On the Spezialschar of Maass

Abstract

Let 𝑀_𝑘^(𝑛) be the space of Siegel modular forms of degree 𝑛 and even weight 𝑘. In this paper firstly a certain subspace Spez(𝑀_𝑘^(2𝑛)), the Spezialschar of 𝑀_𝑘^(2𝑛), is introduced. In the setting of the Siegel threefold, it is proven that this Spezialschar is the Maass Spezialschar. Secondly, an embedding of 𝑀_𝑘⁽²⁾ into a direct sum ⨁^{⌊𝑘/10⌋}_𝜐=0 Sym²𝑀_𝑘+2𝜐 is given. This leads to a basic characterization of the Spezialschar property. The results of this paper are directly related to the nonvanishing of certain special values of L-functions related to the Gross-Prasad conjecture. This is illustrated by a significant example in the paper.

1. Introduction

Maass introduced and applied in a series of papers [1–3] the concept of a Spezialschar to prove the Saito-Kurokawa conjecture [4, 5]. Let 𝑀_𝑘⁽²⁾ be the space of Siegel modular forms of degree 2 and weight 𝑘. Let 𝔸 be the set of positive semidefinite half-integral matrices of degree 2. Hence 𝑇∈𝔸 can be identified with the quadratic form 𝑇=[𝑛,𝑟,𝑚]. A modular form 𝐹∈𝑀_𝑘⁽²⁾ is in the Spezialschar if the Fourier coefficients 𝐴(𝑇) of 𝐹 satisfy the relation 𝐴([𝑛,𝑟,𝑚])=_{()𝑑∣𝑛,𝑟,𝑚}𝑑^𝑘−1𝐴𝑛𝑚𝑑²,𝑟𝑑,1, (1.1) for all 𝑇∈𝔸. The space of special forms is called the Maass Spezialschar 𝑀^Maass_𝑘.

The purpose of this paper is twofold. First we introduce the concept of the Spezialschar Spez(𝑀_𝑘^(2𝑛)) for Siegel modular forms of even degree 2𝑛. This is done in terms of the Hecke algebra ℋ^𝑛 attached to Siegel modular forms of degree 𝑛. Let us fix the embedding 𝑆𝑝_𝑛×𝑆𝑝_𝑚⟶𝑆𝑝_𝑛+𝑚,×𝑎𝑏̃𝑐̃𝑑⟼0𝑏̃0̃𝑑.𝑎𝑏c𝑑𝑎0𝑏0𝑎0𝑐0𝑑0𝑐0 (1.2) Let |_𝑘 be the Petersson slash operator and let 𝑇 be the normalized Heckeoperator 𝑇∈ℋ^𝑛 (see (4.21)). Let ⋈_𝑇=(𝑇×1_2𝑛)−(1_2𝑛×𝑇) and Spez𝑀_𝑘^(2𝑛)∶=𝐹∈𝑀_𝑘^(2𝑛)|𝐹|_𝑘⋈_𝑇=0,∀𝑇∈ℋ^𝑛. (1.3) Then we have the following theorem.

Theorem 1.1. The Spezialschar introduced in this paper is the Maass Spezialschar in the case of the Siegel threefold. Spez𝑀_𝑘⁽²⁾=𝑀^Maass_𝑘. (1.4)

The second topic of this paper is the characterization of the space of Siegel modular forms of degree two and the corresponding Spezialschar in terms of Taylor coefficients and certain differential operators: 𝒟_𝑘,2𝜈∶𝑀_𝑘⁽²⁾⟶𝑀^Sym_𝑘+2𝜈, (1.5) here 𝜈∈ℕ₀ and 𝑀^Sym_𝑘+2𝜈=Sym²(𝑀_𝑘+2𝜈). To simplify the notation, let 𝑀_𝑘∶=𝑀_𝑘⁽¹⁾. It would be very interesting to generalize this approach also to other situations, as to the Hermitian modular forms [6].

Before we summarize the main results, we give an example which also serves as an application. Let 𝐹₁, 𝐹₂, and 𝐹₃ be a Hecke eigenbasis of the space of Siegel cusp forms 𝑆⁽²⁾₂₀ of weight 20 and degree 2. Let 𝐹₁ and 𝐹₂ generate the Maass Spezialschar. Let 𝑓₁ and 𝑓₂ be the normalized Hecke eigenbasis of 𝑆⁽¹⁾₂₄. Then we have 𝒟_20,4𝐹_𝑗=𝛼_𝑗𝑓₁⊗𝑓₁+𝛽_𝑗𝑓₁⊗𝑓₂+𝑓₂⊗𝑓₁+𝛾_𝑗𝑓₂⊗𝑓₂. (1.6) It it conjectured by Gross and Prasad [7] that the coefficients 𝛼_𝑗, 𝛽_𝑗, and 𝛾_𝑗 are related to special values of certain automorphic L-functions. The Gross-Prasad conjecture has been proven by Ichino [8] for the Maass Spezialschar and 𝜈=0. Moreover, we show in this paper that the vanishing at such special values has interesting consequences. We have 𝐹_𝑗∈𝑆^Maass₂₀ if and only if the special value 𝛽_𝑗 is zero.

Theorem 1.2. Let 𝑘∈ℕ₀ be even. Then we have the embedding 𝔻_𝑘=^{⌊𝑘/10⌋}_𝜈=0𝒟_{𝑘,𝑘+2𝜈}∶𝑀_𝑘⁽²⁾⟶𝑀^Sym_𝑘⊕𝑆^Sym_𝑘+2⊕⋯⊕𝑆^Sym_{⌊⌋𝑘+2𝑘/10}. (1.7) For 𝐹∈𝑆_𝑘⁽²⁾, we have 𝒟_𝑘,0𝐹∈𝑆^Sym_𝑘.

