On explicit construction of Hilbert–Siegel modular forms of degree two

LXXXI.3 (1997)

On explicit construction of Hilbert–Siegel modular forms of degree two

by

Hisashi Kojima (Tokyo)

Introduction. Several authors have developed the theory of lifting from the space of modular forms of one variable to that of modular forms on the orthogonal groups attached to quadratic forms over Q (cf. [1, 4–6, 8]).

Shimura [9], [10] dealt with the problem of construction of arithmetic mod- ular forms on orthogonal groups over totally real algebraic number fields.

However, he did not take up the explicit calculation of the Fourier coeffi- cients of lifted modular forms.

On the other hand, in [3], [4] we have established a correspondence Ψ

between the space S

(M, χ) of modular cusp forms of half integral weight (2k − 1)/2 of level M to the space M

(M, χ) of Maass forms of Siegel modular cusp forms of degree two of weight k of level M in such a way that it commutes with the actions of Hecke operators. We evaluated explicitly the Fourier coefficients of Ψ

(f ) with a form f in S

(M, χ), and made clear a coincidence with Shimura’s zeta func- tions attached to f and Andrianov’s zeta functions attached to Ψ

(f ).

We note that these results are closely related to Saito–Kurokawa’s conjec- ture concerning Siegel modular forms of degree two. Using the technique in the theory of group representation of Jacquet and Langlands, Piatetski- Shapiro [7] discussed Saito–Kurokawa’s conjecture in the case of Siegel modular forms on GpSp(2, A

) where A

is the adele ring of an arbi- trary number field F . Unfortunately, it seems that his approach is diffi- cult to use for an explicit calculation of the Fourier coefficients of the lifted forms.

The first purpose of the present note is to show the existence of a corre- spondence Ψ

between Hilbert modular forms f of half integral weight with respect to the principal congruence group and Hilbert–Siegel modular forms Ψ

(f ) of degree two attached to totally real number fields. The second one

1991 Mathematics Subject Classification: 11F27, 11F30, 11F37, 11F41, 11F46.

[265]

is to determine an explicit relation between the Fourier coefficients of f and Ψ

(f ).

Section 1 is a preliminary section. In Section 2, using theta series as- sociated with Weil representations of quadratic forms, we shall construct Hilbert–Siegel modular forms of degree two of integral weight from Hilbert modular forms of half integral weight. In Section 3, we shall derive relations between the Fourier coefficients of those modular forms. Our results can be regarded as a development of those of [3]. We use theta function methods similar to those of Friedberg [2] and Kojima [3] (cf. [1], [5]).

1. Notation and preliminaries. We denote, as usual, by Z, Q, R and C the ring of rational integers, the rational number field, the real number field and the complex number field. For a commutative ring A with the unity 1, we denote by A

the set of m × n matrices with entries in A. Furthermore, we denote by SL

(A) (resp. GL

(A)) the group of all matrices M with det(M ) = 1 (resp. det(M ) ∈ A

), where A

is the group of all invertible elements in A, and set A

= A

and M

(A) = A

for simplicity. Let E

be the unity of GL

(A).

Throughout the paper, we denote by F a totally real algebraic number field of degree l of class number one. We denote by J

the set of all em- beddings of F into C, and by τ

, . . . , τ

the elements of J

. For a ∈ F , we set a

= τ

(a) (1 ≤ i ≤ l). Let r be the ring of all integers in F . Now, we denote by H

the complex upper half space of degree n, i.e.,

H

= {Z ∈ M

(C) |

Z = Z, =(Z) > 0}.

