2 Siegel modular forms of half integral weights and Plus space

A lifting conjecture on Siegel modular forms of half integral weights

Shuichi Hayashida and Tomoyoshi Ibukiyama

1 Conjecture

Our main purpose of this short note is to give the conjecture below on a lifting from all pairs of elliptic modular forms of weight 2k−2 and 2k−4 to Siegel modular forms of half integral weights k−1/2 of degree 2. Here the image of the lifting should be in a certain space of ”new” forms. This kind of ”new forms” was first introduced by Kohnen for degree one case as ”plus space”

to give a refinement of Shimura correspondence (cf. [16]). This notion can be generalized to general degree n. We can prove an exact correspondence between Jacobi forms and plus subspace of Siegel modular forms of half integral weights of degree n(known by Eichler-Zagier [2] when n= 1). Then using Tsushima’s dimension formula for Jacobi forms of degree two, we give an explicit structure theorem on Jacobi forms and Siegel modular forms of half integral weights of degree two. By using these cusp forms, we can make a numerical experiment on Euler factors which supports our conjecture.

Conjecture 1.1

Let k be a natural number. For any pair off ∈S_2k−2(SL(2,Z)) and

g ∈S_2k−4(SL(2,Z))which are common eigen forms of Hecke operators, there exists a Hecke eigen form F ∈S_k⁺_−1/2(Γ⁽²⁾₀ (4)) such that

L(s, F) =L(s, f)L(s−1, g).

up to Euler 2 factors, where L(s, F) is the L-function defined in Zhuravlev [22] for Siegel modular forms of half integral weights, and L(s, f) and L(s−1, g) are the usual L-functions of elliptic modular form. . The precise notation will be explained in the following sections.

It is known that the above L(s, F) satisfies a functional equation with center at k−1, and so does the right hand side. Hence this also supports our conjecture.

2 Siegel modular forms of half integral weights and Plus space

We denote by H_n Siegel upper half space of degree n , and by Sp(n,Z) the Siegel full modular group of size 2n . We define the following subgroup of Sp(n,Z).

Γ⁽ⁿ⁾₀ (4) = {

M =

( A B C D

)

∈Sp(n,Z);C ≡0 mod 4 }

.

To explain an automorphy factor of half integral weights, we introduce notation for theta constants. We write e(z) = e^2πix. For m = (m^′, m^′′) (m^′, m^′′ ∈Zⁿ) and Z ∈H_n, we put

θ_m^′_,m^′′(Z) = ∑

p∈Zⁿ

e (1

2

t(p+m^′

2 )Z(p+m^′

2 ) +^t(p+m^′ 2 )m^′′

2 )

.

We put θ(Z) =θ₀₀₀₀(2Z), and we define a character ψ of Γ⁽ⁿ⁾₀ (4) as follows ψ(M) =

( −4 detD

)

, forM = (

A B C D

)

∈Γ⁽ⁿ⁾₀ (4).

We say that a holomorphic functionF onH_n is a Siegel modular form of weight k−1/2 of Γ⁽ⁿ⁾₀ (4) with character χ if F satisfies

F(M Z) = (θ(M Z)/θ(Z))^2k−1χ(M)F(Z)

for anyM ∈Γ⁽ⁿ⁾₀ (4) (and satisfies the cusp condition whenn= 1.) It is called a cusp form if it vanishes at cusps. We denote the space of these modular forms byMk−1/2(Γ⁽ⁿ⁾₀ (4), χ) and the space of cusp forms bySk−1/2(Γ⁽ⁿ⁾₀ (4), χ).

We write the Fourier expansion ofF(Z)∈M_k−1/2(Γ⁽ⁿ⁾₀ (4), ψ^l) as F(Z) =∑

T≥0

a(T)e^{2πi tr(T Z)}.

We say that F belongs to the plus space M_k⁺_−1/2(Γ⁽ⁿ⁾₀ (4), ψ^l) if a(T) = 0 unless T+ (−1)^k+lλ^tλ∈4L^∗_n for some column vector λ∈(Z/2Z)ⁿ, where L^∗_n is the set of n times n half integral symmetric matrix.

Ifn = 1 andl = 0, the above space is Kohnen’s plus space in [16].

