Automorphic Forms and Zeta Functions: Proceedings of the Conference in Memory of Tsuneo Arakawa Rikkyo University

Proceedings of the Conference

in Memory ofHisuneo tXrakawa

Siegfried Bocherer Tomoyoshi Ibukiyama Masanobu Kaneko Fumihiro Sato

World Scientific

Proceedings of the Conference

in Memory ofl'suneo drakawa

AUTOMORPHIC FORMS AND

ZETA FUNCTIONS

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Proceedings of the Conference

in Memory ofrfsuneo tArakawa

AUTOMORPHIC FORMS AND

ZETA FUNCTIONS Rikkyo University, Japan

4 - 7 September 2004

Editors Siegfried Bocherer Universitdt Mannheim, Germany

Tomoyoshi Ibukiyama Osaka University, Japan

Masanobu Kaneko Kyushu University, Japan

Fumihiro Sato Rikkyo University, Japan

\[p World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONGKONG • TAIPEI • CHENNAI

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AUTOMORPHIC FORMS AND ZETA FUNCTIONS Proceedings of the Conference in Memory of Tsuneo Arakawa Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-256-632-5

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In memory of Tsuneo Arakawa (1949-2003)

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PREFACE Tsuneo Arakawa, an eminent researcher in modular forms in several variables and zeta functions, passed away suddenly on October 3, 2003 by rupture of a cerebral aneurysm. This is the proceedings of the "Conference on Automorphic Forms and Zeta Functions" in memory of Tsuneo Arakawa, which was held at Rikkyo University, Tokyo, on September 4-7, 2004. This volume is dedicated to his memory. Most of the papers are based on the lectures given at the conference. Some of the authors, such as Don Zagier and Aloys Krieg, who could not take part in the conference, contributed their papers at our solicitation. Arakawa's works are reviewed mainly in the first article of this volume. The articles of S. Hayashida and H. Narita may also serve as reviews of Arakawa's achievements on skew holomorphic Jacobi forms and automorphic forms on Sp(l,q), respectively, since they report new results of their own obtained on the basis of Arakawa's results. Many of the other papers are also more or less related to Arakawa's works. The example most notable in this respect is the article of Ibukiyama and Katsurada. The KoecherMaass type Dirichlet series considered in that paper was first introduced by Arakawa almost 30 years ago, and attracted attention only very recently. We believe that this collection of papers illustrates both the fruitfulness of Arakawa's works and current trends in modular forms in several variables and related zeta functions. The conference was supported financially by the Grant in Aid for Scientific Research from JSPS (Japan Society of Promotion of Science) (B) No. 16340012 (Principal Researcher F. Sato) and partly by the same JSPS Grant (A) No. 13304002 (Principal Researcher T. Ibukiyama). We thank JSPS for the support. We also thank all the speakers at the conference and the more than 100 participants.

September, 2005

The Editors: Siegfried Bocherer (Universitat Mannheim) Tomoyoshi Ibukiyama (Osaka University) Masanobu Kaneko (Kyushu University) Fumihiro Sato (Rikkyo University)

vu

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CONTENTS

Preface

v

Tsuneo Arakawa and His Works

1

Estimate of the Dimensions of Hilbert Modular Forms by Means of Differential Operators Hiroki Aoki

20

Marsden-Weinstein Reduction, Orbits and Representations of the Jacobi Group Rolf Berndt

29

On Eisenstein Series of Degree Two for Squarefree Levels and the Genus Version of the Basis Problem I Siegfried Bocherer

43

Double Zeta Values and Modular Forms Herbert Gangl, Masanobu Kaneko and Don Zagier

71

Type Numbers and Linear Relations of Theta Series for Some General Orders of Quaternion Algebras Ki-Ichiro Hashimoto

107

Skew-Holomorphic Jacobi Forms of Higher Degree Shuichi Hayashida

130

A Hermitian Analog of the Schottky Form M. Hentschel and A. Krieg

140

The Siegel Series and Spherical Functions on 0(2n)/(0(n) Yumiko Hironaka and Fumihiro Sato

IX

x 0(n))

150

x

Contents

Koecher-Maafi Series for Real Analytic Siegel Eisenstein Series Tomoyoshi Ibukiyama and Hidenori Katsurada

170

A Short History on Investigation of the Special Values of Zeta and L-Punctions of Totally Real Number Fields Taku Ishii and Takayuki Oda

198

Genus Theta Series, Hecke Operators and the Basis Problem for Eisenstein Series Hidenori Katsurada and Rainer Schulze-Pillot

