# On the Spezialschar of Maass

Authors: Bernhard Heim
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

LetMk(n) be the space of Siegel modular forms of degree n and even weight k. In this paper firstly a certain subspace Spez(Mk(2n)), the Spezialschar of Mk(2n), is introduced. In the setting of the Siegel threefold, it is proven that this Spezialschar is the Maass Spezialschar. Secondly, an embedding of Mk(2) into a direct sum ⨁υ=0⌊k/10⌋
Sym2Mk+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.


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## Body

## 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 Mk(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∈Mk(2) is in the Spezialschar if the Fourier coefficients A(T) of F satisfy the relation
(1.1)A([n,r,m])=∑d∣(n,r,m)dk-1A([nmd2,rd,1]),
for all T∈𝔸.  The space of special forms is called the Maass Spezialschar MkMaass.The purpose of this paper is twofold. First we introduce the concept of the SpezialscharSpez(Mk(2n)) for Siegel modular forms of even degree 2n. This is done in terms of the Hecke algebra ℋn attached to Siegel modular forms of degree n. Let us fix the embedding
(1.2)Spn×Spm⟶Spn+m,(abcd)×(ãb̃c̃d̃)⟼(a0b00ã0b̃c0d00c̃0d̃).
Let |k be the Petersson slash operator and let T̃ be the normalized Heckeoperator T∈ℋn (see (4.21)). Let ⋈T=(T̃×12n)-(12n×T̃) and
(1.3)Spez(Mk(2n)):={F∈Mk(2n)|F|k⋈T=0,∀T∈Hn}.
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.(1.4)Spez(Mk(2))=MkMaass.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:(1.5)Dk,2ν̲:Mk(2)⟶Mk+2νSym,
here ν∈ℕ0 and Mk+2νSym=Sym2(Mk+2ν). To simplify the notation, let Mk:=Mk(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. LetF1, F2, and F3 be a Hecke eigenbasis of the space of Siegel cusp forms S20(2) of weight 20 and degree 2. Let F1 and F2 generate the Maass Spezialschar. Let f1 and f2 be the normalized Hecke eigenbasis of S24(1). Then we have
(1.6)D20,4̲Fj=αjf1⊗f1+βj(f1⊗f2+f2⊗f1)+γjf2⊗f2.
It it conjectured by Gross and Prasad [7] that the coefficients αj, βj, and γj 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 Fj∈S20Maass if and only if the special value βj is zero.Theorem 1.2.
Letk∈ℕ0 be even. Then we have the embedding
(1.7)Dk=⨁ν=0⌊k/10⌋Dk,k+2ν̲:Mk(2)⟶MkSym⊕Sk+2Sym⊕⋯⊕Sk+2⌊k/10⌋Sym.
For F∈Sk(2), we have 𝒟k,0̲F∈SkSym.Surprisingly the Maass Spezialschar property can be recovered inMkSym⊕Sk+2Sym⊕⋯⊕Sk+2⌊k/10⌋Sym in the following transparent way. Let (fj) be the normalized Hecke eigenbasis of Mk. Let us define the diagonal subspaces MkD={∑jαjfj⊗fj∈MkSym} and SkD=SkSym⋂MkD. Then we can state the following theorem.Theorem 1.3.
LetF∈Mk(2).  Then we have
(1.8)F∈MkMaass⟺DkF∈MkD⊕Sk+2D⊕⋯⊕Sk+2⌊k/10⌋D,
and similarly
(1.9)F∈SkMaass⟺DkF∈⨁ν=0⌊k/10⌋Sk+2νD.These two theorems give a transparent explanation of our example from a general point of view. In Section5, we deduce an application related to a multiplicity one theorem of SL2 instead of GL(2).Notation 1.
LetZ∈ℂn,n and tr the trace of a matrix; then we put e{Z}=e2πi(trZ). For l∈ℤ, we define πl=(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/2r/2m) with T=[n,r,m]. The subset of positive-definite matrices we denote with 𝔸2+.


