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arXiv:1711.06619v1 [math.NT] 17 Nov 2017

The Maaß Space for Paramodular

Groups

by

Bernhard Heim1 and _{Aloys Krieg}2

In this paper we describe a characterization for the Maaß space associated with the paramodular group of degree 2 and squarefree levelN. As an appli-cation we show that the Maaß space is invariant under all Hecke operators. As a consequence we conclude that the associated Siegel-Eisenstein series belongs to the Maaß space.

2010 Mathematics Subject classification: 11F55, 11F46

Keywords: Paramodular group; Maaß space; Hecke operator; Eisenstein series

1_{Bernhard Heim, GUtech, Way No. 36, Building No. 331, North Ghubrah, Muscat, Sultanate of Oman,}

bernhard.heim@gutech.edu.om

2_{Aloys Krieg, Lehrstuhl A f¨ur Mathematik, RWTH Aachen University, D-52056 Aachen,}

1 Introduction

Paramodular forms have first been studied by Siegel [20] in the sixties. Recently paramod-ular forms of degree two and squarefree level N came into focus again. For example

• The paramodular conjecture by Brumer and Kramer [4] • New and old form theory by Roberts and Schmidt [18] • The B¨ocherer conjecture for paramodular forms [19] • Borcherds lifts [3]

Most of the results have in common that a certain subspace of generalized Saito-Kurokawa lifts, the Maaß space, plays a significant role. Although several authors have studied these lifts in the orthogonal setting (cf. Sugano [21] or Oda [17] or [14]) or as a direct generalization of the original approach of Maaß by Gritsenko ([10], [11], [12]), not all results are stated in an explicit form such that they can easily be applied in the works above and that certain refined properties had not been worked out.

In this paper we give a characterization of the Maaß space for the paramodular group similar to [13] in the case of the Siegel modular group. This characterization is applied in order to show that the Maaß space is invariant under the full Hecke algebra described in [8]. As an application we can show that the Siegel-Eisenstein series always belongs to the Maaß space.

2 The Maaß space

Let

H2 :=

Z =

τ z

z τ′

∈C2×2; Z =Ztr,ImZ >0

denote the Siegel half-space of degree 2 and

Sp₂(R) :=_{M _∈R4×4;J[M] :=MtrJM =J_},

J =

0 ₋I

I 0

, I =

1 0 0 1

the symplectic group of degree 2. Sp₂(R) acts onH2 in the usual way

Z 7→MhZi:= (AZ+B)(CZ+D)−1, M =

A B C D

,

and on the set of holomorphic functionsf :H2 →Cfork∈Zvia

f(Z)7→f |

k

GivenN ∈Nlet

ΣN :=

      

   

∗ ∗N _{∗ ∗}

∗ ∗ ∗ ∗/N

∗ ∗N _{∗ ∗}

∗N _∗N _∗N _∗

  

∈Sp2(Q); ∗ ∈Z       

be the paramodular group of degree 2 and level N. A modular form f ∈ Mk(ΣN) is a

holomorphic function f :H2→C satisfying

f |

k

M =f for all M ∈ΣN

and possesses a Fourier expansion of the form

f(Z) =X

T>0

αf(T)e2πitrace (T Z), T =

n r/2

r/2 mN

, n, m∈N0, r∈Z,

4nmN−r2>0,

and a Fourier Jacobi expansion of the form

f(Z) = ∞

X

m=0

f_m∗(Z) = ∞

X

m=0

fmN(τ, z)e2πimN τ

′

, Z =

τ z

z τ′

∈H2,

(1)

where

fmN(τ, z) =

X

n,r

αf(T)e2πi(nτ+rz).

We will refer to both fmN and f_mN∗ as Jacobi forms. Next we define the Maaß space

M∗k(ΣN) to consist of all f ∈ Mk(ΣN) such that for all T 6= 0

αf(T) =

X

δ|gcd(n,r,m)

δk−1αf(nm/δ2, r/δ, N),

(2)

whereαf(n, r, mN) denotes the Fourier coefficient of

_{n r/2}

r/2mN

.

If Jk,N denotes the space of Jacobi forms of weight k and index N in the sense of

Eichler-Zagier [5], Gritsenko [10] demonstrated that the map

ϕ:M∗k(ΣN)→Jk,N, f 7→fN,

is an isomorphism, where we have for even k>4

f0(τ, z) =f0∗(Z) =αf(0,0,0)Ek(τ), αf(0,0, N) =−

2k Bk

αf(0,0,0)

with the normalized elliptic Eisenstein series

Consider the embedding

SL2(R)×SL2(R)֒→Sp2(R),

calledJacobi group inside ΣN given by

ΣJ_N :=

Then the elements fmN ∈Jk,m can be characterized by

f_mN∗ _|

k

M =f_mN∗ for all M _∈ΣJ_N.

