On Level p Siegel Cusp Forms of Degree Two

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 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  (mod4).

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 𝑀_𝑘(Γ^) (resp., 𝑆_𝑘(Γ^)) the corresponding space of Siegel modular forms (resp., cusp forms) of weight 𝑘.

We will be mainly concerned with the Siegel modular group Γ² and the congruence subgroup Γ²⁰(𝑁)∶=⎧⎪⎨⎪⎩⎛⎜⎜⎝𝐶𝐴𝐷𝐵⎞⎟⎟⎠∈Γ²∣𝐶≡𝑂(mod𝑁)⎫⎪⎬⎪⎭.(2.2) In both cases, 𝐹∈𝑀_𝑘(Γ^) has a Fourier expansion of the form 𝐹(𝑍)=_{0≤𝑇∈Λ2}𝑎(𝑇;𝐹)exp[2𝜋𝑖tr(𝑇𝑍)],(2.3) where Λ₂∶=𝑇=𝑡_𝑖𝑗∈Sym₂(ℚ)∣𝑡₁₁,𝑡₂₂,2𝑡₁₂∈ℤ,(2.4) 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 𝑓_𝑘∈𝑆_𝑘(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 𝐻(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 𝐻(𝑟,𝑁)=𝐿1−𝑟,𝜒_𝐷_0<𝑑|𝑓𝜇(𝑑)𝜒_𝐷(𝑑)𝑑^𝑟−1𝜎_2𝑟−1𝑓𝑑.(3.1) 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 function 𝐺(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)=𝐺_ℚ(𝑖)(𝑠,𝑁) over ℚ(𝑖) is defined by 𝐺(𝑠,𝑁)∶=⎧⎪⎨⎪⎩11+||𝜒₋₄(𝑁)||𝜎^{𝑠,𝜒−4}−𝐵(𝑁)−𝜎_{𝑠,𝜒−4}(𝑁),if𝑁>0,_{𝑠+1,𝜒−4}2(𝑠+1),if𝑁=0,(3.2) where 𝐵_𝑚,𝜒 is the generalized Bernoulli number with character 𝜒, 𝜎_{𝑠,𝜒−4}(𝑁)∶=_0<𝑑|𝑁𝜒₋₄(𝑑)𝑑^𝑠,𝜎_{𝑠,𝜒−4}(𝑁)∶=_0<𝑑|𝑁𝜒₋₄𝑁𝑑𝑑^𝑠.(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].

Theorem 3.1. There exists a Siegel cusp form 𝑓_𝑘∈𝑆_𝑘(Γ²) whose Fourier coefficients 𝑎(𝑇;𝑓_𝑘) are given as follows: 𝑎𝑇;𝑓_𝑘=_{0<𝑑|𝜀(𝑇)}𝑑^𝑘−1𝛼_𝑘4de𝑑t²(𝑇),(3.4) where 𝛼_𝑘(𝑁)∶=𝐻(𝑘−1,𝑁)−𝐵^2𝑘−2𝐵_{𝑘−1,𝜒−4}_{𝑠𝑠∈ℤ2≤𝑁}𝜀(𝑇)∶=max𝑙∈ℕ∣𝑙₋₁𝐺𝑘−2,𝑁−𝑠²,𝑇∈Λ₂.(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]).

4. 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)}: 𝑓_𝑘=𝑐_𝑘⋅𝐸_𝑘−𝐸_{𝑘,ℚ(𝑖)}||_ℍ2,(4.1) 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(mod4), and {𝑘_𝑚} is the sequence defined by 𝑘_𝑚=𝑘_𝑚(𝑝)∶=2+(𝑝−1)𝑝^𝑚−1.(4.2) Then there exists the 𝑝-adic limit 𝑓^∗𝑝∶=lim_𝑚→∞𝑓_𝑘𝑚,(4.3) and 𝑓^∗𝑝 represents a cusp form of weight 2 with level 𝑝, that is, 𝑓^∗𝑝∈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.
(2) Kikuta and Mizuno studied a similar problem under more general situation. They noted that if we take the sequence {𝑘_𝑚} with 𝑘_𝑚=𝑘+(𝑝−1)𝑝^𝑚−1, 𝑘∈ℕ (𝑘>4), then lim_𝑚→∞𝑓_𝑘𝑚 is no longer a cusp form [6, Theorem 1.7].
(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 Γ².