## 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 polynomialpk,2ν̲. Let k and ν be elements of ℕ0. Let a and b be elements of a commutative ring. Then we put
(2.1)pk,2ν̲(a,b)=∑μ=0ν(-1)μ(2ν)!μ!(2ν-2μ)!(k+2ν-μ-2)!(k+ν-2)!a2ν-2μbμ.
If we specialize the parameters, we have pk,0̲(a,b)=1 and pk,2ν̲(0,0)=0 for ν∈ℕ.Letℍn be the Siegel upper half-space of degree n. Let Mk(n) be the vector space of Siegel modular forms on ℍn with respect to the full modular group Γn=Spn(ℤ). Moreover, let Sk(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,τ̃) for (τzzτ̃)∈ℍ2 and put q=e{τ},ξ=e{z}, and q̃=e{τ̃}. Let dk̂ be the dimension of Sk.Definition 2.1.
Letk,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:
(2.2)Dk,2ν̲F(τ,τ̃)=pk,2ν̲(12πi∂∂z,(12πi)2∂∂τ∂∂τ̃)F|z=0(τ,τ̃).
In the case ν=0, we get the pullback F(τ,0,τ̃) of F on ℍ×ℍ.
LetF∈Mk(2) with Tth Fourier coefficient AF(n,r,m) for T=[n,r,m]∈𝔸2. Then we have
(2.3)Dk,2ν̲F(τ,τ̃)=∑n,m=0∞A2ν̲F(n,m)qnq̃mwithA2ν̲F(n,m)=∑r∈Z,r2⩽4nmpk,2ν̲(r,nm)AF(n,r,m).
Let MkSym=Sym2Mk,SkSym=Sym2Sk, and SkSym=(Sk⊗Sk)Sym. Let us further introduce a related Jacobi differential operator  𝒟k,2ν̲J,m. This is given by exchanging π-1(∂/∂τ̃) with m in the definition of the ultraspherical differential operator given in (2.2). Applying the operator 𝒟k,2ν̲J,m on Jacobi forms Φ∈Jk,m of weight k and index m on ℍ×ℂ matches with the effect of the operator 𝒟2ν introduced in [9, Section 3 formula (2)] on Φ.
SinceF∈Mk(2) has a Fourier-Jacobi expansion of the form
(2.4)F(τ,z,τ̃)=∑m=0∞ΦmF(τ,z)q̃m,withΦmF∈Jk,m,
it makes sense to consider 𝒟k,2ν̲ with respect to this decomposition in a Fourier-Jacobi expansion
(2.5)Dk,2ν̲=⨁m=0∞Dk,2ν̲J,m.Lemma 2.2.
Letk,ν∈ℕ0 and let k be even. Then 𝒟k,2ν̲ maps Mk(2) to Mk+2νSym and to Sk+2νSym if ν≠0. Moreover, the subspace Sk(2) of cusp forms is always mapped to Sk+2νSym.Proof.
We know from the work of Eichler and Zagier [9] since 𝒟k,2ν̲J,m=𝒟k,2ν that 𝒟k,2ν̲J,mΦmF∈Mk+2ν. Let ν>0; then 𝒟k,2ν̲J,mΦmF∈Sk+2ν and for F∈Sk(2) we have 𝒟k,2ν̲J,mΦmF∈Sk+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
(2.6)Dk,2ν̲F(τ,τ̃)=∑m=0∞(Dk,2ν̲J,mΦmF)(τ)q̃m,
where all “coefficients” amF(τ)=𝒟k,2ν̲J,mΦmF(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 𝒟k,2ν̲F(τ,τ̃)=∑i,jαi,jfi(τ)fj(τ̃). Here (fi)i is a basis of Mk+2ν. Finally the cuspidal conditions in the lemma also follow from symmetry arguments.Remark 2.3.
LetF:ℍ2→ℂ be holomorphic. Let g∈SL2(ℝ) and let S=(T00T), where T=(0110). Then we have
(2.7)Dk,2ν̲(F|k(g×12))=(Dk,2ν̲F)|k+2ν(g×12),Dk,2ν̲(F|k(12×g))=(Dk,2ν̲F)|k+2ν(12×g),Dk,2ν̲(F|kS)(τ,τ̃)=(Dk,2ν̲F)|k+2ν(τ̃,τ).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 operators𝒟k,2ν̲ can be seen at this point as somewhat artificial. If we apply 𝒟k,2ν̲ to Siegel modular forms F, we lose information. For example, we know that dimS20(2)=3 and contains a two-dimensional subspace of Saito-Kurokawa lifts. Since dimS20Sym=1, we obviously lose information if we apply 𝒟20,0̲. 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 𝒟20,0̲Fj≠0 for j=1,2,3. The general case seems to be even worse, since for example, dimMk(2)~k3 and dimMkSym~k2. On the other hand, from an optimistic viewpoint we may find about k pieces 𝒟k,2ν̲F 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.Letk∈ℕ0 be even. Let F∈Mk(2) and Φ∈Jk,m. Then we denote by
(3.1)F(τ,z,τ̃)=∑ν=0∞χ2νF(τ,τ̃)z2ν,Φ(τ,z)=∑ν=0∞χ2νΦ(τ)z2ν,
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 χ2ν0 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
(3.2)γk,2ν̲=(12πi)2ν(k+2ν-2)!(2ν)!(k+2ν/2-2)!.
Further we put
(3.3)χ2νμ,μ=∂2μ∂τμ∂τ̃μχ2νF,ξ2νμ,μ=(γk,2ν̲)-1∂2μ∂τμ∂τ̃μDk,2ν̲F.
Then a straightforward calculation leads to the following useful formula.Lemma 3.1.
Letk,ν∈ℕ0 and let k be even. Let F∈Mk(2). Then we have
(3.4)(Dk,2ν̲F)(τ,τ̃)=γk,2ν̲∑μ=0ν(-1)μ(k+2ν-μ-2)!(k+2ν-2)!μ!(∂2μχ2ν-2μF∂τμ∂τ̃μ).
A similar formula is valid for Jacobi forms with normalizing factor γk,2ν̲J,m=γk,2ν̲.Corollary 3.2.
Let2ν0 be the vanishing order of F∈Mk(2). Then we have 𝒟k,2ν̲F=0 for ν<ν0 and
(3.5)Dk,2ν0̲F(τ,τ̃)=γk,2ν̲χ2ν0F(τ,τ̃)∈Mk+2ν0Sym∖{0}.
Similarly, we have for Φ∈Jk,m with vanishing order 2ν0 the properties 𝒟k,2ν̲J,mΦ=0 for ν<ν0 and 𝒟k,2ν0̲J,mΦ=γk,2ν̲χ2ν0Φ∈Mk+2ν0.Example 3.3.
It is well known thatdimS10(2)=1. Let Φ=Φ10∈S10(2) be normalized in such a way that AΦ(1,1,1)=1. Then it follows from 𝒟10,0̲Φ=0 that AΦ(1,0,1)=-2 since dimS10Sym=0. Then Φ has the Taylor expansion
(3.6)Φ10(τ,z,τ̃)=35π2Δ(τ)Δ(τ̃)z2+Δ′(τ)Δ′(τ̃)z4+O(z6),
we can also express the Taylor coefficients χ2νF in terms of the modular forms 𝒟k,2ν̲F. This can be done by inverting the formula (3.4). Finally we get
(3.7)χ2ν=∑μ=0ν(k+2ν-2μ-1)!(k+2ν-μ-1)!μ!ξ2ν-2μμ,μ.
Before we state our first main result about the entropy of the family 𝒟k,0̲F,𝒟k,2̲F,𝒟k,4̲F,… we introduce some further notation. 							(3.8)Wk=SkSym⊕⨁j=1⌊k/10⌋Sk+2jSym,Wkcusp=SkSym⊕⨁j=1⌊k/10⌋Sk+2jSym.
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.
Letk∈ℕ0 be even. Then we have the linear embedding
(3.9)Dk:{Mk(2)↪Wk,F⟼⨁ν=0⌊k/10⌋Dk,2ν̲F.
Since 𝒟k,0̲Sk(2) is cuspidal we have the embedding of Sk(2) into 𝕎kcusp.Remark 3.5.
It can be deduced from [14] that 𝒟k,0̲⊕𝒟k,2̲ is surjective. Hence for k<20 we have  (i)
Mk(2) is isomorphic to Mk for k<10;(ii)
Mk(2) is isomorphic to MkSym⊕SkSym for 10⩽k<20 and Sk(2)≃Sk⊕Sk+2.Proof.
First of all we recall that we have already shown that𝒟k,0̲Mk(2)⊆MkSym and 𝒟k,2ν̲Mk(2)⊆Sk+2νSym for ν>0. Let F∈Mk(2) and suppose that 𝔻kF is identically zero. Then it would follow from our inversion formula (3.7) that
(3.10)F(τ,z,τ̃)=∑ν=⌊k/10⌋+1∞χ2νF(τ,τ̃)z2ν.
For such F the general theory of Siegel modular forms of degree 2 says that the special function Φ10∈Sk(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=tk>0. Hence there exists a Siegel modular form G of weight k-10tk. 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 𝔻kF≡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.
LetEk2,1(f) be a Klingen Eisenstein series attached to f∈Sk. Let Ek denote an elliptic Eisenstein series of weight k. Then it can be deduced from [12] that 𝒟k,0̲Ek2,1(f)=f⊗Ek+Ek⊗f mod SkSym.Remark 3.8.
It would be interesting to have a different proof of the Theorem3.4 independent of the special properties of Φ10.Remark 3.9.
The asymptotic limit of the dimension of the quotient of𝕎k/Mk(2) is equal to 91⁄25. Let us put dk=dimMk.  (i)
The dimension of the target space𝕎k is given by
(3.11)dim⁡⁡Wk~1288∫0k/10(k+2x)2dx~128812⋅39153k3.(ii)
The asymptotic dimension formula ofMk(2) is given by
(3.12)dim⁡⁡Mk(2)~128812⋅3⋅5k3.
(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 𝕎k 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