Surprisingly the Maass Spezialschar property can be recovered in 𝑀^Sym_𝑘⊕𝑆^Sym_𝑘+2⊕⋯⊕𝑆^Sym_{𝑘+2⌊𝑘/10⌋} in the following transparent way. Let (𝑓_𝑗) be the normalized Hecke eigenbasis of 𝑀_𝑘. Let us define the diagonal subspaces 𝑀^𝐷_𝑘={∑_𝑗𝛼_𝑗𝑓_𝑗⊗𝑓_𝑗∈𝑀^Sym_𝑘} and 𝑆^𝐷_𝑘=𝑆^Sym_𝑘⋂𝑀^𝐷_𝑘. Then we can state the following theorem.

Theorem 1.3. Let 𝐹∈𝑀_𝑘⁽²⁾. Then we have 𝐹∈𝑀^Maass_𝑘⟺𝔻_𝑘𝐹∈𝑀^𝐷_𝑘⊕𝑆^𝐷_𝑘+2⊕⋯⊕𝑆^𝐷_{⌊⌋𝑘+2𝑘/10}, (1.8) and similarly 𝐹∈𝑆^Maass_𝑘⟺𝔻_𝑘𝐹∈^{⌊𝑘/10⌋}_𝜈=0𝑆^𝐷_𝑘+2𝜈. (1.9)

These two theorems give a transparent explanation of our example from a general point of view. In Section 5, we deduce an application related to a multiplicity one theorem of 𝑆𝐿₂ instead of 𝐺𝐿(2).

Notation 1. Let 𝑍∈ℂ^𝑛,𝑛 and tr the trace of a matrix; then we put 𝑒{𝑍}=𝑒^{2𝜋𝑖(tr𝑍)}. For 𝑙∈ℤ, we define 𝜋_𝑙=(2𝜋𝑖)^𝑙. Let 𝑥∈ℝ; then we use Knuth’s notation ⌊𝑥⌋ to denote the greatest integer smaller or equal to 𝑥. Let 𝔸₂ denote the set of half-integral positive semidefinite matrices. We parametrize the elements 𝑇=(^𝑛𝑟/2_𝑟/2𝑚) with 𝑇=[𝑛,𝑟,𝑚]. The subset of positive-definite matrices we denote with 𝔸⁺₂.

2. Ultraspherical Differential Operators

The first two sections of this paper follow the strategy of Eichler and Zagier [9]. Nevertheless, there are several topics which are different (e.g., divisors of Jacobi forms and Siegel modular forms).

Let us start with the notation of the ultraspherical polynomial 𝑝_𝑘,2𝜈. Let 𝑘 and 𝜈 be elements of ℕ₀. Let 𝑎 and 𝑏 be elements of a commutative ring. Then we put 𝑝_𝑘,2𝜈(𝑎,𝑏)=^𝜈_𝜇=0(−1)^𝜇(2𝜈)!!!𝜇!2𝜈−2𝜇𝑘+2𝜈−𝜇−2()!𝑎𝑘+𝜈−2^{2𝜈−2𝜇}𝑏^𝜇. (2.1) If we specialize the parameters, we have 𝑝_𝑘,0(𝑎,𝑏)=1 and 𝑝_𝑘,2𝜈(0,0)=0 for 𝜈∈ℕ.

Let ℍ_𝑛 be the Siegel upper half-space of degree 𝑛. Let 𝙼_𝑘^(𝑛) be the vector space of Siegel modular forms on ℍ_𝑛 with respect to the full modular group Γ_𝑛=𝚂𝚙_𝑛(ℤ). Moreover, let 𝚂_𝑘^(𝑛) denote the subspace of cusp forms. If 𝑛=1, we drop the index to simplify notation. We denote the coordinates of the threefold ℍ₂ by (𝜏,𝑧,𝜏) for (𝜏^𝑧𝜏𝑧)∈ℍ₂ and put 𝑞=𝑒{𝜏},𝜉=𝑒{𝑧}, and 𝑞=𝑒{𝜏}. Let 𝑑_𝑘 be the dimension of 𝑆_𝑘.

Definition 2.1. Let 𝑘,𝜈∈ℕ₀ and let 𝑘 be even. Then we define the ultraspherical differential operator 𝒟 on the space of holomorphic functions 𝐹 on ℍ₂ in the following way: 𝒟_𝑘,2𝜈𝐹(𝜏,𝜏)=𝑝_𝑘,2𝜈1𝜕2𝜋𝑖,1𝜕𝑧2𝜋𝑖²𝜕𝜕𝜕𝜏𝜕𝜏𝐹_𝑧=0(𝜏,𝜏). (2.2) In the case 𝜈=0, we get the pullback 𝐹(𝜏,0,𝜏) of 𝐹 on ℍ×ℍ.
Let 𝐹∈𝙼_𝑘⁽²⁾ with 𝑇th Fourier coefficient 𝐴^𝐹(𝑛,𝑟,𝑚) for 𝑇=[𝑛,𝑟,𝑚]∈𝔸₂. Then we have 𝒟_𝑘,2𝜈𝐹(𝜏,𝜏)=^∞_𝑛,𝑚=0𝐴^𝐹_2𝜈(𝑛,𝑚)𝑞^𝑛𝑞^𝑚𝐴with_𝐹2𝜈(𝑛,𝑚)=_{𝑟∈ℤ,𝑟2⩽4𝑛𝑚}𝑝_𝑘,2𝜈(𝑟,𝑛𝑚)𝐴^𝐹(𝑛,𝑟,𝑚). (2.3) Let 𝑀^Sym_𝑘=Sym²𝑀_𝑘,𝑆^Sym_𝑘=Sym²𝑆_𝑘, and 𝑆^Sym_𝑘=(𝑆_𝑘⊗𝑆_𝑘)^Sym. Let us further introduce a related Jacobi differential operator 𝒟^𝙹,𝑚_𝑘,2𝜈. This is given by exchanging 𝜋₋₁(𝜕/𝜕𝜏) with 𝑚 in the definition of the ultraspherical differential operator given in (2.2). Applying the operator 𝒟^𝙹,𝑚_𝑘,2𝜈 on Jacobi forms Φ∈𝐽_𝑘,𝑚 of weight 𝑘 and index 𝑚 on ℍ×ℂ matches with the effect of the operator 𝒟_2𝜈 introduced in [9, Section 3 formula (2)] on Φ.
Since 𝐹∈𝑀_𝑘⁽²⁾ has a Fourier-Jacobi expansion of the form 𝐹(𝜏,𝑧,𝜏)=^∞_𝑚=0Φ^𝐹_𝑚(𝜏,𝑧)𝑞^𝑚,withΦ^𝐹_𝑚∈𝐽_𝑘,𝑚, (2.4) it makes sense to consider 𝒟_𝑘,2𝜈 with respect to this decomposition in a Fourier-Jacobi expansion 𝒟_𝑘,2𝜈=^∞_𝑚=0𝒟^𝐽,𝑚_𝑘,2𝜈. (2.5)