We set H = H

for simplicity. Let Sp(n, R) (resp. Sp(n, F )) be the real symplectic group of degree n (resp. the symplectic group of degree n over F ), i.e.,

Sp(n, R) = {g ∈ GL

(R) |

gJ

g = J

} (resp. Sp(n, F ) = {g ∈ Sp(n, R) ∩ M

(F )}) with

J

=

 0 E

−E

0

 . The group Sp(n, R) acts on H

by

Z → g(Z) = (AZ +B)(CZ +D)



Z ∈ H

, g =

 A B

C D



∈ Sp(n, R)

 . To define Hilbert–Siegel modular forms, we take an arithmetic congru- ence subgroup Γ

(N ) of Sp(n, F ) in the form

Γ

(N ) = {γ ∈ Sp(n, F ) ∩ M

(r) | γ ≡ E

(N )}

with a positive integer N . The group Γ

(N ) can be embedded into Sp(n, R)

(= Sp(n, R) × . . . ×Sp(n, R)) by the mapping γ → (γ

, . . . , γ

) with γ

= (γ

) ∈ Γ

(N ). It is well known that Γ

(N ) acts properly discontinuously on H

by

γ(Z) = (γ

(Z

), . . . , γ

(Z

))

for every γ ∈ Γ

(N ) and Z = (Z

, . . . , Z

) ∈ H

, and the volume of Γ

(N )\H

is finite. We call a holomorphic function F on H

a Hilbert–

Siegel modular form of weight k =

(k

, . . . , k

) (∈ Z

) with respect to Γ

(N ) if the following conditions are satisfied:

(1.1) F (γ(Z)) =

Y

det(C

Z

+ D

)

F (Z) for every γ =

∈ Γ

(N ) and Z = (Z

, . . . , Z

) ∈ H

, and (1.2) F (Z) is finite at each cusp of Γ

(N ).

We denote by M

(Γ

(N )) the space of all such modular forms.

2. Construction of Hilbert–Siegel modular forms of degree two.

For α =

(α

, . . . , α

) ∈ F

, we set α

=

(α

, . . . , α

) (1 ≤ i ≤ l). Then F

can be embedded into R

by α → (α

, . . . , α

) for every α ∈ F

. Let S be a non-degenerate symmetric matrix over F . Let S(R

) be the space of all rapidly decreasing functions on R

. For each f = (f

, . . . , f

) ∈ Q

i=1

S(R

), we define an element γ(σ, S)f in Q

i=1

S(R

) by γ(σ, S)f =

Y

γ(σ

, S

)f

for every σ = (σ

, . . . , σ

) ∈ SL

(R)

, where S

= (S

) (S = (S

)) and γ(σ

, S

) is the Weil representation given in [4]. We consider the following four matrices:

S =



 

 

0 −1 0 0 0

−1 0 0 0 0

0 0 0 −1 0

0 0 −1 0 0

0 0 0 0 −2



 

  , S

=



 0 −1 0

−1 0 0

0 0 −2



 ,

A =



 

 

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 √

2



 

  , A

=



 1 0 0

0 1 0

0 0 √

2



 .

Let χ

be a character of (r/(M ))

with a positive integer M . For each h =

(. . . , h

, . . .) ∈ r

, set

e χ

(h) =

n χ

(h

) if (h

, M ) = 1, 0 otherwise.

For k =

(k

, . . . , k

) ∈ Z

(k

≥ 0) and x = (x

, . . . , x

) ∈ R

, we define an element g

(x) of Q

i=1

S(R

) by g

(x) =

Y

(

x

S

A

(−i, i, 1, −1, 0))

exp(−π

x

AAx

).

We define a theta series (2.1) Θ

(z, g) =

Y

(=(z

))

X

h∈r⁵

e

χ

(h)γ(σ

, S)g

(e %(g)

h) for every z = (z

, . . . , z

) ∈ H

and g = (g

, . . . , g

) ∈ Sp(2, R)

, where e %(g) = (A

%(g

)A, . . . , A

%(g

)A) and % is an isomorphism of Sp(2, R)/{±E

} onto SO(

A

SA

)

given in [3]. Before describing the transformation for- mula for Θ

(z, g), we recall the definition of Hilbert modular forms of half integral weight.