234

The Quadratic Mean of Automorphic L-Functions W. Kohnen, A. Sankaranarayanan and J. Sengupta

262

Inner Product Formula for Kudla Lift Atsushi Murase and Takashi Sugano

280

On Certain Automorphic Forms of Sp(l,q) sults and Recent Progress) Hiro-Aki Narita

(Arakawa's Re314

On Modular Forms for the Paramodular Groups Brooks Roberts and Ralf Schmidt

334

SL(2,Z)-Invariant Spaces Spanned by Modular Units Nils-Peter Skoruppa and Wolfgang Eholzer

365

T S U N E O ARAKAWA A N D HIS W O R K S

1. Tsuneo Arakawa (1949 - 2003) Tsuneo Arakawa was born on April 14,1949 in Yokohama, Japan. In March 1968, he graduated from the senior high school at Komaba that is affiliated with the Tokyo University of Education (currently University of Tsukuba). He then entered the University of Tokyo. There he finished his bachelor's degree as a mathematics major in 1973 and stayed there for his graduate study. He specialized in Siegel modular forms under the supervision of Yasutaka Ihara. In 1975 he completed his master's thesis, which was later published as [3], and obtained his master's degree. Then immediately he was appointed as a lecturer at the Department of Mathematics, Rikkyo University. He was promoted to the rank of assistant professor in 1984, to associate professor in 1986 and then to professor in 1993. He remained there till his untimely death in 2003. He was awarded the doctorate of science from the University of Tokyo in 1982 for his work on the analytic continuation of the Koecher-Maafi zeta functions. In 1978 he married Miyuki Makiyama and later they had two children. His first mathematical achievement after his appointment at Rikkyo was the analytic continuation and the determination of the principal part of the Koecher-Maafi zeta functions ([2], [11])- Some of us still remember the day when he reported this result at the Algebra Colloquium of the University of Tokyo. The audience was awed by the great skill with which he carried out the computation, which seemed quite complicated to them. It is well known that Professor Klingen devoted the final chapter of his book Introductory lectures on Siegel modular forms, which is a standard reference in the field, to this fundamental result. With this result Arakawa established himself as a member of the group of mathematicians at the forefront of work on modular forms of several variables. From January 1980 to March 1981 he stayed in Bonn as a visiting member of the SFB Theoretische Mathematik of Bonn University, which later

1

2

The

Editors

became the Max-Planck-Institut fur Mathematik. His second long term stay in Germany was from April 1988 to March 1990, first in Gottingen as a visiting member of SFB 170 and then in Bonn at the Max-Planck-Institut. He enjoyed his several short term visits to Germany, especially to Oberwolfach and Mannheim. Through his graduate study he became familiar with the works of German mathematicians such as Siegel and Maafi. In fact, the first paper his advisor suggested that he study was Siegel's famous paper "Einfuhrung in die Theorie der Modulfunktionen n-ten Grades". His mathematics follows the tradition of German mathematics. Through his stays in Germany he became acquainted with most of the German mathematicians working in the field. The friendships he nurtured with them are clearly reflected in this memorial volume. Another mathematician who was quite influential to him was Takuro Shintani, who was an assistant professor at the University of Tokyo when Arakawa was a graduate student. One can see Shintani's influence on him in many of his research papers. One of the notable features of Shintani's works was that he reached the deepest mathematical truth through hard but skillful computations. Arakawa's works share this feature with Shintani's. As is seen from the list of publications at the end of this article, his works are approximately classified into three subjects: (1) Siegel modular forms and Jacobi forms, (2) Selberg zeta functions, and (3) special values of zeta and L-functions. His achievements in each subject will be reviewed in each of the following sections. He was very modest and generous. The articles of Hayashida and Narita in this volume clearly show his generosity in sharing his ideas with young mathematicians. He was also very kind to the students in his classes. He carefully prepared his lectures and he was willing to help his students, spending much time outside the classroom. They appreciated his generosity immensely. We were deeply dismayed by the news of his abrupt death on October 3, 2003, only a short time after he talked to us at the beginning of the winter semester about his future plans. His death was a great loss for mathematics and for all of us who knew him. The Editors

