# On Levelp Siegel Cusp Forms of Degree Two

Authors: Hirotaka Kodama; Shoyu Nagaoka; Yoshitsugu Nakamura
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

We give a simple formula for the Fourier coefficients of some degree-two Siegel cusp form with levelp.


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## Body

## 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 anonzero cusp form of weight 2 on Γ02(p) if p>7 and p≡3  (mod4).


## 2. Siegel Modular Forms of Degree 2

We start by recalling the basic facts of Siegel modular forms.TheSiegel upper half-space of degree 2 is defined by
(2.1)H2:={Z=X+iY∈Sym2(C)∣Y>0(positive-definite)}.
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.We will be mainly concerned with the Siegel modular groupΓ2 and the congruence subgroup
(2.2)Γ02(N):={(ABCD)∈Γ2∣C≡O(mod⁡N)}.
In both cases, F∈Mk(Γ′) has a Fourier expansion of the form
(2.3)F(Z)=∑0≤T∈Λ2‍a(T;F)exp⁡[2πitr⁡(TZ)],
where
(2.4)Λ2:={T=(tij)∈Sym2(Q)∣t11,t22,2t12∈Z},
and a(T;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 functionH(r,N) (r,N∈ℤ≥0) in [2]. In the case that r and N satisfy (-1)rN=D·f2 where D is a fundamental discriminant and f∈ℕ, the function is given by
(3.1)H(r,N)=L(1-r,χD)∑0<d|f‍μ(d)χD(d)dr-1σ2r-1(fd).
Here, L(s,χ) is the Dirichlet L-function with character χ, and μ is the Möbius function. For the precise definition of H(r,N), see [2, page 272].Secondly, we introduce Krieg’s functionG(s,N) (s,N∈ℤ≥0) associated with the Gaussian field ℚ(i). Let χ-4 be the Kronecker character associated with ℚ(i). Krieg’s function G(s,N)=Gℚ(i)(s,N) over ℚ(i) is defined by
(3.2)G(s,N):={11+|χ-4(N)|(σs,χ-4(N)-σ~s,χ-4(N)),ifN>0,-Bs+1,χ-42(s+1),ifN=0,
where Bm,χ is the generalized Bernoulli number with character χ,
(3.3)σs,χ-4(N):=∑0<d|N‍χ-4(d)ds,σ~s,χ-4(N):=∑0<d|N‍χ-4(Nd)ds.
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].Theorem 3.1.
There exists a Siegel cusp formfk∈Sk(Γ2) whose Fourier coefficients a(T;fk) are given as follows:
(3.4)a(T;fk)=∑0<d|ε(T)‍dk-1αk(4det⁡(T)d2),
where
(3.5)αk(N):=H(k-1,N)-B2k-2Bk-1,χ-4∑s∈Zs2≤NG(k-2,N-s2),ε(T):=max⁡{l∈N∣l-1T∈Λ2}.
Here, Bm is the mth Bernoulli number.Remark 3.2.
The above result shows that the cusp formfk is a form in the Maass space (cf. [1]).


