(af(1), · · · , af(n)) ∈ Fnp | f ∈ S̃λ+ 1
(Γ0(4N))
,Fp := Op/pOp.
We define L̃S(4N, λ +
;n) to be the orthogonal complement of ṼS(4N, λ +
;n) in Fn
Using Lemma 4.1, we obtain the following proposition.
Proposition 4.3. Suppose that λ is a positive integer and 1 ≤ N ≤ 4. For each e ∈ N,
e ≥ λ
− 1, take r = 2e−λ+1. The linear map ψ̃r,e(4N) : M̃r+ 1
,p(Γ0(4N)) → L̃S(4N, λ+
; e · ω(4N)), defined by
ψ̃r,e(4N)(g) = (b(4N, e, g; 1), · · · , b(N, e, g; e · ω(4N))) ,
is an isomorphism. Here b(4N, e, g; ν) is defined in (3.2).
Proof. Note that dimS 3
(4N) = 0 and that
dimSλ+ 1
(4N) +N + 1 +
= dimMλ+ 1
(see [10]). So, from Lemma 4.1 and Table 1, it is enough to show that ψr,e(4N) is injective.
If g is in the kernel of ψr,e(4N), then
R4N (z)
e · R4N (z)e ≡ 0 (mod p) by Sturm’s formula
(see [21]). So we have g(z) ≡ 0 (mod p) since R4N(z)e 6≡ 0 (mod p). This completes the
proof. �
12 D. CHOI AND Y. CHOIE
Theorem 4.4. Take a prime p,N = 1, 2, 4 and
f(z) :=
af(n)q
n ∈ Sλ+ 1
(Γ0(4N)) ∩ L[[q]].
Suppose that p ⊂ OL is any prime ideal with p|p and that af (n) is p-integral for every
integer n ≥ n0. If λ ≡ 2 or 2 +
(mod p−1
) or p = 2, then there exists a positive
integer b such that
≡ 0 (mod p), ∀n ∈ N.
Proof of Theorem 4.4. i) First, suppose that p ≥ 3: Take positive integers ℓ and b such
(4.3)
3− 2α(p : λ)
p2b +
pm(p:λ) + ℓ(p− 1) = 2.
Note that if b is large enough, that is, b > logp
3−2α(p:λ)
pm(p:λ) − 2
, then there
exists a positive integer ℓ satisfying (4.3). Also note that atf(0) = 0 for every cusp t of
Γ0(4N) since f(z) is a cusp form. So, if r = 2e− α(p : λ) + 1, then Theorem 3.1 implies
that, for g(z) ∈ M̃r+ 1
(Γ0(4N)),
e·ω(4N)∑
b(4N, e, g; ν)af(νp
2b−m(p:λ)) ≡ 0 (mod p),
since
R4N (z)e
f(z)p
m(p:λ)
Eℓp−1(z)
e·ω(4N)∑
b(4N, e, g; ν)q−νp
+ a g(z)
R4N (z)
(0) +
a g(z)
R4N (z)
(n)qnp
af(n)q
npm(p:λ)
(mod p).
So Proposition 4.3 implies that
p2b−m(p:λ)
2p2b−m(p:λ)
, · · · , a
e · ω(4N)p2b−m(p:λ)
∈ ṼS
4N,α(p : λ) + 1
If α(p : λ) = 2 or 2 +
, then
dimSα(p:λ)+ 1
(Γ0(4N)) = dim ṼS
4N,α(p : λ) +
THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 13
ii) p = 2: Note that
∆4N,1(z)
R4N (z)
= q−1+O(1) for N = 1, 2, 4. So, there exists a polynomial
F (X) ∈ Z[X ] such that
F (j4N(z))
∆4N,1(z)
R4N (z)
= q−n +O(1).
For an integer b, 22
> λ+ 2, let
G(z) :=
F (j4N(z))
∆4N,1(z)
R4N(z)
f(z)θ(z)2
1+2b−2λ+3.