Tsuneo Arakawa and His Works

3

2. Arakawa's works on Siegel and Jacobi modular forms Arakawa worked widely in the area of modular forms : (i) Dirichlet series associated with modular forms: [1], [2], [6], [11], [21], [25], [26]. (ii) Dimension formula for modular forms: [3], [4], [7], [12], [14], [15]. (iii) Jacobi forms: [13], [19], [21], [17], [20], [24], [28]. (i) Koecher introduced Dirichlet series corresponding to holomorphic Siegel modular forms and Maass obtained their analytic continuations and functional equations using the method of invariant differential operators. The Dirichlet series are now called the Koecher-Maass series. In his first paper [1], Arakawa investigated several Dirichlet series related to the Fourier coefficients of the Eisenstein series on the Siegel upper halfplane, which can be viewed as an analogue of the Koecher-Maass series for the real analytic Siegel Eisenstein series (cf. the article of Ibukiyama and Katsurada in this volume). He proved their analytic continuations and functional equations using Shintani's method and results on zeta functions associated with quadratic forms. Arakawa investigated the Koecher-Maass series for holomorphic forms very precisely and gave residue formulas. In [2], he treated the case of Siegel modular forms of level one. His method is based on the Klingen Eisenstein series and the structure theorem of the space of Siegel modular forms. In the Procceeding of Taniguchi Symposium in Katata 1983, Klingen suggested the probelm to prove the theorem without help of Klingen Eisenstein series. Arakawa's first proof, which was different from the one in [2], of the residue formula meeted the demand of this problem. He published it later in [11] and extended the residue formulas to the case of Siegel modular forms with arbitrary level. Furthermore he obtained explicit functional equations and residue formulas for Epstein-Koecher zeta functions. After 8 years, Arakawa returned to this theme again in [21]. He introduced Koecher-Maass series for a Jacobi form of degree n, weight k and index S and obtained a meromorphic continuation and a functional equation following Maass' original method. Moreover under the assumption on the maximality of S, he gave the residue formula. Arakawa was also interested in i-functions associated with Hecke eigen modular forms. In [6], along the lines of Andrianov, he succeeded to obtain analytic continuations and functional equations of the spinor i-functions associated with vector-valued Siegel modular forms of degree two (under a certain technical assumption on Fourier coefficients). He also considered the

4

T. Sugano

spinor L-function associated with automorphic forms on Sp(l, 1) belonging to a non-holomophic discrete series (unpublished work). In [25], Arakawa constructed Saito-Kurokawa lifting from cusp forms of weight k — 1/2 (k is an odd positive integer) and level 4 to Siegel modular forms of weight k and level 4 with non-trivial character. He used the method of Duke-Imomoglu, the converse theorem of Imai and some results of Katok-Sarnak. He also gave another construction of the lifting by means of Eichler-Zagier, where Jacobi forms were effectively used. In [26], Arakawa, Makino and Sato extended Imai's converse theorem to noncuspidal case. Consequently, another proof of the Saito-Kurokawa lifting for not necessarily cuspidal Siegel modular forms was given. (ii) In [3], Arakawa gave explicit dimension formulas of the spaces of cusp forms on the Siegel upper half-plane of degree two with respect to some arithmetic groups having only zero dimensional cusps. Such groups are denned from quaternion unitary groups of degree two. The same results had been obtained by Yamaguchi by a different method (using the Hirzeburch-Riemann-Roch theorem). His calculation is based on the Selberg trace formula. He wrote two papers [4] and [7] on dimension formulas of certain nonholomorphic cusp forms on Sp(l,q) (cf. Narita's article in this volume). The papers [12], [14], [15] treat Selberg zeta functions and dimension formulas of lower weights. The results in these papers will be discussed in the next section. (iii) In [13], real analytic Jacobi Eisenstein series Ek,m((T, z), s) of weight k and index m with respect to the Jacobi group were studied. These coincide with the holomorphic Jacobi Eisenstein series introduced by Eichler-Zagier at s = (k — l / 2 ) / 2 . Arakawa proved that -Efc,m((r, z),s) was analytically continued to a meromorphic function in the whole s-plane and satisfied a functional equation between s and 1 — s. Calculating Rankin-Selberg convolution of Siegel Eisenstein series and Siegel cusp forms was developed by Garrett and Bocherer. It was a powerful technique for investigating Siegel modular forms and their L-functions. In [19], Arakawa showed that this technique was applicable to the case of Jacobi forms of degree n. He established Garrett-Bocherer decomposition of real analytic Jacobi Eisenstein series and obtained an integral representation of the standard L-function of a Hecke eigen Jacobi cusp form. In [17] and [20], Arakawa studied Siegel formula for Jacobi forms. Let S be a fixed positive definite half-integral symmetric matrix of size I. Denote

