Bessel functionals and Siegel modular forms

R

AINER

S

CHULZE

-P

ILLOT

Bessel functionals and Siegel modular forms

Journal de Théorie des Nombres de Bordeaux, tome

7, n

o

_{1 (1995),}

p. 15-20

<http://www.numdam.org/item?id=JTNB_1995__7_1_15_0>

© Université Bordeaux 1, 1995, tous droits réservés.

L’accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les condi-tions générales d’utilisation (http://www.numdam.org/conditions). Toute uti-lisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Bessel Functionals and

_Siegel

Modular Forms

par Rainer SCHULZE-PILLOT

The existence and

_uniqueness

of Whittaker models

_plays

an essential role

in the

_theory

of

_automorphic

forms for the _{group GL.~.} In _contrast, it is

well known that

_{holomorphic Siegel}

modular forms of

_{degree n &#x3E;}

2

_(or

the

corresponding automorphic

representations

of the adelic

_symplectic

_group

Spn

of rank

_n)

do not _possess Whittaker models. As a

replacement

in the

case n =

_2,

Novodvorski and

Piatetski-Shapiro

studied Bessel models

(to

be defined

_below)

and

_proved

_{uniqueness [N-PS].}

Their existence is trivial

in the case of

_holomorphic

modular forms. If an

_automorphic

form for the

group

PGSp2(A)

possesses such a model of a so called

special

_type

it was

shown in

_[PS-S]

that it can be related to a

generic

form

(i.e.,

one

having

a

Whittaker

_model)

for the same _group

by

theta

lifting

it to the

_metaplectic

cover of

Sp2

and back

_again

(using

different

additive,

characters).

The

key

ingredient

in this is the

_{explicit computation}

of the Whittaker coefficient of the lifted form in terms of the Bessel functionals of

_special

_type

of the

original automorphic

form. The _purpose of this note is to show that this

procedure

can fail in a nontrivial _way

by exhibiting Siegel

modular forms of

degree

and

_weight

2 which have no Bessel model of

_special

_type

(the

Bessel

functionals of

_special

_type

vanish

_trivially

for

_Siegel

modular forms of odd

weight

for

_{I’o(N)-groups).}

The

_{examples presented}

here came _{up in joint}

work with B6cherer and with Furusawa

_{[B-SP 1, 2, 3, B-F-SP~.}

Let F be a

Siegel

modular form of

_degree

2 for some _congruence

_subgroup

r of level N

_{of Sp2 (Z)}

and

_put

If .~ has Fourier

_expansion

we

_put

for a discriminant A 0

where the summation is over a set of

_{representatives}

of the

_{r’-equivalence}

classes of

_positive

definite half

_integral

_symmetric

matrices of discriminant

L1

and

_(T)

is the number of proper units

(integral automorphs)

in 1~ of T. The Koecher-MaaB series of F is then

Let

_G1

=

SP2 9

GL4

be the

symplectic

_group of rank

2,

G =

GSp2 D

G1

the group of

symplectic similitudes,

both _groups viewed as

algebraic

_groups over

_Q.

For a

_Siegel

modular form F of

_degree

2 and

_weight

r with

_respect

to a _congruence

subgroup

Sp2 (Z)

for some

_integer

N we denote

by

Fs

the

_automorphic

form on

G(A)

corresponding

in the usual _{way to} F.

In

_particular,

_Fs

is left invariant under

_G(Q)

and

_right

invariant under the

groups

,"

.

, /

(embedded

into

_G(A)

_by

_putting

all other

_components

_equal

to

₁₎

for all finite

_primes

_p. The function

_Fs

is invariant under the center of

_G(A)

and hence induces an

_automorphic

form on We use the well known

identification of

_G/Z(G)

=

PGSp2

with the

special orthogonal

group of a

quadratic

form in five variables:

_Following

_[Si]

we let V be the

_Q-vector

space of all matrices

with J =

(~1 ~)

E

_Q,

M E V is

_equipped

with the

_quadratic

form

_q(x)

= detM - with associated bilinear form

B(x, y)

=

q(x

₊

y) -

q(~) - q(y)

and

decomposes

_as V =

(Qei

₊

Qe-i)

1

where

_Qei

+ is a

hyperbolic plane

and the

_embeddings

are the obvious ones. On V the _group of

_symplectic

similitudes

_G(Q)

acts

through

X ~ =:

(where

À(g)

is the similitude

(g

H

t (g)) :

G- H :=

SO(V, q)

_gives

an

isomorphism

from

G/Z(G)

onto

H as

algebraic

_groups over

Q.