3 Plus space and Jacobi forms of index 1

When integerkisevenand the degreen = 1, it was known thatM_k⁺_−1/2(Γ⁽¹⁾₀ (4)) is linearly isomorphic to Jacobi forms of index 1 of weightk as Hecke algebra module. This isomorphism was generalized for higher degree in Ibukiyama [11]. We give here some more generalization, namely for cases of odd k and also “skew” holomorphic cases. Skoruppa [18] defined the notion of skew holomorphic Jacobi forms of degree 1 and for odd integer k, he proved that the skew holomorphic Jacobi forms of weight k is linearly isomorphic to Kohnen’s plus space of weightk−1/2. Arakawa [1] generalized the definition of skew holomorphic Jacobi forms for general degree.

We review Jacobi forms and skew holomorphic Jacobi forms. LetF(τ, z) be a function onH_n×Cⁿ. We sayF is a Jacobi form of weight k of index 1, if F satisfies the following three conditions;

(i) F(τ, z+^tλτ +µ) = e(−^tλτ λ−2^tλz)F(τ, z) for anyλ,µ∈Zⁿ,

(ii) F(M(τ, z)) = det(Cτ +D)^ke(^tZ(Cτ + D)⁻¹CZ)F(τ, z) for any M = (^{A B}_{C D})∈Sp(n,Z), where M(τ, z) = (M τ, ^t(Cτ +D)⁻¹z),

(iii) F satisfies the cusp condition, namely F has Fourier expansion F(τ, z) = ∑

4N− ^trr≥0

A(N, r)e(tr(N τ +^trz)).

Instead of above conditions (i), (ii) and (iii), if F satisfies next three conditions (i), (ii^′) and (iii^′), we say F is a skew holomorphic Jacobi form of weight k of index 1;

(ii^′)F(M(τ, z)) = det(Cτ +D)^k−1|det(Cτ +D)|e(^tZ(Cτ +D)⁻¹CZ)F(τ, z) for any M ∈Sp(n,Z),

(iii^′)F has the Fourier expansion of the following shape:

F(τ, z) = ∑

4N−^trr≤0

A(N, r)e(tr(N τ − i

2(4N −^trr)Im(τ) +^trz)) .

We write the space of Jacobi forms (resp. skew holomorphic Jacobi forms) of weight k of index 1 by J_k,1 (resp. J_k,1^sk).

Theorem 3.1 For general degree, the plus space M_k−1/2⁺ (Γⁿ₀(4), ψ^l) is iso- morphic to Jacobi forms of weight k index 1 or skew holomorphic Jacobi forms of weight k index 1 as Hecke algebra module, according tok+l is even or odd.

M_k⁺_−1/2(Γ₀(4), ψ^l) ∼=

l\k even odd 0 J_k,1 J_k,1^sk 1 J_k,1^sk J_k,1

Moreover, cusp forms of plus space and cusp forms of (skew holomorphic) Jacobi forms are isomorphic by this isomorphism.

When n= 1 the above theorem is due to Eichler-Zagier or Skoruppa.

4 The structure of Plus space of degree 2

Hereafter we treat the case of degree n = 2. All the Siegel modular forms of degree 2 of half integral weights of Γ⁽²⁾₀ (4) are given explicitly, but we omit the description here (See [5].) We explain here only the structure theorem on the plus space of degree two.

Theorem 4.1 The structure of the plus space of degree2 is given as follows,

⊕

l=0,1

⊕

k

M_k⁺_−1/2(Γ⁽²⁾₀ (4), ψ^l) =

⊕16

i=1

A P_ki−1/2 .

where A = C[E₄(4Z), E₆(4Z), χ₁₀(4Z), χ₁₂(4Z)], and E₄, E₆, χ₁₀, χ₁₂ are well known generators of Siegel modular forms of degree 2 of even weights.

The generators P_ki−1/2 ∈M_k⁺

i−1/2(Γ⁽²⁾₀ (4), ψ^l) ( l = 0 for 1≤i≤8 and l = 1 for 9 ≤i≤16 ) are written explicitly by using theta constans. Here weights are given by (k₁, ..., k₁₆) = (1,4,6,7,9,10,12,15,21,24,26,27,29,30,32,35).

We get explicit structure for cusp forms of plus space as follows.