We define Hecke operators just as in [7] without any normalizing factors by

f _| the sum is finite.

assertions are equivalent

(i) f _|

kT

↑

p =f | kT

↓

p.

(ii) αf(n, r, pmN) +pk−1αf(n,r_p,m_pN)

=αf(pn, r, mN) +pk−1αf(n_p,r_p, mN) for all n, r, m.

(iii) αf(pνn, pρr, pµmN) =

min(_Xν,ρ,µ)

δ=0

pδ(k−1)αf(pν+µ−2δn, pρ−δr, mN),

αf(pνn,0, pµmN) =

min(_Xν,µ)

δ=0

pδ(k−1)αf(pν+µ−2δn,0, mN),

αf(pνn,0,0) =σk−1(pν)αf(n,0,0),

αf(0,0, pµmN) =σk−1(pµ)αf(0,0, mN) for all (n, r, m) such that p∤n, p∤r, p∤m.

(iv) f _|

k

ΣJ_N √1_pdiag (p, p,1,1) ΣJ_N

=f |

k



ΣJ_N 1_pdiag (p, p2, p,1) ΣJ_N + X

umodp

ΣJ_NI×

1 u/p

0 1

ΣJ_N

 .

Here and in the following we setαf(n, r, mN) = 0 if nor r or mis not integral.

Proof. One can apply the isomorphism between ΣN and O(2,3) (e.g. [16]) and may the

use [14] and [9]. Alternatively we give a short direct proof.

(i)⇔(ii) Use the representatives √1_p p_{0 1}0, √1_{p 0}1b_p, b = 0, . . . , p−1, of M(p) and calculate the Fourier expansion in (i):

f _|

kT

↑

p(Z) =pk/2f

pτ √pz

√_{pz τ′}

+p−k/2 X

umodp

f

(τ+u)/p z/√p z/√p τ′

=p−1+k/2 X

n,r,m

αf(pn, r, mN) +pk−1αf

n

p,rp, mN

e2πi(nτ+rz/√p+mN τ′),

f |

kT

↓

p(Z) =pk/2f

τ √pz

√_{pz pτ′}

+p−k/2 X

umodp

f

τ z/√p

z/√p (τ′+u/N)/p

=p−1+k/2 X

m,n,r

αf(n, r, pmN) +pk−1αf

n,r_p,m_pN e2πi(nτ+rz/√p+mN τ′).

where the caseµ= 0 is clear. If µ>1 assume both ν >1 and ρ>1. Then (ii) yields

αf pνn, pρr, pµmN=αf pν+1n, pρr, pµ−1mN+pk−1αf pν−1n, pρ−1r, pµ−1mN

−pk−1αf pνn, pρ−1r, pµ−2mN

.

By induction hypothesis this equals

min(ν+1_X,ρ,µ−1)

δ=0

pδ(k−1)αf pν+µ−2ρn, pρ−1r, mN+

min(_Xν,ρ,µ)

δ=1

pδ(k−1)αf pν+µ−2δn, pρ−δr, mN

−

min(ν+1_X,ρ,µ−1)

δ=1

pδ(k−1)αf pν+µ−2δn, pρ−δr, mN

=

min(_Xν,µ,ρ)

δ=0

pδ(k−1)αf pν+µ−2δn, pρ−δr, mN.

The remaining cases are dealt with in the same way. (iii)⇒(i) Inserting (iii) we get

αf(pνn, pρr, pµmN)−pk−1αf(pν−1n, pρ−1r, pµ−1mN) =αf(pν+µn, pρr, mN),

which yields the claim.

(i)⇔(iv) Equation (iv) is equal to (i) applied to

Z

1 0

0 √p

=

τ z√p z√p pτ′

instead ofZ because of

ΣJ_N √1_pdiag (p, p,1,1)ΣJ_N

= ΣJ_N √1_pdiag (p, p,1,1) ˙_∪ [˙

umodp

ΣJ_{N √}1

p

1 u

0 p

×

p 0 0 1

.

Since the remaining double cosets consist of a single right coset, the claim follows. As all the elliptic Hecke operators _Tn are polynomials inTp,p∈P, we get

Corollary 1. Given f _{∈ M}k(ΣN), N ∈N the following assertions are equivalent:

(i) f ∈ M∗

k(ΣN). (ii) f _|

kT

↑

n =f | kT

↓

n for alln∈N.