5. 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≡0or3(mod4), we write 𝑁 as 𝑁=−𝐷⋅𝑓² where 𝐷 is a fundamental discriminant and 𝑓∈ℕ. Then, we define 𝐻^∗𝑝(𝑁)∶=−1−𝜒_𝐷(𝑝)𝐵_{1,𝜒𝐷0<𝑑}_{|𝑓(𝑑,𝑝)=1}𝜇(𝑑)𝜒_𝐷(𝑑)𝜎^∗1𝑓𝑑,(5.1) where 𝜎^∗1(𝑚)=_{0<𝑑|𝑚(𝑑,𝑝)=1}𝑑.(5.2) Secondly, for 𝑁∈ℤ_≥0, we define 𝐺^∗𝑝(𝑁)∶=⎧⎪⎨⎪⎩11−+(||−𝜒1₋₄)(𝑁)||^{ord𝑝(𝑁)}𝜎^{∗0,𝜒−4}1(𝑁),if𝑁>0,2,if𝑁=0,(5.3) where 𝜎^∗0,𝜒₋₄(𝑁)=_{0<𝑑|𝑁(𝑑,𝑝)=1}𝜒₋₄(𝑑).(5.4)

Remark 5.1. From the definition, the following holds: 𝐺^∗𝑝(𝑁)=0if𝑝∤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(mod4). Then the Fourier coefficients 𝑎(𝑇;𝑓^∗𝑝) of 𝑓^∗𝑝∈𝑆₂(Γ²⁰(𝑝)) are given by 𝑎𝑇;𝑓^∗𝑝=_{0<𝑑|𝜀(𝑇)(𝑑,𝑝)=1}𝑑𝛼^∗𝑝4de𝑑t²(𝑇),(5.6) where 𝛼^∗𝑝(𝑁)∶=𝐻^∗𝑝(𝑁)−𝑝−16_{𝑠𝑠∈ℤ2≤𝑁}𝐺^∗𝑝𝑁−𝑠².(5.7) Here, 𝐻^∗𝑝 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_𝑚→∞𝛼_𝑘𝑚(𝑁)=𝛼^∗𝑝(𝑁).(5.8)

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

Lemma 5.3. Consider the following: lim_𝑚→∞𝐻𝑘_𝑚−1,𝑁=𝐻^∗𝑝(𝑁).(5.9)

Proof. Under the description 𝑁=−𝐷⋅𝑓², we can write 𝐻(𝑘_𝑚−1,𝑁) as 𝐻𝑘_𝑚−1,𝑁=−𝐵^{𝑘𝑚−1,𝜒𝐷}𝑘_𝑚−1_0<𝑑|𝑓𝜇(𝑑)𝜒_𝐷(𝑑)𝑑^𝑘𝑚−2𝜎_{2𝑘𝑚−3}𝑓𝑑,(5.10) (cf. (3.1)).
Using Kummer’s congruence, we obtain lim_𝑚→∞𝐵_{𝑘𝑚−1,𝜒𝐷}𝑘_𝑚−1=1−𝜒^𝐷(𝑝)𝐵_1,𝜒𝐷.(5.11) On the other hand, we have lim_𝑚→∞0_<𝑑|𝑓𝜇(𝑑)𝜒_𝐷(𝑑)𝑑^𝑘𝑚−2𝜎_{2𝑘𝑚−3}𝑓𝑑=_{0<𝑑|𝑓(𝑑,𝑝)=1}𝜇(𝑑)𝜒_𝐷(𝑑)𝜎^∗1𝑓𝑑,(5.12) because lim_𝑚→∞lim_𝑚→∞𝑑^𝑘𝑚−2=10,,iiff𝑝𝑝∤∣𝑑𝑑,,𝜎_{2𝑘𝑚−3}(𝑙)=lim_𝑚→∞0_<𝑑|𝑙𝑑^{1+2(𝑝−1)𝑝𝑚−1}=_{0<𝑑|𝑙(𝑑,𝑝)=1}𝑑=𝜎^∗1(𝑙),(𝑙∈ℕ).(5.13) This proves (5.9).