LetJk,m be the space of Jacobi forms of weight k and index m. We denote the subspace of cusp forms with Jk,mcusp. Let |k,m  be the slash operator for Jacobi forms and Vl (l∈ℕ0) the operator, which maps Jk,m to Jk,ml. More precisely, let Φ(τ,z)=∑c(n,r)qnξr∈Jk,m. Then (Φ|k,mVl)(τ,z)=∑c*(n,r)qnξr with
(4.1)c*(n,r)=∑a|(n,r,l)ak-1c(nla2,ra),forl∈N,
and for l=0, we have c*(0,0)=c(0,0)(-2k/B2k) and for l=0 and n>0 we have c*(n,r)=c(0,0)σk-1(n). This includes the theory of Eisenstein series in a nice way [9].Definition 4.1.
The lifting𝒱 is given by the linear map
(4.2)V:{Jk,1⟶Mk(2),Φ⟼∑l=0∞(Φ|k,1Vl)q̃l.
The image of this lifting is the Maass Spezialschar MkMaass of weight k. The subspace of cusp forms we denote with SkMaass.Remark 4.2.
(i)
The lifting is invariant by the Klingen parabolic ofSp2(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)
LetΦ∈Jk,m and l,μ∈ℕ0. Then we have
(4.3)Dk,2μ̲J,ml(Φ|k,mVl)=(Dk,2μ̲J,mΦ)|kTl.
Here Tl is the Hecke operator on the space of elliptic modular forms.(iv)
LetF∈MkMaass be the lift of Φ∈Jk,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.
LetF∈Mk(2) be a Siegel modular form. Then the following properties are equivalent.Arithmetic
LetA(n,r,m) denote the Fourier coefficients of F then
(4.4)A(n,r,m)=∑d|(n,r,m)dk-1A(nmd2,rd,1).Lifting
LetΦ1F be the first Fourier-Jacobi coefficient of F. Then all other Fourier-Jacobi coefficients satisfy the identity
(4.5)ΦmF=Φ1F|k,1Vm.
Let F∈Sk(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 of𝕎k and the Proof of Theorem 1.3