(iii) f _|

kT

↑

p =f | kT

↓

Proof. We get (2) from the application of part (iii) in Lemma 1 to all primes pdividing

m.

The main result of this section is

Theorem 1. A modular formf _{∈ M}k(ΣN),N ∈Nbelongs to the Maaß spaceM∗_k(ΣN)

if and only if there exists a finite setS ⊂P such that

f |

kT

↑

p =f | kT

↓

p for all p∈P, p /∈S.

Proof. “_⇒” Use Corollary 1 withS =_∅.

“⇐” LetM =NQ_q∈Sq. Then Lemma 1 holds for all p∈P,p∤M. IffM denotes the Maaß lift of fN we considerg:=f−fM. It follows from Lemma 1 and (2) thatg has a

Fourier-Jacobi expansion of the form ∞

X

m=0

gmN(τ, z)e2πimN τ

′

,

where at least one prime divisor ofm belongs to S. Hence all the Fourier-Jacobi coeffi-cients gmN vanish, wheneverm is coprime toM. Thus the direct analog of Lemma 3 in

[2] yields g_≡0, hencef =fM _{∈ M}∗

k(ΣN).

3 The maximal normal extension

Σ

b

N

IfN is squarefree andν denotes the number of prime divisors ofN we obtain a maximal normal extension of index 2ν of ΣN given by

b

ΣN =hΣN, Fd;d|Ni ⊂Sp2(R),

where

Fd=

V_d−1 0 0 V_dtr

, Vd=

1 √

d

αd βN

γ δd

, α, β, γ, δ∈Z, αδd−βγN/d= 1.

Note thatF_d2∈ΣN. Instead ofVdwe may choose an arbitrary element of

Γ◦(N)Vd=VdΓ◦(N) =

1 √

d

∗d _∗N

∗ ∗d

∈SL2(R); ∗ ∈Z

,

where

Γ◦(N) =

∗ ∗N

∗ ∗

∈SL2(Z); ∗ ∈Z

.

Lemma 2. Let N ∈N be squarefree and d|N. Then the mapping

M∗k(ΣN)→ Mk∗(ΣN), f 7→f | k

F_d−1(Z) =F(Z[V_dtr]),

is an isomorphism.

Proof. The Fourier coefficient of T in f(Z[Vtr

d ]) is equal to the Fourier coefficient of

T[V_d−1] off, hence byαf(n′, r′, m′N) given by

 

n′ r′

m′

 =

 

d ₋γ γ2N/d

−2N αd+γN/d ₋2αγN

N/d −α α2d

 

 

n r m

 .

(6)

As the latter matrix belongs to SL3(Z) we get

gcd(n′, r′, m′) =gcd(n, r, m).

Then the claim follows from (2).

Next we have a closer look at eigenforms under Fd.

Theorem 2. Let f _{∈ M}∗_k(ΣN), d|N, N ∈N squarefree, ε=±1. Then the following assertions are equivalent

(i) f(Z[V_dtr]) =ε f(Z).

(ii) αf(T[Vd−1]) =ε αf(T) for allT. (iii) αf(n, r, N) =ε αf(n′, r′, N), whenever

4nN₋r2= 4n′N₋r′2, r_{≡ −}r′mod 2d, r _≡r′mod 2N/d.

Proof. Clearly (i) and (ii) are equivalent. In view of (2) it suffices to require (ii) for all

matricesT withm= 1. By

αf

T

1 0

u 1

=αf(T) for u∈Z

we apply (6) in order to get for T =_r/n r/_{2 N}2

αf T[V_d−1]=αf n′′,(αd+γN/d)r, N

where 4n′′N −(αd+γN/d)2r2 = 4nN −r2. In view of

we conclude

(αd+γN/d)r_≡

(

−r mod 2d , r mod 2N/d .

(7)

Thus (ii) becomes equivalent with (iii).

If we haveε= 1 for alldthe consequence is

Corollary 2. A modular form f _{∈ M}∗

k(ΣN), N ∈N squarefree, belongs to Mk(ΣbN) if and only if there exists a functionα∗_f :N0→C such that for all T >0, T 6= 0

αf(T) =

X

δ|gcd(n,r,m)

δk−1α∗_f (4NdetT)/δ2.