Lemma 5.4. Consider the following: lim_𝑚→∞𝐵_{2𝑘𝑚−2}𝐵_{𝑘𝑚−1,𝜒−4}_{𝑠𝑠∈ℤ2≤𝑁}𝐺𝑘_𝑚−2,𝑁−𝑠²=𝑝−16_{𝑠𝑠∈ℤ2≤𝑁}𝐺^∗𝑝𝑁−𝑠².(5.14)

Proof. First, we calculate the factor of Bernoulli numbers. Again by Kummer’s congruence, we obtain lim_𝑚→∞𝐵_{2𝑘𝑚−2}𝐵_{𝑘𝑚−1,𝜒−4}=2lim_𝑚→∞𝐵_{2𝑘𝑚−2}2𝑘_𝑚−2⋅𝑘^𝑚−1𝐵_{𝑘𝑚−1,𝜒−4}=2⋅(1−𝑝)⋅𝐵²2⋅1−𝜒₋₄1(𝑝)𝐵_1,𝜒−4=𝑝−16.(5.15) Here, we used the facts that 𝜒₋₄(𝑝)=−1 and 𝐵_1,𝜒−4=−1/2.
Next we calculate lim_{𝑚→∞𝑠𝑠}_{∈ℤ2≤𝑁}𝐺𝑘_𝑚−2,𝑁−𝑠².(5.16) If 𝑁′∶=𝑁−𝑠²>0, then 𝐺𝑘_𝑚−2,𝑁^=11+||𝜒₋₄𝑁^||𝜎^{𝑘𝑚−2,𝜒−4}𝑁^−𝜎_{𝑘𝑚−2,𝜒−4}𝑁^,(5.17) (cf. (3.2)). Therefore, we need to calculate lim_𝑚→∞𝜎_{𝑘𝑚−2,𝜒−4}𝑁^,lim_𝑚→∞𝜎_{𝑘𝑚−2,𝜒−4}𝑁^.(5.18) We have lim_𝑚→∞𝜎_{𝑘𝑚−2,𝜒−4}𝑁^=lim_𝑚→∞0<_𝑑|𝑁′𝜒₋₄(𝑑)𝑑^{(𝑝−1)𝑝𝑚−1}=_{0<𝑑|𝑁′(𝑑,𝑝)=1}𝜒₋₄(𝑑).(5.19) To calculate lim_𝑚→∞𝜎_{𝑘𝑚−2,𝜒−4}(𝑁^), we write 𝑁′ as 𝑁^=𝑝^𝑒⋅𝑁^, (𝑝,𝑁^)=1, namely, 𝑒=ord_𝑝(𝑁^). Then we have lim_𝑚→∞𝜎_{𝑘𝑚−2,𝜒−4}𝑁^=lim_𝑚→∞0<_𝑑|𝑁′𝜒₋₄𝑁𝑑^𝑑^{(𝑝−1)𝑝𝑚−1}==𝜒_{0<𝑑|𝑁′′−4}𝜒₋₄𝑝^𝑒⋅𝑁^(𝑝)^𝑒_0<𝑑|_{𝑁′(𝑑,𝑝)=1}𝜒⁻⁴(𝑑).(5.20) Combining these formulas, we obtain lim_𝑚→∞𝜎_{𝑘𝑚−2,𝜒−4}𝑁^−𝜎_{𝑘𝑚−2,𝜒−4}𝑁^==11−−(−𝜒1)₋₄(𝑝)^{ord𝑝(𝑁′)}_{ord𝑝(𝑁′)}_{0<𝑑|𝑁′(𝑑,𝑝)=1}𝜒₋₄(𝑑)𝜎^∗0,𝜒₋₄𝑁^.(5.21) If 𝑁′=𝑁−𝑠²=0, then 𝐺𝑘_𝑚−2,0=−𝐵^{𝑘𝑚−1,𝜒−4}2𝑘_𝑚−1.(5.22) Thus, we get lim_𝑚→∞𝐺𝑘_𝑚−2,0=−1−𝜒₋₄(𝑝)𝐵^1,𝜒−42=12.(5.23) Consequently, lim_{𝑚→∞𝑠𝑠}_{∈ℤ2≤𝑁}𝐺𝑘_𝑚−2,𝑁−𝑠²=_{𝑠𝑠∈ℤ2≤𝑁}𝐺^∗𝑝𝑁−𝑠².(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.