Let(fj) be the normalized Hecke eigenbasis of Mk. With this notation, we introduce the diagonal space
(4.6)MkD={∑jαjfj⊗fj∈MkSym},
and the corresponding cuspidal subspace SkD. Now we are ready to distinguish the Maass Spezialschar in the vector spaces 𝕎k and 𝕎kcusp. This leads to Theorem 1.3 stated in the introduction. Before we give a proof we note the following.Remark 4.4.
(i) Theorem1.3 describes a link between Siegel modular forms and elliptic Hecke eigenforms.
(ii) LetF∈M20(2) and let (fj) be a Hecke eigenbasis of S24. Then F∈MkMaass if and only if
(4.7)D20,4̲F=α0E24⊗E24+αf1⊗f1+γf2⊗f2;
here α0,α,γ∈ℂ.Proof.
We first show that ifF is in the Maass Spezialschar then 𝒟k,2ν̲F is an element of the diagonal space. Let ν∈ℕ0 and let Φ1F be the first Fourier-Jacobi coefficient of F. Then we have
(4.8)(Dk,2ν̲(VΦ))(τ,τ̃)=∑l=0∞(Dk,2ν̲J,l(Φ|k,1Vl))(τ)q̃l.
Here we applied the Fourier-Jacobi expansion of the differential operator 𝒟k,2ν̲ acting on Siegel modular forms. Then we used the formula (4.3) to interchange the operators 𝒟k,2ν̲J,l and Vl to get
(4.9)(Dk,2ν̲F)(τ,τ̃)=∑l=0∞(Dk,2ν̲J,lΦ)|kTlq̃l.
Now let (fjk+2ν̲)j=1d̂k+2ν be a normalized Hecke eigenbasis of Sk+2ν. Let 1≤j1,j2≤d̂k+2ν. Then we have
(4.10)〈(Dk,2ν̲F)(*,τ̃),fj1k+2ν̲〉=〈k(Dk,2ν̲J,lΦ),fj1k+2ν̲〉fj1k+2ν̲,
which leads to the desired result
(4.11)〈(Dk,2ν̲F),fj1k+2ν̲⊗fj2k+2ν̲〉=0,forj1≠j2.
It remains to look at the Eisenstein part if ν=0. Since the space of Eisenstein series has the basis Ek and is orthogonal to the functions given in (4.10), we have proven that the Spezialschar property of F implies that 𝔻kF∈𝕎kD.
Now let us assume thatF∉MkMaass. Then we show that 𝔻kF∉𝕎kD. Since the map
(4.12)(Dk,0̲⊕Dk,2̲):MkMaass⟶MkD⊕Sk+2D
is an isomorphism, we can assume that (𝒟k,0̲⊕𝒟k,2̲)(F) projected on MkD⊕Sk+2D is identically zero. Altering F by an element of the Maass Spezialschar does not change the property we have to prove. If 𝒟k,0̲F∉MkD or 𝒟k,2̲F∉Sk+2D, we are done; otherwise we can assume that
(4.13)(Dk,0̲⊕Dk,2̲)(F)≡0.
Then we have the order F=2ν0⩾4 and k⩾20, since F∉MkMaass. Let
(4.14)F(τzzτ̃)=∑ν=ν0∞χ2ν0F(τ,τ̃)z2ν
be the Taylor expansion of F with χ2ν0F(τ,τ̃)∈Sk+2ν0 not identically zero. Let Φ10∈S10(2) be the Siegel cusp form of weight 10 and degree 2. It has the properties that χ0Φ10≡0 and χ2Φ10(τ,τ̃)=cΔ(τ)Δ(τ̃) with c≠0. Since order F=2ν0, we also have
(4.15)Φ10ν0‖F.
This means that there exists a G∈Sk-10ν0 such that χ0G is nontrivial and
(4.16)F=(Φ10)ν0G.
Hence we have for the first nontrivial Taylor coefficient of F, the formula
(4.17)χ2ν0F(τ,τ̃)=(χ2Φ10(τ,τ̃))ν0χ0G(τ,τ̃),=cν0Δ(τ)ν0Δ(τ̃)ν0χ0G(τ,τ̃).
And the coefficient a1(τ̃) of q is identically zero. Now let us assume for a moment that χ2ν0F∈Sk+2ν0D. Then we have
(4.18)χ2ν0F(τ,τ̃)=∑l=1d̂k+2ναlflk+2ν0̲(τ)flk+2ν0̲(τ̃),
and the coefficient of q is given by ∑l=1d̂k+2ν0αlflk+2ν0̲(τ̃). Since (flk+2ν0̲)l=1d̂k+2ν0 is a basis we have α1=⋯=αd̂k+2ν0=0. But since we assumed that orderF=2ν0, we have a reductio ad absurdum. Hence we have shown that χ2ν0F∉Sk+2ν0 which proves our theorem.Corollary 4.5.
Klingen Eisenstein series are not in the Maass Spezialschar.Remark 4.6.
Letk be a natural even number. Let F be a Siegel modular form of degree two and weight k. Then we have
(4.19)F∈MkMaass⟺Dk,2ν̲F∈Mk+2νD,∀ν∈N0.


