The Conference on L-functions: Fukuoka, Japan, 18-23 February 2006

THE CONFERENCE DN

FUNCTIONS Editors

Lin Weng Masanobu Kaneko

This page is intentionally left blank

THE CONFERENCE ON

Z-FUNCTIONS

THE CONFERENCE ON

Z-FUNCTIONS Fukuoka, Japan

18 - 23 February 2006

Editors

Lin Weng Masanobu Kaneko Kyushu University, Fukuoka,

Japan

Y ^ World Scientific NEWJERSEY

• LONDON • SINGAPORE • BEIJING • SHANGHAI • HONGKONG

• TAIPEI • CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 11 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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THE CONFERENCE ON L-FUNCTIONS Copyright © 2007 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 13 978-981-270-504-4 ISBN 10 981-270-504-X

Printed in Singapore by World Scientific Printers (S) Pte Ltd

Preface From February 18 to 23, 2006, with the support from our distinguished speakers and participants, the Conference on L-Functions was successfully held at Nishijin Plaza, Fukuoka, Japan. This volume is an invaluable limited collection of the works related to the conference. The aim of the conference is to provide a common platform for mathematicians working on L-functions to communicate with each other more effectively. To a certain degree, this has been fulfilled: More than 130 participants attended the conference. We are also quite encouraged by the fact that many many young researchers and students were present. In fact, much more has been achieved - Not only our lectures cover numerous topics related to L-functions, they also reflect the current, most advanced and extremely important aspects of L-functions. Each volume should send its clear message to the readers. Ours is as follows: On one hand, the theory of L-functions has already become very much diverse and is naturally embedded into various branches of mathematics; on the other hand, the theory is also pretty much integrated. Accordingly, in this volume we collect the papers of Bruggeman, Ibukiyama and Kim related to automorphic forms; the algebraic theory of Hida on X-invariant of p-adic L-functions; the analytic results of Jutila, Motohashi and Suzuki on the circle method, Sieve method and the Riemann Hypothesis of rank 3 zeta; the conjecture of Kaneko on multiple zeta values; the articles of Matsumoto et al. on zeta functions of root systems and of Wakayama et a/.'s on spectral zeta functions; the paper of Murty on a special class of L-functions, and the geometric approach of Weng to Lfunctions. The conference was sponsored by JSPS and a COE program at the Graduate School of Mathematics, Kyushu University, Japan in relation with Akito Futaki, Ryoichi Kobayashi, Iku Nakamura, and Mitsuhiro T. Nakao, among others. We express our gratitude to them. Clearly, without the help from a large group of students, the conference would not run so smoothly. Thank them all, our dear young fellows. The Editors v

List of Participants S. Akiyama, Niigata University H. Aoki, Tokyo University of Science N. Aoki, Rikkyo University R. Bruggeman, Universiteit Utrecht D. Burns, King's College London J. Choy, Korea Institute for Advanced Study N. Dan, The Romanian Academy S. Djoko, Kyushu University S. Egami, Toyama University I. Fesenko, Nottingham University M. Furusawa, Osaka City University H. Gangl, Durham University Y. Gon, Kyushu University T. Goto, Tokyo University of Science K. Gunji, Tokyo University Y. Hamahata, Tokyo University of Science M. Hanamura, Tohoku University S. Harada, Kyushu University S. Haran, Israel Institute of Technology Y. Hashimoto, Kyushu University T. Hayashi, Kyushu University T. Hibino, Toua University H. Hida, UCLA K. Hiraga, Kyoto University M. Hirano, Ehime University T. Hiranouchi, Kyushu University S. Hirayama, Kyushu University T. Ibukiyama, Osaka University Y. Ichihara, Hiroshima University A. Ichino, Osaka City University K. Ihara, Nagoya University T. Dceda, Kyoto University N. Ishii, Osaka Pref. University Y. Ishikawa, Okayama University K. Itakura, Tokyo University T. Ito, Kyoto University D. Jeon, Korea Institute for Advanced Study D. Jiang, University of Minnesota B. Jun, Korea Institute for Advanced Study M. Jutila, University of Turku J. Kajikawa, Kyushu University

M. Kaneko, Kyushu University S. Kang, Korea Institute for Advanced Study T. Kashio, Osaka University Y. Kato, Tohoku University Y. Kawamoto, Kyushu University H. Kawamura, Hokkaido University H. Kim, University of Toronto K. Kimoto, University of the Ryukyus M. Koike, Kyushu University T. Komatsu, Kyushu University K. Konno, Fukuoka University of Education T. Konno, Kyushu University R. Kozuma, Kyushu University M. Kurihara, Keio University J. Lagarias, University of Michigan E. Lapid, Hebrew University J. Lee, Hongik University M. Lee, Korea Institute for Advanced Study Y. Lee, Seoul National University T. Machide, Hokkaido University Y. Maeda, Hokkaido University K. Matsumoto, Nagoya University S. Matsumoto, Kyushu University K. Matsuno, Tokyo Metropolitan University M. Mera, Kyushu University T. Miyazaki, Keio University Y. Mizuno, Osaka University Y. Mizusawa, Sophia University S. Mochizuki, Tokyo University D. Morishita, Nagoya University M. Morishita, Kyushu University Y. Motohashi, Nihon University S. Muneta, Kyushu University K. Murty, University of Toronto H. Naito, Kagawa University T. Nakahara, Saga University S. Nakajima, Gakushuin University D. Nakajo, Kyushu University T. Nakamura, Nagoya University Y. Nanbara, Kyushu University H. Ochiai, Nagoya University

vii

viii T. Oda, University of Tokyo T. Okazaki, Osaka University K. Onodera, Keio University K. Ota, Kyushu University M. Ozaki, Kinki University Y. Ozeki, Kyushu University K. Prasanna, UCLA N. Ramachandran, University of Maryland Y. Sakai, Kyushu University F. Sato, Rikkyo University M. Shimura, Chuo University M. Spiess, University of Bielefeld K. Suzuki, Kyushu University M. Suzuki, Nagoya University Y. Taguchi, Kyushu University Y. Takakura, Kyushu University A. Tanaka, Kyushu University T. Tanaka, Kyushu University H. Taya, Tohoku University A. Teshima, Kyushu University M. Tien, Tsing Hua University H. Tokitsu, Kyushu University L. Truelsen, University of Aarhus T. Tsuji, Tokai University

List of Participants H. Tsumura, Tokyo Metropolitan University H. Tsutsumi, Osaka Univ. of H. & S. Science S. Uchiyama, NTT A. Umegaki, Sophia University T. Umeno, Kyushu University E. Urban, Columbia University R. Vidunas, Kyushu University C. Virdol, Nagoya University I. Wakabayashi, Seikei University S. Wakatsuki, Kyoto University M. Wakayama, Kyushu University S. Wakiyama, Kyushu University L. Weng, Kyushu University A. Yamagami, Kyoto University M. Yamagishi, Nagoya Inst, of Technology K. Yamanohira, Kyushu University Y. Yamasaki, Kyushu University T. Yamazaki, Tsukuba University S. Yasuda, Kyoto University T. Yasuda, Kyushu University H. Yoshida, Kyoto University M. Yoshida, Kyushu University J. Yu, Tsing Hua University D. Zagier, MPIM Bonn

Contents Preface

v

List of Participants

vii

Quantum Maass Forms R. BRUGGEMAN

1

X-Invariant of />Adic L-Functions H. HIDA

17

'hree Siegel Modular Forms of Weight Three and Conjectural Correspondence of Shimura Type and Langlands Type T. IBUKIYAMA

55

Convolutions of Fourier Coefficients of Cusp Forms and the Circle Method M. JUTILA

71

On an Extension of the Derivation Relation for Multiple Zeta Values M. KANEKO

89

On Symmetric Powers of Cusp Forms on GLi H. H. KIM

. .

95

Zeta Functions of Root Systems Y. KOMORI, K. MATSUMOTO & H. TSUMURA

115

Sums of Kloosterman Sums Revisited Y. MOTOHASHI

141

The Lindelof Class of L-Functions K. MURTY

165

IX

X

A Proof of the Riemann Hypothesis for the Weng Zeta Function of Rank 3 for the Rationals . . . M. SUZUKI Elliptic Curves Arising from the Spectral Zeta Function for Non-Commutative Harmonic Oscillators and r0(4)-Modular Forms K. KIMOTO&M. WAKAYAMA A Geometric Approach to L-Functions L. WENG

Contents

175

201

219

Quantum Maass Forms R. BRUGGEMAN

In this paper, we propose a definition of the concept of quantum Maass forms. On the basis of this definition, we discuss the relation with Maass forms and with cohomology groups. In [9], D.Zagier defined the function

(0.1)

n

*0 = e^^YJ\(l-e2niM)71=0

j=\

For y = ± (" *) e PSL2(Z), c> 0, this function satisfies (0.2)

v (£)-v(rXc£

+
gr(g),

with a multiplier system v of weight - | . The function gy is the restriction to Q of an element of C°°(R). The name quantum modular forms is used for functions on Q that satisfy a modular transformation property modulo simpler functions on R, in Zagier's example the smooth functions gy. The word "quantum" has no physical meaning here. Quantum modular forms are not modular forms, but have some relation to holomorphic modular forms (to the Dedekind eta function in Zagier's example). The subject of this paper is the definition of quantum Maass forms, related to PSL2(Z)-invariant real analytic eigenfunction of the Laplace operator on the upper half plane that are not holomorphic. A definition is proposed in §1.1.5, and extended in §1.2.2. Section 2 gives results that suggest that these definitions are sensible. 1

2

R. Bruggeman

Acknowledgements. Don Zagier introduced me to quantum modular forms and quantum Maass forms. The results in this paper are closely connected to results on the relation between Maass forms and cohomology, in a project of John Lewis, Don Zagier and me. I have profited from Tobias Miihlenbruch's comments on a previous version of this paper. It was a pleasure to attend the conference on L-functions in Fukuoka, where I presented these results. I am grateful for the invitation.

1

Quantum Maass forms associated to Maass cusp forms and Eisenstein series

We work with the discrete subgroup T = PSL2(Z) of G = PSL2(R). By [ ^ ] we denote ± ( ^ ) e G . The space ££ consists of the complex valued functions u on the upper half plane § that satisfy (i) T-invariance: u(yz) = u{z) for all y € T. (ii) Eigenfunction of Laplace operator: -v 2 (d2. + <92) u = s{\ - s)u for some s 6 C, the spectral parameter. Our main interested is in the case 0 < Re s < 1. For convenience, we assume s £ j . The space ££ has infinite dimension. The finite dimensional subspace space Ws of Maass forms is defined by the condition (iii) Polynomial growth: u(z) = CMy4) as y —» oo for some A > 0. The space Mf^ c Mf^ of Maass cusp forms consists of u satisfying (iii') Quick decay: u{z) = O (y~A) as y -* oo for all A > 0. 1.1

Quantum Maass forms associated to Maass cusp forms

The space MfJ is non-zero only for a discrete set of s e | + i R . Each u e Mf^ has a Fourier expansion (1.1)

u(z) =

Yjan^Ks-X/2(27T\n\y)e2ninx.

3

Quantum Maass Forms

A Maass cusp form is even if a_„ = an for all n, and odd if a_„ = -an. This gives a decomposition Mf? = Mf^'+ © Mf^~, where Mf^'+, respectively Mf^'~, is the subspace of even, respectively odd, Maass cusp forms. The main theorem in [3] associates to each u e Mf^* objects of three other types: (a) (See Maass, Chap. IV & V of [5].) With an as in (1.1), the L-function is L(p) = 2ZZi^n-p.

Let ys(p) = I ^ r ( ^ i ) r ( ^ ) . The

completed L-function, L*(p) = ys(p+ ^§M L(p), has a holomorphic continuation top € C, and satisfies L*(l-p) = ±L*(p). (b) The periodic function on C \ R

is holomorphic and 1-periodic. It is not r-invariant. (c) The period function f(T)-r-2sf(-l/T),

Hr) =

which extends holomorphically to C = C \ (-oo, 0]. It satisfies (1.3)

Mr) = ( - l y ) ,

T- 2 > (-M = ±^(r).

The following theorem is not explicitly stated in [3]. Don Zagier pointed it out to me. Theorem 1. The periodic function f associated to a Maass cusp form can be extended to Q such that for all £ € Q: (1-4)

/(r)

= m

+ o(l)

(T-£*0.

It satisfies /(f+1) = /(£) T 2 7 fy )

= f(€) -
for all

^eQ,

/or all £ e Q n (0, oo).

4

R. Bruggeman

Geodesic approach. Here T A £ means that T approaches £ along a geodesic in § or § - . So T tends to £ along a piece of a vertical line through £, or along a euclidean circle with its center on the real axis. The implicit constant in o(l) may depend on the geodesic. Proof. In §1.4 of [3], the integral representation f±(p)e±i7rP'2r-P dp

/(r) = r ^ f l m

JRep=C

is given for ±Imr > 0, with C > 0. Moving the line of integration to Rep = - e for some small e > 0, we pick up a residue at p - 0. The residue gives a constant /(0), which can be expressed in terms of the L-function; see (1.16b) in [3]. An estimate of the L-function shows that the integral over Rep = - e contributes o(l) as T -^ 0. This gives (1.4) for£ = 0. Repeatedly applying the periodicity and the relation ^ ( T ) = f(j) - T~2S/(-1/T), gives (1.4) at all £ e Q. We use that T H T + 1 and T H - map geodesies to geodesies. • 1.1.1

Explicit formula

A direct extension of the reasoning in [3] gives an expression in terms of the twisted L-series (1.5)

Lo(p,0 =

J^nane2*^1-?,

which has an analytic continuation and a functional equation. For £ = f, a, c € Z, c > 0, (a, c) = 1, aa = 1 mod c:

„ 6)

/(f) = -I^*r(- + I)c- I .(, + I.-f).

One needs the meromorphic continuation of both L0 and Le(p,^) = Zn*o a„e2mnf\n\~p. This continuation is obtained from integrals of u(£ + iy) and dxu{i; + iy) over y e (0, oo). The functional equations relate these Lseries for £ = ^ and £ = -c, and imply that Le (±(s - \),gj = 0 for all

5

Quantum Maass Forms 1.1.2

Automorphic distribution

As we shall discuss in §2.1, the periodic function / represents a hyperfunction on R, which for / coming from a Maass cusp form is actually a distribution. The limit behavior in Theorem 1 indicates that this distribution is rather regular. Schmid and Miller, [7], [6], have studied this distribution as the distribution derivative of a continuous function. 1.1.3

Smooth extension of the period function

The proposition in §111.3, [3] implies that there is iff e C°°(R) such that ifr(x) = iff(x) for x > 0, and | X | _ 2 ^ ( - 1 / J C ) = iff(x) for x e R \ {0}. So iff is real analytic on R \ {0}. At x < 0, the value of ij/{x) has no direct relation to the values of ift(T) with T eC near x. We have If | - 2 y ( - l / £ ) = /(£) - &(£) for almost all £ € Q. 1.1.4

Principal series

This leads to a reformulation in terms of the smooth vectors in the principal series representation with spectral parameter s. Let *Y? be the space of / e C°°(R) for which x •-» |x|- 2 V(-l/^) extends as an element of C°°(R). This space is invariant under the following right action of G: (1 -7)

(f\2sg) (x) = \cx + d\-2sf(gx)

for g =

ab eG. cd

The extended period function iff is an element of 'V™. There is a 1cocycle y i-> cy on T with values in
/ ( £ ) - ( / b y ) ( £ ) = cY(g)

for almost all £ e Q .

So indeed, the function on Q induced by the periodic function satisfies a modular transformation property, modulo an element of "V™. Actually, \j/ is real analytic outside 0 and oo. So the cocycle c takes values in the G-invariant subspace \x\~2sf(-\/x) is given by a power series in — that converges on a neighborhood of — = 0.

6

jR. Bruggeman

1.1.5

Definition of quantum Maass forms

Let Us be the space of (equivalence classes of) functions Q \ E —» C where E c Q is a finite set. We consider two such functions to be equal in Hs if their difference is zero outside a finite subset of Q. The formula (1.7) defines a right action of T in Rs. By restriction,
0—•T?'00—»*?,—»0i—*0.

The periodic function / coming from the Maass cusp form u determines an element of Hs. Its class in Qs is T-invariant. The function / is not in f^. Actually, ^ = C is not very interesting. Let us define the space of quantum Maass forms with spectral parameter s as (1.10)

qMf : = # / * $ .

Theorem 1 shows that there is a map qs : Mf^ —> qMfy. 1.1.6

Map to the semi-analytic cohomology

To (1.9) is associated a long exact sequence, of which we use the following part: (1.11)

0—>()-->*£—»££—»ff1(T,'V?).

(The space (*V^°) is zero.) This implies that there is an injective map

Crql^-^CW0).

In [4] it is shown that there is an injective map rs : Mf^ —> H1 (T, 'Vf'™). The image is the parabolic cohomology subgroup 77par (T, 1^'°°). (For the modular group T - PSL2(Z), the parabolic cohomology group H^w (T,A) can be defined as the subgroup of classes in Hx (r, A) that can be represented by a cocycle c that satisfies cj = 0 for T = L j e Y.) It turns out that rs is equal to the composition cs o qs. So the quantum Maass forms constitute a space through which we can factorize the map from Maass cusp forms to cohomology. The injectivity of r^ implies that qs is injective.

7

Quantum Maass Forms 1.2

Eisenstein series, extended quantum Maass forms

The Eisenstein series Es(z) = M2s)ys + A(2s - l)yl~s

(1.12)

+ 2 ^ |nrV2,_!(W) A(M) = w - * r ( ^ £ ( i O ,

^Ks-l/2(2n\n\y)e2ninx,

crB(n) =

J V

gives a meromorphic family of elements of Mfs, complementary to the cusp forms. It is holomorphic in the region 0 < R e 5 < l , s ^ j . A modification of the theory of Lewis and Zagier works for the Eisenstein series; see §IV.l, [3]. The holomorphic periodic function associated to Es can be chosen as (1.13)

/,(r) =

±

VSftXl- 'M2»

±

, T ( 1 _ s) £ cr2s^n)e^

,

with the convention ± 1 = sign Im T. The period function ^ ( T ) = fs(j) T~2sfs(-l/r) is holomorphic on C . However, it has no smooth extension through 0. The function ifrs defined by (/^(JC) = ifss(x) for x > 0, and iffs(x) = |jc|_2'y^(—1/JC) is real analytic on R \ {0}, but it is not an element ofV?' 00 . The expression (1.13) is explicit, and implies that at the rational point 2, a, c € Z, c > 0, (a, c) - 1, we do not have a limit value, but an expansion: (1.14) Mr)

=

l

-c2s-2K(2s-\)-

± y-^Til

l

- s)T {s + I ) c-2*M2s) ( * (r - ^jf2S

-^-T(, + i)|((f))^,,/c ) + 0(l)

T-*f.

where ±1 = sinlmr, and where £(•, •) denotes the Hurwitz zeta function. The computations are similar to those for cusp forms. We use the fact that

8

R. Bruggeman

the twisted L-series associated to (1.13) have expressions in terms of the Hurwitz zeta function. The example of the Eisenstein series shows that there are two difficulties: (1) There are no limit values, but expansions. (2) The cocycle has values in a larger space than rV^,0°. If we do not want to end the study of quantum Maass forms here, generalizations are needed. 1.2.1

Type of quantum Maass forms

'Vf'00 consists of the semi-analytic vectors in
0 —» V , —» K, —» ££ —» 0.

We define the space of quantum Maass forms of type t as qMfj = (Q's) fR^. In §1.1, we have omitted t = (io, oo) from the notations. The inclusions between space of vectors in the principal series induces the following natural maps

qMfffs qMf?

> qMff °° ^"qMfr

1.2.2

Extended quantum Maass forms

Let Rs be the space of systems p that associate to almost all £ e Q an expansion /?(£, T) holomorphic in r near £. Two such systems of expansions

9

Quantum Maass Forms

are equal if p i ( £ r ) = pifor) + o(l) (T ^ £) for almost all £. A_constant is a holomorphic function, so %s c ft*. The group T acts in ^ by (pb*y)(£T) = jy(T)-sp(y%,yT), where (1.16)

7 y ( T r = ((cr + d) 2 )"'

forr

erf

is a holomorphic extension of x i-» |c* + rf| . For any type t we define ££ = RspV's. This is a huge space. We refrain from giving a name to the elements of (Q'sf /Ws. To the Eisenstein series Es, we can associate an element of \Q"'sj /fcs, represented by the system of expansions in (1.14). It turns out that the middle term in (1.14), with ± ((T/(T - a/c))~2s, determines a system of expansions in *R%. So this term may be considered as trivial and should be divided out. It seems sensible to define 4ls c *RS as the systems of expansions of the form (1.17)

p(t,r)

= Jt^

+ ct + oQ.)

(T-Z+&.

It is a T-invariant subspace. We define Q?s = H1 (T,
2 Quantum Maass forms associated to invariant eigenfunctions The aim is to attach extended quantum Maass forms to all invariant eigenfunctions in &rs. 2.1

Invariant hyperfunctions

Up till now, we have used the line model of the principal series. When dealing with hyperfunctions it is more comfortable to use the projective

10

R. Bruggeman

model. In that model, ^Vf and "V^3 correspond to analytic, respectively smooth, functions on P^. The transformation behavior is (2.1) [c d\

I +X

If / is in the line model, then Pr{x) = (x2 + l) f(x) is in the projective model. The results of §1 can be formulated in the projective model: Principal series functions have to be multiplied by (l + x2J . To an (extended) quantum Maass form represented by p we associate the corresponding object in the projective model, given by ppr(g,T) = (1 + T2) p(£,r). When dealing with ppT, there is no problem in also considering p(°°, T). We assume that from now on, this reformulation in the projective model has been done. A drawback of the projective model is that the simple relation /(£+1) = /(£) in Theorem 1 becomes [j^ij

Pr(t + D =

Fit)2

Let &s be the space of all solutions of -y {oi1x + d2)u = s(l - s)u. Helgason, [2], has shown that there is an isomorphism of G-spaces n^ : < V~0) —* &s, the Poisson integral. For further explanations, see [8], [1], or [4]. Under this isomorphism, (
^(anKs-i/2(2n\n\y)

+ bnh/2-s(2n\n\y))eIninx

neZ\{0)

The convergence of this series implies that bn = O (e~AM J for each A > 0, and an = O (ee|n|) for each e > 0. The hyperfunction au such that Usau = u is represented on R by the function fl\r) = (l + T 2 ) /«(T), where /„ is given by

(2.3)

Mr) = ± ^ ^ a o ± £ r < l - s) £

with ±Imr > 0. See Proposition 2.1 in [1].

n^a±ne^,

Quantum Maass Forms

11

A representative g of au is a holomorphic function on U \ P^ for some neighborhood U of P^ in P£. Two representatives differ by a holomorphic function on a full neighborhood of P^. That difference corresponds to an element of "Vf. If y € T, then f^y is an other representative of au. So 7 H r r = g - g\^sy is a cocycle with values in fVf. This induces a map r^ : ££ —> 7/1 (T,
Quantum Maass forms of analytic type

Theorem 2. There is an injective linear map q" : 6^ —> eqMf^ such that c? o q- = r-. Proof. Let w e 6^, a € ( T V ) , u = Hsa. A representative g of a determines p e fes by P(£,T) = g(r) + o(l) (r -^ £). The T-invariance of a implies that pl^T = P + ry, with r r € 0- It turns out that this representative is invariant under T. So we can define an element of fiJ^ by defining mo(£, yr) = ff(T)smo(oo, T), independent of the choice of y e T such that £ = yoo. The remaining part a°° is a hyperfunction supported in {00}. The term boyl~s corresponds to a hyperfunction that is near 00 represented by a multiple o f T H T . This has the behavior of the expansion at 00 of an element of*,.

12

R. Bruggeman

The terms with I\/2-s correspond to a hyperfunction that is near oo represented by IH2

(2.4) B{T) =

S

j-z

£ \n\l-'bnT(l + T~2)S yFx(\;2 - 2s;2ninr).

This last contribution is not explicitly given in §4.3 in [1]; so we give more details. First we have to check that the hyperfunction yS induced by B is sent to the /i/2-i-part of (2.2). The estimate bn = O (e -/4|n| ) for each A > 0 implies the absolute convergence in (2.4). The support of B is contained in (oo). The Poisson map is given by an integral around oo: yQ+T 2 ) *J|TM?

\l~s

\(T-Z)(T-Z)J

dr 1+T2'

with R > 1, R > \z\. After a change of variables: -2s

usBiz) = ^ L £ i „ r v V - f (i + If (i + * V dr • iFi(l; 2 - 2*; 2mn{r + x)) — T

for sufficiently large R. Insert the power series expansion for iFi and the

(

2\i-l

/

\l-2s

1 + ^2 J and (1 + 7) • Some computations lead to the power series expansion of h/2-s, and give the desired terms in (2.2). It turns out that B\%ST(T) = B(r) + o(l) as r ^ 00. To check this, we use again the power series expansion of 1F1 and the binomial formula for powers of 1 + i. So the contribution of B(T) to p(<x>, r) can be transported to all £ € Q to give an element of 9^. Now we have a system of expansions p\ e *R.S given by pi(£, T) = g(r) - m(£, T) with m e f\^, and p(oo, T) = CIQQT 1 Ceo "T" o(l) (T ^> 00). Let

y = [c2] 6 r PI(£T)

such that

^ = r°° e Q- As T -^ ^: = M r ' V f c i © (00, r_1r) =

^"'^^"'^-(ffll^Kr-'T))

13

Quantum Maass Forms = M r M ( g ( r lr) + ry(y

1

r)-m(oo,y ! T ) )

= iy{y~lr)s (ry{y-lT) + pi(oo, y" 1 ^)

rf

1_

= (Anal) + (Anal) (d^ (— -

_ ^ ) + Coo + o(\)] ,

C2(T-0)

c

where (Anal) means a function that is holomorphic in T on a neighborhood of P^. This shows that p\ e
Quantum Maass forms and analytic cohomology

In [4] it is shown that the injective map rs : ££ -* Hl (T, Hl (T, V?) is bijective. Up till now, the assumption s + \ was convenient. Here it is essential. Proof. Let / be holomorphic on a neighborhood of oo. For |Imr| sufficiently large, we consider the one-sided average OO

(2,,

OO

/

..

2

\

S

/SW^/er,^^/^,). n=0

n=0

v

'

14

R. Bruggeman

The sum converges absolutely for Res > \, and defines a holomorphic function that satisfies (2.6)

/l£Av?|£(l - 7-) = / | £ ( 1 - DI^Avf = / .

If /(°°) = Oi the convergence and relation (2.6) hold for Re s > 0. For /O(T) = (l + T~2)S we find (2.7)

/ol£Av?(r) = T2* (l + T" 2 )' (Qs, r ) ,

where £(•, •) is the Hurwitz zeta function. This gives the meromorphic continuation of /ol^AVp to s € C, with a first order pole at s = \ as the sole singularity in 0 < Re s. Writing a general / as / = /(oo)/ 0 + / 1 ? we have /!(oo) = 0, and find /|£Av£ for all s * ±, 0 < Re s < 1. The relation (2.6) extends as well. The asymptotic behavior of the Hurwitz zeta function implies that there are AQ and A\ such that (2.8)

( / l ! | > r ) ( T ) = AIT + A0 + o(l)

(r -** oo).

For £ = 700, the subgroup ^ c T fixing £ is generated by n% = yTy~x. The one-sided average Av+ is obtained from Av£ by conjugation. Let ceZ1 (T, TJ 0 ). We define pe
ptf.T)

= - ( C W | ^ A V + ) ( T ) + O(1)

(T-*•£).

For (J e r , we find 75(T)-V(^,5T)

- ^ ( r r V ^ - . I ^ A v ^ . ^ + od)

- c * | £ ( ^ - l ) | £ A v + ( T ) - c»f |£Av+(T) + o(l) C«(T) + P ( ^ , T ) .

So p represents [p] e eqMf^ and c£[p] = [c].

