Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 149154, 9 pages
doi:10.1155/2012/149154
Research Article
On Level p Siegel Cusp Forms of Degree Two
Hirotaka Kodama, Shoyu Nagaoka, and Yoshitsugu Nakamura
Department of Mathematics, Kinki University, Higashiōsaka 577-8502, Japan
Correspondence should be addressed to Shoyu Nagaoka, nagaoka@math.kindai.ac.jp
Received 9 July 2012; Accepted 6 August 2012
Academic Editor: Aloys Krieg
Copyright q 2012 Hirotaka Kodama et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We give a simple formula for the Fourier coefficients of some degree-two Siegel cusp form with
level p.
1. Introduction
In the previous paper 1, the second and the third authors introduced a simple construction
of a Siegel cusp form of degree 2. This construction has an advantage because the Fourier
coefficients are explicitly computable. After this work was completed, Kikuta and Mizuno
proved that the p-adic limit of a sequence of the aforementioned cusp forms becomes a Siegel
cusp form of degree 2 with level p.
In this paper, we give an explicit description of the Fourier expansion of such a form.
This result shows that the cusp form becomes a nonzero cusp form of weight 2 on Γ20 p if
p > 7 and p ≡ 3 mod 4.
2. Siegel Modular Forms of Degree 2
We start by recalling the basic facts of Siegel modular forms.
The Siegel upper half-space of degree 2 is defined by
H2 : Z X iY ∈ Sym2 C | Y > 0 positive-definite .
2.1
Then the degree 2 Siegel modular group Γ2 : Sp2 Z acts on H2 discontinuously. For a
congruence subgroup Γ ⊂ Γ2 , we denote by Mk Γ resp., Sk Γ the corresponding space
of Siegel modular forms resp., cusp forms of weight k.
2
International Journal of Mathematics and Mathematical Sciences
We will be mainly concerned with the Siegel modular group Γ2 and the congruence
subgroup
AB
2.2
Γ20 N :
∈ Γ2 | C ≡ O mod N .
CD
In both cases, F ∈ Mk Γ has a Fourier expansion of the form
FZ aT ; F exp2πi trT Z,
2.3
0≤T ∈Λ2
where
Λ2 : T tij ∈ Sym2 Q | t11 , t22 , 2t12 ∈ Z ,
2.4
and aT ; F is the Fourier coefficient of F at T .
3. Siegel Cusp Form of Degree 2
In the previous paper 1, we constructed a cusp form fk ∈ Sk Γ2 whose Fourier coefficients
are explicitly computable. We review the result.
First, we recall the definition of Cohen’s function. Cohen defined an arithmetical
function Hr, N r, N ∈ Z≥0 in 2. In the case that r and N satisfy −1r N D · f 2 where
D is a fundamental discriminant and f ∈ N, the function is given by
f
r−1
.
μdχD dd σ2r−1
Hr, N L 1 − r, χD
3.1
d
0<d|f
Here, Ls, χ is the Dirichlet L-function with character χ, and μ is the Möbius function. For
the precise definition of Hr, N, see 2, page 272.
Secondly, we introduce Krieg’s function Gs, N s, N ∈ Z≥0 associated with the
Gaussian field Qi. Let χ−4 be the Kronecker character associated with Qi. Krieg’s function
Gs, N GQi s, N over Qi is defined by
Gs, N :
⎧
1
⎪
⎪
σ
N − σs,χ−4 N ,
⎪
⎪
⎨ 1 χ−4 N s,χ−4
if N > 0,
⎪
⎪
Bs1,χ−4
⎪
⎪
,
⎩−
2s 1
if N 0,
3.2
where Bm,χ is the generalized Bernoulli number with character χ,
σs,χ−4 N :
0<d|N
χ−4 dds ,
σs,χ−4 N :
0<d|N
χ−4
N
ds .
d
3.3
This function was introduced by Krieg 3 to describe the Fourier coefficients of Hermitian
Eisenstein series of degree 2.
The following theorem is one of the main results in 1.
International Journal of Mathematics and Mathematical Sciences
3
Theorem 3.1. There exists a Siegel cusp form fk ∈ Sk Γ2 whose Fourier coefficients aT ; fk are
given as follows:
a T ; fk dk−1 αk
0<d|εT 4 detT ,
d2
3.4
where
αk N : Hk − 1, N −
B2k−2 G k − 2, N − s2 ,
Bk−1,χ−4 s∈Z
s2 ≤N
εT : max l ∈ N | l−1 T ∈ Λ2 .
3.5
Here, Bm is the mth Bernoulli number.
Remark 3.2. The above result shows that the cusp form fk is a form in the Maass space cf.
1.