We write Γ

(N ) for Γ

(N ) for simplicity. Throughout this paper, we treat only a congruence subgroup Γ

(N ) such that every congruence sub- group of Γ

(N ) is generated by its elements

with d

> 0 (1 ≤ i ≤ l) (see [9, Lemma 7.4] for the existence of such Γ

(N )). Let k =

(k

, . . . , k

) be an element of Q

satisfying 2k ∈ Z

. For a Γ

(N ) as above, we denote by S

(Γ

(N )) the vector space of all holomorphic functions f on H

which satisfy

f (γ(z)) =

 2c dr



(cz + d)

f (z) for all γ ∈ Γ

(N )

and which vanish at each cusp of Γ

(N ). We refer to [9] for the definition of the symbols

and (∗)

. By Shimura [9, Prop. 7.1], we can verify that Θ

(z, g) admits the following transformation formula:

(2.2) Θ

(γ(z), g)

=

 2c dr



(cz + d)

Θ

(z, g)

 γ =

 ∗ ∗ c d



∈ Γ

(N )



for a suitable integer N . Furthermore, by [3, p. 67], we can prove the fol- lowing transformation formula:

(2.3) Θ

(z, g

g) = Θ

(z, g) for each g

∈ Γ

(M ).

Let N

be a positive integer satisfying N | N

. For f ∈ S

(Γ

(N )),

we define a function Ψ

on H

by

(2.4) Ψ

(Z)

= Y

det( √

−1 C

+ D

)

\

Γ₁(N⁰)\H^l

Y

v

Θ

(z, g)f (z) dz

with z = (z

, . . . , z

) ∈ H

, g( √

−1 E

, . . . , √

−1 E

) = Z g =

and dz = Q

i=1

v

du

dv

(z

= u

+ √

−1 v

). By Shimura [10, Theorem 6.2], we see that Ψ

is holomorphic on H

. Therefore it follows from (2.3) that Ψ

is a Hilbert–Siegel modular form of M

(Γ

(M )).

Next we shall determine explicitly the Fourier expansion of Ψ

. Put H =

 T =

 t

t

/2 t

/2 t

     t

∈ r

 .

We consider an equivalence relation on H × r defined by (T, t) ∼

(T

, t

) if and only if T

= εT and t

= ε

t for a suitable ε ∈ U (N

) = {ε | ε is an unit element of r and ε ≡ 1 (mod N

)}. For a (T, t) ∈ H × r, we define a coefficient c

(T, t) by

c

(T, t) = X

φ∈R(N⁰)

χ

(tc/δ)W (φ

) X

h∈r³/(c)³

e[tr

(a

hS

h/(2c))]

(2.5)

× Y

((t/δ)

)

e[tr

(det(T )d/c)]a(det(T )) with φ

=

and

f |[φ

]

(z) = (cz + d)

f (φ

(z))

= X

µ∈(1/N⁰)

a(µ)e h X

i=1

µ

z

i

with φ

=

. The various symbols will be explained later.

Now the main theorem of this paper can be stated as

Theorem. Suppose that k ∈ (2Z)

, 2 | N

, M δ

| N

and N

satisfies the condition that f has an expression of the form in (2.5) for each f ∈ S

(Γ

(N )) and for each φ ∈ R(N

). Then Ψ

has the Fourier ex- pansion of the form

Ψ

(Z) = c X

c

(T, t)e h X

i=1

tr((t

/(δ

)

)T

Z

) i

,

where c (6= 0) is a constant not depending upon f , (δ) means the conjugate different from F , the sum P

is taken over H × r/∼

and

T

=

 (t

)

(t

)

/2 (t

)

/2 (t

)

 .

3. Proof of Theorem. In this section, after preparing some theta series, we shall show that for an element

(3.1) g =

 √ Y

0

0 √

Y

 , . . . ,

 √ Y

0

0 √

Y



∈ Sp(2, R)

, Θ

(z, g) can be split into the product of simpler theta functions. Set Y

= y

Y

with

Y

=

 y

y

y

y



, det(Y

) = 1,

y

> 0 and y

> 0. Set also Y = (Y

, . . . , Y

), Y

= (Y

, . . . , Y

) and y = (y

, . . . , y

). For ε =

(ε

, . . . , ε

) ∈ Z

(ε

≥ 0) and z = (z

, . . . , z

) ∈ H

(z

= u

+ √

−1 v

), we define two theta series by Θ

(z, Y

) =

Y

v

X

h∈r³

Y

H

([ √

2πv (y

, −y

, −2y

)h]

)

× e[([u

hS

h + √

−1 v

hR(Y

)h]