Lemma 2.2. Let 𝑘,𝜈∈ℕ₀ and let 𝑘 be even. Then 𝒟_𝑘,2𝜈 maps 𝑀_𝑘⁽²⁾ to 𝑀^Sym_𝑘+2𝜈 and to 𝑆^Sym_𝑘+2𝜈 if 𝜈≠0. Moreover, the subspace 𝑆_𝑘⁽²⁾ of cusp forms is always mapped to 𝑆^Sym_𝑘+2𝜈.

Proof. We know from the work of Eichler and Zagier [9] since 𝒟^𝙹,𝑚_𝑘,2𝜈=𝒟_𝑘,2𝜈 that 𝒟^𝙹,𝑚_𝑘,2𝜈Φ^𝐹_𝑚∈𝑀_𝑘+2𝜈. Let 𝜈>0; then 𝒟^𝙹,𝑚_𝑘,2𝜈Φ^𝐹_𝑚∈𝑆_𝑘+2𝜈 and for 𝐹∈𝑆_𝑘⁽²⁾ we have 𝒟^𝙹,𝑚_𝑘,2𝜈Φ^𝐹_𝑚∈𝑆_𝑘+2𝜈 for all 𝜈∈ℕ₀. We are now ready to act with the ultraspherical differential operator with respect to its Fourier-Jacobi expansion directly on the Fourier-Jacobi expansion of 𝐹 in a canonical way 𝒟_𝑘,2𝜈𝐹(𝜏,𝜏)=^∞_𝑚=0𝒟^𝙹,𝑚_𝑘,2𝜈Φ^𝐹_𝑚(𝜏)𝑞^𝑚, (2.6) where all “coefficients” 𝑎^𝐹_𝑚(𝜏)=𝒟^𝙹,𝑚_𝑘,2𝜈Φ^𝐹_𝑚(𝜏) are modular forms. This shows us, that if we apply the Peterson slash operator |_𝑘+2𝜈𝛾 here 𝛾∈Γ to this function with respect to the variable 𝜏, the function is invariant. The same argument also works for the Fourier-Jacobi expansion with respect to 𝜏. From this we deduce that 𝒟_𝑘,2𝜈𝐹(𝜏,𝜏)=∑_𝑖,𝑗𝛼_𝑖,𝑗𝑓_𝑖(𝜏)𝑓_𝑗(𝜏). Here (𝑓_𝑖)_𝑖 is a basis of 𝑀_𝑘+2𝜈. Finally the cuspidal conditions in the lemma also follow from symmetry arguments.

Remark 2.3. Let 𝐹∶ℍ₂→ℂ be holomorphic. Let 𝑔∈𝑆𝐿₂(ℝ) and let 𝑆=(^𝑇0_0𝑇), where 𝑇=(⁰¹₁₀). Then we have 𝒟_𝑘,2𝜈𝐹|_𝑘𝑔×1₂=𝒟_𝑘,2𝜈𝐹|_𝑘+2𝜈𝑔×1₂,𝒟_𝑘,2𝜈𝐹|_𝑘1₂×𝑔=𝒟_𝑘,2𝜈𝐹|_𝑘+2𝜈1₂,𝒟×𝑔_𝑘,2𝜈𝐹|_𝑘𝑆(𝜏,𝜏)=𝒟_𝑘,2𝜈𝐹|_𝑘+2𝜈(𝜏,𝜏). (2.7)

Remark 2.4. There are other possibilities for construction of differential operators as used in this section (see Ibukiyama for an overview [10]). But since the connection between our approach and the theory developed of Eichler and Zagier [9] is so useful we decided to do it this way. We also wanted to introduce the concept of Fourier-Jacobi expansion of differential operators, which is interesting in its own right.

3. Taylor Expansion of Siegel Modular Forms

The operators 𝒟_𝑘,2𝜈 can be seen at this point as somewhat artificial. If we apply 𝒟_𝑘,2𝜈 to Siegel modular forms 𝐹, we lose information. For example, we know that dim𝑆⁽²⁾₂₀=3 and contains a two-dimensional subspace of Saito-Kurokawa lifts. Since dim𝑆^Sym₂₀=1, we obviously lose information if we apply 𝒟_20,0. But even worse let 𝐹₁ and 𝐹₂ be a Hecke eigenbasis of the space of Saito-Kurokawa lifts and 𝐹₃ a Hecke eigenform of the orthogonal complement; then we have 𝒟_20,0𝐹_𝑗≠0 for 𝑗=1,2,3. The general case seems to be even worse, since for example, dim𝑀_𝑘⁽²⁾∼𝑘³ and dim𝑀^Sym_𝑘∼𝑘². On the other hand, from an optimistic viewpoint we may find about 𝑘 pieces 𝒟_𝑘,2𝜈𝐹 which code all the relevant information needed to characterize the Siegel modular forms 𝐹.

Garrett in his papers [11, 12] introduced the method of calculating pullbacks of modular forms to study automorphic L-functions. We also would like to mention the work of Gelbart et al. at this point ([13]). And recently Ichino in his paper: “Pullbacks of Saito-Kurokawa lifts” [8] extended Garrett’s ideas in a brilliant way to prove the Gross-Prasad conjecture [7] for Saito-Kurokawa lifts. In the new language we have introduced, it is obvious to consider Garrett's pullbacks as the 0th Taylor coefficients of 𝐹 around 𝑧=0. Hence it seems to be very lucrative to study also the higher Taylor coefficients and hopefully get some transparent link.