## 4.p-Adic Siegel Modular Forms

The cusp formfk 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,ℚ(i):
(4.1)fk=ck⋅(Ek-Ek,Q(i)|H2),
for some ck∈ℚ. The p-adic properties of the Eisenstein series Ek and Ek,ℚ(i) 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.
Letp be a prime number satisfying p≡3(mod4), and {km} is the sequence defined by
(4.2)km=km(p):=2+(p-1)pm-1.
Then there exists the p-adic limit
(4.3)fp*:=lim⁡m→∞fkm,
and fp* represents a cusp form of weight 2 with level p, that is,
(4.4)fp*∈S2(Γ02(p)).Remark 4.2.
(1) Thep-adic convergence of modular forms is interpreted as the convergence of the Fourier coefficients.
(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-1)pm-1, k∈ℕ (k>4), then limm→∞fkm is no longer a cusp form [6, Theorem 1.7].
(3) The cuspidality offp* 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 offp*.To describea(T;fp*), we will introduce two functions Hp* and Gp*.First, forN∈ℕ with N≡0or3(mod4), we write N as N=-D·f2 where D is a fundamental discriminant and f∈ℕ. Then, we define
(5.1)Hp*(N):=-(1-χD(p))B1,χD∑0<d|f(d,p)=1μ(d)χD(d)σ1*(fd),
where
(5.2)σ1*(m)=∑0<d|m(d,p)=1d.
Secondly, for N∈ℤ≥0, we define
(5.3)Gp*(N):={1-(-1)ordp(N)1+|χ-4(N)|σ0,χ-4*(N),ifN>0,12,ifN=0,
where
(5.4)σ0,χ-4*(N)=∑0<d|N(d,p)=1χ-4(d).Remark 5.1.
From the definition, the following holds:(5.5)Gp*(N)=0ifp∤N.The main theorem of this paper can be stated as follows.Theorem 5.2.
Letp be a prime number satisfying p≡3(mod4). Then the Fourier coefficients a(T;fp*) of fp*∈S2(Γ02(p)) are given by
(5.6)a(T;fp*)=∑0<d|ε(T)(d,p)=1‍dαp*(4det⁡(T)d2),
where
(5.7)αp*(N):=Hp*(N)-p-16∑s∈Zs2≤NGp*(N-s2).
Here, Hp* and Gp* are the functions defined in (5.1) and (5.3), respectively.From Theorems3.1 and 4.1, the proof of Theorem 5.2 is reduced to show that
(5.8)lim⁡m→∞αkm(N)=αp*(N).We proceed the proof of (5.8) step by step.Lemma 5.3.
Consider the following:(5.9)lim⁡m→∞H(km-1,N)=Hp*(N).Proof.
Under the descriptionN=-D·f2, we can write H(km-1,N) as
(5.10)H(km-1,N)=-Bkm-1,χDkm-1∑0<d|f‍μ(d)χD(d)dkm-2σ2km-3(fd),
(cf. (3.1)).
Using Kummer’s congruence, we obtain(5.11)lim⁡m→∞Bkm-1,χDkm-1=(1-χD(p))B1,χD.
On the other hand, we have
(5.12)lim⁡m→∞∑0<d|f‍μ(d)χD(d)dkm-2σ2km-3(fd)=∑0<d|f(d,p)=1μ(d)χD(d)σ1*(fd),
because
(5.13)lim⁡m→∞dkm-2={1,ifp∤d,0,ifp∣d,lim⁡m→∞σ2km-3(l)=lim⁡m→∞∑0<d|l‍d1+2(p-1)pm-1=∑0<d|l(d,p)=1d=σ1*(l),(l∈N).
This proves (5.9).Lemma 5.4.
Consider the following:(5.14)lim⁡m→∞B2km-2Bkm-1,χ-4∑s∈Zs2≤N‍G(km-2,N-s2)=p-16∑s∈Zs2≤NGp*(N-s2).Proof.
First, we calculate the factor of Bernoulli numbers. Again by Kummer’s congruence, we obtain(5.15)lim⁡m→∞B2km-2Bkm-1,χ-4=2lim⁡m→∞B2km-22km-2⋅km-1Bkm-1,χ-4=2⋅(1-p)⋅B22⋅1(1-χ-4(p))B1,χ-4=p-16.
Here, we used the facts that χ-4(p)=-1 and B1,χ-4=-1/2.
Next we calculate(5.16)lim⁡m→∞∑s∈Zs2≤NG(km-2,N-s2).
If N′:=N-s2>0, then
(5.17)G(km-2,N′)=11+|χ-4(N′)|(σkm-2,χ-4(N′)-σ~km-2,χ-4(N′)),
(cf. (3.2)). Therefore, we need to calculate
(5.18)lim⁡m→∞σkm-2,χ-4(N′),lim⁡m→∞σ~km-2,χ-4(N′).
We have