4. p-Adic Siegel Modular Forms
The cusp form fk introduced in Theorem 3.1 was constructed by the difference between the
Siegel Eisenstein series Ek and the restriction of the Hermitian Eisenstein series Ek,Qi :
fk ck · Ek − Ek,Qi |H2 ,
4.1
for some ck ∈ Q. The p-adic properties of the Eisenstein series Ek and Ek,Qi are studied by
the second author cf. 4, 5. After our work 1 was completed, Kikuta and Mizuno studied
p-adic properties of our form fk . The following statement is a special case in 6.
Theorem 4.1. Let p be a prime number satisfying p ≡ 3 mod 4, and {km } is the sequence defined
by
km km p : 2 p − 1 pm−1 .
4.2
fp∗ : lim fkm ,
4.3
Then there exists the p-adic limit
m→∞
and fp∗ represents a cusp form of weight 2 with level p, that is,
fp∗ ∈ S2 Γ20 p .
4.4
Remark 4.2. 1 The p-adic convergence of modular forms is interpreted as the convergence
of the Fourier coefficients.
4
International Journal of Mathematics and Mathematical Sciences
2 Kikuta and Mizuno studied a similar problem under more general situation. They
noted that if we take the sequence {km } with km k p − 1pm−1 , k ∈ N k > 4, then
limm → ∞ fkm is no longer a cusp form 6, Theorem 1.7.
3 The cuspidality of fp∗ essentially results from the fact that there are no nontrivial
modular forms of weight 2 on the full modular group Γ2 .
5. Main Result
In this section, we give an explicit formula for the Fourier coefficients of fp∗ .
To describe aT ; fp∗ , we will introduce two functions Hp∗ and G∗p .
First, for N ∈ N with N ≡ 0 or 3 mod 4, we write N as N −D · f 2 where D is a
fundamental discriminant and f ∈ N. Then, we define
f
,
μdχD dσ1∗
Hp∗ N : − 1 − χD p B1,χD
d
0<d|f
d,p1
5.1
where
σ1∗ m d.
0<d|m
d,p1
5.2
Secondly, for N ∈ Z≥0 , we define
⎧
1 − −1ordp N ∗
⎪
⎪
⎪
⎨ 1 χ N σ0,χ−4 N, if N > 0,
−4
∗
Gp N :
⎪
⎪
⎪
⎩1,
if N 0,
2
where
∗
σ0,χ
N −4
χ−4 d.
0<d|N
d,p1
5.3
5.4
Remark 5.1. From the definition, the following holds:
G∗p N 0 if p N.
5.5
The main theorem of this paper can be stated as follows.
Theorem 5.2. Let p be a prime number satisfying p ≡ 3 mod 4. Then the Fourier coefficients
aT ; fp∗ of fp∗ ∈ S2 Γ20 p are given by
4 detT dα∗p
,
a T ; fp∗ 5.6
d2
0<d|εT d,p1
International Journal of Mathematics and Mathematical Sciences
5
where
α∗p N : Hp∗ N −
p − 1 ∗
Gp N − s2 .
6 s∈Z
5.7
s2 ≤N
Here, Hp∗ and G∗p are the functions defined in 5.1 and 5.3, respectively.
From Theorems 3.1 and 4.1, the proof of Theorem 5.2 is reduced to show that
lim αkm N α∗p N.
5.8
m→∞
We proceed the proof of 5.8 step by step.
Lemma 5.3. Consider the following:
lim Hkm − 1, N Hp∗ N.
5.9
m→∞
Proof. Under the description N −D · f 2 , we can write Hkm − 1, N as
Hkm − 1, N −
Bkm −1,χD f
,
μdχD ddkm −2 σ2km −3
km − 1 0<d|f
d
5.10
cf. 3.1.
Using Kummer’s congruence, we obtain
Bkm −1,χD 1 − χD p B1,χD .
m → ∞ km − 1
5.11
lim
On the other hand, we have
lim
m→∞
μdχD dd
km −2
0<d|f
f
∗ f
σ2km −3
μdχD dσ1
,
d
d
0<d|f
d,p1
5.12
because
lim d
m→∞
lim σ2km −3 l lim
m→∞
This proves 5.9.
m→∞
0<d|l
km −2
1,
0,
d12p−1p
m−1
if p d,
if p | d,
d σ1∗ l,
0<d|l
d,p1
l ∈ N.