### 4.3. The Spezialschar

LetG+Spn(ℚ) be the rational symplectic group with positive similitude μ. In the sense of Shimura, we attach to Hecke pairs the corresponding Hecke algebras
(4.20)Hn=(Γn,G+Spn(Q)),H0n=(Γn,Spn(Q)).
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 ℋ01. Let g∈G+Spn(ℚ) with similitude μ(g). Then we put
(4.21)g̃=μ(g)-1/2g
to obtain an element of Spn(ℝ). We further extend this to ℋn.Definition 4.7.
LetT∈ℋn. Then we define
(4.22)⋈T=(T̃×12n)-(12n×T̃).
Here × is the standard embedding of (Spn,Spn) into Sp2n.Now we study the action|k⋈T on the space of modular forms of degree 2n for all T∈ℋn or T∈ℋ0n. The first thing we would like to mention is that for F∈Mk(2n) the function F|k⋈T is in general not an element of Mk(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.
Letn and k be natural numbers. Let Mk(2n) be the space of Siegel modular forms of degree 2n and weight k. Then we introduce the Spezialschar corresponding to the Hecke algebras ℋn and ℋ0n. 									(4.23)Spez(Mk(2n))={F∈Mk(2n)|F|k⋈T=0,∀T∈Hn},Spez0(Mk(2n))={F∈Mk(2n)|F|k⋈T=0,∀T∈H0n}.
Moreover, Spez(Sk(2n)) and Spez0(Sk(2n)) are the cuspidal part of the corresponding Spezialschar.It is obvious that these subspaces ofMk(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 Theorem1.1

