International Journal of Mathematics and Mathematical Sciences

Volume 2012 (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

Received 9 July 2012; Accepted 6 August 2012

Academic Editor: Aloys Krieg

Copyright © 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.

Abstract.

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 -adic limit of a sequence of the aforementioned cusp forms becomes a Siegel cusp form of degree 2 with level .

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 if and   ().

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

Then the degree 2 Siegel modular group acts on discontinuously. For a congruence subgroup , we denote by (resp., ) the corresponding space of Siegel modular forms (resp., cusp forms) of weight .

We will be mainly concerned with the Siegel modular group and the congruence subgroup

In both cases, has a Fourier expansion of the form

where

and is the Fourier coefficient of at .

3. Siegel Cusp Form of Degree 2

In the previous paper [1], we constructed a cusp form whose Fourier coefficients are explicitly computable. We review the result.

First, we recall the definition of Cohen’s function. Cohen defined an arithmetical function () in [2]. In the case that and satisfy where is a fundamental discriminant and , the function is given by

Here, is the Dirichlet -function with character , and is the Möbius function. For the precise definition of , see [2, page 272].

Secondly, we introduce Krieg’s function () associated with the Gaussian field . Let be the Kronecker character associated with . Krieg’s function over is defined by

where is the generalized Bernoulli number with character ,

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 form whose Fourier coefficients are given as follows:

where

Here, is the th Bernoulli number.

Remark 3.2. The above result shows that the cusp form is a form in the Maass space (cf. [1]).

4. -Adic Siegel Modular Forms

The cusp form introduced in Theorem 3.1 was constructed by the difference between the Siegel Eisenstein series and the restriction of the Hermitian Eisenstein series :

for some . The -adic properties of the Eisenstein series and are studied by the second author (cf. [4, 5]). After our work [1] was completed, Kikuta and Mizuno studied -adic properties of our form . The following statement is a special case in [6].

Theorem 4.1. Let be a prime number satisfying , and is the sequence defined by

Then there exists the -adic limit

and represents a cusp form of weight 2 with level , that is,

Remark 4.2. (1) The -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 with , (), then is no longer a cusp form [6, Theorem 1.7].

(3) The cuspidality of essentially results from the fact that there are no nontrivial modular forms of weight 2 on the full modular group .

5. Main Result

In this section, we give an explicit formula for the Fourier coefficients of .

To describe , we will introduce two functions and .

First, for with , we write as where is a fundamental discriminant and . Then, we define

where

Secondly, for , we define

where

Remark 5.1. From the definition, the following holds:

The main theorem of this paper can be stated as follows.

Theorem 5.2. Let be a prime number satisfying . Then the Fourier coefficients of are given by

where

Here, and 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

We proceed the proof of (5.8) step by step.

Lemma 5.3. Consider the following:

Proof. Under the description , we can write as

(cf. (3.1)).

Using Kummer’s congruence, we obtain

On the other hand, we have

because

This proves (5.9).

Lemma 5.4. Consider the following:

Proof. First, we calculate the factor of Bernoulli numbers. Again by Kummer’s congruence, we obtain

Here, we used the facts that and .

Next we calculate

If , then

(cf. (3.2)). Therefore, we need to calculate

We have

To calculate , we write as , , namely, . Then we have

Combining these formulas, we obtain

If , then

Thus, we get

Consequently,

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 for .

Corollary 5.5. Assume that . If , then does not vanish identically.

Proof. We calculate the Fourier coefficient at . From the theorem, we have

The assumption implies that

On the other hand, (because ) and . Hence,

Consequently, we obtain

if .

Remark 5.6. We have . These identities are consistent with the fact that (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 be a half-integral, positive-definite symmetric matrix of rank .

We associate the theta series

If we take a symmetric with level , then

In some cases, we can construct cusp forms by taking a linear combination of theta series.

The Case  . Set

and . It is known that (cf. [7]). We can take a nonzero element of as

(Yoshida’s cusp form cf. [8]).

Table 1 gives a first few examples for the Fourier coefficient of and .

Table 1

The relation between and is

Further examples of the Fourier coefficients of can be obtained from Table 2.

Table 2

3
4
7
8
11
12
15
16
19
20
23
24
27
28
31

0
0

0

0

0

32
35
36
39
40
43
44
47
48
51
52
55
56
59
60

0
0
0
0
0
0

0
0
0
0

63
64
67
68
71
72
75
76
79
80
83
84
87
88
91

0

2
0

0

0
0

0
0
0
0

92
95
96
99
100

2
0
0

0

References

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