D

Quantum Maass Forms

15

References [1] R.W.Bruggeman: Automorphic forms, hyperfunction cohomology, and period functions; J. reine angew. Math. 492 (1997) 1-39 [2] S.Helgason: Topics in harmonic analysis on homogeneous spaces; Progr. in Math. 13, Birkhauser, 1981 [3] J.Lewis, D.Zagier: Period functions for Maass waveforms. I; Ann. of Math. 153 (2001) 191-258 [4] R.Bruggeman, J.Lewis, D.Zagier: Period functions for Maass wave forms. II; in preparation [5] H.Maass: Modular Junctions of one complex variable; Notes by S.Lai, Tata Institue of Fundamental research; reprint Springer-Verlag, 1983 [6] S.D.Miller, W.Schmid: The highly oscillatory behavior of automorphic distributions for SL(2); Letter in Math. Phys. 69 (2004) 265-286 [7] W.Schmid: Automorphic distributions for SL(2, R); p. 345-387 in Conference Moshe Flato 1999, Quantization and Symmetries, vol. 1, Math. Phys. Studies 21 (2000), Kluwer Acad. Publ. [8] H.Slichtkrull: Hyperfunctions and harmonic analysis on symmetric spaces; Birkhauser, 1984 [9] D.Zagier: Vassiliev invariants and a strange identity related to the eta function; Topology 40 (2001) 945-960

Roelof Bruggeman Mathematisch Instituut, Universiteit Utrecht, Postbus 80010, NL-3508 TA Utrecht, Nederland Email: [email protected]

X-Invariant of /?-Adic L-Functions H. HIDA1

Contents Introduction

17

Lecture 1: Galois deformation and ^-invariant 1.1 Selmer groups 1.2Greenberg's X-invariant

24 27 29

Lecture 2: Elliptic curves with multiplicative reduction 2.1 Hecke algebras for quaternion algebras 2.2 Proof of Theorem 2.1

. . . .

31 31 34

Lecture 3: X-invariants of CM fields 3.1 Ordinary CM fields and their Iwasawa modules 3.2 Anticyclotomic Iwasawa modules 3.3 The X-invariants of CM fields

37 37 39 40

Appendix: Differential and adjoint square Selmer group . . . .

44

Introduction2 Let Q c C be the field of all algebraic numbers. We fix a prime p > 2 and a p-adic absolute value | • \p on Q. Then C p is the completion of Q under | • \p. We write W = \x 6 K\\x\p < 1} for the p-adic integer ring of 'The author is partially supported by the NSF grant: DMS 0244401 and DMS 0456252. This introduction is the note of the lecture delivered at the conference on L-functions at Kyushu University on February in 2006, and the following four sections are die notes of the lectures at Harvard University, Spring 2006, in dieir eigenvariety semester organized by Mazur and Taylor. We would like to thank the organizers of the conference at Kyushu University and die semester at Harvard University. 2

17

18

H. Hida

sufficiently large extension K/Qp inside Cp. We write Qp for the field of all numbers in Cp algebraic over Qp. Start with a strictly compatible system {pi} of semi-simple Galois representations p\ : Gal(Q/Q) —> GLd(T\) for primes I of the coefficient field T c Q . We suppose that p is associated to a pure (absolute Hodge) motive in the sense of Deligne (see [D]). We assume that p does not contain the trivial representation as a subquotient. We write S for the finite set of ramification of p and p\ is unramified outside S U {oo, (}, where t is the residual characteristic of I. We write p = y e Or||£|p < l} and often write W := OT,V, where Oj is the integer ring of T. Often we just write p for p p which acts on V = T$. For simplicity, we assume that p £ S. Assume that q \ I, and let E((X) = det(l -pq(Frobe)\vi X) € T[X\. We always assume thatp p is ordinary in the following sense: p restricted to Gal(Qp/Qp) is upper triangular with diagonal characters N"' on the inertia Ip for the p-adic cyclotomic character N ordered from top to bottom as a\ > 02 > • • • > 0 > • • • > adThus (N"\ 0

* N"i

P\i„ = V 0

0

••• N"d I

In other words, we have a decreasing filtration T'+lp c T'p stable under Gal(Qp/Qp) such that gr>(p)(-i) := (T'plTMp)(-i), the Tate twists, is unramified. Define d

HP(X) = Y\ det(l - Frobp^wJX)

= [~[(1 - ajX). j=\

i

Then it is believed to be EP(X) = HP(X) if p £ S and EP(X)\HP(X) otherwise. In any case, ord p (a 7 ) e Z. Let us define (07 J

ifordp(or;)> 1,

[pa]

1

if ordp(or;) < 0

and put e = \{j\pj = p)\.

&(p) = f](l -fijp-1) and 8+(p) = f ] (1 -fijp-1).

£-Invariant ofp-Adic L-Functions

19

Then the complex L-function is defined by L{s,p) = Yle E{(£~s)~l. We assume that the value at 1 is critical for L(s,p) (in the motivic sense of Deligne in [D]). We suppose to have an algebraicity result (conjectured by Deligne) that for a well defined period c+(p)( 1)) e C x such that §c+(p(l)) & S € ~~ for all finite order characters e : Z* —» pp°°(Q). Then we should have Conjecture 0.1. Suppose that s = 1 is critical for p. Then there exist a power series Oa"(X) € W[[X]] and a p-adic L-function L^n(s,p) = Q>T{yx~s - 1) interpolating L(l,p ® e)for p-power order character e such that ®pn(e(y) - 1) ~ 8(p ® £ ) § ^ f f with the modifying p-factor £(p) as above (putting £(p ® e) = 1 if s + I). The L-function L^(s,p) has zero of order e + ord i= i L{s,p)for a nonzero constant J?n(p) e C* (called the analytic JL-invariant), we have c + (p(i»

*->i (s - ly

where "lim^i " is the p-adic limit, c + (p(l)) is the transcendental factor of the critical complex L-value L(l,p), and £ + (p) is the product ofnonvanishing modifying p-factors. When e > 0, we call that L™{s,p) has an exceptional zero at s = 1. Here is an example. Start with a Dirichlet character x '• (Z/AfZ) —» Q with^(-l) = - 1 . Then c(p + (l)) = (2ni). If we suppose ;f = (—j for a square free positive integer D, the modifying Euler factor vanishes at s = 1 if the Legendre symbol {—) = 1 <^ (p) = PV in Ooryc™ with p = Ix € OQjyrpjIljclp < l}. By a work of Kubota-Leopoldt and Iwasawa, we have a p-adic analytic L-function Lf,n(s,x) -
-X(p)pm-X)

L?{m,X) = o r (y1-" -1) = e o r / o m - m>Ar) ~ F

S(xNm)^^-

(2m)m for all positive integer m as long as \n - n\p < 1 for all n prime to p. If we have an exceptional zero at 1, it appears that we lose the exact connection of the p-adic L-value and the corresponding complex L-value. However, the conjecture says we can recover the complex L-value via an appropriate m

20

H. Hida

derivative of the p-adic L-function as long as we can compute £,an(p). We may regard^ as a Galois character Gal(Q[//w]/Q) = (Z/iVZ)x -> {±1}, and we remark that x(Erobp) = 1 to have the exceptional zero. For our later use, for the class number h of Q[ y-D], we write the generator of p as w\ so, ph = (jn) for w € Q[ V ^ ] . Though we formulated the conjecture for p g S, if p p is ordinary semistable, we have the same phenomena and can formulate a conjecture similarly. Here is such an example. Start with an elliptic curve E/Q, which yields a compatible system pg := [TeE] given by the £-adic Tate module T{E. Suppose that E has split multiplicative reduction at p. In this case, HP(X) = (1 - X)(l - pX) and EP(X) = (l-X), 6(pE) = 0 and G+(p) = 1. Then by the solution of the Shimura-Taniyama conjecture by Wiles et al, this L-function has p-adic analogue constructed by Mazur such that we have
^M

= S $ | = -l^forqeCp

given by q = m/m (p* = (m)) f

and the class number h ofQ[ ~y -D]; 2. For E split multiplicative at p, writing E(CP) = C^/q2 for the Tate period qe Q*, we have £an(pE) =

^ .

Here logp is the Iwasawa logarithm and \x\p = p_ordpW. The first assertion is due to Gross-Koblitz [GsK] and Ferrero-Greenberg [FG], and the second was conjectured by Mazur-Tate-Teitelbaum [MTT] and later proven by Greenberg-Stevens [3] and by some others further later.

H-Invariant ofp-Adic L-Functions

21

In the first formula of the theorem, the sign " - " of jj— is correct as explained in [G] (10) (and if we evaluate the value at s = 0, the minus sign should be removed as was done in [FG]. For an ordinary p-adic Galois representation p : Gal(Q/Q) -* GL^W), there is a systematic way to create many Galois representations whose eigenvalues of Frobp contain 1. Indeed, let p acts on the d x d matrices Md(W) by conjugation. Since Md(W) = Ad(W) © (scalar matrices} for the trace 0 space Ad(W) which is stable under the conjugation. Then the action of Ad(p)(Frobp) on Ad(W) has eigenvalue 1 with multiplicity > d - 1. However it is easy to check that the system Ad(p) is not critical if d > 2. Thus we assume that d = 2. Now require that pp : Gal(Q/F) -» GLa(W) be a Galois representation associated to a p-ordinary Hilbert Hecke eigenform (belonging to a discrete series at oo) over a totally real field F. We make Ad{pF) and consider the induced representation Indp Ad(pp) whose eigenvalues of Frobp has 1 with multiplicity e for the number e of prime factors of p in F. The system Ind^ Ad(pF) is critical at s=l. Arithmetic X-invariant. Returning to a general ordinary representation p = p p , we describe an arithmetic way of constructing p-adic L-function due to Iwasawa and others. We define Galois cohomologically the Selmer group _ Sel(p) c

tf1(Gal(Q/Q00),p®Qp/zi»)

for the Zp-extension Q«,/Q inside Q0v°) by the subgroup of cohomology classes unramified outside p whose image vanishes in Hl{lp, (fi/'F+p) ®z Qp/Zp). Here T+p is the middle filtration Txp and Ip is each inertia group at p. The Galois group T = Gal(Qoo/Q) acts on //1(Gal(Q/Q0O),p®Qp/Z/7) and hence on Sel(p), making it as a discrete module over the group algebra W[[T]] = lim W[r/rP"]. Identifying T with 1 + pZ p by the cyclotomic character, we may regard y e Y. Then W[[r]] = A by y t~* 1 + X. By the classification theory of compact A-modules of finite type, the Pontryagin dual Sel*(p) has a A-linear map into Yifeci A//A with finite kernel and cokernel for a finite set Q c A. The power series $ p = rj/en f(X)

22

H. Hida

is uniquely determined up to unit multiple. We then define Lp(s,p) = ®P(yl~s - 1)- Greenberg gave a recipe of defining £(p) for this Lp(s,p) and verified in 1994 the conjecture for this Lp(s,p) except for the nonvanishing of X(p) (under some restrictive conditions). For the adjoint square Ad(pf) for pF associated to a Hilbert modular form, the conjecture (except for the nonvanishing of -£(p)) was again proven in my paper [HOO] in the Israel journal (in 2000) under the condition that pF = (pF mod mw) is absolutely irreducible over Gal(Q/F[pip\) and the p-distinguishedness condition for pF\Gal(F IF ) f° r a ^ Pl/> (which we recall later). If there exists an analytic p-adic L-function Lapn(s,p) - O^Xy1-* - 1) interpolating complex L-values, the main conjecture of Iwasawa's theory confirms <J>p = 0£" up to unit multiple. Suppose now that pF is associated to a Hilbert modular Hecke eigenform of weight k > 2 over a totally real field F. Following Greenberg's recipe, we try to compute £{Ad[pF)) = £(Indvp Ad(pF)). By ordinarity, we have PF\GMQ IF) - ( o a ) w ' m t w o distinct diagonal characters av and /3P factoring through 7P -> Gal(Fp[jUpoo]/Fp) for the inertia group / p for all p|p. We consider the universal nearly ordinary deformation p : Gal(Q/Q) —> GLQ,{R) over K with the pro-Artinian local universal Kalgebra R. This means that for any Artinian local AT-algebra A with maximal ideal mA and any Galois representation pA : Gal(Q/F) -> GL2{A) such that 1. unramified outside ramified primes for pF; Z

P^lcaKQ /Fp) - (*0aA» ) W i t h ffA,p = a P

m

°

d

™A S U C h

that the diag

°"

nal characters factor through Ip -» Gal(F p [/y»]/F p ) for all p|p; 3. det(pA) = detpf; 4. PA = PF mod mA, there exists a unique ^-algebra homomorphism (p : R -» A such that (pops pA. Write Tp s Z p for the p-profinite part of Gal(Fp[^p°°]/Fp). Choose a generator y„ of r p and identify W[[TP]] with W[[XV]] by y p <-» 1+XP. Since PlGai(Qp/Fp) = (Stf* ), Mp" 1 : Gal(Fp[pp~]/Fp) -+ /?factors through Tp and induces an algebra structure on R over W[[XP]]. Thus R is an algebra over

23

£-Invariant ofp-Adic L-Functions

^[[-Xp]]p|p- If we write
£(Ind?^(pF)) (d6p([p,Fp]) = det(< dX

V

jx=0Y\^gp(7P)aP([p,Fv]r\ /p,p

where y p is the generator ofTv by which we identify the group algebra W[[Tp]] with W[[XP]]. This result is proved as [HMI] Theorem 3.73 under some redundant hypothesis in order to make the book [HMI] self-contained. In this note, we will sketch the proof in the general case assuming that p totally splits in F/Q and that pp is unramified outside p and co. The proof in the nonsplit case is more complicated, and we will give a full proof in [H06] along with conjectures predicting .^-invariant of symmetric powers of pf. Here are some examples showing usefulness of this theorem: Take a totally imaginary quadratic extension M/F in which all prime factors p\p in F splits as <jB e for e - |S|. Since we can compute explicitely the universal deformation p of p = Indj^ i//, we get from the theorem

24

H. Hida

Corollary 0.5. We have

where Ny is the local norm NM^/QP and e(^3) is the ramification index of A proof of this is given in [HMI] Corollary 5.39, though the sign (-l) e is erroneously omitted there. We will revisit briefly the proof of this corollary in Lecture 3 and correct the sign error. If E/F is an elliptic curve with split multiplicative reduction at all p\p, we write E / F P ( F P ) = Fp/q% for the Tate period qv e F*. Then we have directly from the theorem the following Corollary 0.6. Assume that E/F is associated to a Hilbert modularform on GL(2)/F. Then the Z-invariant -C(Ind^ Ad(pE)) is given by flp onM//^"))' where Nv is the local norm Npf/Qp. The above two corollaries are obtained by explicitly computing the universal representation p. The case where F = Q is treated in [H04]. We will generalize this corollary to Theorem 2.1 dealing with elliptic curves having multiplicative reduction at some prime factors of p and ordinary good reduction at the other prime factors of p and give a proof of Theorem 2.1 limiting ourselves to E having good reduction everywhere outside p and to F/Q in which p totally splits. The general case will be treated in [H06].

1

Lecture 1: Galois deformation and X-invariant

In the following three lectures delivered at the eigenvariety semester at Harvard University in Spring 2006, we give a sketch of the proof of the results mentioned in the introduction, assuming as a simplifying assumption that p completely splits in F/Q (the general case is treated in [HMI] Chapters 3 and 5 and also in [H06]). Let p > 2 be a prime, and fix a totally real finite extension F/Q. As we wrote, for simplicity, hereafter, we assume always that p_splits completely in F/Q. We start with a Galois representation pF : Gal(Q/F) -» GLa{W)

.£-Invariant ofp-Adic L-Functions

25

associated to a discrete series (<=> k > 2) Hilbert modular form / (over F) with coefficients in a finite extension W/Zp (a DVR). We assume the ordinarily of pF: PF\DV

= I QP

a

J with ep * a p , ep|/p = Nk~x and a p (/ p ) = 1

on the decomposition group and the inertial group /„ c D p c Gal(Q/F) for all prime factor p of p in F. Here JV(cr) € Z* is the p-adic cyclotomic character with exp(^) 0 " = e x p ( ^ j ^ ) for all n > 0 and k > 1 is an integer. Again for simplicity, we assume thatp is unramified outside p and oo. Thus for any prime I \ p, writing /|7\l) = a\f, we have Tr(p{Frob\)) = a\ e W. Let K be the quotient field of W (so, K/Qp is a finite extension). We consider the universal nearly ordinary deformation p : Gal(Q/F) -* GL2(R) over K with the pro-Artinian local universal K-algebra R. This means that for any Artinian local AT-algebra A with maximal ideal mA and any Galois representation p A : Gal(Q/F) —> GZ,2(A) such that (Kl) unramified outside p; (K2) PAlGa](Qp/Fp) = (oalv) with aA,p = av mod mA; (K3) det(pA) = detp F ; (K4) pA = pF mod mA, there exists a unique ^-algebra homomorphism : R —> A such that (pop s p A . Note that Af : Gal(F p [/yo]/F p ) a Z£ (by splitting of p in F/Q). Let Tp = 1 + pZ p c Gal(Fp[p;p<*>]/Fp). Choose a generator yv of Tp and identify W[[TV]] with W[[XP]] by y p «-» 1 +X P . Since pl Gal( Q p/Fp) = (o *,), (Spa"1 : Tp —» /? induces an algebra structure on R over WT[XP]]. Thus R is an algebra over ^T[[Xp]]P|p. If we write

AT for the morphism with y> o p s p F , by our construction, Ker(^o) 2 C^p)p|p. Here is the theorem we have seen in the first lecture: Theorem 1.1. Suppose R = K[[Xv]]v\p. Then, if

X(Ad(pF)) = det ( ^ y

D

)

f \x=Q

f [ logp(yp)ap([p, Fp]y*.

26

H. Hida

Greenberg proposed a conjectural formula of the ^-invariant for a general p-adic p-ordinary Galois representation V with an exceptional zero. When V = Ad(pp), his definition goes as follows. Under some hypothesis, he found a unique subspace H c_Hl(F,Ad{pF)) of dimension e = \{v\p}\ represented by cocycles c : Gal(Q/F) -» Ad(pp) such that 1. c is unramified outside p; 2. c restricted to Dp is upper triangular after conjugation for all p\p. By the condition (2), c\iv modulo upper nilpotent matrices factors through the cyclotomic Galois group Gal(Qp[^pco]/Qp) because Fv = Qp, and hence C\DV modulo upper nilpotent matrices becomes unramified everywhere over the cyclotomic Zp-extension Foo/F. In other words, the cohomology class [c] is in Selfoo(.A
P(°")

~(

n

( ^l for a-e Dp'with any p'|p.

Then ap : Dv> -» K is a homomorphism. His X-invariant is defined by JXAd(pF)) = det \(ap(.[p, FP']\ylp

(\ogp(yp>YX ap([yp>, Fv\))p^p)

J.

The above value is independent of the choice of the basis {cp}p. When F = Q, by a result of Kisin [Ki] 9.10, [Kil] and [Ki2] 3.4 (generalizing those of Wiles [W] and Taylor-Wiles [TW]), we always have R a K[[Xp]]. In general, assuming the following two conditions: (ai) p = (PF mod m^) is absolutely irreducible over Gal(Q/F[jUp]); (ds) pss has a non-scalar value over Gal(F p /F p ) for all prime factors p\p, by using a result of Fujiwara (see [Fu] and [Ful]), we can prove R = K[[Xp]]p\p. The following conjecture for the arithmetic L-function is a theorem under the condition (ai) and semi-stability of pp over O except for the nonvanishing £(Ad(pF)) * 0 (see [HMI] Theorem 5.27 combined with (5.2.6) there):

27

.£-Invariant ofp-Adic L-Functions

Conjecture 1.2 (Greenberg). Suppose (ds) and that p is absolutely irrerith x Sx s n ducible. For L'SaIpith (s,Ad(p (s,Ad(pFF)))) = Qf <&ritharith {y(y - - - 1), then Lapprith (s,Ad(pF)) has zero of order equal to d = [F : Q] and for the constant £(Ad(pF)) € Kx specified by the determinant as in the theorem, we have Lar" (s,Ad(pF)) Jim

vj.c W T ^Qyi^ NMI-V^QP] = £(Ad(pF))\\SelQ,(lndp Ad(pF))\\p

up to units. The factor 8+(Ad(p)) does not show up in the above formula, because if we write S F(Ad(pF)) for the Bloch-Kato Selmer group HxAF,Ad{p) <S> Qp/Zp), we have, as explained in [MFG] page 284, ||SelQ(Ind?AJ(pF))||;1/[A::Q>] = G+(Ad(pF))\\SF(Ad(pF))\\'mK'Qp]

up to units,

p and the value \\SF(Ad(pF))|| is directly related the primitive complex L-value L(l,Ad(pF)) up to a period. In the following section, we describe the Selmer group and how to define H.

1.1

Selmer Groups

We recall Greenberg's definition of Selmer groups. Write F^/F for the maximal extension unramified outside p and oo. Put © = G&l(F^/F) and ©M = Gal(F^VM). Let V = Ad{pF) with a continuous action of ©. We fix a W-lattice T in V stable under ©. Write D = Dv c © for the decomposition group of each prime factor p|p. Choosing a basis of pF so that pF\o is upper triangular. We have a 3-step filtration:

(ord)

v D r;v D r;v => {oj,

where taking a basis so that pF\D is upper triangular, T^V is made up of upper triangular matrices, and T£ V is made up of upper nilpotent matrices, and on f^V/T^V, D acts trivially (getting eigenvalue 1 for Frobv). Since V is self-dual, its dual V*(l) = Hom/KV, K)®N again satisfies (ord).

28

H. Hida

Let M/F be a subfield of F(p\ and put ©M = Gal(F (p) /M). We write p for a prime of M over p and q for general primes of M. We put Lp(V) = Ker(Res : H\MV, V) -> tf^/p, ; = ^ r ) ) . Then for a ©^-stable WMattice T of V, we define for the image LP(V/T) of Lp(V) in H\MV,V/T) SelM(A) = K e r t f ^ © * , A) -» ]~[( H w ^ ) ) f o r / l = V> V / r L pvv p The classical Selmer group of V is given by SelM(V/T), equipped with discrete topology. Write F^ for the cyclotomic Zp-extension of F. We define " - " Selmer group SeY^(V/T) replacing LP(A) by (1.1)

Lp-(V) = Ker(Res : H\MV, V) -» ff^/p, ^ 7 7 ) ) . Lemma 1.3. Suppose R = K[[Xp]]PiP. Then we hove S&\~F(V) 2 Horn*(nWm|, /(T) and Sel f (V) = 0. Proof. Consider the space Derx(R, K) of continuous AT-derivations. Let K[E] = K[t]/(t2) for the dual number e = (t mod t2). Then writing Kalgebra homomorphism : R —> A"[e] as 0(r) = 00(0 + \(r)s and sending 0 to 0i € DerK(R, K), we have Hom/f.aig(/?, £[e]) a DerK(R, K) = Hom/f(m/;/m|, £). By the universality of (R,p), we have Hom^aig(/?, AT[e]) {p : © —» GL2(A'[e])|p satisfies the condtions ( K l ^ ) } Pick p as above. Write p(cr) = po(cr) + p\(cr)s. Then cp = pip^.1 can be easily checked to be a 1-cocycle having values in M2{K) D V. Since det(p) = det(pf) => Tr(cp) = 0, cp has values in V. By the reducibility condition (K2), [cp] e Sel^.(V). We see easily thatp ^ p ' « [cp] = [cp>]. We can reverse the above argument starting a cocycle c giving an element of Sel^(V) to construct a deformation pc with values in K[e\. Thus we have {p : Gal(Q/F) -> GL2JK[s])\p satisfies the condtions (Kl^)} = SeVF(V).

29

.£-Invariant ofp-Adic L-Functions

Since the algebra structure of R over W[[Xp]]P|p is given by 6papx, the /^-derivation 5 : R —» K corresponding to a ^T[e]-defonnation p is a W[[^p]]-derivation if and only if pi|Gal(-p /F) ~ (oo)> which is equivalent to [cp] € Self (V), because we already knew that Tr(cp) = 0. Thus we have Self (V) a Derw[[Xv]](R, K) = 0. a We also have Lemma 1.4.

(V)

Self (V) = 0=>H\(S,V)sY\

^J%P-.

Indeed, by the Poitou-Tate exact sequence, the following sequence is exact: Self (V) - • H\f&M, V) -» J~| P

tf (

. ^;;.V) -> Sel F (V*(l))\ ^ w

An old theorem of Greenberg gives dim Self (V) = dim Self (V*(l))* (see [G] Proposition 2); so, we have the assertion (V). • 1.2

Greenberg's .^-invariant

Here is Greenberg's definition of £(V): The long exact sequence of TpV/TpV <^-» V/TpV -» V/TpV gives a homomorphism, noting F p =

H\Fv,r;v/r;v)

= Kom(^p,r;v/rp+v) X z/ 1 ^,

V)/LP(V).

Note that Hom(G^,r p -V/r p + V) - (T;V/TV+V)2 canonically by 0 ^ ( ^ M ( 0 ( [ p , F p ] ) ) . Here local Artin symbol (suitably normalized). Since

LP(T;V/TP+V)

= Ktr(H\Fv,rp-v/rv+v) ^

= K2

[JC,F P ]

= [x,Q p ] is the

H\ip,rp-v/rp+v)),

30

H. Hida

the image of ip is isomorphic to fv VfFp+V s K. By (V), we have a unique subspace H of H1^, V) projecting down onto

nHl(Fv,V)

]-rT , ^ P

p

fV

/

Then by the restriction, H gives rise to a subspace L of

Y\Hom(Gfv,r;v/rv+v) * f[(r p -v/r p + v) 2 p

p

isomorphic to Tlv('Jrv~V/'Fp+V). If a cocycle c representing an element in H is unramified, it gives rise to an element in Self (V). By the vanishing of Self(V) (Lemma 1.3), this implies c = 0; so, the projection of L to the first T'V

factor Yiv ijky (via (f> i-* (~ W ^ V -» n P ? T W^p+ V. We then define £(V) := det(£) e isT. Let p : (5/r -> GLi{R) be the universal nearly ordinary deformation withp| D = I J. Then cp = -^-\x=oPFl is a 1-cocycle (by the argument proving Lemma 1.3) giving rise to a class of H. By Lemma 1.3, H = SelF(V), and {cp}p gives a basis of H over K. We have 6([u, F p ]) = (1 + Xp)iogp(«)/iogp(r) for u e 0 x = z x W r i t i n g /-flp(o-)

C (ff ) =

'" l 0

*

\

i

Pl r)

' fl,( /^(©oo, V) is given by / / ' ( r , ^ © ^ , V ) ) = 0 because H°(<5„,V) = 0. Thus the image of H in SCIF^V/T) gives rise to the order d exceptional zero of ari,h L (s,Ad(pF)) at s = 1. We have proved _1

Proposition 1.5. For the number of prime factors e = [F : Q] of p in F, we have ord i=1 Lfth(s,Ad(pF))>e.

31

£-Invariant ofp-Adic L-Functions

2 Lecture 2: Elliptic curves with multiplicative reduction Let p be an odd prime. Order the prime factors of p in F as P i , . . . , pe. In this lecture, we describe the computation of the X-invariant of Ad(TpE) for a modular elliptic curve E/p with split multiplicative reduction at Vj\p > 2 for j - 1,2,..., k and ordinary good reduction at Vj\p for j > k. Theorem 2.1. Assume that R = QP[[XP]]P\P. Suppose that the Hilbertmodular elliptic curve E has split multiplicative reduction at Pj for j = 1,2,..., k (k < e) with Tate period qj at Vjfor j k. Then for the local Artin symbol [p, Fv] = Frobp and the norm Qj = Npv./qp(qj), we have for p# = TpE £(Ind? Ad{pE)) Id6p([p,Fi])\ /agp([p,F,-])\ - H ord,(<2^

\

dX;

\

V

^J

|I

r-f p r log j p(yPj)

J. , . , lx=ol 1 atv(lP,Fi\y

n>k,i>k

i>k

where, with the identification of the group algebra W[[TV]] and W[[XP]], yp is the generator of the p-profinite part Tp ofN(Ga\(Fp[p.p°°]/Fp)). In the proof, for simplicity, as before, we assume that p is completely split in F/Q. Also, again for simplicity, in the following proof, we assume E has good reduction outside p and k - 1. We put I> = YlpTp. 2.1

Hecke algebras for quaternion algebras

We make some preparation for the proof, gathering known facts. We assume that F * Q (otherwise the theorem is known by Greenberg-Stevens). For simplicity, p splits completely in F/Q. Take first a quaternion algebra fio/F central over F unramified everywhere such that Bo ®Q R. = M2(R)r x B.d~r with 0 < r < 1 (so r = d mod 2). Then we consider the automorphic variety (either a Shimura curve (r = 1) or a O-dimensional point set (r = 0)) given by Xn(pn) = B*\B*A/Su(pn)ZAC<

32

H. Hida

where ZA = F^ is the center of B^, Coo is a maximal compact subgroup of the identity component of B^ and identifying B^f = Bo ®Q F\ with M2(F\) for all primes I, Sn(Pn)

= ((achd) € Gbm\(*e»)

s ( J I)

mod p")

for 0 = r i i ^ i . Consider Mn s Hr(Xn(pn),Zp) which is the Pontryagin dual of Hr{X\\(pn), Q>p/Zp) which is a finite rank free Z^-module with Hecke operator action of 7\n) for all prime ideals outside p and U(p^) = U(pv)n and the diamond operator action (z) coming from ( ^ J for z € Op. Let e = lim„^oo U(p)nl as an operator acting on Mn (U(p) = flp U(pp)). Let M° rd be the direct summand eMn. We have natural trace map Mm -» M„ for m > « compatible with all Hecke operators and all diamond operators. By the diamond operator action, M°^d = lim M°rd naturally become a W[[IY]]-module. Here is an old theorem of mine: Theorem 2.2. The W[[TF]]-module M%d is free of finite rank over W[[rF]]. Let h be the W{[r>]]-algebra generated over WI[r>]] by T(n) for all n prime to p and all U(p). Then we have Corollary 2.3. The W[[TF]]-module h is torsion free of finite type with hF/(Xv)v\phF pseudo isomorphic to the Hecke algebra associated with Hr(Xu(p),W). Actually if p > 5, h is known to be free over WTIT/r]] and the pseudo isomorphism as above is actually an isomorphism. Let T be the local ring of the universal nearly ordinary Hecke algebra h acting nontrivially on the Hecke eigenform associated to E. Let P e Spf(T)(Q/)) corresponding to pF, that is, pj mod P ~ pF. Let T P = lim Tp/P"TP for the localization Tp. Since pE = T„E ® Qp is <

n

absolutely irreducible, by the technique of pseudo representation, we can construct the modular deformation pj : (5 —> GL,2(Tp) which satisfies (Kl-4); in particular, detpx = N, because the central character is trivial. Since E is modular over F, we have the surjective Q^-algebra homomorphism R -> Tp for the localization-completion Tp. Since Tp is integral and of dimension d, we have

33

X-ihvariant ofp-Adic L-Functions Corollary 2.4. IfR = K[[Xp]]v\p, then R s T/>.