Tsuneo Arakawa and His Works

5

by Sym£, + j(S : Z ) + the set of positive definite half-integral symmetric matrices with lower right / x I submatrix S. Then B ro .i(Z) := | Y " ° J

a e SLm{Z),

a; e M , , m ( Z ) |

acts on S y m ^ + i ( 5 : Z ) + naturally. He introduced the notion of 5-calss and S-genus through this action. Each 5-genus consits of finitely many 5-classes. Assume that m > n and let Q € S y m ^ + J ( 5 : Z ) + and T € Sym£ + |(S : Z)+. Denote by A(Q,T) the number of solutions of x = (Xl)

( n e M m , n ( Z ) , z 2 £ Mi, n (Z)) of the equation Q ^ * 1 ° ) ] = 2\

Let Q i , . . . , Qif be a complete set of representatives of the S-classes in the S-gunus of Q. Then Arakawa proved the "arithmetic Siegel formula"

Here E(Qj) = # 0 ( Q j ) n S m ) ; ( Z ) , av(Q,T) are local densities and e = 1 or 1/2. As in the original case, this can be reformulated to the "analytic Siegel formula". He proved that the Jacobi Eisenstein seies coincided with a finite linear sum of theta sereis. In [20], Arakawa obtained a Minkowski-Siegel formula (mass formula) for Q € Sym!^ + 1 (l, Z ) + when det (2Q) = 1. Jacobi forms of weight k give rise to modular forms of weight k by restriction z = 0. More generally, Eichler-Zagier defined a linear mapping Dv : Jk,i{To{N),x) —> Mk+v{To(N),x) by using differential operators. Arakawa and Bocherer investigated the kernel of Do precisely in [24] and [28]. First they showed that Ker Do was isomorphic to Mfc_i(ro(iV),xw), where u> is the character of SL2 (Z) occuring in the transformation law of rf. Second, they characterized D2(Ker Do) in terms of vanishing orders at cusps. In [28], they showed that the restriction map J2,i(^o{N)) —• Af2(r0(Ar)) was injective for any square free N (This was first observed by Kramer for prime levels). As an application of their results, a conjecture of Hashimoto on theta series was proved. Takashi Sugano (Kanazawa University) 3. Arakawa's works on Selberg zeta functions 3.1. Tsuneo Arakawa wrote several articles with focus on the relation between the dimension of the space of various automorphic forms and the vanishing order of Selberg zeta functions. In this review, we will pick up

6

K. Takase

some of his results and will give a general framework in which we can understand Arakawa's achievements more clearly. The topics we will discuss in this section are the followings. Let Sk(T, x) be the space of the cusp forms of weight k with respect to a subgroup T of SX 2 (Z) of finite index with a finite dimensional unitary representation (or a finite dimensional unitary multiplier system) x- On the other hand let Zr(s,x) be the Selberg zeta function defined by oo

Zr(s, X) =

II

II det I 1 - x(7)iV(7)"s-m)

{7},tr7>0m=0 ls

where 11(7} tr7>o the product over the primitive hyperbolic T-conjugacy classes of positive trace. Here we assume that T contains —12. Then the first result of Arakawa's which we will discuss is Theorem 3.1. ([12]) dim52(r, x) — ords=iZr(s,x)

+ dimx • vol(T\G) • - + [elliptic] + [parabolic],

and d i m S i ( r , x ) = „ord s = 1 /2^r(s>x) + [elliptic] + [parabolic] where [elliptic] and [parabolic] are terms coming from the elliptic conjugacy classes and the parabolic conjugacy classes respectively. Here G = SL,2(M.) and vol(r\G) is the volume of the quotient space r \ G with respect to the Haar measure on G which induces the G-invariant measure 1 dxdy (x + V^ly € fl) (1) n y2 on the complex upper half plane Sj (the Haar measure on the maximal compact subgroup K = SO(2) is normalized so that the volume of K is equal to 1). We will also discuss the dimension of the space of Jacobi forms. Let S be a positive definite half-integral matrix of odd size I and Jk,s(F) the space of the Jacobi forms of weight k and index S with respect to T = 5^2(2). On the other hand the transformation formula of the theta series 0sAz>v>)='52e(z-S[q

+ r] + 2S(q + r,w))

(zeSj,w&Cl)

(2)

q€Z'

(r 6 (2S)~1Zl/l}) with respect to V produces a multiplier system x which is canonically decomposed into two sub-multiplier system x± ( s e e §3-5 for

Tsuneo Arakawa and His Works

7

the precise definition). Here e(t) = exp(27rv c lt) and S(X,Y) = XSfY and S[X] = S(X,X) as usual. Then the Selberg zeta function Zr,s,e(s) is denned by oo

Zr.sA") =

II det (J - Xeh)N(>y)—m) .