Proof. It suffices to show that for f _{∈ M}∗

k(ΣN)∩ Mk(ΣbN)

αf(n, r, N) =αf(n′, r′, N)

holds, whenever

4nN₋r2 = 4n′N ₋r′2.

Elementary number theory shows that r2_≡r′2 _{mod 4N implies}

r_{≡ −}r′ modp or r _≡r′modp for alle primesp_|N and p= 2

r≡ −r′ mod 4 or r≡r′mod 4 if N is even.

Thus we define

d:= Y

p|N, p odd r≡−r′

modp

p if N is odd or N is even andr_≡r′mod 4,

d:= 2 Y

p|N, p odd r≡−r′

modp

p if N is even and r_{≡ −}r′ mod 4.

Thus we get

(αd+γt/d)r_≡r′ mod 2N

in (7). Now the claim follows from Theorem 2.

We give a characterization of cusp forms in Mk(ΣbN), i.e. only positive definite T

appear in the Fourier expansion. If φdenotes the Siegel φ-operator we get

Lemma 3. Given a squarefree N ∈ N and f ∈ Mk(ΣbN) the following assertions are equivalent:

(ii) αf(n,0,0) = 0 for alln∈N0.

(iii) f _|φ_≡0.

Proof. Given T =_r/n_2mNr/2> 0 with detT = 0 and T ₆= 0 there exists U _∈Γ◦_(N)Vd for somed|N such that

T[U] =

t 0 0 0

, t=gcd(n, r, m).

Remarks. a) As we may choose VN = ₋01 0N

, we conclude

f _|

k

WN = (−1)kf

for all f _{∈ M}∗_k(ΣN) in view ofαf(n, r, mM) =αf(m, r, nN) in (2).

b) ΣN corresponds to the discriminant kernel in SO(2,3) (cf. [16]). Moreover the

extension ΣN ∪ΣNWN corresponds to the discriminant kernel in O(2,3).

c) WithVN = ₋01 0N

one easily verifies

WN(M×M′)WN =M′×M

forM, M′∈SL2(R) in (4). Thus one gets

f _|

kT

↓

n =f | k

WN | kT

↑

n | k

WN.

4 Hecke operators

Consider the subgroup

G:=

1 √

nM ∈Sp2(R); n∈N, M ∈Z

4×4

of Sp₂(R) as well as its Jacobi subgroup

GJ :=

      

   

∗ 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 _{∗ ∗} 0 0 0 ∗

   ∈G

      

.

Then (ΣN,G), (ΣbN,G) and (ΣJ_N,GJ) are Hecke pairs (cf. [1], [7], [15]). Denote the

associated Hecke algebras by

The structure ofHN and HbN was described in [8] and moreover the

b

ΣN∩GJ =ΣbN, G⊂ΣbN ·GJ.

Thus the mapping

X

induces an injective homomorphism of the Hecke algebras

ι:HbN → HJN

(8)

(cf. [1], Proposition 3.1.6, [15], I(6.4)).

Theorem 3. Let N _∈ N be squarefree. Then the Maaß space _M∗_k(ΣN) is invariant

under the Hecke algebra_HN.

Proof. According to Lemma 2 it suffices to demonstrate the invariance under _HbN. It

was shown in [8] that_HbN is generated by

TN(q) :=ΣbN√1_qdiag (1,1, q, q)ΣbN and TN∗(q) :=ΣbN1_qdiag (1, q, q2, q)ΣbN, q∈P.

According to [8] representatives of the right cosets inTN(q) are given by

1

Ifq _|N additionally by

are given by

Ifq _|N additionally the right cosets

Ifq |N therefore ι(T_n∗(q)) additionally consists of the double cosets

ΣJ_N 1

qdiag (1, q, q

2_{, q) Σ}J

N +

X

umodq

u6≡0 modq

ΣJ_N 1

q

   

q 0 0 1

0 q 1 u/N

q 0

0 q

   ΣJN

+ ΣJ_N 1

q√q

   

q 0 0 1

0 q2 _{q 0}

q2 0 0 q

   ΣJN +

X

cmodq

ΣJ_N 1

q√q

   

q2 0 0 q

d q 1 qc/N

q 0

0 q2

   ΣJN.

A tedious straightforward calculation shows that ι(TN(q)) andι(TN∗(q)) commute with

ΣJ_N√1_pdiag (p, p,1,1)ΣJ_N, ΣJ_N1_pdiag (p, p2, p,1)ΣJ_N

and

X

umodp

ΣJ_NI_×

1 u/p

0 1

ΣJ_N,

as long as p6=q. Thus the claim follows from Theorem 1.