Proof.
LetF∈Mk(2). Then we have F∈MkMaass if and only if 𝒟k,2ν̲F∈Mk+2νD for all ν∈ℕ0. This follows from Remark 4.6. On the other side, the property 𝒟k,2ν̲F∈Mk+2νD is equivalent to the identity
(4.24)(Dk,2ν̲F)|k+2ν⋈T=0,∀T∈H.
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 𝒟k,2ν̲ and the Petersson slash operator |*. This leads to
(4.25)Dk,2ν̲F∈Mk+2νD⟺Dk,2ν̲(F|k⋈T)=0.
So finally it remains to show that if 𝒟k,2ν̲(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(τzzτ̃) with respect to z around 0, we get with the same argument as given in the proof of Theorem 3.4 the desired result.


## 4.1. Basics of the Maass Spezialschar

LetJk,m be the space of Jacobi forms of weight k and index m. We denote the subspace of cusp forms with Jk,mcusp. Let |k,m  be the slash operator for Jacobi forms and Vl (l∈ℕ0) the operator, which maps Jk,m to Jk,ml. More precisely, let Φ(τ,z)=∑c(n,r)qnξr∈Jk,m. Then (Φ|k,mVl)(τ,z)=∑c*(n,r)qnξr with
(4.1)c*(n,r)=∑a|(n,r,l)ak-1c(nla2,ra),forl∈N,
and for l=0, we have c*(0,0)=c(0,0)(-2k/B2k) and for l=0 and n>0 we have c*(n,r)=c(0,0)σk-1(n). This includes the theory of Eisenstein series in a nice way [9].Definition 4.1.
The lifting𝒱 is given by the linear map
(4.2)V:{Jk,1⟶Mk(2),Φ⟼∑l=0∞(Φ|k,1Vl)q̃l.
The image of this lifting is the Maass Spezialschar MkMaass of weight k. The subspace of cusp forms we denote with SkMaass.Remark 4.2.
(i)
The lifting is invariant by the Klingen parabolic ofSp2(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)
LetΦ∈Jk,m and l,μ∈ℕ0. Then we have
(4.3)Dk,2μ̲J,ml(Φ|k,mVl)=(Dk,2μ̲J,mΦ)|kTl.
Here Tl is the Hecke operator on the space of elliptic modular forms.(iv)
LetF∈MkMaass be the lift of Φ∈Jk,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.
LetF∈Mk(2) be a Siegel modular form. Then the following properties are equivalent.Arithmetic
LetA(n,r,m) denote the Fourier coefficients of F then
(4.4)A(n,r,m)=∑d|(n,r,m)dk-1A(nmd2,rd,1).Lifting
LetΦ1F be the first Fourier-Jacobi coefficient of F. Then all other Fourier-Jacobi coefficients satisfy the identity
(4.5)ΦmF=Φ1F|k,1Vm.
Let F∈Sk(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 of𝕎k and the Proof of Theorem 1.3