Since θ(z) ≡ 1 (mod 2), Theorem 3.1 implies that af(2b · n) ≡ 0 (mod p). �
To apply Theorem 4.4, we need the following two propositions.
Proposition 4.5 (Proposition 3.2 in [22]). Suppose that p is an odd prime, k and N are
integers with (N, p) = 1. Let
f(z) =
a(n)qn ∈ Mλ+ 1
(Γ0(4N)).
Suppose that ξ :=
cp2 d
, with ac > 0. Then there exist n0, h0 ∈ N with h0|N, a sequence
{a0(n)}n≥n0 and r0 ∈ {0, 1, 2, 3} such that
(f |Upm|λ+ 1
ξ)(z) =
4n+r0≡0 (mod p
a0(n)q
4n+r0
m , ∀m ≥ 1.
Proposition 4.6 (Proposition 5.1 in [1]). Suppose that p is an odd prime such that p ∤ N
and consider
g(z) =
a(n)qn ∈ Sλ+ 1
(Γ0(4Np
j)) ∩ L[[q]], for each j ∈ N.
Suppose further that p ⊂ OL is any prime ideal with p|p and that a(n) is p-integral for
every integer n ≥ 1. Then there exists G(z) ∈ Sλ′+ 1
(Γ0(4N)) ∩OL[[q]] such that
G(z) ≡ g(z) (mod p),
where λ′ + 1
= (λ+ 1
)pj + pe(p− 1) with eN large.
Remark 4.7. Proposition 4.6 was proved for p ≥ 5 in [1]. One can check that this holds
also for p = 3.
Now we prove Theorem 1.
Proof of Theorem 1. Take
Gp(z) :=
η(8z)48
η(16z)24
∈M12(Γ0(16)) if p = 2,
η(z)27
η(9z)3
∈M12(Γ0(9)) if p = 3,
η(4z)p
η(4p2z)
∈M p2−1
(Γ0(p
2)) if p ≥ 5.
14 D. CHOI AND Y. CHOIE
Using properties of eta-quotients (see [12]), note that Gp(z) vanishes at every cusp of
Γ0(16) except ∞ if p = 2, and vanishes at every cusp ac of Γ0(4Np
2) with p2 ∤ N if p ≥ 3.
Thus, Proposition 4.5 implies that there exist positive integers ℓ,m, k such that
(f |Upm)(z)Gp(z)ℓ ∈ Sk+ 1
(Γ0(16)) if p = 2,
(f |Upm)(z)Gp(z)ℓ ∈ Sk+ 1
(Γ0(4p
2N)) if p ≥ 3.
Note that k ≡ λ (mod p− 1). Using Proposition 4.6, we can find
F (z) ∈ Sk′+ 1
(Γ0(4N))
such that F (z) ≡ (f(z)|Upm)Gp(z)ℓ ≡ (f |Upm)(z) (mod p) and k′ ≡ k (mod p − 1).
Theorem 4.4 implies that there exists a positive integer b such that (F |Up2b)(z) ≡ 0
(mod p). Thus, we have shown so far that if ρ ∈ p \ p2, all the Fourier coefficients of
· F (z)|Upm+2b are p-integral. Repeat this argument to complete our claim. �
4.2. Proof of Theorem 2. Theorem 2 can be derived from Theorem 3.1 by taking a
special modular form.
Proof of Theorem 2. Take a positive integer ℓ and a positive even integer u such that
3− 2α(p : λ)
pm(p:λ) + ℓ(p− 1) = 2.
Let F (z) :=
∆4N,3−α(p:λ)(z)
R4N (z)
and G(z) := Ep−1(z)
ℓf(z)p
m(p:λ)
. Since Ep−1(z) ≡ 1
(mod p), we have
F (z)G(z) ≡
a∆4N,3−α(p:λ)(z)
R4N (z)
(n)qnp
af(n)q
nm(p:λ)
(mod p).