Under absolute irreducibility of pF over Gal(F/F[pp]) with non-scalar semi-simplification of PflGal(7? /F ) for all p|p, the isomorphism R =* K [[Xp]]p|p will be proven by showing R s TP first (see Appendix). Take a quaternion algebra Bi/F such that B i ® Q R s M2(R)9 x ¥Ld~q with # < 1 and B is ramified only at Pi (among finite places). Then at Pi, we have a unique maximal order Rx in BPl. Then we define U\\{pn) to be the product of 5 ii(p") (Pl) and R* and define Yn(p") =

B^BIJUXX^ZACO..

Then we define ex = lim^oo U(p^Vl))nl acting on Nn = Hq(Yxx(p"),Zp), the dual of the cohomology group Hq(Yxx{pn),%IZp). LetTi = n p * P l Tp. We go through all the above process and define hi c EndvnrrniOini e\Nn). <—~n

Since p£ (or corresponding automorphic representation 7r#) is Steinberg at Pi, by the Jacquet-Langlands correspondence, we have a Hecke eigenvector fx in W{Yxx(p), Z p ) giving rise to E. Then we define Ti to be the local ring of hi acting nontrivially on f\. Let Pi € Spf(Ti)(H0 be the point associated to pE- We then have a deformation pr, : © -» GL2(Tx,p) of pg. Since the central character is trivial, we have detpr! = N. Theorem 2.5. We have 1. hi is torsion-free of finite rank over W[[Tx\], and Ti,p, = ^ [ [ X p j , . . . , XVd]];

2. prl restricted to Gal(FPl /F P ! ) is isomorphic to(sff * J, where e = ± 1 is the eigenvalue ofFrobVl on the etale quotient ofTpE; 3. There is a surjective algebra homomorphism T/X Pl T -» Ti inducing an isomorphism Tp/XVlTp s Xi,/*,; 4. There is a surjective algebra homomorphism T/([/(pi) - e)T -» Ti sending T(n) to T(n), where t/(pi) = U(pPl). Here is a sketch of proof. The first assertion follows from construction; in other words, it can be proven by the same way as the proof of Corollary 2.3. By the Jacquet-Langlands correspondence, T covers T\. Any

34

H. Hida

automorphic representation n corresponding to a point of Spf(Ti)(Qp) is Steinberg at pi because B\ ramifies at pi. Since points corresponding classical automorphic representation is Zariski dense in Spf(Ti), the Galois representation has to have the form as in (2). Thus the eigenvalue of t/(pi) of n is ±1 and the corresponding Galois representation has the form as in (2). The assertion (1) implies (3). By (2), C/(pi) is either ±1. Since C/(pi) is a formal function on the connected Spf(Ti), U(p\) = E is a constant, which implies (4). • 2.2

Proof of Theorem 2.1

Write for simplicity, Xj := XPj, Fj = FPj and pj = pVj. By (3) and (4) of Theorem 2.5, C/(Pi) = £ mod X\ is a constant independent of Xj := XPj for all j > 2. Thus ^ J ^ l x ^ o = 0 for a l l ; > 2. Thus

. JdU(vd\ det

dXj

dU(pi) x=o

dXi

xdet

(dU(Pi)\ \v -^ j '

*=°

X=0 ), >i>2j22

Since SPi{[p,Fi]) = t/(p,), we get from the formula we stated in the first lecture: £(lnd®Ad(pE))

the following new formula: (2.1) £(Ind?Arf(p £ )) = d6<>([p>Fl]) dXi

Xi=0

log/yp^ap.Ct/p.F!])- 1

Thus the result follows from the following lemma of Greenberg-Stevens: Lemma 2.6. Let us write y - yPv We have, for 8\ = 6Pl, , dX,

log„(?i)

35

^.-Invariant of p-Adic L-Functions

Proof. Since aVl([p,Fi]) = 1 (split multiplicative reduction), we can forget about this factor. Since the linear operator £ : WfT^V/T^V -> I\vr;V/r;V induces £t : r-V/r+V -» T-V/r+V by our diagonalization of its matrix. This Xi comes from the subspace Li c Hom^,r'V/r^V)

= Hom(Df ,Qp)

for Di = Gal(Q p /Fi) has a generator 00 = ^ / ^ : D? -> Op- Thus Xi=0 by definition dSi([p,Fi]) , , , , , X[p,Fil) logp(r) = logp(r)°^"M[y,Fi]) 5Xj *=o °^" L e t p £ = TpE ®Zp Qp, p £ = (p mod (Xj,X 2 ,.. .,Xd)), and write Qj = Qp[Xi]/(X^). The character (#i mod X^) is an infinitesimal deformation of the trivial character fitting into the following commutative diagram of D\ -modules: Q^(ei) ^ ^

QP(1)

PE —^-*

Qptfi)

» PE

> QP-

x

Twist this diagram by e^ N = 6\, getting a new diagram

QPW ^ ^ 4,

QpK^J P (l)

PE —*-* Optf]) ^

PE '* HE

4,

'

\Lp-

Once this type of diagram is obtained (with leftmost column given by Qp(l) -* Qp(l))> by a general result of Greenberg-Stevens in such a situation (see [GS1] (2.3.4)), we get ^(foi,Qp])|x I = o = 0 => ^ ( f o i . Q , ] ) ^ = 0. Write ^i = p"u for a = ordp(gi) and u € Z*. Then logp(w) = logp(gi). We have <5i([tfi,Qp])=
36

H. Hida

(because N([u,Qp]) = u x). Differentiating this identity with respect to Xi, we get from 6\(\u, Qp])|x,=o = 6i([p, QP])lx,=o = 1 dSiOpuQp]) logptel) d6i(lpi,Qp]) a= adX\ Xi=o \ogp(y) dXx x,=o

log„(u) logp(y)

= 0.

From this, we conclude , , iog„(7)

dS^puQp]) dXi

_ logP(gi) Xi=o ~ ordp(tfi)' n

The fixed field of the kernel of o is a Zp-extension M^ /Qp (Fi = Qp). Since Li 3

log„(r) PV

MiOpuQp]) 3Xi

Xi=o

logp(<7o) ordp(^o)'

Proof. Let 0O = S^1 ff; • Df -» Qp- Let Moo/Qp be the composite of all Zp-extensions of Qp; so, by local class field theory, Gal(Moo/Qp) = Zp. Then we have [
*>«P.Q,])/*=3&5-

^.-Invariant ofp-Adic L-Functions

37

3 Lecture 3: X-invariants of CM fields Let p be an odd prime. Let M/F be a totally imaginary quadratic extension of the base totally real field F. We study the adjoint square Selmer group when the Galois representation is an induction of a Galois character of (5M := GaliM^/M). Put © F := Gal(M (p) /F). For simplicity, we assume that p > 2 totally splits in Af/Q. We relate the Selmer group with a more classical Iwasawa module of a quadratic extension of F, and from the torsion property of the Selmer group already proven, we deduce some (new) torsion property of such classical Iwasawa modules. 3.1

Ordinary CM fields and their Iwasawa modules

Let OM be the integer ring of M. We consider Z = lim CIM(PH) for the ray class group CIM(P") of M modulo pn. Let A be the maximal torsion subgroup of Z, and put YM - Z/A, which has a natural action of Gal(M/F). We split Z = Axr M . We define Y+ = tf °(Gal(M/F), TM) and T~ = TM/T+. Since p > 2, the action of Gal(M/F) splits the extension r + e-» I'M -» T~, and we have a canonical decomposition YM = P" xY~. Write n~ : Z —> T~', n+ : YM —* Y+ and TTA : Z —» A for the three projections. Take a character

Q , and regard it as a character of Z through the projection: Z - » A. Let Mo, be the composite of all Zp-extensions of M. Then by class field theory, Moo is the subfield of the ray class filed of M modulo p°° fixed by A. Let Qoo/Q be the cyclotomic Zp-extension. Let M^c be the composite MQoo/M. Define M^ (resp. M£) for the fixed subfield of T~ (resp. r + ) . Since MCZ° is abelian over F, we have M^c c M£ and a projection 7Tcyc : Y+ —> Ga^M^/M) c 1 + pZ p . The Leopoldt conjecture for F asserts that ncyc is an isomorphism; in other words, M£ = M^c. The extension M^/M is called the anticyclotomic tower over M. Thus if the Leopoldt conjecture holds for F, AfM is the composite of the cyclotomic Zp-extension M^c and the anticyclotomic zjf '^-extension M^. To introduce Iwasawa modules for the multiple Zp-extensions M^/M, we fix a CM type S, which is a set of embeddings of M into Q such that /jif = S U Ec for the generator c of Gal(M/F). Over C, an abelian variety with complex multiplication by M has C-points isomorphic to C s /Z(a) for

38

H. Hida

a lattice a in M (see [ACM] 5.2), where 1(a) = { M a ) ) ^ e Cz\a e a}. By composing ip, we write S p for the set of p-adic places induced by ip o cr for cr e Z. We assume (spt)

E p n 2 p c = 0.

This is to guarantee the abelian variety of CM type S to have ordinary good reduction modulo p (whose Galois representation is hence ordinary at all P\P). Writing M(p°°) for the ray class field over M modulo p°°, we identify Z with Gal(M (p°°)/Af) via the Artin reciprocity law. Fix a character ip of A. We then define MA by the fixed field of T in M(p°°); so, Gal(AfA/M) = A. Since

.£-Invariant ofp-Adic L-Functions

39

the E-Leopoldt conjecture asserts the closure 0£ of 0£ in Ls = ripez,, ^P satisfies dimQ(C>£ ® z Q) = dim Qp (0£ ®Zp Q P ). If X~[^>] is a torsion A--module, we can think of the characteristic element T~{ip) e A" of the module X~[cp]. The anticyclotomic main conjecture (cf. [HT] Conjecture 2.2) predicts the identity (up to units) of T~(
Anticyclotomic Iwasawa modules

A character tj/ of A is called anticyclotomic if \p(ccrc~l) = ~(cr) = ^(co-c_1cr_1) = ^(c)o-| M -. Then we consider Indj^^j) : Gal(F/F) -> GL2(W[[r-]]). We write orM//r for the quadratic character of Gal(F/F) identifying Gal(M/F) with {±1}. Lemma 3.2. We have 1. det(Indj^) = «M/F*IF* and Tr(Indj^(FroZ>,)) = Z bc o M ^ M/f(b)= i X(b) for a prime I ofF unramified for I n d j ^ , identifying a character X o/Gal(F/M) with a character ofM*(oa)/Mx by the Artin symbol, 2. Ad(Ind^(ip)) = «M/F © Indj^(^—) as (&p-modules. Since Indj^(^)|(5M = 7p®lpc with £>c(cr) = 7p{ccrc~l), we then define !FP+Indj^^ = ^for p € S p . In Lecture 1, we have already defined r$AdQnAFM7p) and the Selmer group SelF(AJ(Ind£lp) ®Zp (W[[T-]])*). Since the image of F+iAdQnd^lp)) in CKM/F is trivial in the above decomposition in Lemma 3.2 and the image of !Fp+(Acf(IndJ^^)) is given by

40

H. Hida

r+CIndj^-)), we get (cf. [HMI] Exercise 1.12 and Corollary 3.81) Sel/KAdOnd&G)®^]] ( M P n ] ) * ) = Sel F (OMIF ®z, (WUT]])*) © Sel f (Ind^(^-) ®wurn (W[[T]\)*)

= Hom(crM ®z w[[r~]], qp/zp) e SeiM((^-) ® w n r n M P n r ) , where C/^ is the quotient of CLM by the image of ClF (the order of Cl~M is equal to the order of the a^/^-eigenspace of CIM up to a power of 2). By the definition of the Selmer group, we note that (3.1)

SelM(£~ Swnril (Mfr-]])*) as Hom(X-fcr],Qp/Z„),

which shows Theorem 3.3. Le? J&e notation be as above. Then we have SelF(Ad(IndM(cp))) = (C/^ ®z W[[r"]]) © X " [ f ] as W[[r~]]-modules. Moreover X~[cp~] is a torsion W[{T~]]-module without exceptional zero ifij/:=(p~ satisfies the following conditions: (atl) The character \p has order prime to p. (at2) The local character fly is non-trivial for all ^8 € Ep. (at3) The restriction fl* of if/ to Gal(F/M*)for the composite M* ofM and the unique quadratic extension inside F[pp] is non-trivial. The first assertion follows from the argument given as above. The torsion property follows from the theorem of Taylor-Wiles and Fujiwara and the propositions in the following appendix. In [HMI] Theorem 5.33, it is checked under assumptions milder than (atl-3) imply the assumption of the theorem of Taylor-Wiles and Fujiwara. 3.3

The ^-invariant of CM fields

Consider the universal CRF,Q) deforming pF = I n d j ^ mod mw among W-deformations PA into GL,2(A) for proartinian W-algebras A with residue field W/mw satisfying the following conditions

41

.£-Invariant ofp-Adic L-Functions (Wl) unramified outside p; (W2) PAlGal(Qp/F)j) = (o cl,) with ceAiV = av mod mA and a^pttp1!/,, factoring through Gal(Fv[ixp^]/Fvy, (W3) det(pA) = detp; (W4) pA=P

mod mA,

We know KF - T by Fujiwara (see Appendix). Since dimSpf(W[[r - ]]) = dimSpf(T), Spf(W[[r-]]) gives an irreducible component of Spf(7?F). Write I = W[|T_]] simply. Let nx : T = HF -» I be the projection (which factors through 7t°yc). We would like to compute the .^-invariant of the component I. Thus we need to compute a(p p ) = ni(U(pv)). The following fact follows from the fact Indj^ 0|©M = 0 © ([py, M (A - )* by c H ( O - | M - ) 1 / 2 . Then lp = Ax, and we write K% = n\ o * : Gal(F/M) —> I x . Then, *i restricted to the inertia group /

Gal(Qp[/ip<»]/Qp) s Z x . Since the W[U>]]-algebra structure of I is induced by the nearly ordinary character of Indj^ K\ (restricted to the inertia group /
Kid^My]) = (1 +

Xv)l0^N^))n°^^\

where p = ? fl O, ]VP : M

Qp is the norm map and y p is the generator of Tv := (1 + pZp) n Np(0*). Choose an element tzrOP) e M so that yh = (mW)) for each

Ki{P\)=K^wm-^)=«i(«*) n *i(c7^)r(p))'

42

H. Hida

where T&C$)y' is the ^'-component of vr(ty) e M * c M*. By (3.2), we get KI(P%)

= (1+XP)

logpWd^TO'W^p)) l0

Mrp)

]~~[ (1+XpO

KPjlogp^pK^^P)^')) log (

"V

where p' = W f\ O. Here log p is the Iwasawa p-adic logarithm defined over Qp characterized by logp(p) = 0. In particular, we have logp(Nv(uy))

= log(A^t<70P)-£(p))) =

-e(vnogp(Np(mW)y)).

Thus we have Lemma 3.5. Let the notation be as above. Then we have, for primes ^3' e T.p and v' = W n 0, av = r\ :—r K(pv)(l + Xv,) \ dXp> h\ogp(yp>) We have a(p p ) = cvic(pv) for a nonzero constant c p e W x , because the nearly ordinary character of Indj^ 7p is K times a character of £>$ with values in Wx. We do not need to pay much attention to the constant c p , because the formula of the X-invariant only involve da(p p )

(Ylaip.r'd^y^F^dJi nln p|p

V\ vv

dX

V

/p,P'

in which the constant cp cancels out. Specializing the above formula to the locally cyclotomic point P, we get Theorem 3.6. Let the notation and the assumption be as above and as in Theorem 3.3, including (atl-4). Then we have, for any specialization Ipp of ^p modulo a locally cyclotomic point P 6 Spf(I)(W0, £{Ad{lndFM7pP)) = (-1Y det ({\ogp(NAM^7c))))^

e(v) [1 IT' pip

where p = Onty andp' = Onty'.

43

^.-Invariant ofp-Adic L-Functions Note here ordp(vr(^)L,

c)

) = -h/e(p), taking the valuation ordp asso-

ciated to ty e X. By Lemma 3.2 (2) and Theorem 3.3, we see £(Ad(Indj^ P )) =

£(aM/F),

and this is the reason for the independence of XCAd(Indj^ £>/>)) on the choice of the locally cyclotomic points P. If F = Q, we have TjC$)m(ty)c = ph and hence logp(t3r(^)) = - logp(ar0P)c). Consequently, log^CT^) 1 - 0 ) = 2 log (tzrOP)), and therefore the above formula coincides with the classical analytic ^-invariant formula for aujF of Ferrero-Greenberg. For a given ordinary CM type (Af, Z p ), we can choose iff satisfying the assumptions of Theorems 3.3 and 3.6. Then through the above process, we can compute XXO^MIF) as follows: Corollary 3.7. Suppose that M/F is an ordinary CM-quadratic extension of M satisfying (spt). Choose a p-ordinary CM-type S of M. Then the £.invariant £,{O>MIF) ofGreenbergfor the quadratic Galois character OSM/F ( ^ £ ) is given by

pip

where h is the class number ofM,p' = <^'C\0 and cr(^P) is a generator of ^8 € S„. If the prime p does not split in F/Q, the Ji-invariant ofaM/F does not vanish. A regulator similar to the above determinant was introduced long ago in [FeG] (3.8) in the context of (classical) cyclotomic Iwasawa's theory. Suppose that p does not split in F/Q. Then for w = tuOP), Ny(xn) = I l o - e E ^ = ^ £ - Then we have Irj^lp = 1 and |tcrci:|/, < 1 and hence Ny(ml~c) cannot be of the form £pa for a e Q and a root of unity £. Thus logp(AAp(*n71_c)) ^ 0 as claimed in the corollary. Of course, by Baker's argument (exploited by Brumer in the p-adic case), if M/Q is abelian, we can also confirm the nonvanishing of the determinant in the corollary.

44

H. Hida

4 Appendix: Differential and adjoint square Selmer group Recall the universal nearly ordinary deformation p : Gal(Q/F) —> GL2(R) over K with the pro-Artinian local universal ^-algebra R. This means that for any Artinian local AT-algebra A with maximal ideal m^ and any Galois representation pA : Gal(Q/F) -» GL2(A) such that (Kl) unramified outside p; (K2) PA\Gal(Qp/Fp) - (o al,v) with aA,v = ap mod mA; (K3) det(pA) = detp; (K4) pA=P

mod m^,

there exists a unique ^-algebra homomorphism

A such that yj o p = PA- We write <&K(A) the collection of the isomorphism classes of the deformations PALet p = (p mod mjy), and consider a similar deformation changing base ring from K to W. Then we have a universal couple CR,g) as long as (ai/r) p is absolutely irreducible and (ds) p " is not scalar-values over D p for all v\p (these assumptions we always assume). This means that for any pro-Artinian local W-algebra A with A/mA = W/mw = F for the maximal ideal mA and any Galois representation pA : Gal(Q/F) -* GLaiA) such that (Wl) unramified outside p; (W2) PAlGal(Qp/Fp) = (o al,) with Q;A,P = orp mod tnA; (W3) det(pA) = detp; (W4) pA=p

mod ntA,

there exists a unique W-algebra homomorphism ^> : 'R —> A such that cpog s PA. We write (A) the collection of the isomorphism classes of this finer deformations pA. Thus (A) = Homiy_a/g(7?, A). Let p € <1)(A) acting on L. Define (4.1)

f = { € EndA(L)|Tr(0) = 0J.

45

.£-invariant ofp-Adic L-Functions

We let cr e (5/r act on v € T by conjugation v i-> p(cr)vp(cr) . As before, T has the following three step filtration stable under D p for each prime ideal p|p of F:

T D r;T

(4.2)

D r+T D

{0}.

LetZ* = Qp/Zp = Hom(Zp.Qp/Zp) and A* = Hom(A,Qp/Zp). We thus have Se\F(Y/T) = SdlF(T®Z*p) for V/T := T®Zp Q p /Z p , and we also have SclF(Ad(p))

=SelF(T®AA*) = Ker(i/1((5,f®AA*)^r|H1(/p,

\~A

A

* ))

for p € 0(A) and the inertia subgroup 7P c (5. Note that D p acts trivially on TfV/TpV. We often indicate this fact by writing TpV/T+V ^ K as Dp-modules. Proposition 4.1 (B. Mazur). Suppose $ has a universal couple CRF,gF). Then the Pontryagin dual Sel*F(V/T) is canonically isomorphic to the module of \-differentials fyiF/w[\rF]] ®KF,

A, where cp : Up —* A is the W-algebra homomorphism such that p =

= $" o r d ' v ; so,
46

H. Hida

W, r i-» r © d(r) is obviously a W-algebra homomorphism, and we get (4.3)

(pe(D(7?[X])|p

mo&X =

g}/*x

= {pe C>(7?[X])|p mod X « p | / w a (£ e Hom^ajgC^?, K[X])|f

mod X = id)

= D«v(K,X) a H o m R ( ^ / w , X ) , where "w x " is conjugation under (1 © Af„(X)) n Gl^MX]), and " « " is conjugation by elements in GL2(K[X]). Let p be the deformation in the left-hand side of (4.3). Then we may write p(cr) = g(cr) © u'(cr) (here u'(cr) is a "derivative" of p(cr)). We see Q(O-T)

© H'(OT) = (£>(o-) © M'(O-))(£>(T) © u'(j)) = Q((TT) © (e(cr)H'(T) + Il'(0-)e(T)).

Define w(cr) = u\cr)g((r)~x, which is a cocycle with values on MjiX) by the above formula. Since detp = detp = detp, x(cr) = p(cr)^(a-)_1 has values in SL^CRIX]), u has values in Ad(X) - L(Ad(g)) <S>n X. Hence u : <5SF -* Ad(X) is a 1-cocycle. It is a straightforward computation to see the injectivity of the map: (p 6
mod X * g) / *x^

H\<SSF, Ad(X))

given by p h-> [«]. We put T±{Ad{X)) = T*L(Ad(g)) ®^ X. Since el/, is upper-triangular (up to conjugation), we have w|/p has values in 'F'AdiX). If further we insist on <%(W[[TV]]) = 0, since WflTV]]-algebra structure is given by 6pa~x which is the character of lower right corner of g (restricted to / p ) this means the corresponding cocycle u\if has values in {(oo)}- Since Tr(u) = 0, we conclude u\Ip e T^Ad{X). If L is finite, we may tale X = L, and this gives the desired isomorphism, because T <8>A A* = Ad(L). If L is not finite, then L <2>A A* can be written as a union of Ad(X) for finite X, and by taking the inductive limit, we get the assertion. • If we replace T* ? in the definition of the Selmer group by f~ ?, we get the "minus" Selmer group Sel^(?), and by the same argument SelF(T®A A*)* a

OHT/WOKA.

£-Invariant ofp-Adic L-Functions

47

We can apply the above argument to (R,p). If p e (W), we have a unique P e Spf(RF)(W) such that g mod P = p. JThen R is canonically isomorphic to the P-adic completion-localization Up of 7? at P and p : © F i» GL2(7?) -» GLZWP) = GLaiR). Thus we get Corollary 4.2. Assume R = K[[XV]]. Then we have SIRJK ®R,])v p,|p (log^y,,) -1 a p ( b y , f y ])p,p>) J • Then by the above corollary, putting cp = g£-p~l

, {cP}v\p is a basis of

H. Then writing cp ~ ( Q" a*p )> we can compute the above formula. Note that ap = 6?1§;\X=Q, and 6p([yp, Fp]) = (1+XP) and 6p([Pp, Fv]) = U(Pp). From this we get the formula we stated in the first lecture. We add the following condition to the deformations L satisfying ( W l 4) to make the universal ring small enough to prove Self (V/T) is finite (and SelHV) = 0). Let Z p be the set of all prime factors of p in O. Fix a pair of integers (*xP, /C2,p) for each p e £ p , and write K for the tuple (KI,P, K2,PV We assume that [K] = K\

H. Hida

48

Here a reduced algebra A free of finite rank over W[[JCI, ... ,x,]] is a local complete intersection over R = W[[JCI,... ,xt]] if A is isomorphic to R[[TU..., T r ]]/(/i(T),..., fr(T)) for r power series ^(T), where r is the number of variables in R[[T\,... ,Tr]]. Though the assertion of RK,F being a local complete intersection is technical, as we will see later, this claim is a key to relating the size of the Selmer group with the corresponding Lvalue. In the classical setting of Galois representations associated to elliptic modular forms of weight k (in S k(^\ (N))), we have K = (0, k - 1). Thus the condition K2iP - /CI,P > 1 is equivalent to requiring k > 2. Theorem 4.4 (Wiles, Taylor, Fujiwara). Suppose that the initial representation p is associated to a Hilbert modular form of p-power level (in this case, we call p modular). If (ai^f) holds for M = F[p.p\, Conjecture 4.3 holds. See Fujiwara's papers [Fu] and [Ful]. A more general version of this theorem is proven as Theorem 3.67 and Corollary 3.42 in [HMI]. Proposition 4.5. Assume Conjecture 4.3. Then 1. Se\F(Ad(p) ®w W*) isfinitefor any p e ®k(W) and R = K[[Xv]]pelp ifi<2,v ~ K\,P ^ I far all p 6 JLP, 2. "RF is a reduced local complete intersection free of finite rank over W[[TF]], 3. Se\*F(Ad(gF) <&RFftp)is a torsion Hf-module, 4. For an irreducible component Spf (I) of Spf("RF), write pi = n o gF for the projection n : "RF -» I- Then Se\*F(Ad(pi) ®i I*) is a torsion l-module. Proof. If R is reduced and free of finite rank over W, QR/W is a finite module. Thus the first assertion follows. Note that PK - Ker(/c: W[|T>]] -> W) is generated by ((1 + JC„) - N(yv)Ki*) for p e S. Thus nKPK = {0}. Since HFIPKR-F - RK,F which is free of finite rank s over W, by Nakayama's lemma, ]]-hnear map i : W[\TFW -> RF sending (a\,...,as) to }Zjajrj- Taking another K', we find that KFIP^F - K',F which is free over W; so, it has to be free of

.£-Invariant ofp-Adic L-Functions

49

rank s over W. Thus Ker(i) c PSK, for all id; so, i has to be an isomorphism. This shows the freeness in the second assertion. Let C be the set of all K = (KP)P such that K2,P - *i,p > 1 for all p. Then we still have ( \ e c PK = W- Thus the natural W-algebra homomorphism Up —> YIKSC RK,F is an injection. The right-hand side is reduced (i.e., no nilpotent radical), and Up is reduced. We write

(fx{T),...Jr{T)) Write ~tj e rrt/^ for the image of Tj in RK. Take a lift tj in m^ of tj so that ~tj = (tj modPx'R). Define

)®W[[I>]],K W = (fi,...,fr); so, taking a lift _/) e Ker(^) of / ; , we find Ker(y) = (/i. • • •, /r) by Nakayama's lemma, and hence RF is a local complete intersection over WIUV]]. Since 'Rp is reduced and finite over W[[T>]], QnF/w[[rF]] is a torsion KF-module. From this, the last two assertions follow. As RKF = Hp/P^p is reduced, Spf ("Rp) is etale over Spf(W[|T>]]) around p = P; so, R = ]]. For simplicity, we assume that I = W[[I>]]. In particular, writing K for the field of fractions of I, we have Q = K © X for a complementary ring direct summand X. Let I' be the projection of Kp to X. Then Spf(7?F) = Spf(I) U Spf(I') (and Spf(I') is the union of irreducible components other than Spf(I)). We take the intersection Spf(Co) = Spf(I) n Spf(I'); so, Co = I ®]]-freeness tells us that chari(Co) = (K © 0) n Up is an intersection of a power of prime divisors (cf. [BCM] 7.4.2). Since I = WI[TV]] is regular, and hence char(Co) is a principal ideal generated by ft e I. For this conclusion, we do not need the isomorphism I = W[\Tp]] = W[[jtp]]p but a milder condition that I is a Gorenstein ring over W[|T>]] is enough (that is, HomW[[rF]](I, W[[r>]]) as I as I-modules; see [H88] Theorem 6.8). Note

50

H. Hida

that a local complete intersection over W[[VF]] is a Gorenstein ring (e.g., [CRT] Theorem 21.3). Now by a theorem of Tate (e.g., [MFG] 5.3.4), charCfl^/wnr,]] ® ^ I) = char(C0) = (h). We have for any prime ideal P e Spf(I) with t : I/P s W, writing PP = i ° Pi : ®F -» G^2(W) Sel^(Arf(pp) ® w W*) a n*,/w[[r>n ®KF,/> W a Sel^(Arf(pi) ®i D ®i I/P. This shows that if char(Sel£.(Ad(pi) ®i I*)) = (h) for A € I, we have char(Sel*F(Ad(pP)®w W*)) = (h(P)), where h(P) - (h mod P) e W. Thus we get Corollary 4.6. Wfefazve\Sel*F(Ad(pP) <S>W W*)\ = \h(P)\~[K'Qp] for all P e Spf(I)(W). In this corollary, we do not preclude the case where Sel*F(Ad(pp) <S»w W*) is infinite. In such an extreme case, simply h{P) = 0 and, hence, |fc(P)|p-[*:Q'] - ~ .