Corollary 5.5. Assume that 𝑝≡3(mod4). If 𝑝>7, then 𝑓^∗𝑝 does not vanish identically.

Proof. We calculate the Fourier coefficient 𝑎(𝑇;𝑓^∗𝑝) at 𝑇=¹⁰⁰¹. From the theorem, we have 𝑎⎛⎜⎜⎝⎛⎜⎜⎝1001⎞⎟⎟⎠;𝑓^∗𝑝⎞⎟⎟⎠=𝛼_∗𝑝=𝐻_∗𝑝(4)(4)−𝑝−16𝐺^∗𝑝(4)+2𝐺^∗𝑝(3)+2𝐺^∗𝑝(0).(5.25) The assumption 𝑝≡3(mod4) implies that 𝐻^∗𝑝(4)=−1−𝜒₋₄(𝑝)𝐵_1,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−16𝐺^∗𝑝(4)+2𝐺^∗𝑝(3)+2𝐺^∗𝑝(0)=𝑝−16.(5.27) Consequently, we obtain 𝑎⎛⎜⎜⎝⎛⎜⎜⎝1001⎞⎟⎟⎠;𝑓^∗𝑝⎞⎟⎟⎠=𝛼_∗𝑝(4)=1−𝑝−16=7−6𝑝<0(5.28) if 𝑝>7.

Remark 5.6. We have 𝑓^∗3=𝑓^∗7=0. These identities are consistent with the fact that dim𝑆₂(Γ²⁰(3))=dim𝑆₂(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=𝑆^(2𝑚) be a half-integral, positive-definite symmetric matrix of rank 2𝑚.

We associate the theta series 𝜗(𝑆,𝑍)=_{𝑋∈𝑀2𝑚,2(ℤ)}exp2𝜋𝑖tr^𝑡𝑋𝑆𝑋𝑍,𝑍∈ℍ₂.(6.1) If we take a symmetric 𝑆=𝑆^(2𝑚)>0 with level 𝑝, then 𝜗(𝑆,𝑍)∈𝑀_𝑚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 𝑄¹⁽¹¹⁾=⎛⎜⎜⎜⎜⎜⎜⎜⎝10101200112201030203⎞⎟⎟⎟⎟⎟⎟⎟⎠,𝑄²⁽¹¹⁾=⎛⎜⎜⎜⎜⎜⎜⎜⎝111221121210212104221224⎞⎟⎟⎟⎟⎟⎟⎟⎠,𝑄³⁽¹¹⁾=⎛⎜⎜⎜⎜⎜⎜⎜⎝2111202121122102121212⎞⎟⎟⎟⎟⎟⎟⎟⎠,(6.3) and 𝜗_𝑖=𝜗(𝑄^𝑖(11),𝑍). It is known that dim𝑆₂(Γ²⁰(11))=1 (cf. [7]). We can take a nonzero element of 𝑆₂(Γ²⁰(11)) as 𝐶₂(11)=3𝜗₁−2𝜗₂−𝜗₃(6.4) (Yoshida’s cusp form cf. [8]).

Table 1 gives a first few examples for the Fourier coefficient of 𝑓^∗11 and 𝐶₂(11).

Table 1 

The relation between 𝑓^∗11 and 𝐶₂(11) is 𝑓^∗11=−136𝐶²(11).(6.5) Further examples of the Fourier coefficients of 𝑓^∗11 can be obtained from Table 2.

Table 2