If Fourier coefficients of f(z) at each cusp are p-integral, then
((F ·G)|2γt) (z) ≡
atF (n)q
atG(n)q
atf (n)q
at∆4N,3−α(p:λ)(z)
R4N (z)
(mod p)
for t ≁ ∞. Since
aF (z)G(z)(0) ≡ a∆4N,3−α(p:λ)(z)
R4N (z)
(0)af (0) + af (p
u−m(p:λ)) (mod p) ,
F (z)G(z)
(0) ≡ at∆4N,3−α(p:λ)(z)
R4N (z)
(0)atf (0) (mod p) for t ≁ ∞,
for large u, the Residue Theorem implies Theorem 2 by letting u = 2b. Therefore it
is enough to check a p-integral property of Fourier coefficients of f(z) at each cusp:
take a positive integer e such that ∆(z)ef(z) is a holomorphic modular form, where
THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 15
∆(z) := q
n=1(1− qn)24. Note that the q-expansions of j4N (z) and ∆4N,12e+λ(z) at each
cusp are p-integral. Thus (4.1) implies that
∆(z)ef(z) =
δ12e+λ(4N)∑
cnj4N (z)
n∆4N,12e+λ(z).
Moreover, cn is p-integral since
j4N (z)
n∆4N,12e+λ(z) = q
δ12e+λ(4N)−n +O
qδ12e+λ(4N)−n+1
and f(z) ∈ OL[[q, q−1]]. Note that p ∤ 4N since 1 ≤ N ≤ 4 and p ≥ 5 is a prime.
So Fourier coefficients of j4N (z), ∆N,12e+λ(z) and
at each cusp are p-integral. This
completes our claim. �
5. Proof of Theorem 3
Theorem 3 follows from Theorem 1 and Theorem 2.1.
Proof of Theorem 3. Note that j(z) ∈ MH . Let
g(z) := Ψ−1(j(z)) and f(z) := Ψ−1(F (z)) =
af (n)q
It is known (see §14 in [4]) that
g(z) =
(θ(z))E10(4z)
4πi∆(4z)
θ(z) d
(E10(4z))
80πi∆(4z)
θ(z).
Since the constant terms of the q-expansions at ∞ of f(z), θ(z) and g(z) are 0, a0
(0) =
and a0g(0) =
· 456
, respectively, we have
f(z)− kθ(z)−
a0f(0) + k(1− i)/2
a0g(0)
g(z) ∈ M01
(Γ0(4)).
Applying Theorem 1, one obtains the result. �
6. Proofs of Theorem 4 and 5
We begin with the following proposition.
Proposition 6.1. Let p be an odd prime and
f(z) :=
af (n)q
n ∈Mλ+ 1
(Γ0(4)) ∩ Zp[[q]].
If λ ≡ 2 or 3 (mod p−1
), then
p2b−m(p:λ)
≡ −(14 − 4α(p : λ))af(0) + 28
2−1 − 2−1i
)pb(7−2α(p:λ))
a0f (0) (mod p)
16 D. CHOI AND Y. CHOIE
for every integer b > logp
2α(p:λ)−3
pm(p:λ) + 2
Proof of Proposition 6.1. For ν ∈ Z≥0,
pm(p:λ) := ν · (p− 1) + α(p : λ) + 1
For an integer b with
3− 2α(p : λ)
pm(p:λ) − 2
there exists an ℓ ∈ N such that
3− 2α(p : λ)
p2b +
pm(p:λ) + ℓ(p− 1) = 2,
since
3− 2α(p : λ)
p2b +
pm(p:λ) − 2 = 3− 2α(p : λ)
(p2b − 1) + ν(p− 1).
We have
F (z) ≡
n=0 af (n)q
npm(p:λ) (mod p),
G(z) ≡ q−pb + 14− 4α(p : λ) + aG(1)q + · · · (mod p).
Note that aG(n) is p-integral for every integer n. Moreover, we obtain
F (z)G(z)|2 ( 0 −11 0 ) ≡
a0f (0) + · · ·
−26pb
)pb(7−2α(p:λ))
+ · · ·
(mod p),
where a0f (0) is given in (1.1). Note that
∞, 0, 1
is the set of cusps of Γ0(4), so Theorem
2 implies that
af (p
2b−m(p:n)) + (14− 4α(p : λ))af(0)− 28a0f (0)
)pb(7−2α(p:λ))
≡ 0 (mod p).