5.13
6
International Journal of Mathematics and Mathematical Sciences
Lemma 5.4. Consider the following:
p−1 B2km −2 G km − 2, N − s2 G∗p N − s2 .
m → ∞ Bkm −1,χ−4
6 s∈Z
s∈Z
lim
5.14
s2 ≤N
s2 ≤N
Proof. First, we calculate the factor of Bernoulli numbers. Again by Kummer’s congruence,
we obtain
lim
B2km −2
m → ∞ Bkm −1,χ−4
2 lim
B2km −2
km − 1
·
− 2 Bkm −1,χ−4
m → ∞ 2km
B2
1
·
2· 1−p ·
2
1 − χ−4 p B1,χ−4
p−1
.
6
Here, we used the facts that χ−4 p −1 and B1,χ−4 −1/2.
Next we calculate
lim
G km − 2, N − s2 .
5.15
m→∞
5.16
1
σkm −2,χ−4 N − σkm −2,χ−4 N ,
1 χ−4 N 5.17
s∈Z
s2 ≤N
If N : N − s2 > 0, then
G km − 2, N cf. 3.2. Therefore, we need to calculate
lim σkm −2,χ−4 N ,
m→∞
lim σkm −2,χ−4 N .
5.18
m→∞
We have
lim σkm −2,χ−4 N lim
m→∞
m→∞
χ−4 ddp−1p
0<d|N m−1
χ−4 d.
0<d|N d,p1
5.19
To calculate limm → ∞ σkm −2,χ−4 N , we write N as N pe · N , p, N 1, namely, e ordp N . Then we have
N
m−1
χ−4
lim σk −2,χ−4 N lim
dp−1p
m→∞ m
m→∞
d
0<d|N χ−4 pe · N 5.20
0<d|N e χ−4 d.
χ−4 p
0<d|N d,p1
International Journal of Mathematics and Mathematical Sciences
7
Combining these formulas, we obtain
ordp N lim σkm −2,χ−4 N − σkm −2,χ−4 N 1 − χ−4 p
χ−4 d
m→∞
0<d|N d,p1
5.21
∗
1 − −1ordp N σ0,χ
N .
−4
If N N − s2 0, then
Gkm − 2, 0 −
Bkm −1,χ−4
.
2km − 1
5.22
Thus, we get
B1,χ−4
1
lim Gkm − 2, 0 − 1 − χ−4 p
.
2
2
m→∞
5.23
Consequently,
lim
m→∞
G km − 2, N − s2 G∗p N − s2 .
s∈Z
s2 ≤N
s∈Z
s2 ≤N
5.24
The identity 5.14 immediately follows due to these formulas.
The proof of Theorem 5.2 is completed by combining Lemmas 5.3 and 5.4.
An advantage of the formula 5.6 is that we can prove the nonvanishing property for
the cusp form fp∗ for p > 7.
Corollary 5.5. Assume that p ≡ 3 mod 4. If p > 7, then fp∗ does not vanish identically.
Proof. We calculate the Fourier coefficient aT ; fp∗ at T 1 0
01
. From the theorem, we have
1 0
a
; fp∗ α∗p 4
0 1
Hp∗ 4
p − 1 ∗
Gp 4 2G∗p 3 2G∗p 0 .
−
6
5.25
The assumption p ≡ 3 mod 4 implies that
Hp∗ 4 − 1 − χ−4 p B1,χ−4 1.
5.26
On the other hand, G∗p 3 G∗p 4 0 because p 3, 4 and G∗p 0 1/2. Hence,
p−1
p − 1 ∗
Gp 4 2G∗p 3 2G∗p 0 .
6
6
5.27
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International Journal of Mathematics and Mathematical Sciences
Consequently, we obtain
p−1 7−p
1 0
<0
a
; fp∗ α∗p 4 1 −
0 1
6
6
5.28
if p > 7.
Remark 5.6. We have f3∗ f7∗ 0. These identities are consistent with the fact that
dim S2 Γ20 3 dim S2 Γ20 7 0 see 7.
6. Numerical Examples
In this section, we present numerical examples concerning our Siegel cusp forms. To begin
with, we recall the theta series associated with quadratic forms.
Let S S2m be a half-integral, positive-definite symmetric matrix of rank 2m.
We associate the theta series
ϑS, Z exp 2πi tr t XSXZ ,
Z ∈ H2 .
6.1
X∈M2m,2 Z
If we take a symmetric S S2m > 0 with level p, then
ϑS, Z ∈ Mm Γ20 p .
6.2
In some cases, we can construct cusp forms by taking a linear combination of theta series.