Let 𝑘∈ℕ₀ be even. Let 𝐹∈𝑀_𝑘⁽²⁾ and Φ∈𝐽_𝑘,𝑚. Then we denote by 𝐹(𝜏,𝑧,𝜏)=^∞_𝜈=0𝜒^𝐹_2𝜈(𝜏,𝜏)𝑧^2𝜈,Φ(𝜏,𝑧)=^∞_𝜈=0𝜒^Φ_2𝜈(𝜏)𝑧^2𝜈, (3.1) the corresponding Taylor expansions with respect to 𝑧 around 𝑧=0. Here we already used the invariance of 𝐹 and Φ with respect to the transformation 𝑧↦(−𝑧) since 𝑘 is even. Suppose 𝜒_2𝜈0 is the first nonvanishing Taylor coefficient; then we denote 2𝜈₀ the vanishing order of the underlying form. If the form is identically zero, we define the vanishing order to be ∞. To simplify our notation, we introduce normalizing factor 𝛾_𝑘,2𝜈=12𝜋𝑖^2𝜈(𝑘+2𝜈−2)!(2𝜈)!(𝑘+2𝜈/2−2)!. (3.2) Further we put 𝜒^𝜇,𝜇_2𝜈=𝜕^2𝜇𝜕𝜏^𝜇𝜕𝜏^𝜇𝜒^𝐹_2𝜈,𝜉^𝜇,𝜇_2𝜈=𝛾_𝑘,2𝜈⁻¹𝜕^2𝜇𝜕𝜏^𝜇𝜕𝜏^𝜇𝒟_𝑘,2𝜈𝐹. (3.3) Then a straightforward calculation leads to the following useful formula.

Lemma 3.1. Let 𝑘,𝜈∈ℕ₀ and let 𝑘 be even. Let 𝐹∈𝑀_𝑘⁽²⁾. Then we have 𝒟_𝑘,2𝜈𝐹(𝜏,𝜏)=𝛾_𝑘,2𝜈^𝜈_𝜇=0(−1)^𝜇𝑘+2𝜈−𝜇−2!()𝜕𝑘+2𝜈−2!𝜇!^2𝜇𝜒^𝐹_{2𝜈−2𝜇}𝜕𝜏^𝜇𝜕𝜏^𝜇. (3.4) A similar formula is valid for Jacobi forms with normalizing factor 𝛾^𝙹,𝑚_𝑘,2𝜈=𝛾_𝑘,2𝜈.

Corollary 3.2. Let 2𝜈₀ be the vanishing order of 𝐹∈𝑀_𝑘⁽²⁾. Then we have 𝒟_𝑘,2𝜈𝐹=0 for 𝜈<𝜈₀ and 𝒟_𝑘,2𝜈0𝐹(𝜏,𝜏)=𝛾_𝑘,2𝜈𝜒^𝐹_2𝜈0(𝜏,𝜏)∈𝑀^Sym_𝑘+2𝜈0⧵{0}. (3.5) Similarly, we have for Φ∈𝐽_𝑘,𝑚 with vanishing order 2𝜈₀ the properties 𝒟^𝙹,𝑚_𝑘,2𝜈Φ=0 for 𝜈<𝜈₀ and 𝒟^𝙹,𝑚_𝑘,2𝜈0Φ=𝛾_𝑘,2𝜈𝜒^Φ_2𝜈0∈𝑀_𝑘+2𝜈0.

Example 3.3. It is well known that dim𝑆⁽²⁾₁₀=1. Let Φ=Φ₁₀∈𝑆⁽²⁾₁₀ be normalized in such a way that 𝐴^Φ(1,1,1)=1. Then it follows from 𝒟_10,0Φ=0 that 𝐴^Φ(1,0,1)=−2 since dim𝑆^Sym₁₀=0. Then Φ has the Taylor expansion Φ₁₀(𝜏)=3𝜏,𝑧,5𝜋₂Δ(𝜏)Δ(𝜏)𝑧²+Δ^′(𝜏)Δ^′(𝜏)𝑧⁴+𝑂𝑧⁶, (3.6) we can also express the Taylor coefficients 𝜒^𝐹_2𝜈 in terms of the modular forms 𝒟_𝑘,2𝜈𝐹. This can be done by inverting the formula (3.4). Finally we get 𝜒_2𝜈=^𝜈_𝜇=0𝑘+2𝜈−2𝜇−1!𝜉𝑘+2𝜈−𝜇−1!𝜇!^{𝜇,𝜇2𝜈−2𝜇}. (3.7) Before we state our first main result about the entropy of the family 𝒟_𝑘,0𝐹,𝒟_𝑘,2𝐹,𝒟_𝑘,4𝐹,… we introduce some further notation. 𝕎_𝑘=𝑆^Sym_𝑘⊕^{⌊⌋𝑘/10}_𝑗=1𝑆^Sym_𝑘+2𝑗,𝕎^cusp_𝑘=𝑆^Sym_𝑘⊕^{⌊⌋𝑘/10}_𝑗=1𝑆^Sym_𝑘+2𝑗. (3.8) These spaces will be the target of our next consideration. More precisely we define a linear map from the space of Siegel modular forms of degree 2 into these spaces with remarkable properties. The following result is equivalent to Theorem 1.2 in the introduction, given with a slightly different notation.

Theorem 3.4. Let 𝑘∈ℕ₀ be even. Then we have the linear embedding 𝔻_𝑘∶𝑀_𝑘⁽²⁾↪𝕎_𝑘,𝐹⟼^{⌊⌋𝑘/10}_𝜈=0𝒟_𝑘,2𝜈𝐹. (3.9) Since 𝒟_𝑘,0𝑆_𝑘⁽²⁾ is cuspidal we have the embedding of 𝑆_𝑘⁽²⁾ into 𝕎^cusp_𝑘.