)/2]

and

Θ

(z, y) = Y

v

X

m∈r

X

n∈r

χ

(m) Y

exp(−2π √

−1 [mnu]

− π[v(y

m

+ y

n

)]

)H

([ √

2πv (my − ny

)]

), where

[ √

2πv (y

, −y

, −2y

)h]

means √

2πv

(y

, −y

, −2y

)h

, [u

hS

h+ √

−1 v

hR(Y

)h]

means u

h

S

h

+ √

−1 v

h

R(Y

)h

, other symbols [∗]

are similar symbols,

H

(x) = (−1)

exp(x

/2) d

dx

(exp(−x

/2)), R(Y e

) = A

R(Y

)A

and R(Y

) is the matrix in [3]. By the definition, we may derive

Θ

(z, g) = Y

( √ 2π)

k1

X

ε1=0

. . .

kl

X

εl=0

k1

C

. . .

C

× (− √

−1)

Θ

(z, Y

)Θ

(z, y) for g in (3.1).

Now the Poisson summation formula gives an important expression of

Θ

(z, y):

Θ

(z, y) = (1/ √ d)

Y

( √

−2π)

X

m∈r

X

n∈(1/δ)

χ

(m) Y

(m

z

+ n

)

× y

v

exp(−πy

|m

z

+ n

|

/v

),

where d is the discriminant of F . Pulling out greatest common divisors in the pair (m, n), we have

{(m, n) | m ∈ r, n ∈ (1/δ)}

= {t(m

/δ, n

/δ) | (m

, n

) = 1, m

, n

, t ∈ r and δ | tm

}.

Take a pair (c, d) ∈ r × r such that (c, d) = 1 and (c, d) ∈ r/(N

) × r/(N

).

Then, for every pair (m

, n

) ∈ r × r ((m

, n

) = 1, m

≡ c(N

), n

≡ d(N

)), there exists a unique σ ∈ Γ

\Γ

(N

) satisfying (m

, n

) = (c, d)σ. Note that Γ

is the stabilizer of the cusp −d/c of Γ

(N ). Throughout this paper, for the pair (c, d) as described above, we fix a matrix φ ∈ SL

(r) with (c, d) = (0, 1)φ and we denote by R(N

) the set of all such φ. Therefore, by the above arguments, we obtain the following lemma which plays an essential role in our later discussion.

Lemma 3.1. Notations being as above, the theta series Θ

(z, y) coin- cides with

(1/ √ d)

Y

( √

−2π)

Y

y

v

× X

χ

(tc/δ)k(φσ(z); t/δ, y) Y

(t/δ)

J(φσ, z

)

, where the sum P

is taken over all (t, φ, σ) ∈ r × R(N

) × Γ

\Γ

(N

) under the condition that tc/δ ∈ r with

φ =

 ∗ ∗ c d



, k(z, n, y) = exp



−π X

y

|n

|

v

 ,

J(g, z

) = (g

z

+ g

)

 g =

 ∗ ∗ g

g



∈ SL

(r)

 . Put

W (σ) = (d

N

(c)

2

)

(c/i)

icα(σ), α(σ) = {(i/(c

z + cd))

}

{(c/i)

}

{(cz + d)

}

σ =

∈ SL

(r) (c 6= 0)

, where the symbol (∗)

is the same as in [9].

By a method similar to that in Shimura [9] and Friedberg [2], we can derive the following lemma.

Lemma 3.2. Suppose that N is the same integer as in (2.2). Then

Θ

(z, Y

) admits the following transformation formula:

(3.2) Θ

(γ(z), Y

) =

 2c dr



(cz + d)

Θ

(z, Y

) for every γ =

 ∗ ∗ c d



∈ Γ

(N ), (3.3) Θ

(σ(z), Y

) = W (σ)(cz + d)

× X

h⁰∈r³/(c)³

e[tr

(a

h

S

h

/(2c))] e Θ

(z + d/c, Y

)

for every σ =

 a ∗ c d



∈ SL

(r) (c 6= 0) and

Θ e

(z, Y

) = Y

v

X

h∈S⁻¹₁ (1/δ)³

Y

H

([ √

2πv (y

, −y

, −2y

)h]

)

× e[[u

hS

h + √

−1 v

h e R(Y

)h]

/2].