We write

for the induced

_automorphic

form on

(the

index o

_standing

for

or-thogonal).

We need a few more notations

_(see

_[PS-S]).

For T E

M2’’m(Q)

C V let

DT

=

{h

_E H

hel

= _el, hT =

T},

let S denote the

unipotent

radical of the

parabolic

subgroup

P

_(with

Levi

_{decomposition}

P =

MS)

of H

fixing

the

line

_through

_el,

_put

_RT

=

DTS.

We fix the standard additive

character ’0

of

_A/Q,

let _XT be the character of S

_{given by}

extend

XT

trivially

to

RT

and consider the Bessel functional of

_special

_type

(More

general,

the Bessel

_{functionallT,II,1/J(Fo,}

_h)

is defined for a character

1/ of

by

inserting

v(r)

:=

v(d) (for

r = ds E E

DT(A), s

E

_S(A))

into the

_defining

_integral

_above.)

PROPOSITION 1. Let F be a

Siegel modular form of weight

r

for

the _group

and

_Fo

the associated

_{automorphic form}

on

H(~).

Let

{"II, ... ,

be a set

_of

_{representatives}

_of

the double cosets

_Sp2(Z)

(with

r~~~ _ ~ C 0

Sp2(7G)}).

Then the Bessel

_{functionals of}

spe-cial

_type

are zero

_for

all

_binary

_symmetric

T

_if

and

_{only if}

= 0

for

i =

_{1, ...}

n.

Proof.

This is a standard

_computation:

Denote

by

Kp

the

_image

of

under L in

_{H(Qp) ,}

consider the Iwasawa

_{decomposition}

_{H(Qp) =}

and

_decompose

the finite

_part

_hf

of h

_accordingly

as

h f

= We write

m f =

rrirrv fkf

with m in

_M(Q)

_commuting

with

_DT,

the

_components

_mp

of of the form

,( (9Ó (9-1))

B

with

p

/

gp E and

kí

E

_Kp.

_Using

_strong

_{approximation}

for

_SL2

and

= for _m

= ~~C p ~9 )-1 ~ ~

with 9 E

we see that we have to check whether

_{with k f}

E

_{K f}

are zero. The invariance

_property

of

_Fo

on the

right implies

that it is suf-ficient to consider

_kf

with all

_components

_kp

in and

_by

_strong

approximation

for

_Sp2

and the

_right

invariance of

_Fo

under the

_t(Kp)

we can

_{assume k f}

=

(,yi,

... )

for _some 1 i _n

(absorbing

an element

of into the

_integration

over

S).

From formula 1-26 of

[Su]

it follows

then as in

[Fu]

that for h =

7t?

... )

with and a

binary

symmetric

matrix

_Td

=

§

discriminant A = -4d =

4(t2 -

tlt3)

(

t2 _{t3 )} 2 one has 3 _/ one has

Here we write ] and the

summation is as in

_(1).

This _proves

the assertion.

To come to our

examples

we have to recall some notations and results

from

_{[B-SP 1, 2, 3].}

Let N be

_prime,

_{fl, f2 E}

_{s2 (ro (N))}

be two _cusp forms of

_weight

2 for the group

ro(N)

C

SL2 (Z),

normalized

eigenforms

of all

Hecke

_operators

with the same

eigenvalue

under the Fricke involution _wN,

f2.

Let D be the definite

_quaternion

_algebra

_{over Q}

unramified at all

primes # N,

let R be a maximal order in D.

With

_{R£ =}

_{(D 0}

_{R) x}

x

_fl

_R~

consider a double coset

_{decomposition}

with’

To our _cusp forms

_{fl, f2}

there

_{correspond by}

the work of Eichler

[E]

two

functions _{cp2 in} the _space

let the

_{Yoshida lifting ~}

be defined

_by

Then from

_{[B-SP 1]}

we know that F =

y(2)

is a nonzero

_Siegel

cusp form of

degree

2 for the group

rð2)

(N) .