Remark 3.5. It can be deduced from [14] that 𝒟_𝑘,0⊕𝒟_𝑘,2 is surjective. Hence for 𝑘<20 we have (i)𝑀_𝑘⁽²⁾ is isomorphic to 𝑀_𝑘 for 𝑘<10; (ii)𝑀_𝑘⁽²⁾ is isomorphic to 𝑀^Sym_𝑘⊕𝑆^Sym_𝑘 for 10⩽𝑘<20 and 𝑆_𝑘⁽²⁾≃𝑆_𝑘⊕𝑆_𝑘+2.

Proof. First of all we recall that we have already shown that 𝒟_𝑘,0𝑀_𝑘⁽²⁾⊆𝑀^Sym_𝑘 and 𝒟_𝑘,2𝜈𝑀_𝑘⁽²⁾⊆𝑆^Sym_𝑘+2𝜈 for 𝜈>0. Let 𝐹∈𝑀_𝑘⁽²⁾ and suppose that 𝔻_𝑘𝐹 is identically zero. Then it would follow from our inversion formula (3.7) that 𝐹(𝜏,𝑧,𝜏)=^∞_{⌊⌋𝜈=𝑘/10+1}𝜒^𝐹_2𝜈(𝜏,𝜏)𝑧^2𝜈. (3.10) For such 𝐹 the general theory of Siegel modular forms of degree 2 says that the special function Φ₁₀∈𝑆_𝑘⁽²⁾, which we already studied in one of our examples, divides 𝐹 in the ℂ-algebra of modular forms. And this is fulfilled at least with a power of ⌊𝑘/10⌋+1=𝑡_𝑘>0. Hence there exists a Siegel modular form 𝐺 of weight 𝑘−10𝑡_𝑘. But since this weight is negative and nontrivial Siegel modular forms of negative weight do not exist, the form 𝐺 has to be identically zero. Hence we have shown that if 𝔻_𝑘𝐹≡0 then 𝐹≡0. And this proves the statement of the theorem.

Remark 3.6. The number ⌊𝑘/10⌋ in the Theorem is optimal. This follows directly from properties of Φ₁₀.

Remark 3.7. Let 𝐸_𝑘^2,1(𝑓) be a Klingen Eisenstein series attached to 𝑓∈𝑆_𝑘. Let 𝐸_𝑘 denote an elliptic Eisenstein series of weight 𝑘. Then it can be deduced from [12] that 𝒟_𝑘,0𝐸_𝑘^2,1(𝑓)=𝑓⊗𝐸_𝑘+𝐸_𝑘⊗𝑓 mod 𝑆^Sym_𝑘.

Remark 3.8. It would be interesting to have a different proof of the Theorem 3.4 independent of the special properties of Φ₁₀.

Remark 3.9. The asymptotic limit of the dimension of the quotient of 𝕎_𝑘/𝑀_𝑘⁽²⁾ is equal to 91⁄25. Let us put 𝑑_𝑘=dim𝑀_𝑘. (i)The dimension of the target space 𝕎_𝑘 is given by dim𝕎_𝑘∼1288₀^𝑘/10(𝑘+2𝑥)²∼1𝑑𝑥12882⋅3915³𝑘³. (3.11)(ii)The asymptotic dimension formula of 𝑀_𝑘⁽²⁾ is given by dim𝑀_𝑘⁽²⁾∼11288𝑘2⋅3⋅5³. (3.12) (see, [1, Introduction]).

4. The Spezialschar

In this section, we first recall some basic facts on the Maass Spezialschar [5]. Then we determine the image of the Spezialschar in the space 𝕎_𝑘 for all even weights 𝑘. Then finally we introduce a Spezialschar as a certain subspace of the space of Siegel modular forms of degree 2𝑛 and weight 𝑘. Then we show that in the case 𝑛=1 this Spezialschar coincides with the Maass Spezialschar.

4.1. Basics of the Maass Spezialschar

Let 𝐽_𝑘,𝑚 be the space of Jacobi forms of weight 𝑘 and index 𝑚. We denote the subspace of cusp forms with 𝐽^cusp_𝑘,𝑚. Let |_𝑘,𝑚 be the slash operator for Jacobi forms and 𝑉_𝑙 (𝑙∈ℕ₀) the operator, which maps 𝐽_𝑘,𝑚 to 𝐽_{𝑘,𝑚𝑙}. More precisely, let Φ(𝜏,𝑧)=∑𝑐(𝑛,𝑟)𝑞^𝑛𝜉^𝑟∈𝐽_𝑘,𝑚. Then (Φ|_𝑘,𝑚𝑉_𝑙)(𝜏,𝑧)=∑𝑐^∗(𝑛,𝑟)𝑞^𝑛𝜉^𝑟 with 𝑐^∗(𝑛,𝑟)=_{()𝑎|𝑛,𝑟,𝑙}𝑎^𝑘−1𝑐𝑛𝑙𝑎²,𝑟𝑎,for𝑙∈ℕ, (4.1) and for 𝑙=0, we have 𝑐^∗(0,0)=𝑐(0,0)(−2𝑘/𝐵_2𝑘) and for 𝑙=0 and 𝑛>0 we have 𝑐^∗(𝑛,𝑟)=𝑐(0,0)𝜎_𝑘−1(𝑛). This includes the theory of Eisenstein series in a nice way [9].

Definition 4.1. The lifting 𝒱 is given by the linear map 𝐽𝒱∶^𝑘,1⟶𝑀_𝑘⁽²⁾,Φ⟼^∞_𝑙=0Φ|_𝑘,1𝑉_𝑙𝑞^𝑙. (4.2) The image of this lifting is the Maass Spezialschar 𝑀^Maass_𝑘 of weight 𝑘. The subspace of cusp forms we denote with 𝑆^Maass_𝑘.