References Books [ACM]

G. Shimura, Abelian Varieties with Complex Multiplication and Modular Functions, Princeton University Press, Princeton, NJ, 1998.

[BCM]

N. Bourbaki, Algebre Commutative, Hermann, Paris, 196183.

[CRT]

H. Matsumura, Commutative Ring Theory, Cambridge studies in advanced mathematics 8, Cambridge Univ. Press, 1986.

[HMI]

H. Hida, Hilbert Modular Forms and Iwasawa Theory, Oxford University Press, 2006.

^.-Invariant ofp-Adic L-Functions [MFG]

51

H. Hida, Modular Forms and Galois Cohomology, Cambridge Studies in Advanced Mathematics 69, 2000, Cambridge University Press. Articles

[D]

P. Deligne, Valeurs des fonctions L et periodes d'integrales, Proc. Symp. Pure Math. 33.2 (1979), 313-346.

[FeG]

L. Federer and B. H. Gross, Regulators and Iwasawa Modules, Inventiones Math. 62 (1981), 443-457.

[FG]

B. Ferrero and R. Greenberg, On the behavior of p-adic Lfunctions at s = 0, Inventiones Math. 50 (1978), 91-102.

[Fu]

K. Fujiwara, Deformation rings and Hecke algebras in totally real case, preprint, 1999 (arXiv.math.NT/0602606).

[Ful]

K. Fujiwara, Galois deformations and arithmetic geometry of Shimura varieties, ICM proceedings, 2006.

[G]

R. Greenberg, Trivial zeros of p-adic L-functions, Contemporary Math. 165 (1994), 149-174.

[GS]

R. Greenberg and G. Stevens, /?-adic Z^-functions and p-adic periods of modular forms, Inventiones Math. I l l (1993), 407447.

[GS1]

R. Greenberg and G. Stevens, On the conjecture of Mazur, Tate, and Teitelbaum, Contemporary Math. 165 (1994), 183211.

[GsK]

B. H. Gross and N. Koblitz, Gauss sums and the p-adic Tfunction, Ann. of Math. 109 (1979), 569-581.

[H88]

H. Hida, Modules of congruence of Hecke algebras and Lfunctions associated with cusp forms, Amer. J. Math. 110 (1988) 323-382.

52

H. Hida

[HOO]

H. Hida, Adjoint Selmer groups as Iwasawa modules, Israel Journal of Math. 120 (2000), 361-427 (a preprint version downloadable at www. math. u c l a . edu/ hida).

[H04]

H. Hida, Greenberg's ^-invariants of adjoint square Galois representations, IMRN. 59 (2004), 3177-3189 (a preprint version downloadable at www.math. u c l a . edu/ hida).

[H06]

H. Hida, X-invariants of Tate curves, preprint, 2006, (downloadable at www. math. u c l a . edu/ hida).

[HT]

H. Hida and J. Tilouine, Katz p-adic L-functions, congruence modules and deformation of Galois representations, LMS Lecture notes 153 (1991), 271-293.

[HT1]

H. Hida and J. Tilouine, Anticyclotomic Katz p-adic Lfunctions and congruence modules, Ann. Sci. Ec. Norm. Sup. 4th series 26 (1993), 189-259.

[HT2]

H. Hida and J. Tilouine, On the anticyclotomic main conjecture for CM fields, Inventiones Math. 117 (1994), 89-147.

[K]

N. M. Katz, p-adic L-functions for CM fields, Inventiones Math. 49 (1978), 199-297.

[Ki]

M. Kisin, Overconvergent modular forms and the Fontainemazur conjecture, Inventiones Math. 153 (2003), 373^4-54.

[Kil]

M. Kisin, Geometric deformations of modular Galois representations, Inventiones Math. 157 (2004), 275-328.

[Ki2]

M. Kisin, Moduli of finite flat group schemes, and modularity, preprint, 2005.

[MTT]

B. Mazur, J. Tate and J. Teitelbaum, On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Inventiones Math. 84 (1986), 1-48.

[TW]

R. Taylor and A. Wiles, Ring theoretic properties of certain Hecke algebras, Ann. of Math. 141 (1995), 553-572.

£,-Invariant ofp-Adic L-Functions [W]

53

A. Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. 141 (1995), 443-551.

Haruzo Hida Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, U.S.A. Email: [email protected]

Siegel Modular Forms of Weight Three and Conjectural Correspondence of Shimura Type and Langlands Type T. IBUKIYAMA

In this short article, we give conjectures on explicit dimensions of Siegel modular forms of degree two of weight 3 in several cases. We also explain much deeper conjectures, a conjecture on correspondence between Siegel modular forms of half-integral weight and integral weight of degree two, and a conjecture on correspondence between the split group of type C2 and its compact twist, from which the conjectures on dimensions come. Also some relations of these conjectures to super-singular abelian surfaces will be explain. The dimension of the space of Siegel modular forms can be calculated, in principle, either by Riemann-Roch theorem or by Selberg trace formula, as far as the weight is big enough. Of course in actual calculation, there are difficult technical details, and there is no explicit dimension formula for general degree except for conjectural one [12], [13], but at least there is a general way to calculate it, and it is often well handled when the degree is two. On the other hand, if the weight is very small, e.g. if weight (= the representation at the real place) is not integrable, then there is no known general way to calculate it. For example, for Siegel modular forms in the case of degree two, i.e. for automorphic forms of Sp(2,R) (split rank 2), the weight 1, 2, or 3 is a difficult weight of this type. On the other hand, these are interesting weights for geometry. The weight three case corresponds to the canonimal divisor, so this is very interesting case. The weight two case has some conjectural connection with a degree two version of Shimura-Taniyama conjecture as conjectured by Hiroyuki Yoshida. As for weight one, it rarely exists (cf. [15]). This article concerns about weight 3. 55

56

1

T. Ibukiyama

Definition of Siegel modular forms

1.1 We briefly review the definitions and notation. Let Hn be the Siegel upper half space of degree n and 5 p{n, R) the symplectic group of rank n. H2 ={Z = ' Z e M2(C);7m(Z) > 0} 5/K2.R) ={g = {i

* ) e M4(R); V * = / } , / =

Now we define vector valued Siegel modular forms of degree two. We denote by pkj the following irreducible representation of GL^C) pkJ : GL2(C) -> pkJ(g) = det(g)kSymj(g) e,GL,+1(C), where Syrrij is the symmetric tensor representation of GL2(C) of degree j . Let T c Sp(2, R) be a discrete subgroup with vol(r\// 2 ) < °°A holomorphic Siegel modular form of degree two of weight pkj belonging to T is a holomorphic vector valued function F : Hi —> C + 1 such that F(yZ) = pkJ(CZ + D)F(Z) for any y = l £

^ j e T and Z 6 // 2 - The

space of these functions is denoted by Akj(T). The functions which vanish on cusps of T (i.e. the boundaries of the Satake compactification of r \ / / 2 ) are called cusp forms. The space of cusp forms is denoted by 5^, ; (r). When j = 0, we say that the weight is k and we write 5^(0 = Sic,o(J')1.2

Dimension formula of Siegel modular forms

To calculate the dimension of Siegel cusp forms, we can use Selberg trace formula or Riemann-Roch theorem when k>2n for weight k and degree n. If n = 2, we can manage to calculate explicit values of dimensions in many cases. As for general degree n, there is an explicit conjectural formula for torsion free principal congruence groups of any degree n (cf. [12], [13]). But this is a conjecture only for k > In. Main obstruction for the case k < In is non-convergence of trace formula with some representation theoretic problems, or non-vanishing of cohomology. If k < n/2, all Siegel modular forms are singular modular forms, and there are no cusp forms.

57

Siegel Modular Forms of Weight Three

When k satisfies n/2 < k < 2n, we have no general way to calculate the dimension. When n = 2, mysterious weights are k - 1, 2, 3. When k = 1, there exist no Siegel cusp forms for TQ(N) for any natural number N (cf. [15]). When k = 2, the dimension is really mysterious, and we have no conjecture at all. The case k = 3 is the subject of this article. 1.3

Discrete subgroups

We prepare discrete subgroups we shall treat. We define (1) The full modular group Sp(2, Z) := Sp(2, R) n M4(Z). (2) The paramodular group for a prime p.

K(p):=Sp(2,Q)n

This is isomorphic to the group defining the moduli of polarized abelian varieties having {1, p] as elementary divisors of the polarization. (3) The Iwahori subgroup for a prime p.

B(p) := Sp(2,Z) n

(4) A congruence subgroup eSp(2,Z);

C = 0mod4.

2 Conjectures on dimensions of weight 3 We first give conjectural dimensions of vector valued Siegel modular forms of weight det3 Sym2j ofSp(2, Z). Conjecture 2.1. We should have the following generating function of dimensions v^i

2_JdimS3,2j(Sp(2,Z))sJ y'=0

-18

= (1 - s 3 )(l - ^ X l - s5)(l - s6)

58

T. Ibukiyama

where s is a variable. Numerical Examples of conjectured dimensions: dimS 3>; (Sp(2,Z)) should be 0 for any j < 36 and we should have

36 38 40 42 44 46 48 50 52 54 56 dim 1? 0? 0? 1? 1? 1? 2? 1? 2? 3? 3? j

This conjecture comes from much deeper conjecture on Shimura type correspondence, which will be explained later. Next we fix a prime p and we give conjectures on dimensions related with the paramodular group K(p) and the Iwahori subgroup B(p). We can show that dimS 3 (£(p)) = 0 for p < 5. Conjecture 2.2. Assume that p is a prime with p > 5. Then the dimension of Siegel paramodular forms of weight 3 should be given as follows. dimS3(K(p)) = -1 +

l^-i>+i
2880

+, ( 2 / 5 ififp = 2,2> mod 5 , otherwise, \ 0 oi

4K)M

1/6 0

ifp = 5 mod 12, otherwise.

Numerical examples. p 2 3 5 7 11 13 17 19 23 dim 0 0 0 0 0 1 1 1 1

29 31 37 41 43 47 53 59 dim 2? 2? 4? 3? 4? 3? 5? 4? P

Here the numbers without question marks are true values checked by several ad hoc methods. The numbers with question marks are conjectured values.

59

Siegel Modular Forms of Weight Three

We denote by S™w(B{p)) the space of Siegel cusp forms of weight k with respect to B(p) whose local admissible representation at p is the Steinberg representation. Equivalently, S™w(B(p)) is the orthogonal complement of the space spanned by Siegel cusp forms belonging to some discrete subgroup r * B such that B(p) c T c Sp(2, Q). We can show that dim5^ w (B(p)) = 0 for p = 2. Conjecture 2.3. Conjectural dimension of the space S"ew(B(p)) of Siegel "new" cusp forms of weight 3 belonging to the Iwahori subroup B(p) is 0 for p = 3 and given for p > 5 as follows. dim S"ew(B(p)) = 1 +^(P-WP

3

-D-^(P-1)2

-^-4-(T))-^-4-(7))

1/5 2/5 4/5 0

ifp = 5 , ifp = 2,3 mod 5 , ifp = 4 mod 5 , otherwise

1/2 0

ifp = 1 mod 8 , otherwise,

1/6 0

ifp= 11 mod 12, otherwise.

Numerical examples. p 2 3 5 7 11 13 17 19 23 dim 0 0? 0? 0? 2? 8? 23? 38? 83? 29 31 37 41 P dim 225? 298? 618? 935?

43 1136?

47 1625?

60

3

T. Ibukiyama

Conjecture on Eichler type correspondence

Conjecture 2.2 or 2.3 comes from an Eichler-Jacquet-Langlands type (conjectural) correspondence between Sp(2,R) and its compact form USp(2) which the author proposed in the early eighties. The prototype of Jacquet-Langlands correspondence is due to Eichler. So we review this first. Let D be the definite quaternion algebra over Q with discriminant p. Let O be a maximal order of D and for any prime q, put Oq = O ® z Z p . Let D* be the adelization of Dx and let D*, = H be the division quaternion algebra over R. We denote by Av(0%) the space of automorphic forms of D^ with respect to O^ = D£ Y[q O* of "weight" v. Here "weight" v means that the representation at the real place is the representation of D£, induced from the v-th symmetric tensor representation of S U(2) trivial on the center (i.e. v is even) through the isomorphism SU(2)/{±1] a D^/center. When v > 0, we put S v (0*) = A v (0*). When v = 0, we denote by So(0^) the orthogonal complement in AQ(0^) of the one-dimensional subspace of AQ(0%) spanned by constant functions. Denote by rj, (p) the usual congruence subgroup of S Z,2(Z) consisting of elements y = I

,1 e SLa(Z) such that c = 0 mod p. We denote by

SJt^Co \p)) the space of new forms of weight k belonging to I^ \p) in the usual sense of Atkin-Lehner. This is equivalent to say that the local representation at p is the Steinberg representation. We note that the algebras generated by Hecke operators T(q) (q is a prime such that q* p) for GL(2) and D^ are isomorphic with each other. Theorem 3.1 (Eichler). For any integer k>2, there exists an isomorphism

sT^ip))

= sk-2w*A)

as Hecke algebra modules. In particular, d i m S ^ r o ^ ) ) + 1 is equal to the class number ofD. Now we would like to consider the same sort of precise statement for the degree two case. This kind of problem was raised around in 1963 by

Siegel Modular Forms of Weight Three

61

Y. Ihara [16] and now included in the general problem of Langlands. We gave several precise conjectures of this kind of correspondence before (cf. [6], [7]). We shall explain these here. We first explain Q-form of the compact real form of Sp(2, R). We put USp(2) = {g€M2m);gtg

= h}

where H is the division quaternion algebra over R and the bar means to take the main involution of each component. This is the compact real form of 5 p(2, R). Let D be the definite division quaternion algebra D with discriminant p as before. We introduce the following Q-form of GSp(2,C) whose semi-simple part is the Q form of USp(2). G = {ge M2{D); g'g = n(g)l2, n(g) e Q*}. We denote by GA the adelization of G and by Go, the real component of GA- The p-adic completion Gp of G is isomorphic to G* given by

Gp*G;:={seM2(Dp);'l(° j)g = (° J)} . In case of the split symplectic group, for any prime p, Sp(2,Qp) has 7 standard parahoric subgroups, excluding Sp(2,Qp) itself. On the other hand, the group G*p has three standard parahoric subgroups, excluding G* itself. Namely, there are two maximal compact subgroups U\tP and U2^p of G*p and the minimal parahoric subgroup Uo,p = U\tPC\ U2tP, where Uhp

=

GpnGL2(Op),

Here n is the prime element of Op. We note that U\tP and U2,p are very different and not conjugate with each other. For any q + p, we have Gq =

62

T. Ibukiyama

GSp(2, Qq) where GSp(2, Qq) is the group of symplectic similitudes and we put Uq = GSp(2,Zq). Put Ui = GooUi,p Tlq±p Uq c GA- A weight of automorphic forms of GA means an irreducible representation of Goo. For any integer v, we denote by pv the irreducible representation of USp{2) corresponding to the Young diagram parameter (v, v), namely to the Young diagram 1 2 1 2

V V

This representation pv is trivial on the center {±12} of USp(2) and we regard pv as a representation of Goo through the isomorphism G^ I center s USp(2)/[±12}. We denote by Vv the representation space of pv. A Vvvalued function / on GA is said to be an automorphic form of GA of weight v with respect to £/, when it satisfies f(axu) = pv(Woo)/(^) for any a e G, x e GA, and u e [/,, where w^, is the component at the real place of u. We denote by Av(f/,) = the space of automorphic forms of weight v with respect to £/,-. By the way, when v = 0, dimAo(£/i) or dimAo(t/2) is nothing but the class number of the principal genus, or non-principal genus of maximal 0-lattices in D2, respectively. When v = 0, Ao(U{) contains constant functions, so it is reasonable to consider the subspace So(Ut) of Ao(£/,)) which is the orthogonal complement of the one dimensional space spanned by constant functions in Ao(£/,). When v > 0, we put Sy(Ui) = Ay(Ui). The dimensions of AV(C/,) is known for any v and i and p except for the case (/, p) = (0,3) (cf. [1], [2]). We denote by S"ew(Uo) the orthogonal complement of the space spanned by SV(U\) and S v (t/ 2 ) in Sv(£/0)Conjecture 3.2 (cf. [6]). For k > 3, there should exist an linear isomorphism

srwp))=s™(u0)

which preserves L functions. As evidence of this conjecture, we have a dimensional equality for k > 4 (cf. [2]) and a lot of numerical examples of Euler factors supporting the

63

Siegel Modular Forms of Weight Three

conjecture (cf. [6]). If we believe that the same isomorphism holds even for k = 3, we get our precise conjectural dimension which we gave before, since the dimension of the right-hand side is rigorously known for any v (including the case where p = 3 and i = 0). Conjecture 3.3 ([7]). For k>3, there should exist an isomorphism Snkew(K(p)) = Snk%{U2) which preserves L functions. Here the definition of "new forms" is not representation theoretic at moment. We shall give the definition later. As evidence for this conjecture, we have the following dimensional equality for k > 4 dimSk(K(p)) - 2dimSk(Sp(2,Z))

=

dimSk-3(U2) - dimA2(Tj(p)) x dimS2jt-2(SL2(Z)), where ^ ( r j , (/?)) is the space of holomorphic modular forms of one variable of weight 2 of T^\p)) and S ik-iiSLQ.{Z)) is the space of holomorphic cusp forms of one variable of weight 2k-2 of 5L2(Z) (cf [7]). If we believe that the same equality should also hold for k = 3, then since it is well known that 5 4(5L2(Z)) = 0andS 3 (S/>(2,Z)) = 0, the above equality means that dim Si(K(p)) = dim S0(^2)- Since the dimension of the right-hand side is equal to dimAoiUz) - 1, and dimAo(£/2) is already known (cf. [1] (II)), this leads to a precise dimension formula for S3(K(p)). This is our conjecture on the dimension. Now for general k, we define new forms. Put p=

(o P°h)-

The groups K(p), Sp(2,Z) andpSp(2,Z)p~1 contains B(p) as a subgroup and we can define mappings from S k(S p(2, Z)) and from S k(pS p(2, Z)p~i) to Sk(K(p)) by taking the "trace" from B(p) to K{p). Call the subspace of S k(K(p)) spanned by the images of Sk(S p(2, Z)) and Sk(fiS p(2, Z)p~!) the space of old forms. We denote by Snkew(K(p)) the orthogonal compliment of the space of old forms. On the other hand, for U\ or U2, there exists an analogue of the Saito-Kurokawa lifting (actually discovered much

64

T. Ibukiyama

earlier than Saito-Kurokawa lifting by Ihara [16] and later generalized by Ibukiyama and Ihara[9]). As for the group U2, there exists a lifting from the pair of forms in A2(T0(p)) and S 2k-2(.S L2(Z)) to Sk-^{U2), which might vanish in general. We denote by S^(U2) the orthogonal complement of the space obtained by this lifting in Sk-^{U2). The actual interpretation of the dimension of new forms is a little more subtle. If k is even, we should read dimSf w(K(p)) = dimSk(K(p))

-2dimSk(Sp(2,Z)) + dimS 2 *- 2 (SL 2 (Z)),

n

dimS k™(U2) =Sk-3(U2) - dimS 2 (r 0 (p)) x dimS2)t_2(SL2(Z)) • If k is odd, we should read dim Snkew{K(p)) = d\m Sk(K(p)) -

2dimSk{Sp{2,Z)),

dimS^(C/ 2 ) = dimS k . 3 (U 2 ) - dimA 2 (r 0 (p)) x

S2k.2(SL2(Z)).

The reason is as follows. If k is even, there is the Saito-Kurokawa lifting from S 2k-2(S L^Z)) to S p(2, Z) and for these, the two trace maps from Sp(2, Z) andpSp(2, Z)p -1 are not linearly independent. So, it is very plausible that the dimension of old forms in Sk(K(p)) is 2 dim 5 k(S p(2,Z)) S2k-2(SL2(Z)) if ^ is even and 2dimSk(Sp(2,Z) if A: is odd. Also for the compact twist, this formulation fits examples well.

4 Geometric interpretation The lowest weight automorphic forms has good relation with some geometry. We review this in this section. Let K be an algebraically closed field of characteristic p. An elliptic curve E defined over K is called super-singular if it has no p-torsion point. An abelian variety A is called super-singular if A is isogenous to En for some super-singluar elliptic curve E. Abelian surface is super-singular if and only if it has no p-torsion point. (For general dimension, this is false.) The prototype of the relation we shall explain later is due to Deuring and Eichler. Theorem 4.1 (Deuring and Eichler). The class number of the definite quaternion algebra D with disc(D) - p is equal to the number of

65

Siegel Modular Forms of Weight Three

isomorphism classes of super-singular elliptic curves. This is also equal to d i m ^ O ^ C p ) ) = dim52(T(01)(p)) + 1. For abelian surfaces, we shall suggest the following conjectures. Conjecture 4.2. dOmSi(K(p)) + 1 should be equal to the number of irreducible components of the locus of principally polarized super-singular abelian surfaces in the moduli space. Conjecture 4.3. dim S™w(B(p)) should be equal to the (generalized) arithmetic genus of the super-singular locus of principally polarized abelian surfaces in the moduli space. Here the locus is in general a reducible singular curve, but we have a notion of an arithmetic genus of such object defined by J. P. Serre (I learned this from Ekedahl). The reason for these conjectures is as follows. Each dimension which appears in each of the above conjectures should be equal to the dimension of automorphic forms of weight 0 of U2 or UQ, respectively by the conjectures we gave before. But it was rigorously proved that if we replace the dimension by the one for U2 or UQ, these conjectures are true. (As for Conjecture 4.2, this is due to Katsura and Oort [17], and as for 4.3, due to the author [10]).

5

Conjecture on Shimura type correspondence

Here we briefly explain about conjectures on Shimura type correspondence of degree two. More precise details will be published elsewhere [14]. We first review the definition of Siegel modular forms of half-integral weight. To define an automorphic factor of weight 1/2, for Z € Hi we put 0(Z) := YJ peZ

For 7 = [t

e2ni

'pZp •

2

*) e T0(4), we define 71/2(7, Z) = e(yZ)/6(Z). We also

define a character \]/ of To(4) by t//(j) = (^SW)- It is well known that 7i/2(y,Z)2 =
66

T. Ibukiyama

A Siegel modular form of degree two of weight det*~1/2 Syntj with character i// with respect to To(4) is defined to be a holomorphic vector valued function F : H2 -> Ci+1 such that FiyZ) = if,(y)jl/2(y,Z)2k-lSymj(CZ

+ D)F(Z)

for all y e To(4). Such F automatically vanishes on all cusps of ro(4), so they are cusp forms. (Without character, this is false.) We denote by 5,i-i/2,;(Fo(4), iff) the vector space of such vector valued cusp forms. Now, in order to extract the "level one" part, i.e. a kind of new forms from this space, we introduced the plus space S%_l/2 (To(4), iff) of Kohnen type which is characterized by a certain condition on Fourier coefficients. For F € S/t_i/2(ro(4), iff), we have the Fourier expansion

F(Z) = YJ

atnf^iTZ)

T

where Z e Hi and T runs over half-integral symmetric matrices T I

1

V12

) Ci, h, hi e Z). We say that F is in the plus space if a(T) = 0 h

)

unless T satisfies the condition that tx = (-1)* + V mod 4,f2 = (-1)* + V 2 mod 4 and tn = 2vfi(-l)k+l mod 4 for some v,y. € Z. Note that the plus space is exactly the subspace of SVi/2(ro(4),^) which correspond to the holomorphic, or skew-holomorphic Jacobi forms of index 1 of S p(2, Z)J = S p(2, Z) • Z 4 for odd k, or for even k, respectively (cf. [11], [3], [4]). But we omit the details here. Now we can state our conjecture. Conjecture 5.1. For any natural number k>3 and even integer j > 0, we should have a linear isomorphism cr. °- • 5 *-i/2,/ r o( 4 ).^) = 5;+3>2*-6(5p(2,Z)) such that for a common Hecke eigen function F e S~£_i/2 ATo(4),if/), we have L(s, F) = L(s, cr(F)). Here L(s, F) is the L function defined by Zhuravlev [20] and L(j, o~(F)) is the usual spinor L function. In particular, if j = 0, then we should have Sj£_1/2(To(4),
SX2k-6(Sp(2,Z)).

Siegel Modular Forms of Weight Three

67

Since the dimension of the left-hand side is known in Tsushima [19], this leads to the conjecture on dimension we gave before. Remarks. (1) The above conjecture is false for odd j . (2) The weight correspondence is reasonable (we omit the explanation here). (3) For k > 4, we have dimensional equality under standard conjecture on vanishing of cohomology. (We used conjectural results by Tsushima). (4) There are a lot of numerical examples supporting this conjecture.

6

Concluding remarks

If k > 4, all dimSfcjGSp(2,Z)), dim Sk(K(p)) and dim S™w(B(p)) are known and these are elementary functions of k. So if we put k = 3 formally in these formulas for big k > 4, we get some numbers. Now if our conjecture on dimensions for k = 3 is true, then each of such numbers we mentioned above is equal to dim 5 3j(S p(2, Z)), dim S^iKip)) - 1, or dim S"ew(B(p)) + 1, respectively. This means that the usual trace formula should almost work in these cases too. This seems very interesting. (Such easy description cannot be expected in general setting as is noticed in Hiraga [5]) We also note that the appearance of super-singular abelian surfaces might suggest that dimensions of Siegel cusp forms of weight 3 might have some relation with geometry of the reduction of Shimura varieties mod p.

References [1] K. Hashimoto and T. Ibukiyama, On class numbers of positive definite binary quaternion harmitian forms (I) J.Fac.Sci.Univ.Tokyo SectlA Math., 27(1980), 549-601, (II) ibid. 28(1982), 695-699, (III) ibid. 30(1983), 393-^01. [2] K. Hashimoto and T. Ibukiyama, On relations of dimensions of automorphic forms of S p(2, R) and its compact twist Sp(2) (II). Advanced Studies in Pure Math., 7(1985), 30-102.