(5.19)lim⁡m→∞σkm-2,χ-4(N′)=lim⁡m→∞∑0<d|N′‍χ-4(d)d(p-1)pm-1=∑0<d|N′(d,p)=1χ-4(d).
To calculate limm→∞σ~km-2,χ-4(N′), we write N′ as N′=pe·N′′, (p,N′′)=1, namely, e=ordp(N′). Then we have
(5.20)lim⁡m→∞σ~km-2,χ-4(N′)=lim⁡m→∞∑0<d|N′‍χ-4(N′d)d(p-1)pm-1=∑0<d|N′′χ-4(pe⋅N′′)=(χ-4(p))e∑0<d|N′(d,p)=1∑‍χ-4(d).
Combining these formulas, we obtain
(5.21)lim⁡m→∞(σkm-2,χ-4(N′)-σ~km-2,χ-4(N′))=(1-(χ-4(p))ordp(N′))∑0<d|N′(d,p)=1χ-4(d)=(1-(-1)ordp(N′))σ0,χ-4*(N′).
If N′=N-s2=0, then
(5.22)G(km-2,0)=-Bkm-1,χ-42(km-1).
Thus, we get
(5.23)lim⁡m→∞G(km-2,0)=-(1-χ-4(p))B1,χ-42=12.
Consequently,
(5.24)lim⁡m→∞∑s∈Zs2≤NG(km-2,N-s2)=∑s∈Zs2≤NGp*(N-s2).
The identity (5.14) immediately follows due to these formulas.The proof of Theorem5.2 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 thatp≡3(mod4). If p>7, then fp* does not vanish identically.Proof.
We calculate the Fourier coefficienta(T;fp*) at T=(1001). From the theorem, we have
(5.25)a((1001);fp*)=αp*(4)=Hp*(4)-p-16(Gp*(4)+2Gp*(3)+2Gp*(0)).
The assumption p≡3(mod4) implies that
(5.26)Hp*(4)=-(1-χ-4(p))B1,χ-4=1.
On the other hand, Gp*(3)=Gp*(4)=0 (because p∤3,4) and Gp*(0)=1/2. Hence,
(5.27)p-16(Gp*(4)+2Gp*(3)+2Gp*(0))=p-16.
Consequently, we obtain
(5.28)a((1001);fp*)=αp*(4)=1-p-16=7-p6<0
if p>7.Remark 5.6.
We havef3*=f7*=0. These identities are consistent with the fact that dimS2(Γ02(3))=dimS2(Γ02(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.LetS=S(2m) be a half-integral, positive-definite symmetric matrix of rank 2m.We associate the theta series(6.1)ϑ(S,Z)=∑X∈M2m,2(Z)‍exp⁡[2πitr⁡(XtSXZ)],Z∈H2.
If we take a symmetric S=S(2m)>0 with level p, then
(6.2)ϑ(S,Z)∈Mm(Γ02(p)).
In some cases, we can construct cusp forms by taking a linear combination of theta series.The Case  p=11. Set
(6.3)Q1(11)=(10120010121203001203),Q2(11)=(112121212101212042121224),Q3(11)=(2112121201212021121212),
and ϑi=ϑ(Qi(11),Z). It is known that dimS2(Γ02(11))=1 (cf. [7]). We can take a nonzero element of S2(Γ02(11)) as
(6.4)C2(11)=3ϑ1-2ϑ2-ϑ3
(Yoshida’s cusp form cf. [8]).Table1 gives a first few examples for the Fourier coefficient of f11* and C2(11).Table 1
T
(
1
1
/
2
1
/
2
1
)
(
1
0
0
1
)
(
2
1
/
2
1
/
2
1
)
(
2
0
0
1
)
(
3
1
/
2
1
/
2
1
)
a
(
T
;
f
11
*
)
2
/
3
-
2
/
3
0
0
-
2
/
3
a
(
T
;
C
2
(
11
)
)
-
24
24
0
0
24
T
(
3
0
0
1
)
(
2
1
1
2
)
(
2
1
/
2
1
/
2
2
)
(
2
0
0
2
)
(
4
0
0
1
)
a
(
T
;
f
11
*
)
-
2
/
3
2
/
3
2
/
3
0
4
/
3
a
(
T
;
C
2
(
11
)
)
24
-
24
-
24
0
-
48The relation betweenf11* and C2(11) is
(6.5)f11*=-136C2(11).
Further examples of the Fourier coefficients of f11* can be obtained from Table 2.Table 2
N
3
4
7
8
11
12
15
16
19
20
23
24
27
28
31
α
11
*
(
N
)
2
/
3
-
2
/
3
0
0
-
2
/
3
-
2
/
3
2
/
3
4
/
3
0
2
/
3
-
2
/
3
0
-
2
/
3
0
-
2
/
3
N
32
35
36
39
40
43
44
47
48
51
52
55
56
59
60
α
11
*
(
N
)
0
0
0
0
0
0
2
/
3
0
0
0
0
2
/
3
4
/
3
-
2
/
3
-
2
N
63
64
67
68
71
72
75
76
79
80
83
84
87
88
91
α
11
*
(
N
)
0
-
4
/
3
2
0
2
/
3
0
4
/
3
0
0
-
4
/
3
0
0
0
0
-
8
/
3
N
92
95
96
99
100
α
11
*
(
N
)
2
0
0
4
/
3
0



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