On Level p Siegel Cusp Forms of Degree Two

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 Γ0 p if p > 7 and p ≡ 3 mod 4 .


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 Γ 2 0 p if p > 7 and p ≡ 3 mod 4 .

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 Γ 2 : Sp 2 Z acts on H 2 discontinuously.For a congruence subgroup Γ ⊂ Γ 2 , we denote by M k Γ resp., S k Γ the corresponding space of Siegel modular forms resp., cusp forms of weight k.

International Journal of Mathematics and Mathematical Sciences
We will be mainly concerned with the Siegel modular group Γ 2 and the congruence subgroup In both cases, F ∈ M k Γ has a Fourier expansion of the form where and a T ; F is the Fourier coefficient of F at T .

Siegel Cusp Form of Degree 2
In the previous paper 1 , we constructed a cusp form f k ∈ S k Γ 2 whose Fourier coefficients are explicitly computable.We review the result.First, we recall the definition of Cohen's function.Cohen defined an arithmetical function H r, N r, N ∈ Z ≥0 in 2 .In the case that r and N satisfy −1 r N D • f 2 where D is a fundamental discriminant and f ∈ N, the function is given by where

3.5
Here, B m is the mth Bernoulli number.
Remark 3.2.The above result shows that the cusp form f k is a form in the Maass space cf. 1 .

p-Adic Siegel Modular Forms
The cusp form f k introduced in Theorem 3.1 was constructed by the difference between the Siegel Eisenstein series E k and the restriction of the Hermitian Eisenstein series E k,Q i : for some c k ∈ Q.The p-adic properties of the Eisenstein series E k and E k,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 f k .The following statement is a special case in 6 .
Theorem 4.1.Let p be a prime number satisfying p ≡ 3 mod 4 , and {k m } is the sequence defined by

4.2
Then there exists the p-adic limit 3 The cuspidality of f * p essentially results from the fact that there are no nontrivial modular forms of weight 2 on the full modular group Γ 2 .

Main Result
In this section, we give an explicit formula for the Fourier coefficients of f * p .To describe a T ; f * p , we will introduce two functions H * p 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 where Secondly, for N ∈ Z ≥0 , we define where

5.4
Remark 5.1.From the definition, the following holds: The main theorem of this paper can be stated as follows.
We proceed the proof of 5.8 step by step.

5.11
On the other hand, we have

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Lemma 5.4.Consider the following:

5.14
Proof.First, we calculate the factor of Bernoulli numbers.Again by Kummer's congruence, we obtain

5.15
Here, we used the facts that χ −4 p −1 and

5.16
If

5.18
We have lim

5.20
International Journal of Mathematics and Mathematical Sciences 7 Combining these formulas, we obtain lim

5.21
If N N − s 2 0, then Thus, we get

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 f * p for p > 7.
Proof.We calculate the Fourier coefficient a T ; f * p at T 1 0 0 1 .From the theorem, we have

5.25
The assumption p ≡ 3 mod 4 implies that

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 S 2m be a half-integral, positive-definite symmetric matrix of rank 2m.We associate the theta series If we take a symmetric S S 2m > 0 with level p, then ϑ S, Z ∈ M m Γ 2 0 p .

6.2
In some cases, we can construct cusp forms by taking a linear combination of theta series.
The Case p 11. Set

≥0 associated with the Gaussian field Q i . Let χ −4 be the Kronecker character associated with Q
There exists a Siegel cusp form f k ∈ S k Γ 2 whose Fourier coefficients a T ; f k are given as follows: Kikuta and Mizuno studied a similar problem under more general situation.They noted that if we take the sequence {k m } with k m Remark 4.2. 1 The p-adic convergence of modular forms is interpreted as the convergence of the Fourier coefficients.2

Table 2 N 3
Further examples of the Fourier coefficients of f *11 can be obtained from Table2.