Let(fj) be the normalized Hecke eigenbasis of Mk. With this notation, we introduce the diagonal space
(4.6)MkD={∑jαjfj⊗fj∈MkSym},
and the corresponding cuspidal subspace SkD. Now we are ready to distinguish the Maass Spezialschar in the vector spaces 𝕎k and 𝕎kcusp. This leads to Theorem 1.3 stated in the introduction. Before we give a proof we note the following.Remark 4.4.
(i) Theorem1.3 describes a link between Siegel modular forms and elliptic Hecke eigenforms.
(ii) LetF∈M20(2) and let (fj) be a Hecke eigenbasis of S24. Then F∈MkMaass if and only if
(4.7)D20,4̲F=α0E24⊗E24+αf1⊗f1+γf2⊗f2;
here α0,α,γ∈ℂ.Proof.
We first show that ifF is in the Maass Spezialschar then 𝒟k,2ν̲F is an element of the diagonal space. Let ν∈ℕ0 and let Φ1F be the first Fourier-Jacobi coefficient of F. Then we have
(4.8)(Dk,2ν̲(VΦ))(τ,τ̃)=∑l=0∞(Dk,2ν̲J,l(Φ|k,1Vl))(τ)q̃l.
Here we applied the Fourier-Jacobi expansion of the differential operator 𝒟k,2ν̲ acting on Siegel modular forms. Then we used the formula (4.3) to interchange the operators 𝒟k,2ν̲J,l and Vl to get
(4.9)(Dk,2ν̲F)(τ,τ̃)=∑l=0∞(Dk,2ν̲J,lΦ)|kTlq̃l.
Now let (fjk+2ν̲)j=1d̂k+2ν be a normalized Hecke eigenbasis of Sk+2ν. Let 1≤j1,j2≤d̂k+2ν. Then we have
(4.10)〈(Dk,2ν̲F)(*,τ̃),fj1k+2ν̲〉=〈k(Dk,2ν̲J,lΦ),fj1k+2ν̲〉fj1k+2ν̲,
which leads to the desired result
(4.11)〈(Dk,2ν̲F),fj1k+2ν̲⊗fj2k+2ν̲〉=0,forj1≠j2.
It remains to look at the Eisenstein part if ν=0. Since the space of Eisenstein series has the basis Ek and is orthogonal to the functions given in (4.10), we have proven that the Spezialschar property of F implies that 𝔻kF∈𝕎kD.
Now let us assume thatF∉MkMaass. Then we show that 𝔻kF∉𝕎kD. Since the map
(4.12)(Dk,0̲⊕Dk,2̲):MkMaass⟶MkD⊕Sk+2D
is an isomorphism, we can assume that (𝒟k,0̲⊕𝒟k,2̲)(F) projected on MkD⊕Sk+2D is identically zero. Altering F by an element of the Maass Spezialschar does not change the property we have to prove. If 𝒟k,0̲F∉MkD or 𝒟k,2̲F∉Sk+2D, we are done; otherwise we can assume that
(4.13)(Dk,0̲⊕Dk,2̲)(F)≡0.
Then we have the order F=2ν0⩾4 and k⩾20, since F∉MkMaass. Let
(4.14)F(τzzτ̃)=∑ν=ν0∞χ2ν0F(τ,τ̃)z2ν
be the Taylor expansion of F with χ2ν0F(τ,τ̃)∈Sk+2ν0 not identically zero. Let Φ10∈S10(2) be the Siegel cusp form of weight 10 and degree 2. It has the properties that χ0Φ10≡0 and χ2Φ10(τ,τ̃)=cΔ(τ)Δ(τ̃) with c≠0. Since order F=2ν0, we also have
(4.15)Φ10ν0‖F.
This means that there exists a G∈Sk-10ν0 such that χ0G is nontrivial and
(4.16)F=(Φ10)ν0G.
Hence we have for the first nontrivial Taylor coefficient of F, the formula
(4.17)χ2ν0F(τ,τ̃)=(χ2Φ10(τ,τ̃))ν0χ0G(τ,τ̃),=cν0Δ(τ)ν0Δ(τ̃)ν0χ0G(τ,τ̃).
And the coefficient a1(τ̃) of q is identically zero. Now let us assume for a moment that χ2ν0F∈Sk+2ν0D. Then we have
(4.18)χ2ν0F(τ,τ̃)=∑l=1d̂k+2ναlflk+2ν0̲(τ)flk+2ν0̲(τ̃),
and the coefficient of q is given by ∑l=1d̂k+2ν0αlflk+2ν0̲(τ̃). Since (flk+2ν0̲)l=1d̂k+2ν0 is a basis we have α1=⋯=αd̂k+2ν0=0. But since we assumed that orderF=2ν0, we have a reductio ad absurdum. Hence we have shown that χ2ν0F∉Sk+2ν0 which proves our theorem.Corollary 4.5.
Klingen Eisenstein series are not in the Maass Spezialschar.Remark 4.6.
Letk be a natural even number. Let F be a Siegel modular form of degree two and weight k. Then we have
(4.19)F∈MkMaass⟺Dk,2ν̲F∈Mk+2νD,∀ν∈N0.