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[3] S. Hayashida, Skew-holomorphic Jacobi forms of index 1 and Siegel modular forms of half-integral weight. J. Number Theory 106 (2004), no. 2, 200-218. [4] S. Hayashida and T. Ibukiyama, Siegel modular forms of half integral weight and a lifting conjecture. J. Math. Kyoto Univ. 45 (2005), no. 3,489-530. [5] K. Hiraga, On the multiplicities of the discrete series of semisimple Lie groups. Duke Math. J. 85 (1996), no. 1, 167-181. [6] T. Ibukiyama, On symplectic Euler factors of genus two. J.Fac.ScLUniv.Tokyo Sec.lA Math. 30(1984), 587-614. [7] T. ibukiyama, On relations of dimensions of automorphic forms of S/?(2,R) and its compact twist Sp(2) (I). Advanced Studies in Pure Math., 7(1985), 7-29. [8] T. Ibukiyama, T. Katsura, and F. Oort. Supersingular curves of genus two and class numbers. Compositio Math. 57(1986), 127-152. [9] T. Ibukiyama and Y. Ihara. On automorphic forms on the unitary symplectic group Sp(n) and SL2(R). Math. Ann., 278(1987), 307-327. [10] T. Ibukiyama, Arithmetic theory of supersingular abelian varieties, RIMS K = oky = uroku No. 775, in Japanese Einstein metric and Yang-Mills connection Ed. By T. Ochiai (1992), 154-166. [11] T. Ibukiyama, On Jacobi forms and Siegel modular forms of half integral weights. Comment. Math. Univ. St. Paul. 41 no. 2(1992), 109124. [12] T. Ibukiyama and H. Saito, On zeta functions associated to symmetric matrices and an explicit conjecture on dimensions of Siegel modular forms of general degree, Internt. Math. Res. Notices 1992, no.8, 161169. [13] T. Ibukiyama and H. Saito, On zeta functions associated to symmetric matrices II, Functional equations and special values MPI preprint series 97-37 (1997).

Siegel Modular Forms of Weight Three

69

[14] T. Ibukiyama, Conjecture on Shimura correspondence on Siegel modular forms of degree two, in preparation. [15] T. Ibukiyama and N-P. Skoruppa, Appendix to the article by C. Poor and D. S. Yuen, "Dimensions of cusp forms of TQ(P) in degree two and small weights", preprint. [16] Y. Ihara, On certain arithmetical Dirichlet series. J. Math. Soc. Japan 16(1964), 214-225. [17] T. Katsura and F. Oort, Supersingular abelian varieties of dimension two or three and class numbers. Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math. 10, Kinokuniya, Japan and North-Holland, Amsterdam, (1987), 253-281. [18] R. Tsushima, An explicit dimension formula for the spaces of generalized automorphic forms with respect to Sp(2, Z). Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 4, 139-142. [19] R. Tsushima, On the dimension formula for the spaces of Jacobi forms of degree two. Automorphic forms and L-functions (Kyoto, 1999). RIMS K = oky = uroku No. 1103 (1999), 96-110. 11F50. [20] V. G. Zhuravlev, Euler expansions of theta transforms of Siegel modular forms of half-integral weight and their analytic properties, Math, sbornik 123 (165) (1984), 174-194.

Tomoyoshi Ibukiyama Department of Mathematics, Osaka University, Machikaneyama 1-16, Toyonaka, Osaka, 560-0043 Japan Email: [email protected]

Convolutions of Fourier Coefficients of Cusp Forms and the Circle Method M. JUTILA 1

0

Introduction

The underlying idea of the classical circle method is to express a coefficient of a Fourier series by use of a Farey dissection of the interval of integration in the analytic formula for the coefficient. In particular, the constant term of the Fourier series 00

(0.1)

2 bne(nx) 71=-oo

of a periodic function \f/ with period 1 is (0.2)

b0 = I \J/(x)dx, Jo

and bn can be reduced to this on replacing i/r(x) by i/f(x)e(-nx). Now a decomposition of the unit interval gives rise to a linear decomposition of its characteristic function x(x)' a n < i t n u s t 0 a decomposition of the integral (0.2). In the special case iff(x) = e(Nx) with N an integer, the integral (0.2) amounts to the S-function S(N) which equals 1 or 0 for N = 0 or N * 0, respectively. Thus the circle method gives a decomposition of the 5-function. There are various other formulae for the (5-function in literature, for instance one due to W. Duke, J. Friedlander and H. Iwaniec [1]. We are going to discuss an approximate version 6*(N) of 6(N) such that 8*(0) = 1 and S*(N) is "small" for N * 0. A notorious obstacle in applications of the circle method is the irregularity of the lengths of the Farey intervals. In his lecture at ICM 1962 'The author was supported by the grant 8205966 from the Academy of Finland.

71

72

M. Jutila

in Stockholm, Yu. V. Linnik [14] considered what he called the levelling problem, namely replacing the Farey intervals by intervals of uniform length. A well-known device for this purpose is the Kloosterman refinement of the circle method; its idea is to introduce a finer subdivision of the Farey intervals related to rational numbers with a given denominator. In his lecture, Linnik suggested the following argument: to work with intervals of the same length with an attempt to cope with the error term. Interestingly, he pointed out that overlapping of the intervals is not excluded. We presented in [4] a variant of the circle method based actually on ample overlapping of the intervals. In this context, it is essential that the multiplicity of the overlapping turns out to be mostly close to its statistical expectation. This means that if^*(x) is a suitable linear combination of the characteristic functions (possibly with a smooth weight) of neighborhoods of a system of rational numbers, then the mean square of x(x) ~x*(x) over the unit interval is small. In [7], we applied this method to convolutions of Fourier coefficients of Maass forms. In the present paper, we give a variant of the argument of [7] on the basis of another version of the method of overlapping intervals developed in [10]. Its main novelty is an explicit formula (stated in (1.5) below) for the error when bo in (0.2) is approximated by

(0.3)

b*0= f

x\xMx)dx.

Jo

In particular, the above mentioned approximation to the ^-function is (0.4)

6*(N)= \

x*ix)e(NX)dx.

Jo

The general scheme of the argument is expounded in Sec. 2, and applications to convolutions of Fourier coefficients of Maass forms are considered in Sec. 3. Notation. As to the spectral theory of automorphic functions, we follow the notation of [17]. We let s stand for a small positive constant not necessarily the same at each occurrence. The constants implied by the symbols «: or 0(- • •) may depend on e without mentioning. Further, we write A x B to mean that A and B are positive and A <sc B <s: A. The standard notation is used for the Bessel functions / v (x) and Kv(x), as well

Convolutions of Fourier Coefficients and the Circle Method

73

as for the arithmetic functions (p(n) and /j.(ri). In sums V , the asterisk denotes the coprimality condition (a, q) = 1.

1 An arithmetic formula for Fourier coefficients Following [10], we first describe the construction of x*(x) a s a n approximation of the characteristic function^(x) of the interval / = [0,1]. Let Q be a large parameter and let w(q) be a function satisfying 0 < w(q) < 1 for natural numbers q and w(q) = 0 for q $. [Q,2Q]. Further, let A € (0,1 /3] be another parameter and let v(x) be a non-negative continuous function with support in [-A, A] which is bounded by an absolute constant. Also, for normalization, we suppose that •A

— f (x) dx = 1. 2AJ_ A VI

The function v(x) will play the role of a continuous analogue of the characteristic function of the interval [-A, A]. In [10] we supposed that v(x) be even, but this assumption, made just for convenience, is redundant. Putting oo

(1.1)

L = YJw(q)f(q)

and A = 2AL, we define

as a tentative approximation to;^(;c) in the interval /. Alternatively, this can be written as

(i.2)

1 °°

*

q=\

a=l

v

/

\

^"

'

*«^Z^Z iH>

where v*{x) is the extension of v(x) to a periodic function with period 1; to verify this, consider separately the ranges 0 < J C < A , A < J C < 1 - A , and

74

M. Jutila

1 - A < x < 1. The function x*(x) is periodic with period 1 and its integral over / equals 1. We proved in [10] (Theorem 1) that (1.3)

Jo Jo

(x*(x) -x(x)Y dx « A"1 L-lQL logJ'(1 /A)

as a slight sharpening of an estimate in [4]. Note that typically L x Q2, and then the right hand side is «: (Ag 2 ) - 1 log 3 (l/A). In particular, if A x Q~l (as we are going to choose in Sec. 3), this is «: Q~l log3 Q, which is small if Q is large. Let now ^ be a continuous function with period 1 and Fourier coefficients bn. Denote the Fourier coefficients of the function v*{x) by a„. As an approximation of bo, we now have b*0 defined by (0.3). Thus, by (1.2), q 1 °° l = l YMV) Y,

b

r^ I \ I v*[--x)iff(x)dx

(1.4)

to verify the latter expression, note that the range of integration in the first integral can be replaced by [a/q - A, a/q - A + 1]. Now the formula (3.3) in [10] gives the following connection between bo and b*Q: 1

(1-5)

oo

oo

bo = b*0 - - ^ d ^ admbdm 2^ w(dr)n{r), d=\

m±0

r=\

where fi(r) stands for the Mobius function. This is apparently a relation of arithmetic nature. In typical applications, the difference bo-by is estimated as an error term, and b*Q is treated as in the circle method, with conveniently "levelled" intervals. If we interpret bo -b*Q as an integral by (0.2) and (0.3), then by Cauchy's inequality and (1.3) we get (1.6)

bQ - b*0 « L-lQA-l'2M\2

log 3 / 2 (l/A);

this is (3.4) in [10] and represents the scope of the method of [4]. There should be more potential in the method of [10], at least in principle, since

Convolutions of Fourier Coefficients and the Circle Method

75

(1.5) reveals also the structure of bo - b*0, not only its order of magnitude. Moreover, the occurrence of the bn - coefficients on both sides of the relation (1.5) is a new phenomenon indicating possibilities of recursions or iterative arguments. If we are interested in the other coefficients in addition to bo, we simply interpret bn as the constant term for if/(x)e(-nx), and then (1.5) takes the form oo

1

(1.7)

oo

bn = b*n - - ^ d ^ admbn+dm ^ d=\

m±0

w{dr)n{r),

r=\

where b*n is defined as b*Q in (1.4) with if/(x)e(-nx) standing in place of if/(x). Also, we get a formula for linear combinations F

(1.8)

Bn = YjWn+f /=o

and for analogous expressions B*n involving b* + ,, with arbitrary complex coefficients/?/. In fact, it follows immediately from (1.7) that oo

1

(1.9)

Bn = B

d

oo

*n-jTj Yu

a

d=\

m*0

B

*™ "+*™ YJ w(dr)n{r). r=\

The subsequent argument will run, in a nutshell, as follows. Our final goal is the estimation of the coefficients bn, either individually or in mean. Since b*n is more easily tractable, we are interested in the approximation bn ~b*n. Of course, an approximation like this is feasible only under more specific assumptions, and such a case will be discussed in the next section. However, by the following theorem, we may estimate Bo in terms of a suitable sum B*n under fairly general conditions. Now, in the application to convolutions we have in mind, the latter sum can be expressed in terms of Kloosterman sums, and an estimate for it will be obtained by spectral methods. Then get an estimate for Bo as well, and this, in turn, implies an estimate for bn-b*n. Theorem 1. Let if/ be a continuous function with period 1 and Fourier coefficients b„. Suppose that (1.10)

l
76

M. Jutila

for all x. Let Q and F be integers with Q large and 0
(1.11)

for some constant c > 0. Let w(q) be a weight function such that L » Q2 for the quantity L defined in (1.1). Further, let A = D/Q, where D is a sufficiently large constant, and let v(x) be a suitable smooth weight function satisfying the conditions imposed in the construction of the function x* in (1.2). Suppose 1/8/1 < 1 in (1.8). Then we have (1.12)

B0<^bQ~A + max \B*\ |n|
for any fixed numbers A > 0 and e > 0. Proof. We may suppose that b > 0 and |Bol » bQ~A, for otherwise there is nothing to prove. Note that \bn\ < b by (1.10). The function v(x) can be chosen in such a way that its Fourier coefficients an are of rapid decay, say (1.13)

an «: min(A,«-1("A)-A) « (nA+lAA + A - 1 ) - 1

for a large constant A. For instance, a suitable "Vinogradov function" (see [12], Ch. I, §2) gives an example of such a function. By our choices of A and w(q), we have A = 2AL » DQ, where the implied constant is independent of D. By (1.9), we now have 1 °°

(1.14)

0°

B 0 = Bl - - J^ d ^ admBdm J ] w(dr)n(r).

Consider the following possibilities: either (1.15)

\Bn\ < 2|B0| for all \n\ < Ql+E/2,

or else there exists an index n\ with |ni| < Q1+£/2 such that (1.16)

|fi Bl |>2|B 0 |.

Here e is the same as in (1.12). In the case (1.15), we estimate in (1.14) the contribution of the terms with \dm\ > <21+e/2 as an error term which is

Convolutions of Fourier Coefficients and the Circle Method

77

at most bQ~A in absolute value, in view of (1.10), (1.11), and (1.13). Then \Bdm\ < 2\BQ\ in the remaining terms. Further, the relevant values of d lie in the interval 1 < d < 2Q, and for given d, we have f Q/d V w(dr)n(r) « — ^ jr( WOQ/d) by the known estimate for the sum function of the Mobius function; we may suppose that the weight function w is smooth enough to allow effective summation by parts. Also, by (1.13) with A = 1,

J] l**»l « ^((dm)2A + A"1)"1 « 2|B„J>4|B 0 |. In the former case, we get \B0\<\Bni\/2<\B*ni\

+ bQ-A

as above, and (1.12) follows; otherwise we repeat the procedure. After k steps, the latter of the two alternatives means that there exists n* with \nic\ < kQl+E/2 such that \Bnk\ > 2k\B0\. This is clearly impossible if k/ log Q is sufficiently large, by our initial assumption on B0, and then kQl+s/2 «; Q1+E. Thus, in any case, we end up with the inequality (1.12). • Remark. Returning to the remarks we made before stating Theorem 1, we see now that if we manage to bound the sums B*n nontrivially in some sense, Theorem 1 implies a good estimate for a weighted average of bn as n ranges over an interval. This information can be fed back to the identity (1.7) to show eventually that bn and b*n lie close to each other.

78

M. Jutila

2 Applications to convolutions In [7] we studied sums of the form N

T(N;f)

= J^Kn)Kn + f), n=\ N-\

T(N) =

J^tinXN-n), n=\

where the coefficients t{n) are the Hecke eigenvalues of a Maass form u for the full modular group. Then u is an eigenfunction of the hyperbolic Laplacian for an eigenvalue 1 /4 + K2 with K > 0. The Fourier series of u is of the form u(x + yi) =ymYj

P(n)KiK(27r\n\y)e(nx) (y > 0),

where p(n) = p( 1 )t(ri) for n > 1. Further, let e be the parity sign of u, that is u(-x+yi) = eu(x+yi). We refer to [2] or [17] for basic properties of Maass forms. Let us consider the sum T(N; f) as an example of our modification of the argument of [7]. It is convenient to insert a smooth weight function to our sum. Let W(x) be a function supported in the interval [N - M,N], where M and N are positive integers such that (2.1)

Nl/2+e <M<

N/4.

We suppose that (2.2)

Wu\x) <£j M~j

for sufficiently many derivatives. We are going to study the sum 00

(2.3)

T(W; / ) - J ] W(n)t(n)t(n + f) n=\

by use of the exponential sum oo

(2.4)

S(W;a) = Yj W(n)t(n)e(na). n=l

Convolutions of Fourier Coefficients and the Circle Method

79

One may consider the sum (2.3) either for an individual shift f > 1 or in a mean value sense for / varying over an interval; these problems are in fact closely related in view of the identity (1.7). To clarify this point, we first introduce the relevant periodic function \J/ in this context, in the sense of the preceding section. Let WQ(X) be an auxiliary smooth weight function of support in the interval [N/3, cN] with c a sufficiently large constant. Suppose that W0(x) = 1 for N/2 <x< cN/2 and define (2.5)

i//(a) = S(WQ; a)S(W; -a).

If this is written as a Fourier series with coefficients bf, then bf = Yj W(n)W0(ri + f)t(n)t(n + / ) , thus (2.6)

bf = T(W;f) for -N/4

< f < (£ - l)jV;

note that here / is not restricted to positive values. Or if we prefer to interpret T(W; f) as the constant term bo, then the factor e(-fa) should be inserted to our function (2.5). Now (1.4), for suitable weight functions w and v and with this modified function playing the role of if/, gives an approximation, say T*(W;f), to T(W;f). That is, by definition, T*(W;f) equals b*Q in the present case. The following lemma, concerning the average of \T*(W; f)\ over / , is the main step in our approach to the sum T(W; / ) . Lemma 1. Let M and N be positive integers as in (2.1), and Q = MN~6,

(2.7)

A = D/Q,

where D and 6 are positive constants with 6 small and D large. Let the weight function v be as in Sec. 2, and suppose moreover that its support lies in the interval [-A, -A/2]. Also, suppose that the weight function w is such that L » Q2 in (1.1). Then, for 1 < F «: M, we have (2.8)

YJ i
\T*(W;f)\
+

F1/2M-mN)Ns.

80

M. Jutila Proof. By definition,

(2.9)

T*(W;f) q

i °°

=1X

w(

^ S*

r x

v

(-^ 5 ( W o; - + ij)S(W; - - -17)«(-/(- + r,))dr,,

where /I = 2AL » gNext we transform the exponential sums here by the following summation formula of the Voronoi type (see [15], Theorem 2): if F is a smooth function with compact support in (0, oo), then

(2.10)

JT , ( „ )e fcW) = *—i x I

+

\ q I

-*%«*[-!*)

q sinh KK *—i

| 7 2 K I — yffvc\- J-2iK\ —

\

qI

^nx\>F(x)dx

Ae cosh JTK 2 ?(w)e (—J J tf2k f — Vn*l F(x) dx,

where da = 1 (mod g); recall that e is the parity sign of the Maass form u and K is the related spectral parameter. We apply here the asymptotic formula (see [13], eq. (5.11.6)) (2.11)

/ 2 \1/2 n JuAx) ~ I — cos(x - inn - - ) \nx) 4

for x > NE, and otherwise we use the formulae (see [13], eqs. (5.10.4) and (5.10.23)) (2.12)

Jv(x) =

(

*/2)" r c os(*cos0)(sin0) 2v
and (2.13)

Kv(x)=

f°°

e-xcoshucosh(.vu)du.

Now, since rj * NSM~X by (2.7) and our assumptions on the support of the v-function, the factor e(±xrf) in the integrands in the formula (2.10),

Convolutions of Fourier Coefficients and the Circle Method

81

applied to our 5-sums in (2.9), is a genuinely oscillating function in the interval [N - M,N]. Therefore, applying (2.13) and repeated integration by parts or simply direct estimations, we see that the terms involving the X"-Bessel functions are negligibly small. Next, as to the terms involving the 7-Bessel functions, we apply (2.11) for n > Q^N6'1 and (2.12) for n < Q2Ns~l. Again, repeated integration by parts shows that the contribution of the last mentioned terms is negligible. Otherwise the integrand has a saddle point only if n x NQ2n2 x N, and the other terms can be neglected. The formula (2.11) contains just the leading term of the asymptotic expansion, the subsequent terms being similar but smaller, so it will suffice to work with the leading term. Denote by x\ and x2 the variables of integration in the Voronoi formula just applied to our exponential sums. Then the 77-integral in (2.9) is I v(-T])e((xi -x2-

f)rf)dT],

which is small unless x\ - x2 «: M; recall that 1 < / «: M. Therefore we may suppose in the sequel that |x; - N\ «c M; this is clear for i = 2 owing to the factor W(x2). We substitute the exponential sums in the transformed shape into (2.9), summing over the same values of / as in (2.8) with suitable coefficients /3f on the unit circle such that

YJPfT\W;f) = YJ\T\W;f)\. f

f

Then it remains to prove the estimate (2.8) for the sum 00

q=\

X

TJ TJ ^lW"2)("l"2r1/42^/5(n2-"l,/;9) x ff

WQ(x1)W(x2)e4nK^^-^^)/t'(xlX2ry4

XI I v(-r])e((xi -x2-

f)ri)dr]\ dx1 dx2.

There appear also integrals involving the sum of the square roots in the exponent, but these are seen to be small on integrating over jq for a given

82

M. Jutila

value of x\ - X2. The Kloosterman sums 5(«2 _ n\,f;q) appeared after the summation over a. We fix x\, X2, and 77 for a moment, estimating the corresponding integrals trivially in the end of the argument. The above expression should be compared with (6.4) and (6.6) in [7], putting «2 _ "l = P and «2 = "• Namely, if we replace x\ and X2 heuristically by N in the exponential, then the coincidence is perfect except that we now have a sum over / to deal with instead of an individual value of / . Moreover, replacing the numbers XJ by N does not make an essential change, for yjh~ix~ilq is then changed only by an amount <s. M/Q <sc Ns. Thus, if we consider the case of individual / , we get again the result of [7] in a somewhat simplified way (in [7], we used the automorphy of the Maass form rather than the Voronoi formula, so that we had to work out the transformation by an ad hoc argument). In particular, we have T(N;f) « N2/3+E for 1 < / «

N2/3,

which is (1.15) in [7]. The presence of the /-sum entails only a minor modification. Applying Kuznetsov's sum formula for sums of Kloosterman sums, we end up with a spectral expression involving factors tj(f)tj(p), where the fy-numbers are Hecke eigenvalues of the jth Maass form with a spectral parameter KJ «: (N/M)Ne. Moreover, there is a factor depending on KJ, n/p, and / . As in [7], we split up the double sum over n and p into subsums such that n/p is essentially constant in each. Then p is separated from Kj and / . Moreover, the parameters Kj and / can be separated (as in [8], or in [6], p. 454) if we restrict KJ for a moment to a short interval k < KJ < k + 1 to eliminate, essentially, the dependence on KJ for a moment. Then we apply Cauchy's inequality to the y'-sum and invoke the spectral large sieve inequality F

Z

a

j

«(A: + F 1+£ )2|c / | 2 ,

where ctj = | p / l ) | 2 / c o s h ( ^ ) . After that, we sum over k, again by Cauchy's inequality, and apply the "long range" version of the spectral large sieve to the /7-sum. Now, in [7] we had the square root of the sum Yjafj(J)«K2 + f<2+E Kj
Convolutions of Fourier Coefficients and the Circle Method

83

with K «: (N/M)NS as factor in the final estimate, while in the present case this is to be replaced by (K(K + F)F)1/2Ne, and then the estimate (2.8) follows. • In particular, it follows from (2.8) that (2.14)

YJ

\T*(W;f)\
l/l«M

note that the term for / = 0 can be included because T*(W; 0) «: N1+e. In the following lemma, we show that (2.14) holds even for the sums T(W; f) if we restrict the summation a bit. Lemma 2. Let M and N be as in (2.1). Then (2.15)

J]

|T(W;/)|«MAT 1 / 2 + e .

-M
Proof. We may interpret the sum (2.15) as Bo defined in (1.8) with F x M for i//(a) = S(W0;a)S(W;-a)e(Ma) and suitable coefficients /?/. with F x M. Next we estimate Bo by (1.12) noting that Ql+e « M if e is chosen sufficiently small. Then (2.15) follows from (2.14). a We are now in a position to estimate the accuracy of the approximation T(W; f) » T*(W; f) for an individual value of/. Theorem 2. Let M and N be as in (2.1) and 1 < / «: M. Then (2.16)

T(W;f) = T*(W;f) +

0(Nl/2+e).

Proof. We interpret T(W; f) and T*(W; f) as bf and b*f in the notation of (1.7) with (J/(a) as in (2.5). By the estimate (1.13) for the coefficients a„, we may restrict the summation over d and m in (1.7) to values such that \dm\ < M. Since the mean value of |fo/+„| over \n\ < M is «: Nl)2+£ by Lemma 2, we see that by straightforward estimations in (1.7) that bf-b*f «: N1,2+e. • Finally, combining this theorem with Lemma 1, we obtain a statistical result on T(W;f).

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M. Jutila

Theorem 3. With M and N as in (2.1), and F
YJ \T(W;f)\
+

Fl/2M-l/2N)NE.

\
Remark 1. We may apply (2.17) to the mean value of \T(N; f)\ over / . Suppose, for simplicity, that the Ramanujan -Petersson conjecture t{n) «. ne holds. Then T(W; f) «; MNe by a trivial estimation. We may approximate T(N; / ) , up to an error «; M0Ne, by at most «: log N sums T(W; f) with M > Mo » N1/2+e. Hence, by (2.17), ^

\T(N; f)\ « (FM 0 +

Fl/2M~l/2N)Ne.

1
We minimize the right hand side on choosing Mo = F~1/3N2/3. Then, for F « Nxl2~e, we have M0 » Nl/2+e and (2.18)

J]

|7XAr;/)|«(FA02/37V-£.

i
This holds unconditionally for the analogous sum in the case of holomorphic cusp forms, which can also be treated by a spectral identity (see [11]) for T(W; / ) , an analogue for an identity of Y. Motohashi [16] in the additive divisor problem. The mean value estimate (2.18) should be compared with the following a mean value result due to A. Ivic and Y. Motohashi [3] for the error term E(N; f) in the additive divisor problem: for F «c Nl/2~E, ^

E(N;f)2«Fl'3N4/3+E;

\
this implies an analogue for (2.18) by Cauchy's inequality. Remark 2. In applications, the unconditional result (2.17) may serve as a substitute for (2.18). For instance, this is the case in [11], where a certain spectral mean square estimate for Rankin-Selberg zeta-functions is studied. Remark 3. The main new ingredient in the proof of Theorem 2 is a comparison principle used in the proof of Theorem 1; that is, we compare different Fourier coefficients to draw conclusions on a single coefficient, and utilize the familiar phenomenon that an average is usually easier to

85

Convolutions of Fourier Coefficients and the Circle Method

deal with than a single term. The argument of [7] yields the same if we assume the Ramanujan-Petersson conjecture. Indeed, with ^(or) as in (2.5), we deduce from (1.6) that (2.19)

T(W;f)-T*(W;f) « <2"1/2dog 0 3 / 2 If

\S(Wo; a)S(W; -a)\2 da)

.

We have here S(W0; a) « N1/2 log W (see [2], Theorem 8.1) and

X

I

\S (W; -a)\2 da<&

J]

\t(n)\2 « min (MN26, M + JV3/5),

N-M
where 6 is a constant such that t(n) «: ne. The exponent 3/5 here is the classical Rankin-Selberg exponent in the error term for the mean square of Fourier coefficients of a cusp form. Now (2.19) reduces to (2.16) if 8 = s is admissible. Remark 4. An alternative approach to convolutions makes use of the generating function of t{n)t(n + f) (see, e. g., [5], [6] or [9]). Then we encounter a non-trivial problem, namely the estimation of the inner products (u2, Uj). In [5], [6], this was reduced, by a variant of the Rankin-Selberg unfolding method, to convolutions analogous to those which occurred in the proof of Lemma 1. Thus these inner products are hidden in a disguised form even in the present argument.

References [1] W. Duke, J. Friedlander, and H. Iwaniec: Bounds for automorphic L-functions, Invent. Math. 112 (1993), 1-8. [2] H. Iwaniec: Introduction to the Spectral Theory of Automorphic Forms, Bibl. de la Revista Mathematica Iberoamericana, Madrid 1995; 2nd ed. Amer. Math. Soc. 2002. [3] A. Ivic and Y. Motohashi: On some estimates involving the binary additive divisor problem, Quart. J. Math. Oxford 46 (1995), 471-483.

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[4] M. Jutila: A Variant of the Circle Method, Sieve Methods, Exponential Sums, and their Applications in Number Theory, Cambridge Univ. Press 1996, 245-254. [5] M. Jutila: The additive divisor problem and its analogs for Fourier coefficients of cusp forms. I, Mat. Z. 223 (1996), 435-461. [6] M. Jutila: The additive divisor problem and its analogs for Fourier coefficients of cusp forms. II, Math. Z. 225 (1997), 625-637. [7] M. Jutila: Convolutions of Fourier coefficients of cusp forms, Publ. de l'lnstitut Math. Nouv. ser. 65 (79) (1999), 31-51. [8] M. Jutila: On spectral large sieve inequalities, Funct. Approx. Comment. Math. 28 (2000), 7-18. [9] M. Jutila: Sums of the additive divisor problem type and the inner product method, Zap. nauchn. sem. POMI322 (2005), 239-250. [10] M. Jutila: Distribution of rational numbers in short intervals, The Ramanujan J. (to appear). [11] M. Jutila and Y. Motohashi: Uniform bounds for Rankin-Selberg Injunctions, Proc. of the Workshop on Multiple Dirichlet Series, Amer. Math. Soc. (to appear). [12] A. A. Karatsuba: Basic analytic number theory, Springer-Verlag , Berlin 1993. [13] N. N. Lebedev: Special Functions and their Applications, Dover Publications, New York 1972. [14] Yu. V. Linnik Additive problems and eigenvalues of modular operators, Proc. Internat. Congr. Mathematicians (Stockholm 1962), Inst. Mittag-Leffler, Djursholm 1963, 270-284. [15] T. Meurman: On exponential sums involving Fourier coefficients of Maass wave forms, J. reine angew. Math. 384 (1988), 192-207. [16] Y. Motohashi: The Binary Additive Divisor Problem, Ann. Sci. l'Ecole Norm. Sup. 27 (1994), 529-572.