P r o o f. First we consider the case ε

= ((ε

)

, . . . , (ε

)

) ((ε

)

= 0, 1).

By Shimura [9, (7.19) and (7.20)], we obtain the above statements. For an arbitrary ε, applying continuously several types of Maass differential operators λ/(2 √

−1 v

) + ∂/∂z

(λ ∈ Z, 1 ≤ i ≤ l) to Θ

(z, Y

), we get the required assertion.

A direct calculation shows that Ψ

(iY ) =

Y

y

√

2π

X

ε1,...,εl

k₁

C

. . .

C

( √

−1)

(3.4)

× \

Γ₁(N⁰)\H^l

Y

v

f (z)Θ

(z, Y

)Θ

(z, y) dz.

Hence, by Lemma 3.1, we derive

(3.5) \

Γ1(N⁰)\H^l

Y

v

f (z)Θ

(z, Y

)Θ

(z, y) dz

= (1/ √ d)

Y

(− √

−2π)

Y

y

× \

Γ1(N⁰)\H^l

Y

v

f (z)Θ

(z, Y

)

× X

χ

(tc/δ)k(φσ(z); t/δ, y) Y

((t/δ)

)

J(φσ, z

)

dz.

Therefore, the left hand side of (3.5) is equal to (1/ √

d) Y

(− √

−2π)

Y

y

X

t∈r

X

φ∈R(N⁰)

χ

(tc/δ)

× Y

((t/δ)

)

\

φ⁻¹Γ∞(N⁰)φ\H^l

f (z)Θ

(z, Y

)

× Y

v

J(φ, z

)

k(φ(z); t/δ, y) dz.

It is a consequence of Lemma 3.2 that the left hand side of (3.5) can be transformed into the following integral:

(1/ √ d)

Y

(− √

−2π)

Y

y

\

Γ∞\H^l

Y

v

W (φ

)

× X

h⁰∈r³/(c)³

e[tr

(a

h

S

h

/(2c))]k(z; t/δ, y) e Θ

(z + d/c, Y

)

× X

µ∈(1/N⁰)

a(µ)e h X

i=1

µ

z

i

dz.

In order to perform further evaluation, we need to consider an equivalence relation on S

(1/δ)

× r:

(h, t) ∼

(h

, t

) if and only if h

= εh and t

= ε

t for some ε ∈ U (N

).

Hence, by easy evaluation, we can verify that X χ

(tc/δ)

Y

((t/δ)

)

W (φ

) \

Γ_∞\H^l

Y

v

k(z; t/δ, y)

× e Θ

(z + d/c, Y

) X

µ∈(1/N⁰)

a(µ)e h X

i=1

µ

z

i

dz

= X

X

φ∈R(N⁰)

W (φ

)χ

(tc/δ)e[tr

(d

hS

h/(2c))]a(

hS

h/2)

× \

(R⁺)^l

Y

v

H

([ √

2πv (y

, −y

, −2y

)h]

)

× exp n

− π X

[v(

hS

h +

h e R(Y

)h) + y

|t/δ|

(1/v)]

o Y

i=1

dv

,

where the sum P

(resp. P

) is taken over all (t, φ) ∈ r×R(N

) (resp. (h, t) ∈ S

(1/δ)

× r/∼

) under the condition that tc/δ ∈ r with φ

=

. Observe that

∞

\

0

v

exp(−αv − βv

)H

( √

2αv) dv = β

√

2

π exp(−2 p αβ) for any α, β ∈ R

. Consequently, we conclude that

Ψ

(iY ) = c X

c

(T, t) exp



− 2π X

tr((t

/(δ

)

)T

Y

)

 , where the sum P

is the same as in the Theorem and c is a constant. There- fore, by the same method as in [3, p. 72], we conclude the proof of the Theorem.

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Department of Mathematics

The University of Electro-Communications Chofu-shi, Tokyo 182, Japan

E-mail: kojima@prime.e-one.uec.ac.jp

Received on 18.11.1996 (3077)