II

{7},tr(7)>0m=0

Then Arakawa gives the following dimension formula; Theorem 3.2. ([14], [15]) dim •/(C(+S3)/2,s(^) =

or(

is=3/4^r1s,£:(s)+dim Xe-vol(T\G)--+[elliptic]+[parabolic],

and dim J(i + i)/ 2 ,s(r) = o r d s = 3 / 4 Z r , s , - £ ( s )

e

= r_i)('+3)/a

=

/1 \-l

l m l

1= 3

(mod4) (mod 4 ) '

Here ^ T J w 2 ^ ( r ) denotes the space of cuspidal Jacobi forms and [elliptic] and [parabolic] are the contributions from the elliptic conjugacy classes and the parabolic conjugacy classes of T respectively. Actually, as we will see in §3.5, the space of Jacobi forms appearing in the theorem above are closely related with the space of the cusp forms of weight 3/2 or 1/2 with respect to F with multiplier system Xe- So these results (Theorem 3.1 and Theorem 3.2) of Arakawa's are to give dimension formulae of the space of the cusp forms of low weight by means of the Selberg zeta functions. In the proof of these results, Arakawa used Fischer's Selberg trace formula [F] which involves multiplier systems. In the rest of this section, we will review these results from representation theoretic point of view using a standard trace formula. 3.2. Let us start with an abstract trace formula. Let G be a locally compact unimodular group, T a discrete subgroup of G, x a finite dimensional unitary representation of I\ For the sake of simplicity, we will assume that T\G is compact. Let us denote by 7r£ the unitarily induced representation IndpX °f G- Taking a / e Ll(G) such that 7r£(/) is trace class operator,

8

K. Takase

we have a trace formula

Y, m(7r,7r^)tr7r(/) = d i m * • vol(T\G) • / ( l ) 7T6G

+

£ {7}r#l

f{x-l1X)dx.

trX(7)vol(r7\G7)- f

(3)

JG G

^

Here G denotes the set of the unitary equivalence classes of the irreducible unitary representations of G and m(n, 7r*) denotes the multiplicity of IT in 7Tp which is finite because T\G is compact. J2{~/\r^i ls t n e summation over the non-trivial conjugacy classes of F. G 7 denotes the centralizer in G of 7 e r and T 7 = T f~l G 7 . Since T\G is compact, the centralizer G 7 is unimodular, and G~f\G has a right G-invariant measure dx. Now let K be a compact subgroup of G and J an irreducible unitary representation of K such that m(S, 7T\K) < oo for all n € G (for example, if G is a connected semi-simple real Lie group of finite center and K a maximal compact subgroup of G, then m(S, TT\K) < d i m 5 for all 5 € K and 7r £ G). Let us denote by G(5) the set of n € G such that m(0 7TJ

0

*g2Z+l,J 2. Now the Plancherel formula gives (r)dr

/(I) 1

7reG d

J=°
0 only if e = (-l)«+ 3 >/ 2 . Put Q = 25. Let us consider a R-vector space V = MiQ,J

(lndf=! ( R ) x + «^Q,j)®

• ( l
± <S> UQ,J, which is an irreducible unitary representation of SL2(M) t* H[V], or equivalently an irreducible unitary representation of SL,2(M.) IK H[V, D]. The contragredient representations 7f+ and fr_ have the minimal if-types r (F) = 0 if r < n } .

Lemma 2.1. 7 / F e Afc + (n), i/ien DnF € M2A:+2nProof. For (acbd) £ SL 2 (Z), we have Fr1;r2

=(cri+d

cr 2 + A++(2n)

0

• A+~(2n + 2)

• A+-(2n)

0

• i4fe+(2n + 3)

• A f c + ( 2 n + l ) -^=±i» M 2 f c + 4 „ + 2,

0

> Afc-(2n + 3)

• i4fc"(2n + l) - ^ ^

2.2. Fourier coefficients

- ^ = - » M2k+4n, -

of Hilbert modular

^

M2k+4n,

M2k+4n+2.

forms

Now we investigate these exact sequences more precisely. Assume F G A~£+ and set its Fourier coefficients by F{TI,T2)=

]T

c{y) e (vn

+ V'T2) •

v>0, i / > 0

From the transformation formula of F £ - 4 j + , we have c{v) — c{y') and c(y) = c{ev). Now, for the sake of simplicity, we put c(u, v) := c(^u+ ^-v). Put A := {(u,v) £ Z 2 | |i>| ^ V3u}. Then F has a Fourier expansion ir(ri1r2)=

V3

5Z c(u>u)e ( 2 U + _ 6 ~ U ) T l + ( 2 u - — « (u,«)eA \V / V

r2 i .