Convolutions of Fourier Coefficients and the Circle Method

87

[17] Y. Motohashi: Spectral Theory of the Riemann Zeta-Function, Cambridge Univ. Press 1997.

Matti Jutila Department of Mathematics, University of Turku, FI-20014 Turku, Finland E-mail: [email protected]

On an Extension of the Derivation Relation for Multiple Zeta Values M. KANEKO

In this paper, we propose a conjectural generalization of the derivation relation for multiple zeta values. This extension was inspired by works of Alain Connes and Henri Moscovici on a certain Hopf algebra of transverse geometry [1], [2], and is thought of as a first attempt to materialize the suggestion given at the end of Section 7 in [4]. The multiple zeta value (MZV) is a real number defined for each index set k = (£i, k2,. . ., kn) of positive integers with k\ > 1, by the convergent series

m = «kuk2,...,kn)=

£ mi>m2>->m„>0

- ^ — - . m

\

m

2

'''

m

n

One of the fascinating features of these numbers is that the structures of (linear or algebraic) relations over Q among them reflect various other structures in mathematics and mathematical physics. To describe the derivation relation of MZV's, we use the algebraic setup introduced by Hoffman [3]. Let § = Q(x, v) be the non-commutative polynomial algebra over the rationals in two indeterminates x and y, and § ' and §° its subalgebras Q + % and Q + x§>y, respectively. Let Z : ° -> R be the Q-linear map defined by Z(l) = 1 and Z(**>-V 2_1 V • • • **"_1y) = f (*1, *2, • • • , k„). This "evaluation map" arises naturally from the iterated integral representation of MZV's. To find a linear relation among MZV's is to find an element in the kernel of the map Z. For each integer n > 1, define the derivation dn on § by dn{x) = x(x + y)n-xy,

dn(y) = -x(x + v)"" 1 ?. 89

90

M. Kaneko

Since § is the polynomial algebra on x and y, this uniquely determines the derivation dn. (A derivation on § is a linear map d : £> —> § satisfying Leibniz's rule 3(MV) = d{u)v + ud(v) for any u,v e §.) Note that the derivation dn preserves &1 and §°. The derivation relation of MZV's is then the following. Theorem. ([4, Corollary 6]) One has Z(<9„(w)) = Ofor all n > 1 and all we§°. This relation supplies a large class of linear relations of MZV's. To add the derivation relation to (finite) double shuffle relation is equivalent to extend the double shuffle relation by using divergent MZV's. Also, the derivation relation is equivalent to Ohno's relation (minus duality). See [4] for the details. We now proceed to generalize this. In [4, p.328], it is noticed that the derivation dn has an alternative description: Let 9 be the derivation on § defined by 9{x) = x2 + (xy + yx)/2,

9(y) = y2 + (xy + yx)/2.

Then we have dn = &mn-\dx)l{n

- 1)!,

where ad(0)(d) := [9, d] = 6d- 89. Let H be the "Euler operator", namely, the linear endomorphism of %> which sends each homogeneous element w to deg(w)w. Definition. Let c be a rational number. For any n > 1, we define a linear endomorphism dnof& by flic) = ad(0 (c) y , - 1 (5i)/(/i-l)!, where 0 ^ is the linear endomorphism defined by the rules 9(c\x) = 9(x),

0(c)(y) = 9(y)

and 0(C)(MV)

for any u, v £ §.

= 0(c)(M)v + «0(c)(v) + cdi(u)H(y)

91

Derivation Relation for Multiple Zeta Values

Remark. This definition is modeled after the "twisted" Hopf algebra of Connes-Moscovici (see [1], [2], [4]). Note that 0 1, c e Q, and w e §°, we have

Z{df(w)) = 0. When we view c as a parameter, we see that d„ (w) is a polynomial in c of degree « - 1. Hence, the above conjecture is equivalent to the following. Conjecture V. For alln>\

and i with 0 < i < n - 1 and w € §°, we /zave

Z(coefficient of J in <9<,c)(w)) = 0. We have checked this by using Mathematica for all n, i and w with n + deg(w) < 9. Tatsushi Tanaka has extended the experiments by using Risa/Asir and verified the conjecture up to weight 12. Below we give a table of various numbers of independent relations of MZV's obtained by d^\ detailed descriptions of the table being given in the remarks after the table. Recall the weight of the MZV £{k\, k2,..., kn) is ki + k2 + • • • + kn. Table 1: (Conjectural) Numbers of Independent Relations weight

3

dnM fi\w),c± -2 fixed

1 1 1 1

«l?Vi)uC'W

S^\w) for all c U duality

4 2 2 2 2

5 5 5 5 5

6 10 10 10 10

7 22 22 23 23

8 44 44 46 46

9 90 90 97 98

10 181 181 196 200

11 363 363 396 413

12 727 727 838

Remarks, i) The first line of the data gives the number of linearly independent elements in § among all dn{w) with n e N and w e §° varying under the condition n + deg(w) = weight. (The weight of each term of Z(dn(w)) isn + deg(w).) ii) The second line is the same experiment with various fixed c € Q. The result says that, at least for tens of randomly chosen values of c ^ - 2 ,

92

M. Kaneko

the elements d„ (w) as n and w vary give the same number of independent relations as d„(w), regardless of the choice of c. We have no proof of this (nor of Z(d^\w)) = 0, except in low weights). The reason why the value c = - 2 is exceptional is explained by the identity d£~2) = ( - l ) " - 1 ^ , which, as Seidai Yasuda and Kentaro Diara pointed out, is easily proved. iii) The experiments for various single c as in ii) may suggest that d„ only gives the same set of relations as the original derivation dn does. However, we do have new relations, as the third line of the above table show. The third line of the table gives the number of linearly independent elements supplied by the union of d„ (w) and d„ (w) for randomly chosen different values of c\ and C2 (£ -2). It is remarkable here again that any choice of c\ and C2 seems to give the same number of independent relations. This too is only experimental and is not yet proved. iv) The final line is the result of taking all c into account and moreover adding the duality relations. Up to weight 9, we used the form of Conjecture 1' to ensure the "for all c" part, but from weight 10 on we only used the data for several rational values of c for computational reason. Thus the number given may increase if we add more values of c, but we believe that does not occur, in light of Conjecture 2 below. The reason why we add the duality is also explained by Conjecture 2. If the duality is not added, the numbers obtained are strictly less at weights 11 and 12 (and probably for higher weights). v) Seidai Yasuda pointed out the identity 0 M = 0(<» + £(0<°>-0(-2)), or in general c'

for any c' + 0. Hence, to consider d^ for all c is also equivalent to consider ad(0(ci))ad(0(C2)) • • • ad(6'(c"-l))(3i) with each a = 0 and 1 (say). Now let ak be the entry for weight k of the last line of the above table (e.g., a% = 46, a9 = 98, aio = 200,...). Then, we have observed the following remarkable (conjectural) identity.

Derivation Relation for Multiple Zeta Values

93

Conjecture 2. For k>3,we have

d\k-l

Here, fi is the Mobius function. Remark. Let Li be the free graded Lie algebra of ] rank 2 over Q generated by x and y of degree 1, and LM the free graded Lie algebra over Q with infinitely many generators z,- (i = 1,2,3,...) whose degrees are given by deg(z,) = i. The number on the right-hand side of the conjecture is equal to the dimension of degree k - 1 part of both Li and Loo, as Witt's formula shows. We note that the universal enveloping algebra of Li is identified with §, whereas that of Loo is identified with S 1 under the identification Zi = 1 l x JC , ' - V (The algebra S is freely generated by zt = x ~ y, i = 1,2,3,....) This observation suggests intimate connections between these Lie algebras and the relations of MZV's. Note that the number 2k~2 on the left is the total number of MZV's of weight k. Thus the number on the left-hand side of the conjecture is the maximal number of independent MZV's, under the (c)

conjectural relations Z(3„ (w)) = 0 and the duality. The most curious feature of Conjecture 2 is that the weights concerned on both sides differ by 1. The conjecture implies the possibility of constructing linearly independent (under the conjectural relations and the duality) elements of MZV's of weight k from a basis of homogeneous part of degree k - 1 of Li or Loo. But how to construct such elements? We tried several ways at random and found one way (using the derivation 6y in [4, §4] and the regularization) which bears the numerical test up to weight 13, but there is no theoretical support to believe that is the right way. It is most desirable to prove (or disprove) this conjecture and clarify the suggested connection of the MZV to the Lie algebras La and L^. Acknowledgment. The author would like to thank Seidai Yasuda at RIMS, Kyoto, for valuable comments and suggestions.

94

M. Kaneko

References [1] Connes, A. and Moscovici, H., Modular Hecke algebras and their Hopf symmetry, Moscow Mathematical Journal 4 (2004), 67-109. [2] Connes, A. and Moscovici, H., Rankin-Cohen brackets and the Hopf algebra of transverse geometry, Moscow Mathematical Journal 4 (2004), 111-130. [3] Hoffman, M., The algebra of multiple harmonic series, J. Algebra 194 (1997), 477^195. [4] Ihara, K., Kaneko, M., and Zagier, D., Derivation and double shuffle relations for multiple zeta values. Compositio Math. 142-02 (2006), 307-338.

Masanobu Kaneko Graduate School of Mathematics, Kyushu University, Fukuoka 812-8581, Japan Email: [email protected]

On Symmetric Powers of Cusp Forms on GL2 Henry H. KIM

Introduction Let A be the ring of adeles of a number field F. Let n = ®v7rv be a cuspidal automorphic representation of GZ,2(A) with central character a>n. Let Symm: GZ^(C) —> GLm+i(C) be the rath symmetric power representation of GLa(C) on symmetric tensors of rank m. By the local Langlands correspondence, Symm{nv) is a well-defined irreducible admissible representation of GL m+ i(F v ) for every v (see [Kil] for the details). Let Symm(n) = ®vSymm{nv). It is an irreducible, admissible representation of GLm+i(A). Langlands functoriality conjecture is that Symm(7r) is an automorphic representation of GLm+i(A). It is one of the most important problems in the Langlands program, and its solution would imply the Ramanujan conjecture and Sato-Tate conjecture for n [Ki-Shl]. So far it is proved only up to ra = 4, even though we can prove the functoriality for all m for cuspidal representations attached to Galois representations [Ki3]. In this note, we give a new proof of the cuspidality criterion of the symmetric fourth [Ki-Sh2], and supplement various results on Symm{n) in [Kil, Ki-Shl, Ki-Sh2]. For convenience, let Am(n) = Symm{n) ®u~x. Suppose n is an isobaric automorphic representation of GL„(A) of the form II = n\ m • • •fflnk, n\ + • • • + rik - n, where 7r,- is a cuspidal representation of GLni(A). Then we say n is of type (n\,..., «*)• We refer [Kil] for unexplained notations.

1

Symmetric fourth

In this section, we assume that Sym3(n) is a cuspidal representation of GL4(A). Then by the result in [Ki-Shl], Sym2{n) * Sym2(n) ®x for any x * 1- Note that Sym2(n) = Ad(n) <8> a>„. In [Kil], we proved that 95

96

H. H. Kim

Symr{n) is an automorphic representation of GLs(A), either cuspidal or of type (3,2). We gave the cuspidality criterion of SymA(n) in [Ki-Sh2]. Namely, Theorem 1.1. A4(n) is not cuspidal if and only if there exists a non-trivial quadratic character T] of F such that A3(n) =* A3(7r) ® rj. It is equivalent to: AdiriE) - Ad{n£) ®xfor a grossencharacter x of E, where E/F be the quadratic extension, determined by r\, and TTE is the base change of n to GLi(AE). In that case, A\n) - cri m cr2, where c\ = Ad{n) ® (tonn), and 0"2 = 7T(X~l) ® 0)n-

We had to use the following unpublished result on the descent of cuspidal representations of GL4 to GSp4 of Jacquet, Piatetski-Shapiro and Shalika: Theorem 1.2 (Jacquet, Piatetski-Shapiro and Shalika). Let n be a cuspidal automorphic representation ofGL^A) such that there exist a grossencharacter x and a finite set S of places as above for which L$ (s, n, A2<8>x~l) has a pole at s = 1. Then there exists a globally generic cuspidal automorphic representation r of GS p4(A) with central character x such that n is the Langlands' functorial lift ofr under the natural embedding LGSp4 = GSp4(C) ^-* GL4(Q. Here we prove Theorem 1.1 without using the above theorem. Throughout this note, S always denotes a set of places of F such that for v g S, every representation is unramified. First we note that for any cuspidal representation o- of GL*(A), (1.1)

Ls(s,o-xnxA3(7r))

=

Ls(s,o-xA\n))Ls(s,(TxSym2(n)).

Note also that A4(n) is not cuspidal if and only if Ls(s,o- x A4(n)) has a pole at s = 1 for some cuspidal representation cr of GL2(A). Since Ls(s,o-x Sym2(n)) is invertible at s = 1, by (1.1), A4(n) is not cuspidal if and only if LS(S,CTX7TX A*(n)) has a pole at s = 1. However, (1.2)

Ls{s,o- x n x Ai{n)) = Ls(s,((r H n)xA3(n)),

where cr H n is the functorial product of cr, n which is an automorphic representation of GL4(A) [R, TheoremM]. Since A3(^) is cuspidal, Ls(s,(o-®

On Symmetric Powers of Cusp Forms on GLQ.

97

n) x A3(n)) has a pole at s = 1 if and only if cr B 7r is cuspidal, and cr H 7r =* A3(^-) =* A3(TT) ® OJ"1 . Consider A2(o-H TT) =; A2(A3(7r)®co-1).

(1.3)

Both are automorphic representations of GL^iA) [Kil]. By direct calculation, we see that the left hand side of (1.3) is Sym2(cr)®conffiSym2(n)®0Ja-. (We see that for almost all v, A2(o-vH7rv) ^ Sym2(o-v)®ionvmSym2(7Tv)®cOo-v. Since both A2(cr E 7r) and Sym2(cr) ®co„s Sym2(n) ® ov are isobaric automorphic representations of GLf,{K), by the strong multiplicity one theorem, A2(cr B i ) ^ Sym2(o~) ® ton m Sym2(7r) ® coa.) By the main result in [Kil], the right hand side of (1.3) is (A4(n) ® co~2)ffico~l. Hence Sym2(cr) cannot be cuspidal, i.e., cr is monomial. Let cr = /£y ® io~l for some quadratic extension E/F, and x is a grossencharacter of E. Let 77 be the quadratic character attached to E/F by class field theory. Then Ad(cr) = nixx*1) a ^ an^ ^a- - (X\A*)W- Hence we see that^U* = 1, and A\n) = (Ad(n) ® conrf) m {lFEx ® OJ'1)Note that since cr = /£y ® w"1, by [R, Lemma 3.1.1], cr ta n = /£(n E ® * ) ® a ^ . So by (1.2), Ls(s,crxnx

A3(TT))

= Ls(s,I%(nE ®x) XA3(n) ® to'1).

By the adjointness property of the base change [R, Proposition 2.3.1], Ls(s, fE(nE ®x) x A3(?r) ® w"1) = Ls(s, {nE ®x) x ^ V k ® w^1), where o>£ = a>n o Ar£//r = a»^£. By (1.1), the right hand side has a pole at s = I. Hence A3(n)E is not cuspidal. By the cuspidality criterion of the base change (see [R, Proposition 2.3.1].), A3(n) =* A3(n) ® rj. On the other hand, A3(n)E = A 3 (^ £ ). Since Ad(n) is cuspidal and E/F is a quadratic extension, Ad(n)E is cuspidal. Hence by the cuspidality criterion of A3(JTE) in [Ki-Shl], Ading) - Ad(jiE)®n for a grossencharacter fi of £. Here fj3 = 1 and ju ^ 1. Then A3(7r£) = (7r£ ® //)ffl(nE ® ju2). We have L 5 (s, (7T£ « * ) x A3(TT)£ ® co-Ex) = Ls (s, 7r£ x TTJ ®XH)LS (s, nExn^

®xt*2)-

Since it has a pole at s = 1, ,y = n or // - 1 . This finishes the proof of Theorem 1.1.

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2 Symmetric mth powers In this section, we assume that Symm{n) = ®vSymm{nv) is an automorphic representation of GLm+i(A) for m > 5. In [Ki3], we proved that it is the case if n is of icosahedral type, attached to a holomorphic cusp form of weight 1. It is also true if n is of dihedral, tetrahedral or octahedral type, attached to a holomorphic cusp form of weight 1. Let us briefly explain them. We assume F = Q. By Deligne-Serre theorem [DS], n is attached to an odd irreducible 2-dimensional representation p : WQ —> GLz(C) ("p is odd" means that det{p)(c) = - 1 where c is the complex conjugate. Conjecturally Maass forms give rise to even irreducible 2-dimensional representations.) By twisting by a character of WQ, we can assume that p factors through GQ = Ga/(Q/Q), denoted again by p. Let p : GQ —> PGL2(C) be the composition of p and the projection GLi{C) —> PGLadC). Then the image of p is Din, A4, S4, or A5. We say n is of dihedral, tetrahedral, octahedral, or icosahedral type if the image of p is G = D2„,A4, S4,As, resp. Then p factors through G', which is a central extension of G. Note that H2(G, C*) = 1 if G = D2n for n odd. Otherwise, # 2 (G, C*) = Z/2Z [Ka]. (We used the fact that H2(A5,C*) = Z/2Z in [Ki3].) Hence p factors through G if G = D2„, n odd, and otherwise through G, the unique central extension of G by {±1}. Then G' is an amalgamated product GX(±i) C2/1, where C2„ is the cyclic group of order In [Ki3]. Hence irreducible representations of G' are equivalent to a twist of irreducible representations of G. (In the case of £>2„, n odd, irreducible representations of G' are equivalent to a twist of those of £>2„.) Hence p is equivalent to a twist of an irreducible 2-dimensional representation of G. More explicitly, G = 5,L2(F3),GL2(F3) for G = A4,S4, resp. When G = D2n, n even, G = Q4n, the dicyclic group of order An, defined by Q4n =< a,b\a2n = l,b2 = cf,b~xab = a'1 > . Note that Q4n/ < an >^ Din- It has four 1-dimensional characters, and n - 1 two-dimensional representations. On the other hand, the irreducible representations of the dihedral group, D2n, n odd, are: two 1-dimensional characters, and ^ twodimensional representations. Hence by Langlands-Tunnell theorem [La, Tu], all irreducible representations of Di„, Q4n are automorphic. Next consider G = GL2(F3). It has order 48. The character table is

On Symmetric Powers of Cusp Forms on GLQ.

99

given as follows. (See [FH] for character tables of SLi(Fi) and GL2(F3).)

size *i

*; *2 *2 *2 *3

4 *4

1 1 1 2 2 2 3 3 4

Here / = I Q

1 1 1 -2 2 -2 3 3 -4

(1 1) (» 3 ft -•) (f -.') G 5) (> 3 8 1 1 -1 -1 -1 0 0 1

8 1 1 1 -1 1 0 0 -1

12 1 -1 0 0 0 1 -1 0

6 1 1 0 2 0 -1 -1 0

6 1 -1

6 1 -1

-V=2

V=2

0

0

V=2

-V=2

-1 1 0

-1 1 0

J . Note the relations: X3 = Sym2(x2), Sym2^)

= X>2+Xu

X% =*2 Qx'vX'i =X3 ®x'v and^ 4 = X2 ®x'2Also xi is the 2-dimensional representation which comes from the natural map GL2(F3) —• GLa(C), and has been used in the proof of Fermat's last theorem by Andrew Wiles (see [Ge, page 160]): More explicitly, xi is the character of the representation \ft : GL2(F3) —> GL2(Z[ V-2]) c GLa(C), defined by

*(-\ o) = (-l o)' *(1 "/j^-V^ -i + W)-

Here I

and I

are two generators of GL2(F3).

Note that GL2(F3)/{±/} ^ 54, and hence the representations that factor through representations of 54 arex\,x\,X2,X3^X^• Consider G = SL2(F3). It has order 24. The character table is

100

H. H. Kim I size X\ Ml V?x X2

m *i

X3

-/

1 1 1 1 1 1 1 1 2 -2 2 -2 2 -2 3 3

(o I ) (o I ) (o 2) (0 2) (1 0 ) 4 4 4 4 6 1 1 1 1 1 u 2

u - 1 s -s2 0

u2

-u

-u2

u -s2 s 0

2

-u

-u 1 1 -s s2 0

1

1 s2 -s 0

1 0 0 0 - 1

Here u is a primitive cube root of 1, i.e., u = -j + \ V^3, and s = \ + j V 1 ! = « + 1. Note that^ 4 | sz , 2(F3) = rj2 + TJ2, and ^ S L2(¥3) = J"i + /*iNote that 5L2(F3)/{±/} =* A4, and hence the representations that factor through representations of A4 are those taking positive values at - / , namely, xu Hi, n'vX3Hence by Langlands-Tunnell [La, Tu], Gelbart- Jacquet [Ge-J], and Ramakrishnan [R], all irreducible representations of SLviF-}), GL2(F3> are automorphic: lixi,X,2 §i v e rise t o cuspidal representations 7r,7r', resp., then X3>X4 gi v e rise t o Sym2(n),7r H n', resp. Any representation of a finite group is a direct sum of irreducible representations of the group. Hence Symm(p) is equivalent to a twist of a direct sum of irreducible representations of G. Since all irreducible representations of G are automorphic, and so is Symm(p). Namely, Symm{n) is an automorphic representation of GLOT+i(A) for all m. It is clear from the above calculation that if n is of dihedral type, Symm{n) is not cuspidal for all m > 2; if n is of tetrahedral type, Symm(n) is not cuspidal for all m > 3; if n is of octahedral type, Symm{n) is not cuspidal for all m > 4. We prove here that if Sym4(n) is cuspidal, then Sym5(n) is always cuspidal. We can assume that F is an arbitrary number field. Theorem 2.1. Suppose SymA(n) is a cuspidal representation ofGL${A). Then Syms{n) is always a cuspidal representation ofGL^A) Proof. First note (2.1)

Ls(s, cr x n x A4(n)) = Ls(s,
On Symmetric Powers of Cusp Forms on GLa

101

Since A4{n) is cuspidal, Ls(s,crX7TX A4(n)) = Ls(s, (cr a TT) x A4(n)) cannot have a pole at 5 = 1 if cr is a cuspidal representation of GL2(A). Let cr be a cuspidal representation of GL^A). We use Proposition 2.2. [R-W] Let a-be a cuspidal representation ofGL^{A) and n is a non-monomial cuspidal representation ofGLjiA). Then o-\Sn is not cuspidal if and only ifcr^ Ad(n) ®xIn order that Ls(s, (cr H TT) x A4(n)) has a pole at s = 1, cr B n is not cuspidal. Hence cr = Ad(n)®x f° r some^. Then o-®n = (A3(7r)®x)m(n<S>x)Hence Ls(s, a x n x A4(TT)) = L 5 (s, (A3(;r) ®*) x A4(;r))L5(5, (n ®X) x A4(TT)), has no pole at s = 1.

n

By Theorem 2.1, we can recover Corollary 2.3. /W7 Let n be a cuspidal representation ofGL2(A) oficosahedral type. Then Sym5(n) is a cuspidal representation ofGL(,{A). Proof. By [Ki3], Sym5(n) is an automorphic representation of GL^A), and Sym4(n) is a cuspidal representation of GLs(A). We apply Theorem 2.1. o Corollary 2.4. Suppose Sym4(n) is a cuspidal representation ofGLs(A). Then there does not exist n + \ such that Sym4{n) ^ Sym4(n) ® 77. Proof. Consider the following: By setting cr = A3(;r) ® &£2// in (2.1), we have Ls (s, (A3(n) ® w;2??) X TT X A4in)) = Ls (s, A\n) ® w" 2 ^ x A4(TT))L5 (5, Ad(n) Qco^nx A4 in)) = Ls(s,A3(n) ® w; 2 ^ x A5(TT))L5(S,A3(;r) ® co^n x A3(TT)). Here A4(n) ® w~2 =* A4(7r). Hence we can write Ls (s, A 4 ^) X A4(^) ® ij)Ls (s, (Ad(n) ® a*;117) x A4(?r)) = L 5 (s, (A3(TT) ® t^ 2 //) x A\n))Ls(s, (A\TT) ® W^IJ) X A3(TT)).

102

H. H. Kim

Since Ls(s, (A3(n) <S> co~2rf) x A5(n)) has no pole at s = 1, there does not exist J] + 1 such that A4(n) <8> 77 =* A4(JT). We can prove this fact directly as follows: Suppose that A4(n) ® 77 =* 4 A (n) for 77 * 1. Then 77s = 1, and 77 determines a cyclic quintic extension K/F, and A\n) = l£x (automorphic induction) for a grossencharacter x of K. Let Gal(K/F) = (6). Hence A4(7rk = A4(TT*) = x *)? ®X^ o X^ B ; r > where ^ = ^ o f i . Since A4(TT) is cuspidal, x * V • However, according to our calculation in [Ki-Sh2], since A4(nf[) is of type (1,1,1,1,1), one constituent occurs with multiplicity. Contradiction. a Remark 2.1. It is remarkable that there is no cuspidal representation n of Gbi{A) such that Sym4(n) is cuspidal and Sym4(n) ^ Sym4(n) ® 77 for 77 + 1. Consider Ls(s, Sym4(n) x Sym4(n)) = Ls(s, n, Sym*)Ls(s, n, Sym6 <8> io„) X Ls (s, Sym4(n) ® co2n)Ls(s, Sym2(n) ® a>l)Ls(s, co4n) and Ls(s,Sym3(7r) x Sym5(n)) = Ls (s,n,Sym%)Ls (s,7T,Sym6 ® co„) X Ls (s, Sym4(n) ® afyL?(s, Sym2(n) ® col). Hence if Sym5(n) is cuspidal, Ls(s,n,Syms) is always holomorphic at s = 1. On the other hand, if to4 + 1 and Sym4(ri) is self-contragredient, i.e., Sym4(n) =* 5ym4(7r) <8> to~4, then Ls(s,n,Syms) has a pole at s = 1. However, we showed above that there does not exist such n. More generally, suppose that Symm(n) is an automorphic representation of GLm+i(A) whose irreducible constituents are cuspidal representations of GZ*(A) for k > 4. Consider, for any cuspidal representation cr of GLk(A), (2.2) Ls(s,o-X7rxAm(n))

=

Ls(s,crxAm+1(7r))Ls(s,cTxSymm-\n)).

Suppose cr is a cuspidal representation of GLq,. Consider cr E n. Then by our assumption, Ls(s,cr x n x Am(n)) cannot have a pole at s = 1. Therefore, Am+1(n) cannot have a cuspidal representation of GL2(A) as an irreducible constituent. This implies that Am+1(n) cannot have a cuspidal representation of GLm(A) as an irreducible constituent, either. So Ls (s, crx

On Symmetric Powers of Cusp Forms on GLQ

103

n x Am(n)) does not have a pole at s = 1 for cr = Am 1(7r) ® u)„ (2.2). Then we have

'rj in

Ls(sXAm-\n)®u>~{m~2)Ti) XnX Am(n)) = Ls {s, Am{n) ® o£m-2\ X Ls{s,Am~\n)

X Am(n)) ® o)n(m~3)V X Am(n))

= Ls(s, Am-X (*) ® cj^m~Z)T] X Am+1 (*)) x Ls{s,Am-\n)

® ^ ( m _ 3 ) /7 x Am_1(»r)).

Here Am(n) ® w~(m_2) =* A">0T). Hence we can write Ls (s, A^)

X Am(7r) ® rf) Ls (s, (Am-\n) ® ^ ( m _ 3 ) ^ ) X Am(7r))

= Ls(s, (Am-\7i) ® c^(m_2)77) X Am+1(7r)) X L s (s, (A"1"1^) ® ^ (m-3) 77) X A"1"1 (*•))• Hence if 77 * 1, L 5 (s, A™in) x (Am(7r) ® 77)) has no pole at 5 = 1, i.e., A (n) ® 77 * Am(/r). We have proved the following: m

Proposition 2.5. Suppose m > 5. Suppose Symm(n) is of type (ni, ...,w*)> where ni > 4 for each i. Then there does not exist T) £ 1 such that Symm(n) ®TJ=* Symm{n). Since Sym*_1(7r)H7T = Symk(n)mSymk~2(n)®u)„, if cr = Sym\n)&T, then cr la ^ is not cuspidal. One wonders if the converse is true. It would be a natural generalization of Proposition 2.2.