Remark 4.2. (i)The lifting is invariant by the Klingen parabolic of 𝑆𝑝₂(ℤ). Since the Fourier coefficients satisfy 𝐴(𝑛,𝑟,𝑚)=𝐴(𝑚,𝑟,𝑛), the map 𝒱 is well defined. (ii)If we restrict the Saito-Kurokawa lifting to Jacobi cusp forms, we get Siegel cusp forms. (iii)Let Φ∈𝐽_𝑘,𝑚 and 𝑙,𝜇∈ℕ₀. Then we have 𝒟^{𝐽,𝑚𝑙}_𝑘,2𝜇Φ|_𝑘,𝑚𝑉_𝑙=𝒟^𝐽,𝑚_𝑘,2𝜇Φ_𝑘𝑇_𝑙. (4.3) Here 𝑇_𝑙 is the Hecke operator on the space of elliptic modular forms. (iv)Let 𝐹∈𝑀^Maass_𝑘 be the lift of Φ∈𝐽_𝑘,1. Then 𝐹 is a Hecke eigenform if and only if Φ is a Hecke-Jacobi eigenform.

From this consideration we conclude [9] the following.

Proposition 4.3. Let 𝐹∈𝑀_𝑘⁽²⁾ be a Siegel modular form. Then the following properties are equivalent.

Arithmetic
Let 𝐴(𝑛,𝑟,𝑚) denote the Fourier coefficients of 𝐹 then 𝐴(𝑛,𝑟,𝑚)=_{()𝑑|𝑛,𝑟,𝑚}𝑑^𝑘−1𝐴𝑛𝑚𝑑²,𝑟𝑑,1. (4.4)

Lifting
Let Φ^𝐹₁ be the first Fourier-Jacobi coefficient of 𝐹. Then all other Fourier-Jacobi coefficients satisfy the identity Φ^𝐹_𝑚=Φ^𝐹₁|_𝑘,1𝑉_𝑚. (4.5) Let 𝐹∈𝑆_𝑘⁽²⁾ be a Hecke eigenform. Then 𝐹 is a Saito-Kurokawa lift if and only if the spinor L-function 𝑍(𝐹,𝑠) of degree 4 has a pole (see Evdokimov [15]). Similar results are also obtained by Oda [16].

4.2. The Diagonal of 𝕎_𝑘 and the Proof of Theorem 1.3

Let (𝑓_𝑗) be the normalized Hecke eigenbasis of 𝑀_𝑘. With this notation, we introduce the diagonal space 𝑀^𝐷_𝑘=_𝑗𝛼_𝑗𝑓_𝑗⊗𝑓_𝑗∈𝑀^Sym_𝑘, (4.6) and the corresponding cuspidal subspace 𝑆^𝐷_𝑘. Now we are ready to distinguish the Maass Spezialschar in the vector spaces 𝕎_𝑘 and 𝕎^cusp_𝑘. This leads to Theorem 1.3 stated in the introduction. Before we give a proof we note the following.

Remark 4.4. (i) Theorem 1.3 describes a link between Siegel modular forms and elliptic Hecke eigenforms.
(ii) Let 𝐹∈𝑀⁽²⁾₂₀ and let (𝑓_𝑗) be a Hecke eigenbasis of 𝑆₂₄. Then 𝐹∈𝑀^Maass_𝑘 if and only if 𝒟_20,4𝐹=𝛼₀𝐸₂₄⊗𝐸₂₄+𝛼𝑓₁⊗𝑓₁+𝛾𝑓₂⊗𝑓₂; (4.7) here 𝛼₀,𝛼,𝛾∈ℂ.