Theorem 4.2

⊕

l=0,1

⊕

k

S_k⁺_−1/2(Γ⁽²⁾₀ (4), ψ^l) = A^cuspP_1/2⊕A^cuspP_7/2⊕A^cuspP_11/2⊕A^cuspP_13/2

⊕A^cuspP_17/2⊕AP_19/2⊕AP_23/2⊕BP_25/2⊕AP_29/2

⊕ ( ₁₆

⊕

i=9

AP_ki−1/2 )

.

where A^cusp is the ideal of cusp forms ofA generated byχ₁₀(4Z)andχ₁₂(4Z), and we put B =C[E₄(4Z), E₆(4Z)]. The form P_25/2 is written by a certain linear combination of generators of theorem 4.1.

For the proof, we use Tsushima’s dimension formula forJk,1 andJ_k,1^sk. We note that by Theorem 3.1, we also have a structure theorem of J_k,1, J_k,1^cusp, J_k,1^sk,J_k,1^sk,cusp from the above theorems. We omit the details here.

5 Examples of Euler factors

In this section, we describe some examples of Euler factors supporting the conjecture. Let F be an element ofM_k−1/2(Γ⁽²⁾₀ (4)), and assumeF is a Hecke eigen form. For each odd prime p, we denote eigen value ofF for the Hecke operatorT(p, p, p³, p³) orT(1, p, p², p) byω(p) orλ(p). Zhuravlev defined the L-function of F by

L(s, F) = ∏

(p,2)=1

H_p(p^−s, F)⁻¹, where

H_p(T, F) = 1−λ(p)T + (pω(p) +p^2k−5(1 +p²))T²−λ(p)p^2k−3T³+p^4k−6T⁴. A numerical examples of the dimensions of plus space of degree 2 and elliptic modular forms are given in the following table.

k :even 0∼6 8 10 12 14 16 18 20 22 24 dimS_k⁺_−1/2(Γ⁽²⁾₀ (4)) 0 0 1 1 2 4 4 6 9 10 dimS_2k−2(SL(2,Z)) 0 0 1 1 1 2 2 2 3 3 dimS_2k−4(SL(2,Z)) 0 1 1 1 2 2 2 3 3 3 We give some examples of Euler factor H3(T, F) at 3.

weight 19/2 (k = 10)

When k = 10, then k−1/2 = 19/2, the dimension ofS_19/2⁺ (Γ⁽²⁾₀ (4)) is 1.

We take P_19/2 ∈S_k⁺_−1/2(Γ⁽²⁾₀ (4)), then we get

H₃(T, P_19/2) = (1 + 10044T + 3¹⁷T²)(1 + 4284T + 3¹⁷T²) .

We denote by ∆₁₆ or ∆₁₈ the normalized Hecke eigen form belonging to SL(2,Z) of weight 16 or 18, respectively. We denote the eigen value atp= 3 by λ(3,∆16), or λ(3,∆18), respectively. We see

λ(3,∆₁₆) =−3348 . λ(3,∆₁₈) = −4284 . So, we get

H₃(3^−s, P_19/2) =L₃(s,∆₁₈)L₃(s−1,∆₁₆)

where L₃ denotes the Euler 3 factors of ∆₁₈, ∆₁₆ respectively. The result for k = 12 (weight 23/2) is similar and we skip it here.

weight 27/2 (k = 14)

When k = 14, then k−1/2 = 27/2, and dimS_27/2⁺ (Γ₀(4)) = 2. We take two Hecke eigen forms χ⁺_27/2, χ⁻_27/2 ∈ S_27/2⁺ (Γ⁽²⁾₀ (4)). Then Euler factors of these forms were given as follows

H₃(T, χ^±_27/2) = (

1−(

509220 ±1728√

144169 )

T + 3²⁵T²) (

1 + 195804T + 3²⁵T²) . We denote two Hecke eigen forms of weight 24 belongs to SL(2,Z) by ∆⁺₂₄

and ∆⁻₂₄. We denote Hecke eigen form of weight 26 belongs to SL(2,Z) by

∆₂₆. Then Euler-3 factors of these forms are given L₃(s,∆^±₂₄) = 1−(

169740 ±576√

144169 )

3^−s+ 3²³3^−2s , L3(s,∆26) = 1 + 195804·3^−s+ 3²⁵3^−2s .

so we have

H3(3^−s, χ^±_27/2) = L3(s,∆26)L3(s−1,∆^±₂₄) . weight 21/2 (k = 11)

We have dimS_21/2⁺ (Γ₀(4)) = 1 and θ χ₁₀ is a cusp form of plus space of weight 21/2.