3 First occurences of poles of symmetric power L-functions We summarize results on poles of symmetric power L-functions [Ki-Sh2]. Let n be a cuspidal representation of GLi with the central character 10. First, n is of dihedral type. Then L(s, n, Sym2) has a pole at s = 1 if a> + 1, and 0? = 1. If 0) - 1, L(s, n, Sym4) has a pole at 5 = 1 for the first time.

104

H. H. Kim

Second, n is of tetrahedral type. Then L{s,n,SymA ® o r 1 ) has a pole at s = 1 if <x> + 1. Here a> = 77 or ^ 2 , where 7/ is a cubic character such that Ad(n) =* Arf(^) ® 77. If o» = 1, L(s, n, Sym6) has a pole at s = 1 for the first time: Consider Ls(s,Sym2(n)

Ls\s,7T,Syms)Ls\s,Sym3(n))Ls\s,n).

X Sym?(n)) =

Since Syrrv'in) = (n ® fi)ffi(TT ®/U2), where // is a cubic character, the left hand side is holomorphic at s = 1, and Ls(s,n,Sym5) is holomorphic at s = 1. Consider L5(s,Sy/n3(7r) X Sym3(n)) = Ls (j,», Sym6)Ls (s, Sym*(n))Ls (s, Sym2{n))Ls (s, 1). The left hand side has a double pole at s = 1, hence L 5 (s, ;r, Sym6) has a pole at s - 1. Third, «• is of octahedral type. Then by [Ki-Sh2, Theorem 3.3.7], Ls(s,n,Sym6 <8> OJ~3T]) has a pole at s = 1. Here 77 is a quadratic character such that A3(7r) = A3(n) ® 77. If OJ = 1, Ls(s, n, Sym%) has a pole at s = 1 for the first time: We can show as above that Ls {s, n, Symm) does not have a pole at s = 1 for m < 7. Consider Ls{s, Sym4in) x Sym\n)) =

Ls(s,n,Syms)

x Ls(s, n, Sym6)Ls(s, Sym*(n))Ls (s, Sym2(n))Ls(s, 1). Since SymA(n) = n(x~x) ffl Sym2{n) ® 77 (cf. Theorem 1.1), the left hand side has a double pole at s - 1. Hence Ls (s, n, Sym%) has a pole at s = 1. Fourth, 7T is of icosahedral type. Note that Ls(s,n,Symm ®x) has no pole at s = 1 for all m < 11 for any^. For example, for m - 11, consider Ls(s, Sym5(n) X Sym6(n) ®x) 5 s

= L (s,7r,Sym

n

® Ar )]~[L s (j,;r,Syro 11 ~ 2 '«*"')• 1=1

Since Sym5(7r) is cuspidal, and Syrrfi(n) is equivalent to a twist of Sym2(nT) ffl (TT a 7rT), where if is a certain cuspidal representation of GLq(A) [Ki3, Theorem 6.4], the left hand side is holomorphic at s = 1. Hence Ls(s,n, Sym11 ®x) is holomorphic at s = 1.

105

On Symmetric Powers of Cusp Forms on GLi

Ifo» = l,Ls(s,7r, Sym12) has a pole at s = 1 for the first time: Consider Ls(s, Sym6(n) X Sym6(n)) 5

= Ls (s,n,Symn)Y\Ls

(s,7r,Symn-2i)Ls

(s, 1).

i=i

If <x> = 1, Sym6(n) - Sym2(nT) ffi (n ta if), and the left hand side has a double pole at s = 1. So Ls (s, n, Symn) has a pole at s = 1. If n is not of Weil type, i.e., n is not attached to any of the homomorphisms 4> : VV> —> GL2CC), it is expected that L{s, n, Symm ®x)iS entire for all m, and for all^. At the moment, we know that it is the case only for m < 4 [Kil]. For 5 < m < 9, we only know that they are meromorphic in all of C, and invertible for Re(s) > 1 [Ki-Sh2]. For m > 10, no such results are available.

4 Descent to cuspidal representations on classical groups For g e GL2(C), Symm(g) e GSpm+i(C) if m is odd, and Symm(g) e GSOm+\(C) if m is even. (See [G-W, page 244].) For m = 1, we need to identify GSp2(C) = GLoiC). Moreover, if m is even, Symm(g)(detg)~!% e SOm+\(C). Let m = 2k and B(n) = Symm(n) ® (j~k. Then B(n) is a self-dual representation of GL,2k+\(A) with the trivial central character. Then Ls(s,B(7r),Sym2) always has a pole at 5 = 1, since Ls(s,B(n), A2) is holomorphic at s = 1 [Ki2]. However, we can see directly as follows: Ls (s, Bin), Sym2) = Ls (s, n, Sym2m ® of™) Ls (s, n, Sym2"1'4 ® u>~m+2) • • • Ls (s, n, Sym4 ® U)~2)LS (s, 1). Since all the L-functions on the right side have no zeros at s - 1, our result follows. Hence by the result of Ginzburg-Rallis-Soudry [GRS], B(n) descends to a generic cuspidal representation T of 5p2*(A). Let m = 2k - 1, and B(n) = Symm(n) ® u~M. Then B(TT) is a representation of GL2fc(A) with the central character a>* and B(n) 2>oin - B(n). We

106

H. H. Kim

claim that Ls(s, B(n), A2®tonx) has a pole at s = 1: Consider Ls(s,B(n), A 2 ® ^ 1 ) =

Ls(s,n,Sym2m-2®(j;m+l)

x Ls(s, n, Sym2m-6 ® to'"1*3) • • • Ls(s, n, Sym4 ® io~2)Ls (s, 1). Since all the L-functions on the right side have no zeros at s - 1, our result follows. It is conjectured that B(n) descends to a generic cuspidal representation of GSpinik+i (A). When k = 2, GSpin$ =* GSp4, and it is the unpublished result of Jacquet, Piatetski-Shapiro, and Shalika (Theorem 1.2). If con = 1, then it is expected that B(n) descends to a generic cuspidal representation ofS0 2 * + i(A). We state the result for m = 4 as Theorem 4.1. Suppose Sym4(n) is cuspidal. Then Sym4{n) ® (j~2 descends to a generic cuspidal representation T o/S/74(A). Hence L(S,T) = L(s,Sym4(n)®aj-2). We examine the relationship between the infinity components ^00 and TTO. Let n be a cuspidal representation of GZ^(AQ) attached to a holomorphic cusp form of weight I. Then 7Too is a discrete series £>/ of lowest weight I. Recall that the discrete series £>/ of GZ^(R) is the subrepresentation of „ ,ti , ,_tu , , „ (l> if/is even tffl I 2 > e l I 2 )» where / is an integer, / > 2, and e = < I sgn, if / is odd. When / = 1, it is called the limit of discrete series. In particular, Di is parametrized by 4>i : WR = C x U jC x —» GL2(C), /jeu-i)

where *,(«») = ( e

Q

0

\

/0

^ . ^ j , 0 ; (;) = ("

l

/iy-i\

J

]•

Here 0/ = Indwc co, where w(z) = (zz)~~zf~l, and L(s,Di) = L{s, fa) = L(s, co) = rc(s + l-^-) = 2(27r)-(*+^>r(s + ^ ) . Then Sy7n4(0,) = diagie416^,

e216^,

l . e " 2 ^ ' - " ^ - 4 ^ ' - 1 ) ) , and

L(s,Sym4(D,)) = r R (s)r c (s + / - l ) r c ( s + 2(/ - 1)),

On Symmetric Powers of Cusp Forms on GLi

107

where TR(s) = ;r~sr(§). Hence Sym4(Di) = 1fflD2(i-i)+ifflD 4( /_i )+ i. Discrete series of S P4(R) are parametrized by the infinitesimal character x(k,l), k > I > 0, k,l e Z [Ad]. The lowest weight is (it + 1,/ + 2). It is the infinity component of a vector-valued Siegel cusp form of weight (k + 1,1 + 2). The Langlands parameter is k,i ie

k,i(re )

:

WR^S05(C),

= diag(t(2k6),t(2W),\),

0 (f>hl(j) = J 0= (-1

,m / c o s 0 sin
zfr1, 1, z~¥, z ~ ¥ ) ,

0 r 1 0 0 o, 0\ ^ _, , d> can e Wfl) ' k t

z e C.

If I = 0, it is the limit of discrete series. By comparing the parameters, we can see that SymA ({) parametrizes an L-packet of representations of Sp = n(\, sgn), by [DS], n gives rise to an icosahedral representation

~2 descends to a generic cuspidal representation T of S/?4(AQ), which is parametrized by Sym4((f>) ® a>~2 : W Q — • 505(C).

108

H. H. Kim

Let n be a cuspidal representation of GLa(A) with the central character o)n. Consider Sym3(7t),Sym4(n). Assume that they are cuspidal. Then by Theorem 4.1, Sym4(n) <8>u>~2 descends to a generic cuspidal representation r of Sp 4 (A). On the other hand, by the unpublished result of Jacquet, Piatetski-Shapiro and Shalika (Theorem 1.2), Sym3(n) ® a>~1 descends to a generic cuspidal representation n on GS pt(A) with the central character un. Let T' be any irreducible constituent of Tl\sP4(A)Proposition 4.2. T is nearly equivalent to T1, i.e., TV ^ T'vfor almost all v. Proof. By construction, Ls(s,cr x T) = Ls(s,o- X A4(n) <8> io~x) for any cuspidal representation o~ of GL^(A). Since A2(A3(;r)) = A4(;r)fflw^ [Kil], L s (5,o-xAV)) = Ls(s,(o-®a)n)XT') [Ki-Sh2, (4.2)]. Hence Ls (s, O-XT) = Ls(s,crx T'), and our result follows. • Remark 4.1. It is expected that T =* T'. It would follow from the strong multiplicity one result for generic cuspidal representations of S/74(A).

5 Remark on the images of functorial lift We are interested in the image of Syrn^in) and other functorial lifts to 2 GJU(A). For Sym (n), we have Proposition 5.1. (Flicker, Ramakrishnan) Let II be a cuspidal representation ofGL,2(A) which is essentially self-dual, i.e., n ®x = n ®xfor some X- Then II = Ad{n) ®x~X>for some cuspidal representation n ofGLjiA). There are six ways to generate a cuspidal representation of GL4(A) in the literature. 1. The symmetric cube Symr'in), where n is a cuspidal representation of GL2 [Ki-Shl]. In this case, A2(Sym?(n)) = Sym\n) ® ionffl0% [Kil]; 2. The tensor product 7rB7r\ where n, n' are cuspidal representations of GLi [R]. This corresponds to the tensor product map GZ^(C)xGL2(C) —* GL4(C); (g,g') 1—> g®g'. In this case, A2(7rH7r') = Ad(7r)®a>EAd(7r')®w, where a> = (onu)n>;

3. The Asai lift As(n), where ^ is a cuspidal representation of GL,2(AE), and E/F is a quadratic extension [R2]. This corresponds to the L-group homomorphism

109

On Symmetric Powers of Cusp Forms on GLq. r : LREIFGL1

= (GL^C) x GLi(C)) » Gal(E/F) —• GL(C2 ® C2) = GL4(C), r(g,g'; l)(x<8>y) = g(x)®g'(y);

r(l, l;0)(x®>>)

=y®x.

In this case, A.s(7r)£ = n E (n o 9), and A204s(7r)) = IF(Ad(n)) ® ( ^ U p 4. The automorphic induction I^n, where TT is a cuspidal representation of GL,2(AE), and F / F is a quadratic extension [AC]. This corresponds to the L-group homomorphism r : LREIFGLI

= (GZ^C) x GZ^C)) x Gal(E/F) —» GL(C2 e C2) = GL4(C), /•(£,*'; l ) ( x © y ) = s t o © s ' O 0 .

r(l,l;0)(x©;y) = y © * .

In this case, (I^TT)E = nm(no9), and /\2(IFn) = As(n)®OJE/F fflI^(ojn) [Ki4], where o»£//r is the quadratic character attached to F / F by class field theory. This includes lFKx, where \ is a grossencharacter of AT, and AT/F is a quartic extension (normal or non-normal). In [Ki4], we assumed that I^(con) is cuspidal. However it is not necessary: I^(,ojn) is not cuspidal if and only if CJ„ = CJ„ o 9. It is the case if and only if a>n = p. o NE/F for a grossencharacter // of F. Then Ig(u>n) = p a fMOE/F\ 5. The functorial lift from GSp4, corresponding to the L-group homomorphism : GSp 4 (C) —> GL4(C) [As-Sh], where 0 is the embedding. Namely, given a generic cuspidal representation r of GSp4(A), there exists an automorphic representation 0(T) of GL4(A) corresponding to 0. This includes the functorial lift from SO5 to GL4, corresponding to the embedding LS05 = Sp4(C) <^-> GL4(C), since S05 =* PGSp4. In this case, A 2 (0(T)) = n m coT, where II is an automorphic representation of GLs(A). By the unpublished result of Jacquet, Piatetski-Shapiro and Shalika (Theorem 1.2), we know the image of this lift completely, namely, n comes from GSp4 if and only if L(s,n, A2 ®x) n a s a P°l e at s = 1 for some^. If cr is a cuspidal representation of GL2(A), then Sy/n3(<x) is contained in this image, since L(s, Sym3(cr), A2 ® a>~3) has a pole at s = 1. 6. The stable base change from U(2,2)F/F0 [Ki-Kr], where F/FQ is a quadratic extension. This corresponds to the L-group homomorphism BC : GL4(C) x Gal(F/F0) —> (GL4(C) X GL4(C)) x Gal(F/F0), (g, 1) i-» (g, 0(g), 1);

(g, 0) H ^ (g, 9(g), 9),

where 0(g) = / 4 _ 1 'g _ 1 J 4 , 74 is the matrix that defines U(2,2)F/Fo. Note that GL4(C) x Gal(F/F0), (GL4(C) x GL 4 (Q) x Gal(F/F0) are L-groups

110

H. H. Kim

of U(2,2)F/FQ, RF/FoGLt, resp. and RF/FoGL4(AFo) = GL4(A). Let n be a generic cuspidal representation of U(2,2)(AFo). Then the stable base change lift BC(jr) is an automorphic representation of GL4(A) of the form ,where n\ + • • • + n* = 4 and for each i, cr, is a cuspidal representation of GL„,.(A), <x,- a; cr; o 0, cri # CTJ for / =£ 7, and L(s, cr^As <8> u>F/Fo) has a pole at 5 = 1. Here WF/F0 is the quadratic character attached to F/FQ by class field theory. Note that if x is a character of At, then L(s,x,As) = L(S,X\A' ). Hence "L(s,y,As) has a pole at 5 = 1" means simply that^Uj. = 1. If cr is a cuspidal representation of GL2(A), L(s, cr, As ® u>F/Fo) has a pole at s = 1. In this case, it is most likely that A2(TT) is cuspidal. Let n be a cuspidal representation of GL4(A). We want to know when it is in the image of the above lifts. Let's summarize our discussions above as follows: Suppose L(s,n, A2 ®x) has a P°l e at 5 = 1 for a grossencharacter x of F. Then n comes from GSp*. Now n may be either Sym3(cr) for a cuspidal representation cr of GLa(A), or /|cr for a cuspidal representation cr of GL2(A£), or neither. (If con = fio NE/F for a grossencharacter ^ of F, then L(s, /£;r, A2 ® /T 1 ) has a pole at s = 1, and /^7r descends to a generic cuspidal representation of GS/74(A).) We need to consider the base change 7Tf. If TTE is not cuspidal for a quadratic extension E/F, then 7r = /^cr for some cuspidal representation cr of GL2(A£). We do not know when n is of the form Sym3(cr). Suppose A27r is not cuspidal of type (3,3), then n is of the form cr E cr', where <x, cr' are cuspidal representations of GL2(A). Suppose A27r is cuspidal. If there exists a quadratic extension E/F such that (A2;r)£ is not cuspidal, then n is of the form As{cf) for some cuspidal representation cr of GLI(AE). If there exists a quadratic extension F/Fo such that L{s,n,As® u>F/Fo) has a pole at s = 1, n is the stable base change lift from U(2,2)F/Fo. Otherwise, n is not in the image of any lifts.

On Symmetric Powers of Cusp Forms on GLq.

Ill

References [AC]

J. Arthur and L. Clozel, Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Annals of Math. Studies 120, Princeton, 1989.

[Ad]

J. Adams, THETA - 1 0 in Contributions to Automorphic Forms, Geometry and Number Theory, 39-56, Johns Hopkins Press, 2004.

[As-Sh] M. Asgari and F. Shahidi, Generic Transfer from GSp(4) to GL(4), Compos. Math. 142 (2006), 541-550. [DS]

P. Deligne and J.P. Serre, Formes modulaires de poids 1, Ann. scient. Ec. Norm. Sup. 7 (1974), 507-530.

[FH]

W. Fulton and J. Harris, Representation Theory, A First Course, Springer Verlag, 2000.

[lb]

T. Ibukiyama, Numerical example of a Siegel cusp form having the cubic zeta function, preprint (2006).

[Ge]

S. Gelbart, Three lectures on the modularity of PE,3 and the Langlands reciprocity conjecture, in Modular forms and Fermat's last theorem, 155-207, Springer, 1997.

[Ge-J]

S. Gelbart and H. Jacquet, A relation between automorphic representations ofGL(2) andGL(3), Ann. Sci. Ecole Norm. Sup. 11 (1978), 471-552.

[G-W]

R. Goodman and N. Wallach, Representations and invariants of the classical groups, Cambridge University Press, 1998.

[GRS]

D. Ginzburg, S. Rallis and D. Soudry, On explicit lifts of cusp forms from GLm to classical groups, Ann. of Math. 150 (1999), 807-866.

[Ka]

G. Karpilovsky, The Schur Multiplier, Oxford University Press, 1987.

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H. H. Kim

[Kil]

H. Kim, Functoriality for the exterior square of GL4 and symmetric fourth ofGLa, J. Amer. Math. Soc. 16 (2003), 139-183.

[Ki2]

H. Kim, Langlands-Shahidi method and poles of automorphic L-functions: application to exterior square L-functions, Can. J. Math. 51 (1999), 835-849.

[Ki3]

H. Kim, An example of non-normal quintic automorphic induction and modularity of symmetric powers of cusp forms oficosahedral type, Inv. Math. 156 (2004), 495-502.

[Ki4]

H. Kim, An application of exterior square functoriality ofGL\; Asai lift, in Number theory, CRM Proc. Lecture Notes, 36 (2004), 197-202.

[Ki-Kr]

H. Kim and M. Krishnamurthy, Stable base change lift from unitary groups to GLN, IMRP 1 (2005), 1-52.

[Ki-Shl] H. Kim and F. Shahidi, Functorial products for GLi x GLj and the symmetric cube for GL2, Ann. of Math. 155 (2002), 837-893. [Ki-Sh2] H. Kim and F. Shahidi, Cuspidality of symmetric powers with applications, Duke Math. J. 112 (2002), 177-197. [La]

R.R Langlands, Base Change for GUI), Annals of Math. Studies 96, Princeton, 1980.

[R]

D. Ramakrishnan, Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2), Ann. of Math. 152 (2000), 45111.

[R2]

D. Ramakrishnan, Modularity of solvable Artin representations ofGO{A)-type, IMRN, No. 1 (2002), 1-54.

[R-W]

D. Ramakrishnan and S. Wang, A cuspidality criterion for the functorial product on GL{2) X GL(3) with a cohomological application, IMRN (2004), 1355-1394.

[Tu]

J. Tunnell, Artin's conjecture for representations of octahedral type, Bulletin AMS 5 (1981), 173-175.

On Symmetric Powers of Cusp Forms on GLa [W]

113

S. Wang, On the symmetric powers of cusp forms on GL{2) of icosahedral type, IMRN (2003), 2373-2390.

Henry H. Kim Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada Email: [email protected] and Korea Institute for Advanced Study, Seoul, Korea

Zeta Functions of Root Systems Y. KOMORI, K. MATSUMOTO & H. TSUMURA

In this paper, we introduce multi-variable zeta-functions of roots, and prove the analytic continuation of them. For the root systems associated with Lie algebras, these functions are also called Witten zeta-functions associated with Lie algebras which can be regarded as several variable generalizations of Witten zeta-functions defined by Zagier. In the case of type Ar, we have already studied some analytic properties in our previous paper. In the present paper, we prove certain functional relations among these functions of types Ar (r = 1,2,3) which include what is called Witten's volume formulas. Moreover we mention some structural background of the theory of functional relations in terms of Weyl groups.

0

Introduction

Let N be the set of natural numbers, No = N U {0}, Z the ring of rational integers, Q the field of rational numbers, R the field of real numbers, and C the field of complex numbers. Let g be any semisimple Lie algebra. In [25], Zagier defined the Witten zeta-function associated with g by (0.1)

{w(s;Q) = Yj(dim
(j€C),

where ip runs over all finite dimensional irreducible representations of g. Using Witten's result [24], Zagier noted that (0.2)

{w(2k;Q)eQ7r2kl

for k e N, where / is the number of positive roots of g, which is called Witten's volume formula (see [25, Section 7]). In particular, £V(.y;sl(3)) 115

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Y. Komori, K. Matsumoto & H. Tsumura

was already studied by Tornheim [19] and Mordell [15] in the 1950's. The above result for £w(s; sl(3)) was showed by Mordell and its explicit formula was given by Subbarao and Sitaramachandrarao in [18]. Recently Gunnells and Sczech [3] evaluated £V(2&; g) for k e N by means of the generalized higher-dimensional Dedekind sums, and gave certain evaluation formulas for £w(2k; sl(3)) and £w(2k; sl(4)) for k e N in terms of £(2;) for j g N, where £(s) is the Riemann zeta-function. In [10], the second-named author defined multi-variable Witten zetafunctions by oo

(0.3)

&i(3)(s\, s2, s3) = J ] m~s'n~S2{m + n)~s\ oo

(0.4)

^o(5)(5i,s2, 5 3 ,5 4 )= J ] m~Sln-S2(m + nyS3(m + 2nyH. m,n=l

Note that £w(s; sl(3)) = 2s^0)(s, s, s) and £w(s; so(5)) = 6s^o(5)(s, s, s, s). The second-named author proved the analytic continuation of them by the method using the Mellin-Barnes integral formula (see [7, 8, 9]) and calculated their possible singularities. The function £si(3)(si,52>53) 1S ^so called the Tornheim double sum or the Mordell-Tornheim double zetafunction denoted by T(s\, s2,53) or £MT,2(S\,S2, 53), because special values ^si(3)(«i. «2,0.3) for aj € N (j = 1,2,3) were studied by Tornheim in [19]. Recently the third-named author [23] has proved certain functional relations between £si(3)(j"i, s2,53) and £(s). These can be regarded as certain continuous relations including the known formulas between Tornheim's double sums and values of £{s) at positive integers obtained in [15, 19]. For example, we proved that (0.5)

2&0)(2, s, 2) +
for any 5 € C except for singularities of each function on the both sides. In particular, putting s = 2 in (0.5), we obtain Witten's volume formula &P)(2,2,2) = TT 6 /2835 given by Mordell (see [15]). More recently, in [12], the second-named and the third-named authors defined multi-variable Witten zeta-functions £si(r+i)(s) (r e N) associated with sl(r +1) and proved the analytic continuation of them. Furthermore, for r < 3, true singularities of them were determined, certain functional

117

Zeta Functions of Root Systems

relations for them were given and new evaluation formulas for their special values were introduced. However, in that paper, we were unable to prove functional relations which can be regarded as continuous relations including Witten's volume formulas (0.2). It seems possible to generalize these results for sl(r + 1) in [12] to those for any general semisimple Lie algebra g. Indeed, in [4], we define the Witten zeta-function associated with g denoted by £r(s; g) or £r(s; Xr) if g is of type Xr (X = A,B,C,D,E,F,G), for s e C , where n is the number of positive roots of g. For example, fafaM) and £2(s;2?2) c o m _ cide with (S\Q)(S\, S2, S3) defined by (0.3) and £50(5)(si, s2, *3, ^4) defined by (0.4), respectively, except for the order of variables. More generally we further define zeta-functions of roots. Considering their properties, we study the analytic continuation of £r(s; g), and certain recursive structures among them. In [5], we prove certain functional relations among them, in particular, of types Br and Cr (r < 3), which include Witten's volume formulas. Furthermore, in [6], we will study &(s; G2) associated with the exceptional Lie algebra of type G2. Indeed, we will construct a certain functional relation for fr(s', G2) which includes Witten's volume formula. In the present paper, we give an overview of our previous result [12] and our recent results [4,5], and prove some related facts. In Section 1, we introduce the definition of zeta-functions of roots which include the multivariable Witten zeta-functions. We further study certain recursive relations for them. In Section 2, we study some structural background of functional relations for these functions. As its concrete examples, in Section 3, we explicitly give certain functional relations for £r(s;Ar) (r < 3) which can be regarded as continuous relations including Witten's volume formula. For example, we can give a functional relation including (0.6)

£,(2,2,2,2,2,2;A3) ,

1 , V-m n\l + m) (m + n)2(l + m + n)2

l,m,n=l

2

2

v

'

v

23 12 2554051500

'

(see [3]) as a special value-relation. As mentioned above, we were unable to treat Witten's volume formulas in our previous paper ([12]), because we were only able to consider some limited types of relations for &(si, *2, S3,54,0,55; A3) and &(s\, s2, S3,0, s4,55; A3), that is, the case onevariable is equal to 0. In order to remove this limitation, we will study cer-

118

Y. Komori, K. Matsumoto & H. Tsumura

tain multiple polylogarithms of type A3 (see (3.18) and (3.19)). By making use of them, we will be able to construct some functional relations including Witten's volume formula. This method can be applied to zeta-functions of root systems of other types (see [5, 6]).

1

Zeta functions of root systems

In this section, we give an overview of [12] and [4]. First, following [4], we introduce the definition of zeta-functions of root systems associated with semisimple Lie algebras, and more generally, zeta-functions of roots. We quote some notation and results about semisimple Lie algebras from [2,17]. Let g be a complex semisimple Lie algebra of rank r. We denote by A = A(g) the set of all roots of g, by A+ = A+(g) (resp. A_ = A_(g)) the set of all positive roots (resp. negative roots) of g, and by *F = *F(g) = {ori,..., ar] the fundamental system of A. For any a € A, we denote by av the associated coroot. Let X\,..., Ar be the fundamental weights satisfying (aY,A.j) = Aj(af) - <5(y (Kronecker's delta). Then we see that any dominant weight can be written as (1.1)

A = n\A.i H

+ nrAr

(ji\,... ,nr e No).

In particular, the lowest strongly dominant form is p = A\ H + Ar. Let d\ be the dimension of the representation space corresponding to the dominant weight X. Then Weyl's dimension formula (see [17, Section 3.8]) gives that

Putting ntj = rij + 1, it follows from (0.1) that

(1.2,

W.«>-2;n( fe»„,,--*> ) 00

s

= K(a) 2 m\=\

00

''' Z

Y\(av,mlA1

mr=\ aeA+

+ --- + mrArys,

Zeta Functions of Root Systems

119

where the sum on the second member of the above runs over all dominant weights of the form (1.1), and

(1.3)

K(&)=

Y\(a\Xl

+

--- + Ar).

aeA +

Let A* be a subset of A+ = A+(g) which satisfies the condition that for any A.j (1 < j < r), there exists an element or € A* such that (orv, A,) £ 0. Then we define the zeta-function of A* by OO

(1.4)

00


mr-\ aeA*

where s = s(A*) = (sa)ae^ e C"* with n* = |A*|. In particular when A* = A+(g), we denote £r(s; A+(g)) by £.(s; g) or £r(s; Xr) if g is of type Xr, and call it the multi-variable Witten zeta-function associated with g or Xr. Namely we have OO

00

(1.5) £(s; g) = &(s;Xr) = £

''' Z

m\=\

I I (a^m^

+ ''' +

m

r^rSa,

mr=\ aeA +

where s = (sa)ae&+ e C" and n = |A+| is the number of positive roots of g. Note that K(Q)st;r(s,..., s; g) coincides with the one-variable Witten zeta-function £W(S;Q) defined by (0.1). The series (1.4) and (1.5) can be continued meromorphically to the whole space (see [10, Theorem 3]). In the case Xr = Ar, we see that OO

(i.6)

00

u*\Ar) = 2 • • • Z m\=\

II

(m +

' ''' + m J-ir s ">

mr=\ l
where s = s(Ar) = (*y) € C (see [4, (2.3)] or [12, (1.5)]). In Section 3, we prove some functional relations among £r(s;Ar) (r < 3) which can be regarded as continuous relations including Witten's volume formulas for sl(4). Moreover, in [5], we give some functional relations of types Br and Cr for r < 3. Now we consider g = sl(r +1). Let Sj be the y'-th coordinate function. It is well-known that positive roots are (1.7)

AA++ == AA++(A = {<£iEJ = ^J ] <*k 1 < i < j < r + 1 (Arr)) = Ei-Ej= [

i
120

Y. Komori, K. Matsumoto & H. Tsumura

(see the list at the end of [2]). We further define (1.8)

A*(Ar) = {£i -Ej\2
1};

(1.9)

HHAr) = {ei -Sj\2<j
A*(Ar)

(2 < h < r + 1) which are subsets of A+(Ar). For example, putting r = 3, we have the relation A*(A3) c A£(A3) c AJ(A3) C A;(A3) = A+(A3). We can see that s(A*(A3)) = (512,..., sih, s23, s24, S34) (h = 2,3,4). We recall the Mellin-Barnes integral formula

= - j '- f(c)

(1.10)

ns+

(1^)- = - ^ = ,

rf™x
r(5) 2n ^f-[ J(c) where s, A are complex numbers with %s > 0, | arg A\ < n, A £ 0, c is real with -%s < c < 0, and the path (c) of integration is the vertical line 2lz = c. Using this formula, we have (1.11) £3(512,..., s\h, sn, 524,534; A£(A3)) OO

OO

OO

=EZE

II (m + .-. + mj.!)-*

m\ = \ mi=\ m3=l 2
n

. , ,-su 1 f R^IA + Zft)r(-zft) (mi + • • • + m/_i) ' " —— z— 2^V TJta) T(5 U ) 2
x (mi + • • • + mh-2YSih~Zhmz*_xdzh =

7=

=7—r

6 ( s (^3,^);A^_ 1 (A 3 ))* A

for h = 3,4, where S*(A 3 ,Z 3 ) = (512 + *13 + Z3, S23 ~ Z3, «24, «34); S*(A3, Z4) = (S\2, S13 + 514 + Z4, S23, S24, S34 ~ &0-

Hence we find the recursive structure which we denote by (1.12)

£,(• ;A3) = £,(• ;AJ(A3)) -» &(• ; A^(A3)) -> 6 0 ;A$(A3)).