This proves Proposition 6.1. �
6.1. Proof of Theorem 4. Now we prove Theorem 4.
Proof of Theorem 4. Take
f(z) := θ2λ+1(z) = 1 +
r2λ+1(ℓ)q
af(n)q
Note that f(z) ∈Mλ+ 1
(Γ0(4)). Since (θ| 1
( 0 −11 0 ))(z) =
, we obtain
af(0) = 1 and a
f (0) =
)2λ+1
THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 17
Since λ ≡ 2, 3 (mod p−1
) and
, we have
)p2u(7−2α(p:λ))
a0f (0)
pm(p:λ)
)p2u(7−2α(p:λ)) (
)pm(p:λ)(2α(p:λ)+(p−1)(2[ λp−1 ]+m(p:λ))+1)
)(7−2α(p:λ))(p2u−1)(
)8+2(p−1)[ λp−1 ]+m(p:λ)pm(p:λ)(p−1)+(pm(p:λ)−1)(1+2α(p:λ))
)8+2[ λp−1 ](p−1)+2α(p:λ)(pm(p:λ)−1)
)[ λp−1 ]+α(p:λ)m(p:λ)
(mod p),
for some u ∈ N. Applying Proposition 6.1, we obtain the result. �
6.2. Proof of Theorem 5. Consider the Cohen Eisenstein seriesHr+ 1
(z) :=
N=0H(r,N)q
of weight r + 1
, where r ≥ 2 is an integer. If (−1)rN ≡ 0, 1 (mod 4), then H(r,N) = 0.
If N = 0, then H(r, 0) = −B2r
. If N is a positive integer and Df 2 = (−1)rN , where D is
a fundamental discriminant, then
(6.1) H(r,N) = L(1− r, χD)
µ(d)χD(d)d
r−1σ2r−1(f/d).
Here µ(d) is the Möbius function. The following theorem implies that the Fourier coeffi-
cients of Hr+ 1
(z) are p-integral if p−1
Theorem 6.2 ([6]). Let D be a fundamental discriminant. If D is divisible by at least two
different primes, then L(1−n, χD) is an integer for every positive integer n. If D = p, p >
2, then L(1−n, χD) is an integer for every positive integer n unless gcd(p, 1−χD(g)gn) 6= 1,
where g is a primitive root (mod p).
Proof of Theorem 5. Note that E10(z) = E4(z)E6(z). So, E10(z)F (z), E10(z)G(z) and
E10(z)W (z) are modular forms of weights, 8 · 12 , 7 ·
and 8 · 1
respectively. Moreover, the
Fourier coefficients of those modular forms are 11-integral, since the Fourier coefficients
of H 5
(z), H 7
(z) and H 9
(z) are 11-integral by Theorem 6.2. We have
E10(z)F (z) =
+O(q),
E10(z)F (z)| 17
( 0 −11 0 ) =
(1 + i)(2i)−5 +O
E10(z)G(z) =
+O(q),
E10(z)G(z)| 15
( 0 −11 0 ) =
(1− i)(2i)−7 +O
E10(z)W (z) =
+O(q),
E10(z)W (z)| 17
( 0 −11 0 ) =
(1 + i)(2i)−9 +O
18 D. CHOI AND Y. CHOIE
where B2r is the 2rth Bernoulli number. The conclusion now follows from Proposition
6.1. �
6.3. Proof of Theorem 6. We begin by introducing some notations (see [17]). Let
V := (F2np , Q) be the quadratic space over Fp, where Q is the quadratic form obtained
from a quadratic form x 7→ T [x](x ∈ Z2np ) by reducing modulo p. We denote by < x, y >:=
Q(x, y)−Q(x)−Q(y), x, y ∈ F2np , the associated bilinear form and let
R(V ) := {x ∈ F2np : < x, y >= 0, ∀y ∈ F2np , Q(x) = 0}
be the radical of R(V ). Following [14], define a polynomial
Hn,p(T ;X) :=
1 if sp = 0,∏[(sp−1)/2]
j=1 (1− p2j−1X2) if sp > 0, sp odd,
(1 + λp(T )p
(sp−1)/2X)
∏[(sp−1)/2]
j=1 (1− p2j−1X2) if sp > 0, sp even,
where for even sp we denote
λp(T ) :=
1 if W is a hyperbolic space or sp = 2n,
−1 otherwise.