H₃(T, θ χ⁽²⁾₁₀) = (

1 + 12852T + 1162261467T²) (

1−50652T + 1162261467T²) . λ(3,∆₁₈) = −4284 .

λ(3,∆₂₀) = 50652 . so we get

H₃(3^−s, P_21/2) = L₃(s−1,∆₁₈)L₃(s,∆₂₀).

weight 25/2 (k = 13) We define

χ⁺_25/2 := 217432719360 (

119−√

144169 )

θ χ₁₂(4Z) +P_25/2. χ⁻_25/2 := 217432719360

(

119 +√

144169 )

θ χ₁₂(4Z) +P_25/2.

then χ⁽_25/2^±) are Hecke eigen forms. Euler factor are given

H₃(T, χ⁽_25/2^±) ) = (

1 + 386532T + 94143178827T²)

×( 1−(

169740±576√

144169 )

T + 94143178827T² )

. λ(3,∆₂₂) = −128844.

λ(3,∆⁽₂₄^±)) = 169740±576√

144169. so we get

H₃(3^−s, P_25/2^(±)) = L₃(s−1,∆₂₂)L₃(s,∆⁽₂₄^±)).

All these examples support the validity of our conjecture.

We made the same kind of experiment for all even k with 10 ≤ k ≤ 20 (and k = 11 and 13 as above).

All fit our conjecture.

Here we give samples of the Fourier coefficients we used in the above calculation.

weight 19/2, 23/2

wt19/2, 23/2 P_19/2 P_23/2

(3,3,2) 1 1

(11,8,8) −861 64827

(19,4,4) −3423 188649 (24,3,0) −5022 −115182 (27,27,18) 23088645 −2926756395 weight 27/2

wt27/2 χ₁₀(4τ)P_7/2 E₄(4τ)P_19/2

(3,3,2) 0 1

(11,8,8) −7872 −993261

(19,4,4) 23424 525297

(24,3,0) 0 759618

(27,27,18) 4528944576 281757016485

(4,4,4) 1 14

(12,12,12) 462195 −1345014 (28,4,4) −45822 5392092 (36,36,36) 164646611496 4840269943536

weight 21/2 (k = 11)

wt21/2 θ χ₁₀(4Z)

(4,4,4) 1

(12,12,12) 13959 (28,4,4) 1386 (36,36,36) 14006520 weight 25/2 (k = 13)

wt25/2 θ χ12(4Z) P25/2

(4,4,0) 10 6088116142080

(20,8,8) −838986 −51781277705966714880 (36,4,0) −595758 −30081817593186877440 (36,36,0) 156778877538 18420323633463561415557120

(4,4,4) 1 434865438720

(12,12,12) 63 6170162639268741120

(28,4,4) −154566 −7531343211449548800 (36,36,36) 35411540472 3427786444070184309227520

6 Φ-operator

Finally we add a remark on the commutativity of the action of Hecke oper- ators and the Siegel Φ operator on M_k−1/2(Γ₀(4)). Let F be a Hecke eigen form in M_k−1/2(Γ⁽²⁾₀ (4)) and Φ is the usual Φ-operator defined by

(ΦF)(τ) = lim

λ→∞F (

τ 0 0 iλ.

)

If Φ(F)̸= 0, it is also a modular form of weightk−1/2, and there exists a modular formf of integral weight 2k−2. As an analogue to Zarkovskaya’s result, we get the following theorem.

Theorem 6.1 If F ∈ M_k−1/2(Γ0(4) is a Hecke eigen form and Φ(F) ̸= 0, then Φ(F) is also a common eigen form of weight k−1/2 of all the Hecke operatorsT₁(p²), where T₁(p²)is the usual Hecke operators on modular forms of half integral weights one variable.

We also have

L(s,Φ(F)) = L(s, f)ζ(s−1)ζ(s−2k+ 4) up to Euler 2 factors,

Here we can interpret the factor ζ(s−1)ζ(s−2k+ 4) =L(s−1, E_2k−4) where E_2k−4 is the Eisenstein series of weight 2k−4. This is also similar to our conjecture.

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Department of Mathematics, Graduate School of Science, Osaka University,

Machikaneyama 1-16,

Toyonaka, Osaka, 560 Japan.

ibukiyam@math.wani.osaka-u.ac.jp hayasida@gaia.math.wani.osaka-u.ac.jp