Moreover,

if the

eigenvalue

of

_/1,12

under the Fricke involution WN is

+1,

the Koecher-Maali series

s)

is

_identically

zero

[B-SP

_3,

Remark

2.1] (This

remark mentions

only

the for fundamental discriminants A. The

proof

for

_arbitrary

discriminants

_proceeds

as the

proof

of

Corollary

2.2 in

[B-SP 3],

using

Proposition

3 of

_{[B-SP 2]}

and the Remark on _page 373 of

[B-SP 2]).

In

fact,

more is true:

PROPOSITION 2. Let F and N be as above. Then all Bessel

functional of

special

type

associated to the

_{corresponding f unction}

~’o

on

H(A)

are zero.

Proof.

As in

_[B-SP

_1,

Lemma

_8.1]

we have to consider for _’1’1 =

-y2

= . -t n n n , ’"

The Koecher-Maaf3 series for F is

zero from the

_argument

_{given above,}

and

FI-y3(NZ)

is

proportional

to

_F(Z)

by

[B-SP 1,

Lemma

_9.1].

For

_FIT2

we recall from

[B-SP

_1,

Lemma

8.2]

that

with some constant c

(where

I#

is the dual lattice of Each

_pair

_{Xl, X2} as in the summation

_generates

a sublattice K of discriminant in N-1Z of

The

_completion

at the

_prime

N of the latter lattice is the set of vectors

of

_{integral length}

in It is therefore clear that _any

_binary

sublattice of discriminant in of

_7~

has a basis with one basis vector in

_Iij.

But this

_implies

that any

binary

sublattice of discriminant A E of

7~

is

counted

_by

the coefficient at A of the Koecher-Maaii series of

~9~2~

_~~2,

and

it is

_easily

seen that it is counted

2(S~2(7~) :

r~)

=

2N (N + 1)

times. Since

by

definition it is counted twice

_by

the coefficient at A of the Koecher-MaaB series of the theta series of

_degree

2 of

_I#

and since we

_already

know

that the Koecher-Maafl series of

(which

is

_proportional

to is _zero, it follows that

_DKM(FI12,

_s)

is zero as

_well,

which

_by

_Proposition

1

implies

the assertion.

REFERENCES

[B-F-SP]

S. Bocherer, M. Furusawa and R. Schulze-Pillot, On Whittaker _{coefficients of}

some _{metaplectic forms,} Duke Math. Journal (to

appear).

[B-SP 1]

S. Böcherer and R. Schulze-Pillot, Siegel modular forms and theta series

at-tached to quaternion algebras, Nagoya Math. J. 121 _(1991), 35-96.

[B-SP 2]

S. Böcherer and R. Schulze-Pillot, On a theorem of Waldspurger and on

Eisen-stein series _{of Klingen type,} Math. Annalen 288

_(1990),

361-388.

[B-SP 3]

S. Böcherer and R. Schulze-Pillot, The Dirichlet series _of Koecher and Maa03B2 and modular _{forms of weight 3/2,} Math. Zeitschr. 209 _(1992), 273-287.

[E]

M. _Eichler, The basis problem for modular _forms and the trace _of the Hecke

operators, Modular functions of one variable 1, Lecture Notes in Math. _320,

Springer Verlag, Berlin, Heidelberg, New York, 1973, pp. 75-151.

[Fu]

M. Furusawa, On _{L-functions for GSp(4)} x _GL(2) and their special values, J. f. d. reine u. _angew. Math. 438 (1993), 187-218.

[N-PS]

M. E. Novodvorski and I. I. _{Piatetski-Shapiro,} Generalized Bessel models _for

a _{symplectic group of} rank _2, Math. USSR Sbornik 19 (1973), 243-255.

[PS-S]

I. I. Piatetski-Shapiro and D. Soudry, On a _{correspondence of automorphic}

forms on _orthogonal groups of order five, J. Math. pures et _appl. 66 _(1987), 407-436.

[Si]

C. L. _{Siegel, Symplectic geometry,} Gesammelte _{Abhandlungen,} Bd. II, Springer Verlag, Berlin, Heidelberg, 1966, pp. 274-359.

[Su]

T. _Sugano, On _holomorphic cusp forms on _{quaternion unitary groups of degree}

2, J. Fac. Sci. Univ. Tokyo, Sect. IA, Math. 31 _(1984), 521-568.

Rainer SCHULZE-PILLOT Mathematisches Institut der Universitat zu Koln

Weyertal 86-90

D-50931 K61n