## 4.3. The Spezialschar

LetG+Spn(ℚ) be the rational symplectic group with positive similitude μ. In the sense of Shimura, we attach to Hecke pairs the corresponding Hecke algebras
(4.20)Hn=(Γn,G+Spn(Q)),H0n=(Γn,Spn(Q)).
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 ℋ01. Let g∈G+Spn(ℚ) with similitude μ(g). Then we put
(4.21)g̃=μ(g)-1/2g
to obtain an element of Spn(ℝ). We further extend this to ℋn.Definition 4.7.
LetT∈ℋn. Then we define
(4.22)⋈T=(T̃×12n)-(12n×T̃).
Here × is the standard embedding of (Spn,Spn) into Sp2n.Now we study the action|k⋈T on the space of modular forms of degree 2n for all T∈ℋn or T∈ℋ0n. The first thing we would like to mention is that for F∈Mk(2n) the function F|k⋈T is in general not an element of Mk(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.
Letn and k be natural numbers. Let Mk(2n) be the space of Siegel modular forms of degree 2n and weight k. Then we introduce the Spezialschar corresponding to the Hecke algebras ℋn and ℋ0n. 									(4.23)Spez(Mk(2n))={F∈Mk(2n)|F|k⋈T=0,∀T∈Hn},Spez0(Mk(2n))={F∈Mk(2n)|F|k⋈T=0,∀T∈H0n}.
Moreover, Spez(Sk(2n)) and Spez0(Sk(2n)) are the cuspidal part of the corresponding Spezialschar.It is obvious that these subspaces ofMk(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 Theorem1.1

Proof.
LetF∈Mk(2). Then we have F∈MkMaass if and only if 𝒟k,2ν̲F∈Mk+2νD for all ν∈ℕ0. This follows from Remark 4.6. On the other side, the property 𝒟k,2ν̲F∈Mk+2νD is equivalent to the identity
(4.24)(Dk,2ν̲F)|k+2ν⋈T=0,∀T∈H.
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 𝒟k,2ν̲ and the Petersson slash operator |*. This leads to
(4.25)Dk,2ν̲F∈Mk+2νD⟺Dk,2ν̲(F|k⋈T)=0.
So finally it remains to show that if 𝒟k,2ν̲(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(τzzτ̃) with respect to z around 0, we get with the same argument as given in the proof of Theorem 3.4 the desired result.


## 4.3.1. Proof of Theorem1.1

Proof.
LetF∈Mk(2). Then we have F∈MkMaass if and only if 𝒟k,2ν̲F∈Mk+2νD for all ν∈ℕ0. This follows from Remark 4.6. On the other side, the property 𝒟k,2ν̲F∈Mk+2νD is equivalent to the identity
(4.24)(Dk,2ν̲F)|k+2ν⋈T=0,∀T∈H.
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 𝒟k,2ν̲ and the Petersson slash operator |*. This leads to
(4.25)Dk,2ν̲F∈Mk+2νD⟺Dk,2ν̲(F|k⋈T)=0.
So finally it remains to show that if 𝒟k,2ν̲(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(τzzτ̃) 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 algebrasℋ and ℋ0 related to elliptic modular forms. For the corresponding Spezialschar Spez(Mk(2)) and Spez0(Mk(2)), we obtain the following theorem.Theorem 5.1.
Letk be an even natural number. Then the even Spezialschar Spez0(Mk(2)) related to the Hecke algebra ℋ0 which is locally generated by T(p2) is equal to the Spezialschar Spez(Mk(2)) related to the Hecke algebra ℋ which is locally generated by T(p). 							(5.1)Spez0(Mk(2))=Spez(Mk(2)).Proof.
LetF∈Mk(2). We proceed as follows. In the proof of Theorem 1.1, it has been shown that
(5.2)F∈Spez(Mk(2))⟺(Dk,2ν̲F)|k+2ν⋈T=0,∀T∈H,ν∈N0.
Now we show that
(5.3)(Dk,2ν̲F)|k+2ν⋈T(p)=0⟺(Dk,2ν̲F)|k+2ν⋈T(p2)=0,
for all ν∈ℕ0 and prime numbers p. This would finish the proof since
(5.4)F∈Spez0(Mk(2))⟺(Dk,2ν̲F)|k+2ν⋈T=0,∀T∈H0,ν∈N0.
(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. Letϕ∈MkSym and let ϕ|k⋈T(p2)=0. Let (fj) be a normalized Hecke eigenbasis of Mk. Then we have
(5.5)ϕ=∑i,jαi,jfi⊗fj.
Let us assume that there exists an αi0,j0≠0 with i0≠j0. Let us denote λl(p2) to be the eigenvalue of fl with respect to the Hecke operator T(p2). Then we have
(5.6)0=ϕ|k⋈T(p2)=∑i,jαi,j(λi(p2)-λj(p2))fi⊗fj.
From this follows that λi0(p2)=λj0(p2) for all prime numbers p. It is easy to see at this point that fi0 and fj0 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 fi0=fj0. Since this is a contradiction we have ϕ∈MkD. In other words, we have ϕ|k⋈T(p)=0.



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*Source: 726549-2010-12-05.xml*