Following [16], for a nonnegative integer µ, define ρT (p
µ) by
ρT (p
µ)Xµ :=
(1−X2)Hn,p(T ;X), if p|fT ,
1 otherwise.
We extend the functions ρT multiplicatively to natural numbers N by defining
ρT (p
µ)X−µ :=
((1−X2)Hn,p(T ;X)).
D(T ) := GL2n(Z) \ {G ∈M2n(Z) ∩GL2n(Q) : T [G−1] half-integral},
where GL2n(Z) operates by left-multiplication and T [G
−1] = T ′G−1T . Then D(T ) is
finite. For a ∈ N with a|fT , let
(6.2) φ(a;T ) :=
G∈D(T ),|det(G)|=d
ρT [G−1](a/d
Note that φ(a;T ) ∈ Z for all a. With these notations we state the following theorem:
Theorem 6.3 ([17]). Suppose that g ≡ 0, 1 (mod 4) and let k ∈ N with g ≡ k (mod 2).
A Siegel modular form F is in SMaassk+n (Γ2g) if and only if there exists a modular form
f(z) =
c(n)qn ∈ Sk+ 1
(Γ0(4))
THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 19
such that A(T ) =
ak−1φ(a;T )c
|DT |
for all T . Here,
DT := (−1)g · det(2T )
and DT = DT,0f
T with DT,0 the corresponding fundamental discriminant and fT ∈ N.
Remark 6.4. A proof of Theorem 6.3 given in [17] implies that if A(T ) ∈ Z for all T ,
then c(m) ∈ Z for all m ∈ N.
Proof of Theorem 6. From Theorem 6.3 we can take
f(z) =
c(n)qn ∈ Sk+ 1
(Γ0(4)) ∩ Zp[[q]]
such that
F (Z) =
A(T )qtr(TZ) =
ak−1φ(a;T )c
|DT |
qtr(TZ).
By Theorem 1, there exists a positive integer b such that, for every positive integer m,
c(pbm) ≡ 0 (mod pj),
since k ≡ 2 or 3 (mod p−1
). Suppose that pb+2j ||DT |. If pj|a and a|fT , then
ak−1φ(a;T )c
|DT |
≡ 0 (mod pj).
If pj ∤ a and a|fT , then pb
∣∣∣ |DT |a2 and a
k−1φ(a;T )c
|DT |
≡ 0 (mod pj). �
Acknowledgement
We thank the referee for many helpful comments which have improved our exposition.
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School of Liberal Arts and Sciences, Korea Aerospace University, 200-1, Hwajeon-
dong, Goyang, Gyeonggi, 412-791, Korea
E-mail address : choija@postech.ac.kr
Department of Mathematics and Pohang Mathematical Institute, POSTECH, Pohang,
790–784, Korea
E-mail address : yjc@postech.ac.kr
	1. Introduction and Statement of Main Results
	2. Applications: More Congruences 
	2.1. p-adic Limits of Borcherds Exponents
	2.2. Sums of n-Squares
	2.3. Quotients of Eisenstein Series
	2.4. The Maass Space
	3. Linear Relation among Fourier Coefficients of modular forms of Half Integral Weight
	4. Proofs of Theorem ?? and ??
	4.1. Proof of Theorem ??
	4.2. Proof of Theorem ??
	5. Proof of Theorem ??
	6. Proofs of Theorem ?? and ??
	6.1. Proof of Theorem ??
	6.2. Proof of Theorem ??
	6.3. Proof of Theorem ??
	Acknowledgement
	References