Thus, easily we have (D2rF)(r) =

2 3 u 2 r c(u,i;)e(uT). (•U,V)GA

Lemma 2.2. If F € A£+(2n), t/ien c(w, v) = 0 /or any u < n. Proof. We show this lemma by induction on n. If n = 0, this lemma is trivial. Now we assume that this lemma holds for n < r. Let F £ A~£+(2(r + 1)). From the assumption, c(u,v) = 0 for any u < r. If v > r, then 1

c(r,v)=c(e

f- r + ^—v ] J = c(2r - v, - 3 r + 2w) = 0

because 2r — v < r. If v < —r, then c(r, u) = c(r, —v) = 0. Hence F G A£ + (2(r + l)) means r

J2v2j2T.

• A+-(2n + 2)

A+"(2n) - ^ = - > M 2fc+4 „(n + l),

+

> j4* (2n + 3)

V ( 2 n + 1)

_

> AjT (2n + 3)

£>2n + l

M 2 f c + 4 „ + 2 (n + 2), M 2 fc + 4 n + 2 (n + l ) .

A r ( 2 n + 1)

From this proposition, we have the upper bounds for the dimensions of At ,A+-,Ai+and A*-. +

Theorem 2.1. In the following table, each dimension of the left hand side is not greater than the coefficient of xk on the formal power series development of the right hand side.

At+

Af A~k

A:

+

1 2

(1 - z )(l - x 3 )(l - z 4 ) X? 2 (1 - x )(l - z 3 )(l - x 4 ) I11

(1 - x 2 )(l - x 3 )(l - x4) X5

(l-z2)(l-:c3)(l-z4)

Proof. From the previous proposition, the dimension of A~£+ is not greater than oo

oo

oo

^dimM2fc+4„(n) = ^ d i m M 2 f c _ 8 „ = ^dimM2(fc_4n). n=0

n=0

26 H. Aoki

This is the coefficient of xk on the formal power series development of ^

(1 - x2)(l - x3)

(1 - x 2 )(l - x3)(l -

x4)'

We can prove the other three cases in an analogous way.

•

Hence, if we construct the algebraically independent modular forms G2 G A%+, G3 G A%+ and G 4 G A%+, the upper bound of the dimension of A%+ in Theorem 2.1 equals to the true dimension of A~£+. And additionally, if we construct the non-zero modular forms G§ G A§~ and G5 G A$~, all upper bounds in Theorem 2.1 equal to the true dimensions of A%+, A%~, A^+ and -A^" ~. Hence, if we assume the existence of these forms, we have given a new method of the determination of the dimension of Hilbert modular forms on Q(\/3)- In fact, Gundlach [4] constructed these forms G2, G3, G4, G5 and G6. We remark that he also determined all generators of the ring of ©fc€z Ak, using dimension formula of Hilbert modular forms. T h e o r e m 2.2. We have A:=@A++=CIG2,G3,G4}, fcez fcez

fcez

©4T=GBA fcez

Gundlach did not refer to the structures of A~£+, A~£~, A^+ and A^~. But the author believes that he knew these structures, because he constructed G5 and GQ in his paper [4]. 3. A differential o p e r a t o r of R a n k i n - C o h e n - I b u k i y a m a t y p e Rankin-Cohen type differential operators were extended to the Siegel modular forms of general degree by Ibukiyama [6]. Especially, Aoki and Ibukiyama [1] showed that this differential operator gives a very simple relation among the generators of Siegel modular forms of degree 2. In this section, we prove the analogous result on Hilbert modular forms.

Dimensions

of Hilbert Modular Forms

27

For Fi 6 Akl, F2 £ Aki and F3 € Ak3, put / hF!

k2F2

k3F3 \

£ * /-«^ F '

[FuF2,F3]:=det

OTi

OTi

C/Ti

A F l A F2 A Fa \6V 2 [Fi,F

By direct calculation, we have [G2,G3,GA]eA^+.