Proof. We first show that if 𝐹 is in the Maass Spezialschar then 𝒟_𝑘,2𝜈𝐹 is an element of the diagonal space. Let 𝜈∈ℕ₀ and let Φ^𝐹₁ be the first Fourier-Jacobi coefficient of 𝐹. Then we have 𝒟_𝑘,2𝜈(𝒱Φ)(𝜏,𝜏)=^∞_𝑙=0𝒟^𝐽,𝑙_𝑘,2𝜈Φ|_𝑘,1𝑉_𝑙(𝜏)𝑞^𝑙. (4.8) Here we applied the Fourier-Jacobi expansion of the differential operator 𝒟_𝑘,2𝜈 acting on Siegel modular forms. Then we used the formula (4.3) to interchange the operators 𝒟^𝐽,𝑙_𝑘,2𝜈 and 𝑉_𝑙 to get 𝒟_𝑘,2𝜈𝐹(𝜏,𝜏)=^∞_𝑙=0𝒟^𝐽,𝑙_𝑘,2𝜈Φ_𝑘𝑇_𝑙𝑞^𝑙. (4.9) Now let (𝑓^𝑘+2𝜈_𝑗)𝑑_{𝑘+2𝜈𝑗=1} be a normalized Hecke eigenbasis of 𝑆_𝑘+2𝜈. Let 1≤𝑗₁,𝑗₂≤𝑑_𝑘+2𝜈. Then we have 𝒟_𝑘,2𝜈𝐹(∗,𝜏),𝑓^𝑘+2𝜈_𝑗1=𝑘𝒟^𝐽,𝑙_𝑘,2𝜈Φ,𝑓^𝑘+2𝜈_𝑗1𝑓^𝑘+2𝜈_𝑗1, (4.10) which leads to the desired result 𝒟_𝑘,2𝜈𝐹,𝑓^𝑘+2𝜈_𝑗1⊗𝑓^𝑘+2𝜈_𝑗2=0,for𝑗₁≠𝑗₂. (4.11) It remains to look at the Eisenstein part if 𝜈=0. Since the space of Eisenstein series has the basis 𝐸_𝑘 and is orthogonal to the functions given in (4.10), we have proven that the Spezialschar property of 𝐹 implies that 𝔻_𝑘𝐹∈𝕎^𝙳_𝑘.
Now let us assume that 𝐹∉𝑀^Maass_𝑘. Then we show that 𝔻_𝑘𝐹∉𝕎^𝙳_𝑘. Since the map 𝒟_𝑘,0⊕𝒟_𝑘,2∶𝑀^Maass_𝑘⟶𝑀^𝐷_𝑘⊕𝑆^𝐷_𝑘+2 (4.12) is an isomorphism, we can assume that (𝒟_𝑘,0⊕𝒟_𝑘,2)(𝐹) projected on 𝑀^𝐷_𝑘⊕𝑆^𝐷_𝑘+2 is identically zero. Altering 𝐹 by an element of the Maass Spezialschar does not change the property we have to prove. If 𝒟_𝑘,0𝐹∉𝑀^𝐷_𝑘 or 𝒟_𝑘,2𝐹∉𝑆^𝐷_𝑘+2, we are done; otherwise we can assume that 𝒟_𝑘,0⊕𝒟_𝑘,2(𝐹)≡0. (4.13) Then we have the order 𝐹=2𝜈₀⩾4 and 𝑘⩾20, since 𝐹∉𝑀^Maass_𝑘. Let 𝐹𝑧𝜏=𝜏𝑧^∞_𝜈=𝜈0𝜒^𝐹_2𝜈0(𝜏,𝜏)𝑧^2𝜈 (4.14) be the Taylor expansion of 𝐹 with 𝜒^𝐹_2𝜈0(𝜏,𝜏)∈𝑆_𝑘+2𝜈0 not identically zero. Let Φ₁₀∈𝑆⁽²⁾₁₀ be the Siegel cusp form of weight 10 and degree 2. It has the properties that 𝜒^Φ10₀≡0 and 𝜒^Φ10₂(𝜏,𝜏)=𝑐Δ(𝜏)Δ(𝜏) with 𝑐≠0. Since order 𝐹=2𝜈₀, we also have Φ^𝜈0₁₀‖𝐹. (4.15) This means that there exists a 𝐺∈𝑆_{𝑘−10𝜈0} such that 𝜒^𝐺₀ is nontrivial and 𝐹=Φ₁₀^𝜈0𝐺. (4.16) Hence we have for the first nontrivial Taylor coefficient of 𝐹, the formula 𝜒^𝐹_2𝜈0(𝜏,𝜏)=𝜒^Φ10₂(𝜏,𝜏)^𝜈0𝜒^𝐺₀(𝜏,𝜏),=𝑐^𝜈0Δ(𝜏)^𝜈0Δ(𝜏)^𝜈0𝜒^𝐺₀(𝜏,𝜏). (4.17) And the coefficient 𝑎₁(𝜏) of 𝑞 is identically zero. Now let us assume for a moment that 𝜒^𝐹_2𝜈0∈𝑆^𝐷_𝑘+2𝜈0. Then we have 𝜒^𝐹_2𝜈0(𝜏)=𝑑𝜏,^𝑘+2𝜈_𝑙=1𝛼_𝑙𝑓^𝑘+2𝜈0_𝑙(𝜏)𝑓^𝑘+2𝜈0_𝑙(𝜏), (4.18) and the coefficient of 𝑞 is given by ∑𝑑_{0𝑘+2𝜈𝑙=1}𝛼_𝑙𝑓^𝑘+2𝜈0_𝑙(𝜏). Since (𝑓^𝑘+2𝜈0_𝑙)𝑑_{0𝑘+2𝜈𝑙=1} is a basis we have 𝛼₁𝑑=⋯=𝛼_0𝑘+2𝜈=0. But since we assumed that order𝐹=2𝜈₀, we have a reductio ad absurdum. Hence we have shown that 𝜒^𝐹_2𝜈0∉𝑆_𝑘+2𝜈0 which proves our theorem.

Corollary 4.5. Klingen Eisenstein series are not in the Maass Spezialschar.

Remark 4.6. Let 𝑘 be a natural even number. Let 𝐹 be a Siegel modular form of degree two and weight 𝑘. Then we have 𝐹∈𝑀^Maass_𝑘⟺𝒟_𝑘,2𝜈𝐹∈𝑀^𝐷_𝑘+2𝜈,∀𝜈∈ℕ₀. (4.19)

4.3. The Spezialschar

Let 𝐺⁺𝑆𝑝_𝑛(ℚ) be the rational symplectic group with positive similitude 𝜇. In the sense of Shimura, we attach to Hecke pairs the corresponding Hecke algebras ℋ^𝑛=Γ_𝑛,𝐺⁺𝑆𝑝_𝑛(ℚ,ℋ_𝑛0)=Γ_𝑛,𝑆𝑝_𝑛(ℚ). (4.20) We also would like to mention that in the setting of elliptic modular forms the classical Hecke operator 𝑇(𝑝) can be normalized such that it is an element of the full Hecke algebra ℋ¹, but not of the even one ℋ¹₀. Let g∈𝐺⁺𝑆𝑝_𝑛(ℚ) with similitude 𝜇(𝑔). Then we put 𝑔=𝜇𝑔^−1/2𝑔 (4.21) to obtain an element of 𝑆𝑝_𝑛(ℝ). We further extend this to ℋ^𝑛.

Definition 4.7. Let 𝑇∈ℋ^𝑛. Then we define ⋈_𝑇=𝑇×1_2𝑛−1_2𝑛×𝑇. (4.22) Here × is the standard embedding of (𝑆𝑝_𝑛,𝑆𝑝_𝑛) into 𝑆𝑝_2𝑛.

Now we study the action |_𝑘⋈_𝑇 on the space of modular forms of degree 2𝑛 for all 𝑇∈ℋ^𝑛 or 𝑇∈ℋ^𝑛₀. The first thing we would like to mention is that for 𝐹∈𝑀_𝑘^(2𝑛) the function 𝐹|_𝑘⋈_𝑇 is in general not an element of 𝑀_𝑘^(2𝑛) anymore. Anyway at the moment we are much more interested in the properties of the kernel of a certain map related to this action. In particular, in the case 𝑛=1 we get a new description of the Maass Spezialschar.

Definition 4.8. Let 𝑛 and 𝑘 be natural numbers. Let 𝑀_𝑘^(2𝑛) be the space of Siegel modular forms of degree 2𝑛 and weight 𝑘. Then we introduce the Spezialschar corresponding to the Hecke algebras ℋ^𝑛 and ℋ^𝑛₀. Spez𝑀_𝑘^(2𝑛)=𝐹∈𝑀_𝑘^(2𝑛)|𝐹|_𝑘⋈_𝑇=0,∀𝑇∈ℋ^𝑛,Spez₀𝑀_𝑘^(2𝑛)=𝐹∈𝑀_𝑘^(2𝑛)|𝐹|_𝑘⋈_𝑇=0,∀𝑇∈ℋ^𝑛₀. (4.23) Moreover, Spez(𝑆_𝑘^(2𝑛)) and Spez₀(𝑆_𝑘^(2𝑛)) are the cuspidal part of the corresponding Spezialschar.