Now we apply the Hecke operator TN(q) to the Fourier Jacobi expansion.

Corollary 3. Let f ∈ M∗

k(ΣN) for squarefree N ∈N and q∈P,q ∤N. Then

g:=f _|

k

TN(q)∈ M∗k(ΣN)

is characterized by

g_N∗ =f_N∗ _|

k

ΣJ_N 1_qdiag (1, q, q2, q)Σ_NJ + (q2+q)f_N∗.

Proof. It follows from Gritsenko’s work [10] that

f_qN∗ = 1_qf_N∗ _|

k

ΣJ_N √1_qdiag (q, q,1,1)ΣJ_N.

Thus the application of (9) leads to

g∗N =f_qN∗ |

k

ΣJ_N √1_qdiag (1,1, q, q)ΣJ_N = 1_qf_N∗ _|

k

ΣJ_N √1_qdiag (q, q,1,1)ΣJ_{N ·}ΣJ_N √1_qdiag (1,1, q, q)ΣJ_N.

If we use the representatives in (9) and (10), the calculation of the product leads to the result.

Note thatΣbNMΣbN =ΣbNM−1ΣbN for allM ∈Gdue to [8]. Hence the same arguments

Petersson scalar product. As HbN is commutative due to [8], the standard procedure

yields

Corollary 4. Let N _∈Nbe squarefree. Then the space _M∗_k(ΣN)∩ Mk(ΣbN) contains a

basis consisting of simultaneous eigenforms under all Hecke operators in _HbN.

5 The Eisenstein series

Let

ΣN,∞:=

A B

0 D

∈ΣN

and define for evenk >3 the Siegel-Eisenstein series

Ek,N(Z) :=

X

M:ΣN,∞\ΣN

det(CZ+D)−k.

We have

W_d−1ΣN,∞Wd= ΣN,∞ and Wd−1ΣNWd= ΣN

and thereforeEk,N | k

Wd=Ek,N for alld|N. HenceEk,N ∈ Mk(ΣbN) holds. We use an

analog of Elstrodt’s result [6] in order to obtain

Theorem 4. Let N _∈N be squarefree and k >3 be even. Then

Ek,N ∈ M∗k(ΣN)∩ Mk(ΣbN).

Proof. AsJk,N contains a non-cusp form, we conclude the existence of anf ∈ M∗k(ΣN)

with non-zero constant Fourier coefficient from [10]. Due to Lemma 2 we may assume

f ∈ Mk(ΣbN) if we replace f by P d|N

f |

k

Wd. By Corollary 4 we may even assume

αf(0,0,0) = 1 and f | k

b

ΣN √1_qdiag (1,1, q, q)ΣbN =λf

for some prime q ∤N and eigenvalue λ. As Ek,N is an eigenform, too, (cf. [7], IV.4.7)

with the same eigenvalue (cf. [7], IV.4.6)

λ=qk+q2+q+q3−k,

this is also true for g:=f −Ek,N. Using Lemma 1 we conclude that f |φand Ek,N |φ

coincide with the normalized elliptic Eisenstein series.

Hence g is a cusp form by Lemma 3. Ifg₆= 0, we get (cf. [7], IV.4.8)

|λ_|6♯ΣN\ΣN√1_qdiag (1,1, q, q)ΣN

= 1 +q+q2+q3 < q4.

If k>4 is even we define the Jacobi Eisenstein series by

is the translation subgroup of ΣJ

N. Note that

Proof. By construction e∗

k,N is invariant under ΣJN. IfKi,i∈I, denotes a set of

repre-sentatives of ΣJ

n\ΣJN 1qdiag (1, q, q2, q)ΣJN contained in (10) and Mj, j ∈J, of ΓN\ΣJN,

Now observe that

˙

Hence the claim follows.

Finally we derive a connection with Ek,N and the classical Siegel Eisenstein series

Ek,1.

Corollary 6. Let k>4 be even and N _∈Nsquarefree.

a) The N-th Fourier Jacobi coefficient ofEk,N is given by

−B_2kke∗_k,N.

Proof. a) Combine Corollary 4, Theorem 4 and (3).

b) Note that ΣJ_N √1

N diag (1, N, N,1)Σ J

N and ΣJN 1qdiag (1, q, q2, q)ΣJN commute forq∈P,

q ∤N and use the representatives above in order to compute the Fourier coefficient of

0 0

0N

on the right hand side.

Hence the Fourier coefficients of Ek,N can be found in [5], §2.

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