The Case p 11. Set
⎛
11
Q1
⎜1
⎜
⎜
⎜0
⎜
⎜
⎜
⎜1
⎜
⎜2
⎜
⎝
0
⎞
1
0
0⎟
2 ⎟
1⎟
1 0 ⎟
⎟
2 ⎟,
⎟
⎟
0 3 0⎟
⎟
⎟
⎠
1
0 3
2
⎞
1 1 1
⎜1 2 2 2⎟
⎟
⎜
⎜1
1⎟
⎟
⎜
⎜ 1 0 ⎟
⎟
⎜2
2
⎟,
⎜
⎟
⎜1
⎜ 0 4 2⎟
⎟
⎜2
⎟
⎜
⎟
⎜1 1
⎝
2 4⎠
2 2
⎛
11
Q2
⎛
11
Q3
⎜2
⎜
⎜
⎜
⎜1
⎜
⎜
⎜1
⎜
⎜2
⎜
⎜1
⎝
2
⎞
1 1
1
2 2⎟
⎟
1⎟
⎟
2 0 ⎟
2⎟
⎟,
⎟
0 2 1⎟
⎟
⎟
⎟
1
1 2⎠
2
6.3
11
and ϑi ϑQi , Z. It is known that dim S2 Γ20 11 1 cf. 7. We can take a nonzero
element of S2 Γ20 11 as
C2 11 3ϑ1 − 2ϑ2 − ϑ3
Yoshida’s cusp form cf. 8.
∗
Table 1 gives a first few examples for the Fourier coefficient of f11
and C2 11.
6.4
International Journal of Mathematics and Mathematical Sciences
⎛
⎝ 1
1/2
T
∗
aT ; f11
aT ; C2 11
T
∗
aT ; f11
aT ; C2 11
⎞
1/2⎠
⎛
⎞
1
0
⎠
⎝
0 1
1
Table 1
⎛
9
⎞
2
1/2
⎝
⎠
1/2 1
⎛
⎞
2
0
⎝
⎠
0 1
⎛
⎞
3
1/2
⎝
⎠
1/2 1
0
0
−2/3
0
⎞
⎛
2
0
⎠
⎝
24
⎞
⎛
4
0
⎠
⎝
0 2
0 1
2/3
−2/3
−24
⎞
⎛
3
0
⎠
⎝
24
⎞
⎛
2
1
⎠
⎝
0 1
1 2
−2/3
2/3
2/3
0
4/3
24
−24
−24
0
−48
0
⎛
⎝ 2
⎞
1/2⎠
1/2
2
Table 2
N
α∗11 N
N
α∗11 N
N
α∗11 N
N
α∗11 N
3
2/3
32
0
63
0
92
2
4
−2/3
35
0
64
−4/3
95
0
7
0
36
0
67
2
96
0
8
0
39
0
68
0
99
4/3
11
−2/3
40
0
71
2/3
100
0
12
−2/3
43
0
72
0
15
2/3
44
2/3
75
4/3
16
4/3
47
0
76
0
19
0
48
0
79
0
20
2/3
51
0
80
−4/3
23
−2/3
52
0
83
0
24
0
55
2/3
84
0
27
−2/3
56
4/3
87
0
28
0
59
−2/3
88
0
31
−2/3
60
−2
91
−8/3
∗
The relation between f11
and C2 11 is
∗
−
f11
1
C2 11.
36
6.5
∗
can be obtained from Table 2.
Further examples of the Fourier coefficients of f11
References
1 S. Nagaoka and Y. Nakamura, “On the restriction of the Hermitian Eisenstein series and its applications,” Proceedings of the American Mathematical Society, vol. 139, no. 4, pp. 1291–1298, 2011.
2 H. Cohen, “Sums involving the values at negative integers of L-functions of quadratic characters,”
Mathematische Annalen, vol. 217, no. 3, pp. 271–285, 1975.
3 A. Krieg, “The Maass spaces on the Hermitian half-space of degree 2,” Mathematische Annalen, vol. 289,
no. 4, pp. 663–681, 1991.
4 S. Nagaoka, “On p-adic Hermitian Eisenstein series,” Proceedings of the American Mathematical Society,
vol. 134, no. 9, pp. 2533–2540, 2006.
5 T. Kikuta and S. Nagaoka, “On a correspondence between p-adic Siegel-Eisenstein series and genus
theta series,” Acta Arithmetica, vol. 134, no. 2, pp. 111–126, 2008.
6 T. Kikuta and Y. Mizuno, “On p-adic Hermitian Eisenstein series and p-adic Siegel cusp forms,” Journal
of Number Theory, vol. 132, no. 9, pp. 1949–1961, 2012.
7 C. Poor and D. S. Yuen, “Dimensions of cusp forms for Γ0 p in degree two and small weights,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 77, pp. 59–80, 2007.
8 H. Yoshida, “Siegel’s modular forms and the arithmetic of quadratic forms,” Inventiones Mathematicae,
vol. 60, no. 3, pp. 193–248, 1980.
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