It is obvious that these subspaces of 𝑀_𝑘^(2𝑛) are candidates for finding spaces of modular forms with distinguished Fourier coefficients. Further it turns that these spaces are related to the Maass Spezialschar and the Ikeda lift [17]. This leads to Theorem 1.1 stated in the introduction.

4.3.1. Proof of Theorem 1.1

Proof. Let 𝐹∈𝑀_𝑘⁽²⁾. Then we have 𝐹∈𝑀^Maass_𝑘 if and only if 𝒟_𝑘,2𝜈𝐹∈𝑀^𝐷_𝑘+2𝜈 for all 𝜈∈ℕ₀. This follows from Remark 4.6. On the other side, the property 𝒟_𝑘,2𝜈𝐹∈𝑀^𝐷_𝑘+2𝜈 is equivalent to the identity 𝒟_𝑘,2𝜈𝐹_𝑘+2𝜈⋈_𝑇=0,∀𝑇∈ℋ. (4.24) This follows from the fact that the Hecke operators are self-adjoint and that the space of elliptic modular forms has multiplicity one. To make the operator well defined, we used the embedding ℍ×ℍ into the diagonal of ℍ₂. We can now interchange the differential operators 𝒟_𝑘,2𝜈 and the Petersson slash operator |_∗. This leads to 𝒟_𝑘,2𝜈𝐹∈𝑀^𝐷_𝑘+2𝜈⟺𝒟_𝑘,2𝜈𝐹|_𝑘⋈_𝑇=0. (4.25) So finally it remains to show that if 𝒟_𝑘,2𝜈(𝐹|_𝑘⋈_𝑇)=0 for all 𝜈∈ℕ₀ then it follows 𝐹|_𝑘⋈_𝑇=0. By looking at the Taylor expansion of the function 𝐹|_𝑘⋈_𝑇(𝜏^𝑧𝜏𝑧) with respect to 𝑧 around 0, we get with the same argument as given in the proof of Theorem 3.4 the desired result.

5. Maass Relations Revised

We introduced two Hecke algebras ℋ and ℋ₀ related to elliptic modular forms. For the corresponding Spezialschar Spez(𝑀_𝑘⁽²⁾) and Spez₀(𝑀_𝑘⁽²⁾), we obtain the following theorem.

Theorem 5.1. Let 𝑘 be an even natural number. Then the even Spezialschar Spez₀(𝑀_𝑘⁽²⁾) related to the Hecke algebra ℋ₀ which is locally generated by 𝑇(𝑝²) is equal to the Spezialschar Spez(𝑀_𝑘⁽²⁾) related to the Hecke algebra ℋ which is locally generated by 𝑇(𝑝). Spez₀𝑀_𝑘⁽²⁾=Spez𝑀_𝑘⁽²⁾. (5.1)

Proof. Let 𝐹∈𝑀_𝑘⁽²⁾. We proceed as follows. In the proof of Theorem 1.1, it has been shown that 𝐹∈Spez𝑀_𝑘⁽²⁾⟺𝒟_𝑘,2𝜈𝐹_𝑘+2𝜈⋈_𝑇=0,∀𝑇∈ℋ,𝜈∈ℕ₀. (5.2) Now we show that 𝒟_𝑘,2𝜈𝐹_𝑘+2𝜈⋈_𝑇(𝑝)=0⟺𝒟_𝑘,2𝜈𝐹_𝑘+2𝜈⋈_𝑇(𝑝2)=0, (5.3) for all 𝜈∈ℕ₀ and prime numbers 𝑝. This would finish the proof since 𝐹∈Spez₀𝑀_𝑘⁽²⁾⟺𝒟_𝑘,2𝜈𝐹_𝑘+2𝜈⋈_𝑇=0,∀𝑇∈ℋ₀,𝜈∈ℕ₀. (5.4) (this can also be obtained by following the procedure of the proof of Theorem 1.1).
To verify (5.3), we show that to be an element of the kernel of the operator |⋈_𝑇(𝑝2) implies already to be an element of the kernel of |⋈_𝑇(𝑝).
To see this we give a more general proof. Let 𝜙∈𝑀^Sym_𝑘 and let 𝜙|_𝑘⋈_𝑇(𝑝2)=0. Let (𝑓_𝑗) be a normalized Hecke eigenbasis of 𝑀_𝑘. Then we have 𝜙=_𝑖,𝑗𝛼_𝑖,𝑗𝑓_𝑖⊗𝑓_𝑗. (5.5) Let us assume that there exists an 𝛼_𝑖0,𝑗0≠0 with 𝑖₀≠𝑗₀. Let us denote 𝜆_𝑙(𝑝²) to be the eigenvalue of 𝑓_𝑙 with respect to the Hecke operator 𝑇(𝑝²). Then we have 0=𝜙|_𝑘⋈_𝑇(𝑝2)=_𝑖,𝑗𝛼_𝑖,𝑗𝜆_𝑖𝑝²−𝜆_𝑗𝑝²𝑓_𝑖⊗𝑓_𝑗. (5.6) From this follows that 𝜆_𝑖0(𝑝²)=𝜆_𝑗0(𝑝²) for all prime numbers 𝑝. It is easy to see at this point that 𝑓_𝑖0 and 𝑓_𝑗0 have to be cusp forms. In the setting of cusp forms, we can apply a result on multiplicity one for 𝑆𝐿₂ of D. Ramakrishnan [18, Section 4.1] and other people obtain 𝑓_𝑖0=𝑓_𝑗0. Since this is a contradiction we have 𝜙∈𝑀^𝐷_𝑘. In other words, we have 𝜙|_𝑘⋈_𝑇(𝑝)=0.

Acknowledgments

The author thanks the two referees for valuable suggestions.