Note that (1.13) £?(s*(A3,z3); AJ(A3)) = 6(523 - Z3, *24, J34; A*(A3))f(*i2 + «13 + Z3)-

Zete Functions of Root Systems

121

Substituting (1.13) into (1.11), we have (1.14) 6(s;A 3 (A 3 )) =

—

——

X 6O23 - Z3, *24, «34J A*(A3))6>12 + $13 + Z3)*3-

Therefore we can rewrite (1.12) as (1.15)

6 0 \M) = £»(• ; A ; ( A 3 ) ) - » 6 0 ; A ; ( A 3 ) ) - » 6 0 ; A*(A 3 )),

by neglecting the Riemann zeta factor. Renaming e,- as e,_i (2 < / < r + 1), we find that 6 0 ; A*(A3)) is equal to 6 0 ; A2). Consequently we can write (1.15) as (1.16)

6 0 ;A3) -» 6 0 ; A ; ( A 3 ) ) - » 6 0 ;A 2 ).

The above argument can be applied to Ar and Ar_i for any r > 2. Hence we can obtain the following result which can be viewed as a refinement of [12, Theorem 2.2]. Theorem 1.1 ([4, Theorem 1]). Between 6 0 ;A r ) and 6-10 ;Ar_i) (far each r > 2) there is the recursive relation like (1.16), which can be summarizingly written as (1.17)

6 0 \Ar) -» 6 - i 0 \Ar-i) - » • • • - » 6 0 ;A2) - » 6 0 ; Ai) = 6

For zeta-fiinctions of other types, we can also give certain recursive relations similar to (1.17). For the details, see [4, Sections 4 and 5].

2 Structural background of functional relations Hereafter we discuss the topic on functional relations for zeta-functions of root systems. First, in this section, we study the structural background in a general framework. For the details, see [5]. As we mentioned in Section 0, we have already obtained some functional relations for 6(*i. ^2, s 3 ;M) like (0.5) (For the details, see [23], also

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Y. Komori, K. Matsumoto & H. Tsumura

[13, 14]). Recently, in [16], Nakamura gave an alternative proof of functional relations given in [23]. Inspired by Nakamura's method, we aim to generalize these functional relations of type Aj to those for zeta-functions of general root systems. We use the same notation as in Section 1. Let g be a complex simple Lie algebra of type Xr and V be the vector space generated by the roots of g. Let W be the Weyl group acting on V. We denote by P and P+ the lattice of weights and the set of dominant weights, respectively. Let (2.1)

C : = { v e V| > Ofor allar e A}

be a fundamental domain of W, which we call the fundamental Weyl chamber. Let

H* = V\{JwC, weW

where by C we denote the interior of C, and (2.2)

S ((sa)aeA+;Xr) := £

=

PJteVr"

AeP\HA aeA+

zz n^'^-^""

weW XeP+ tt€A+

weW AeP+ waeA+

where sa for a e A_ is defined by sa = s-a and p is the lowest strongly dominant form. Let Aw = A+ n w _1 A_. Divide the product on the righthand side of (2.2) into two parts according to a e A+ or a e A_, and in the latter part replace a by -a. Then we have

(2.3)

s«say,xr)= E ( n ( - i r * " ) £ Y\{a^A+prSwa weW aeA w

XeP+ <*eA+

( ir , fr(( w):Xr)

=z(ri " * ") *' weW

'

aeAw

which is a 'Weyl group symmetric' linear combination of our zeta-function.

Zeta Functions of Root Systems

123

Now we consider the Lerch-type series T(s + 1) Y-» e2nimx s (2m) *-i ms v 7 meZ

m?tO

for 5 € C with 2^5 > 1, x e R and / = V-T. Then we obtain the following general form of functional relations for zeta-functions of root systems (for the proof, see [5]). Theorem 2.1. (2.5)

s ((say, *»•) = 2 (II (-ir'""^ ((**); *.-) ~

k U r<*«+1) Jo

ia( a)

a

|A + \>F| rimes

dxa i=l

aeA+VF

aeA+VF

holds for any sa in a certain region. Remark 2.1. When sa e N, we see that LSa(xa) = BSa([xa}), where [x] = x- [x] denotes the fractional part of x and B„( •) is the Bernoulli polynomial (see [1]). When Xr = Az, the idea of Theorem 2.1 is essentially included in Nakamura's proof of certain functional relations for £2(^1»*2» ^3^2) ([16, Theorem 1.2]) which coincide with those given by the third-named author in [23]. For example, see (0.5). Example 2.1. In the case of type 2?2(= C2X £2(^1, $2,53,54; B2) coincides with £SO(5)(s2,53.^4, «i) (see (0.4) and (1.5)). As well as in [16], direct calculations on the right-hand side of (2.5) for g = so(5) show that, for

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Y. Komori, K. Matsumoto & H. Tsumura

example, (2.6) 6d,2,

s, 2; S 2 ) + 6 d , 2,2, s; S 2 ) + 6(2,1,2, s; B2) - 6 ( 2 , 1 , *, 2; S 2 ) = 3f(2)f(s + 3 ) - y f ( s + 5);

(2.7) 3 39 6(2,2, 5 ,2; B2) + 6(2,2,2,5; B2) = r^(2)^(s + 4) - —^s + 6). z

16

Putting s = 2 in (2.6) and (2.7), we have (2.8)

3 39 6 d , 2,2,2; B2) = -f(2)f(5) - —£(7); z to

(2.9)

6(2,2,2,2; B2) = ^(2)^(6) - ~ f ( 8 ) = * 32"' 302400 4 , v „v Note that (2.8) has been obtained in [22] by a different method, and (2.9) is just equal to Witten's volume formula (0.2) for g = so(5). However it seems to be hard to calculate (2.5) for £r(-;Xr) directly if r is large. Hence we need another technique. In [5], we introduced a certain technique using multiple polylogarithms (which may be called the 'polylogathm technique') to prove general formulas for zeta-functions of types B 2 , S3 and C3 including (2.8) and (2.9). In Section 3, using this technique, we treat (r{-;Ar) and prove certain functional relations for them. At the end of this section, we consider a generating function of values of S{{sa);Xr) at positive integers. Let k = (ka)ae&+ e N ^ ' and t = (ta)aeA+ e C|A+I with \ta\ < In for each a. Then we define (2.10)

F(t,Xr) := (-1)IA+I £ s ( k ; X r ) f [ (^" k

,

aeA+

where

aeA +

"

r» ^ " » , aeA+\>P |A + \*| times

r

(UB^({- z ^i)) n d x a i=l

aeA + VF

aeA+\>F

125

Zeta Functions of Root Systems

(see (2.5) and Remark 2.1). For example, we can explicitly calculate that ( 4

(2.11)

F(tut2,t3,t4;B2)

=

\

\-r-7

(P(t) + Q(t)-R(t)),

x l e'j - 1

V;=i

J

where fi = / 01 , t2 = ta2, f3 = tai+a2, U = *a,+2a2 and 2 ( g f l - 1)(1 + e('2+'3)/2)(g(f2+<3)/2 _

P(t) = =

g f4)

(2*1 + *2 - ftXfe + *3 - 2.4) (g /2

_

1)(en+»

_

gf3)

=

(2?j + r2 - *3)(*1 ~ *3 + *4>

(e &

_

1)(en+*

_ g*)

(2*1 + *2 " *3)(*1 + *2 - U)

Hence we can calculate the Taylor expansion as (2.12)

F(tx,t2,tzM,B2)

2 = 1 + -^^(Aht^ 2880

+ 2t\t2i\ - Ahfy?, + t\i^u ~ 2*i*3*2. - *i*f*4

+ 4/?f2*3 - 2*?*2*4 -

tfhU

~ 4f2*3*4 - 4/2*3*4 - 4^*3*4)

+ 241Q20( 3?1 * 2 *3*4 ~~ 3*1*2*3*4 - 3^2*5*4 + 8
9676800^^

+

-

3*1*5*3*4

'" •

Comparing the coefficients of i\i!jtffi in (2.10) and in (2.12), we can also obtain (2.9), namely a2,2,2,2;Z,2) = (-l)4(27"')8 ' * 2!-2 2 9676800 302400'

However it is also hard to evaluate the zeta-function of type Xr by this method when r is large.

3 Functional relations for £3(s; A3) In this section, we consider zeta-functions of type A3. Though we have already considered certain special cases of this type in our previous paper

126

Y Komori, K. Matsumoto & H. Tsumura

([12]), we will study this type more generally and in detail, by considering a certain multiple polylogarithm of this type. From (1.5), we have h(Sn,5i3,

S23; A2)=

°° V

—J—

1

m\,m2=\

1

v

jrr\ l

L

'

2

6 ( ^ 1 2 . *13. ^14, 523, 5 2 4, *34J A 3 ) 00

=m y^ 3 = i

1

1

OT U m

1)

i ( i + ™2)sHm\ +m2 + m3y»mS2\m2

+

m3)^mf'

In our previous paper [12], we proved a closed form of functional relation for

(-Ifbisu

si,b, s2,0,a;A3) + (-l)bfr(su

s3,a,

s2,0,b;A3)

+ &(a, su 53,0, b, 5 2 ; A3) + £3(51, a, S3,0, s2,fc;A3) in terms of fr(si,s2, sy,Aj) and ^1(5^2) = £(s), where a,b e No and 51,52, S3 e C. For example, we have -£2(si,S3,2,5 2 ,0,1; A 3 ) + £2(s\, s3, l , s 2 , 0 , 2 ; A3) + £ 2 (1,5i, 53,0,2,5 2 ; A 3 ) + £ 2 (5i, 1,53,0,5 2 ,2; A 3 ) = - 3 & ( 5 i , 53 + 3, s2;A2) - £2(5i + 1,53,5 2 + 2;A 2 ) + 2^(2)^(51,53 + 1,5 2 ;A 2 ) (see [12, Theorems 5.9 and 5.10]). However our previous result does not include Witten's volume formula (for example, (0.6)). Therefore we aim to construct some functional relations for £r(s;Ar) (r < 3) which include Witten's volume formulas as value-relations. Let

Q-»

= (2l" - O w

«*) = Z ^ m=\

We recall that

^(-i)'cos(^)

A

l=\

v=0

f, (-lysine) ^ (=1

v=0

m

„ A-iytf* v

'

(-ir^ 1

127

Zeta Functions of Root Systems

for p e N, q e No and 6 e (-n,n) c R (see, for example, [20, Lemma 2]). Note that 0(0) = £(0) = -\. We see that the both sides of (3.2) and of (3.3) are continuous for 6 e [-n, n] in the case p,q e N. Fix p e N and s,t,u,x,y e C with %s > l,%t> 1, %u > 1, |JC| < 1 and \y\ < 1. Put (3.4) F0(d;2p,s,t,u;x,y)

=2

A (-iycos(W) v;=i

x

Z

f,

(-l)^ v

v=o

(2v)!

(-iy"+"xwy"eli(m+n)9 msn'(m + n)u

m,n=l

By (3.2), we see that (3.5)

Fo(0; 2p, s, f, u; x, y) = 0 (-TT < 0 < TT).

By the assumptions on p, s, t, u, the right-hand side of (3.4) is absolutely convergent. Hence, by cos 0 = (ew + e~l6)/2, we have (3.6) /_i\/+m-t-n„m. v n (ei(l+m+n)8

F o (0; 2p, s, t, u; x,y) = 2_j

pPmsn'(m + ri)u

/,m,n=l

2

Z^-»^Z (2v)! ^

v=U

• _i(-/+m+n)0\

/ _x\m+n jjn^ji J(m+n)8

m*n'(m + n)«

m,n=l

Note that in our previous paper ([12]), we only considered the case (x,y) = (1,1). Hence we were only able to treat some limited case of &(•; A3). In this paper, we consider Fo(0; 2/7, s, t, u; x, v) which is a certain triple polylogarithm. Considering the integration of Fo with respect to the variable 0 in the numerator of each summand, we can produce a certain factor in the denominator of each summand, which corresponds to some relevant root. This is the 'polylogarithm technique' mentioned in the preceding section. We will execute this procedure as follows.

128

Y. Komori, K. Matsumoto & H. Tsumura Corresponding to FQ(6; 2p, s, t, u; x, y), we define

(3.7) GA6; 2p, s, t, u; x, y) - —

>

—

r

l,m,n=l

z

( _i\l+m+n

Jn^n

ei(-l+m+n)8

Z J ppmsn'(m + n)u(-l + m + n)d

v=0

u=0

V

M

'

^

^ , msn'(m + n)u+d+2y-» 1 m,n=l

for 9 e [-7T, 7r] and d € No, where

o-fr*

[*(*-iM*-"+i>

(aeN);

(a = 0).

In the case
G0(9; 2p, s, t, u\ x, y) = Fo(0; 2p, s, t, u; x, y) + Co = C0,

where (3.9)

Co = C0(2p, s, t, u; x, y) = - > _,_ ,u+2ps „ ^—' msn'(m + n)u+tp m,n=l

Note that the definition of GdiQ; 2p, s, t, w;x,y) seems to have been given abruptly. However, by using the same consideration as in our previous papers [12, 23], we can presume the expression of Gd(0; 2p, s, t, u; x, y) as (3.7). Since we fix p, s, t, u, x, y, we regard Gd(0; 2p, s, t, u; x, y) as a continuous function for 0 e [-n, n]. We can show (3.10)

-^Gd(.9;2p, s, t, u;x,y) = Gd-i(0; 2p, s, t, u;x,y) dd

(d e N).

129

Zeta Functions of Root Systems

Hence, by integrating the both sides of (3.8) and multiplying by i on the both sides, we have iGi (9; 2p, s, t, u; x, y) = C0(i9) + C\ for some C\ = C\{2p, s, t, u;x,y). By repeating the integration, and by (3.10), we obtain Lemma 3.1. With the above notation, for d € No, we have

.d

v^

(iey

2_lcd-j—^-' J j=o ' where Ck = Ck(2p, s, t, u; x, y) (0 < k < d) are certain constants. (3.11)

i Gd(6;2p,s,t,u;x,y)

=

We can determine {Q} as follows. Putting 9 = ±n in (3.11) with d+l, we have j-flf+l

(3.12)

—— {Gd+i (n; 2p, s, t, w; x, y) - Gd+i (-/r; 2p, s, t, u; x, y)} 2(OT)

ld/2]

= L=0

{in?> "n^+'(2/i + iy.'

Cd 2

-

M

From (3.7), we see that the left-hand side of (3.12) is equal to

K

)

ij vv v

H\

v=0

cr=0

V

2v-2o--l

(2o-+iy. '

xY , ... .J(m +x^y" Sn)

u+d+2v 2
- '

*—*, msn'(m +' n) x

m,n=l m,n=l

x

'

Applying [11, Lemma 2.1] to (3.13) with a = 2p and

d l +x \V e(X) = 2l ~ 8K \ x - 1 / 2-i N

^ msn'(m

+ n)"+d+x'

' m,n=l

we have

(3 14)

pp+d-i\

y

\ 2p-l

/ Zu

*-/ msnt(m

+ ny+2P+d

[

y

AJ

c

a*p d2

^(2fi+iy.

130

Y. Komori, K. Matsumoto & H. Tsumura

for d € N 0 . Applying [21, Lemma 3.3] to (3.14), we have +V 2 (3.15) ft - 2^ j > - v»-i 2\ "2p t ms-TT 2 -7 1 /')•^-' n'{m + n} ! x

v=0

^

v

' m,n=l

'

for * 6 No, where Aj = (1 + (-iy'}/2 (; e Z). Substitute (3.15) into (3.11). Then we have (3.16) /dGrf(0;2/j,s,f,K;.K,)O

= 2XUi-;-v^.J "1 ^ ^

V —— •"

V 2p-l

)i£Jlm*n!(m

+2 +y

+ ny P

., j \

pry-'-1) f — ^ , .»gi.

.iftA-v^f

y=0 v

T]=Q

r

v

' m,n=l

'

•*

by putting TJ = v + j . Combining (3.7) and (3.16) with d = 2q for q e N, we obtain J22.

f\\l+m+n

ym^ji J(l+m+n)8

, 2J, pPm'n'im + n)u(l + m + n)^ l,m,n=l

, A pPm'n'im + n)u(-l + m + n)2** H> l±m+n

y I2q - I + 2v - ny-i6y //=0

x

v=0

X

r-

/i=0

i

v

r

(-l) w+n JC m /g'' (m+n)e

y m,n=l

M

'

7 T A m>n>(m + nyW-* = °(* * ^ ' ^ r

m,n=\

Now we replace y by -ye~'e in (3.17). This is an important procedure to produce a factor in the denominator of each summand, which corresponds

131

Zeta Functions of Root Systems to the root. In fact, we consider (3.18)

F0(0;2p,s,t,u,2q;x,y)

•=, ^y,

l,m,n=\

» >

+

I

i \l+m jJtiyn

P-Pmsn'(m + n)u(l + m + n)2i ( _ 1) / + « Jt «y, e i(-/ + »)fl T;—-—

l,m,n=l

x

J(l+m)6

\

P T- - 2 >
/ \

/

Y ^ / 2 g - l + 2v-)u\(-i6>yj

2

£l y

-^ J^~

t-u*y«

—2Y^-2v)

m,n=l

v=0

* H>\ = 0

y=Q

2v-n

J fi\

^ , msn\m + n)u+2P+2y-P-

(6€[-7T,n]).

We can apply the same argument about FQ as mentioned above to Fo. In fact, for p, q, s, t, u, x, y as fixed above and for d e No, we put (3.19)

Gd(6;2p,s,t,u,2q;x,y) 1 r

A

^_1-)l+mxmynei(l+m)e

'~ idY,2-J, P-Pmsn'(l + m)d{m + n)u(l + m + n)2i l,m,n=l

v

' »

i^£\ ^pmsn'(-l

' v ' ^_ly+mxmyneK-l+m)e

+ m)d(rn + n)"(-/ + m + n)2
Itm+n

v=0

/j=Q

2

v

^

s

d+

I

p=Q

p! ^-< (-iey A m + P--Pn (-l) (m x y+en) m,n=\

\

H-

u+2 +2v mt m n ime

v

'

4

-P-

H

I

132

Y. Komori, K. Matsumoto & H. Tsumura

v=0

n=0

(ifff

V

H

n m n

y

ine

(-\) x y e-

m,n=l

v

' P=0 V

M

y

I

1

'

Let Co = C0(2p, s, t, u, 2q;x,y) = - V , + 2 p n2a, ^ .„• s+z t+ ^—' m Pn ^(m + n)u m,n=\

By (d/d6)Gd(6; 2p, s, t, u, 2q; x, y) = Gd-\(6; 2p, s, t, u, 2q\ x, y) (d e similarly to Lemma 3.1, we can obtain the following lemma. Lemma 3.2. With the above notation, for d € No, we have

(3.20)

i Gd(0;2p,s,t,u,2q;x,y)

= > Q_/——, 7=0

J'

where Q = Q(2p, s, t, u, 2q; x, y) (0 < k < d) are certain constants. In the third and the fourth terms on the right-hand side of (3.19), we change the running indices p and p (0 < p < 2v, 0 < p < p) to p and <x (0 < p < 2p, 0 < o- < 2v - p), by putting cr = 2v - p. Then we have (3.21)

Gd(G; 2p, s, t, u, 2q; x, y) , t-i M.2

ppmsn'(l + m)d(m + n)u(l + m + n)2
l,m,n=\

2

i_ i

\l+mjjn-vnJ(-l+m)8

+ ;^=i ^pmsn'(-l + m)d(m + «)"(-/ + m + n)2^ tem+n

Zeta Functions of Root Systems

133

2z^-->zzrrn:r2v Zv-p

v=0

p=0
x

'

z

pr !

Z-J ms+d+2v-o-pnt(m m,n=l

9

2v 2v-p ,

-2(-l/Z^-2v)ZZ v=0 p=0 o-=0 X

P

x

2v-cr-p

v

'j y

J p\ 2 J

r ^ — -

?—

@e

For simplicity, we denote by £2(si,s2,sy,x,y;A2) rithm of type A2 defined by (3.22)

v

+ n\u+2q+cr

v

/rf-l+2v-CT-p\(^

\

-o--p p

fo(si,s2,sy,x,y;A2)=

[-/T,

;r]).

the double polylogax"1/1

^ mtn—\

By (3.20) and (3.21) with d = 2r + 1 for r e N 0 , we have (3.23) ,•2*4-1

2(in)

\G2r+i(K 2p, s, t, u, 2q; x,y) - G2r+\(.-n; 2p, s, t, u, 2q; x, v)} p

v-l

2v-2fi-l

+

)i=Q


x

I

2

2

-z^--zzr-: f- _; -;_- 1 v=0

(OT) 2 ^

X

(2

+ 1);

q

'

h(s + 2r + 2v-o--2p,u

X

r~

1

+ 2q + o-,t;x,y;A2)

v - l 2v-2//-l

+" v=0

p.=0

X

(2JU+1)!

cr=0

V

'

X

r~

1

£2(5, w + 2p + o-, t + 2r + 2v - cr - 2p; x, y;A2)

(2'+1)!

134

Y. Komori, K. Matsumoto & H. Tsumura

Applying [11, Lemma 2.1] to the second member of (3.23), we have (3.24) 2p-l

(2q-l+ cr\(2p + 2r-cr-\ cr )\ 2p-o--l

-z(

x £2{s + 2p + 2r - o-, u + 2q + cr, t; x, v; A2) 2q-\

_ y i (2p - 1 + cr\(2q +
\

cr

)\

2r-cr-l

2q-a-\ r

X h(s,u + 2p + cr,t + 2q + 2r-o~,x,y,A2)

_

, . -.2v

= V C2r-2v/0, „ < 2 v ++ 11) )! •^ (2v

On the other hand, by (3.20) and (3.21) with d = 2r, we have (3.25) -y [G2r(n\ 2p, s, t, u, 2q\ x, y) + G2r(-n; 2p, s, t, u, 2q; x, v)} **/

=L-i1 pPm^n'O + m) (m + n) (l + m + nfi 2r

+

l,m,n= 1 xoo

u

xmyn + m)2r(m + n)u(-l + m + nft

Z

/^=i PWn'i-1 l*m

-^--fzrrirr v=0 2 (OT) ''

//=0 o-=0

v

x — — Ci(s + 2r + 2v-o--2p,u (2//)!

v=0

/i=0
x

2v - cr - 2p\

cr-2p

+ 2q + o;f, x,y; A2)

'

x

f

2

(in) '' X ——- &(•*, u + 2p + o;t + 2r + 2v-o--2p;x, {2p.)\

-Yc2

2

^

J

M

y;A2)

I

Zeta Functions of Root Systems

135

Applying [11, Lemma 2.1] to the third and the fourth terms on the second member of (3.25), and applying [23, Lemma 4.4] to (3.24), (3.25), we obtain the following. Proposition 3.3. With the above notation,

(326)

f

s

2r

*v

, ^ , fPm n'(l + m) (m + n)u(l + m + n)2i l,m,n=\

•Z \

pPmsn'(-l + m)2r(m + «)"(-/ + m + n)2
l*m tem+n

2p-2p

x

(2q-l+

cr\(2p +

2r-2p-o--l\

&(s + 2p + 2r - 2p - cr, u + 2q + cr, t; x, y; A2)

X £2(s, u + 2q + cr, t + 2q + 2r - 2p - o",x,y; A2)

p=0

cr=0

X

/ \

1-

1

x &(s + 2p + 2r - 2p - o-,u + 2q + or,t;x,y;A2)

Let (x,y) = (1,1) xand calculate second the- left-hand side. fr(s,u + 2q the + o-,t + 2qterm + 2r on - 2p o",x,y;A2). Then we obtain the main result in this section.

136

Y. Komori, K. Matsumoto & H. Tsumura

Theorem 3.4. For p, q e N and r e No, (3.27)

£,(2/7,2r, 2q, s, u, t;A3) + £3(2p, s, u, 2r, 2q, t;A3) + &(2q,2r,2p,t,u, s;A3) + &(2q,t,u,2r,2p,

s;A3)

x £2(5,u + 2q + o-,t + 2q + 2r - 2p - o",A2)

X fr(s + 2p + 2r - 2p - o-,u + 2q + o-,t;A2)

p=0


X

' '

'

holds for all (s, t, u) 6 C 3 except for singularities of all functions on the both sides of (3.27). Example 3.5. Putting (p, q, r) = (1,1,1) and (s, t, u) = (2,2,2) in (3.27), we have (3.28) 4£,(2,2,2,2,2,2; A3) = 8£(2) {£,(4,4,2; A2) + 6(3,5,2; A2)} - 6 {2£2(6,4,2;A2) + 26(5,5,2; A2) + 6(4,6,2; A 2 )}. Using £2(k,m,l;A2) = b(k,m -1,1+ [19]) repeatedly, we see that

1;A2) - ft(* - l,m,/ + 1;A2) (see

6(4,4,2; A2) + 6(3,5,2; A2) = ± {26(4,3,3; A2) - 6(3,4,3; A 2 )}.

137

Zeta Functions of Root Systems

From [20, Theorem 2], the right-hand side is equal to 7r10/935550. Similarly we have 2ft (6,4,2; A2) + 2ft (5,5,2; A2) + ft (4,6,2; A2) = 2ft(6,3,3;A 2 )-ft(3,6,3;A 2 )-^ft(4,4,4;A 2 ). From [20, Theorem 2] and [18, Theorem 4.1], we see that the right-hand side is equal to 887TT 1 2 /3831077250. Therefore we obtain Witten's volume formula (3.29)

6(2,2,2,2,2,2^,)

=

_ | _ - ,

r

. ; .

This implies that (3.27) can be regarded as a continuous relation including Witten's volume formula. Remark 3.1. If we start from (3.3) instead of (3.2) in (3.4), then we can construct a general functional relation for ft (a, c, b, s, u, t;A3) + (-l) a ft(a, s, u, c, b, f; A3) + (-l) fl+fc+c ft(fc,c,a,t,u,s: A3) + (-l) fl+c ft(&,t,u,c,a, s;A3) in terms of ft(si, s 2 ,53; A2) and fts), where a, b € N, c e No and s,t,u e C. For example, in the case (a, b, c) = (1,3,0), we can obtain ft(l, 0,3, s, u, t; A3) - ft(l, s, u, 0,3, t; A3) + ft(3,0,1, t, u, s; A3) - ft(3, f, H, 0,1, s; A3) = 4ft(s, M + 4,f;A2) + ft(5 + 1 , M + 3;A2) - 2^(2)ft(5)M + 2,r;A 2 ). Remark 3.2. As described in this section, we produced relation formulas for ft(•; A3) by uniting those for ft( •; A2) and ft (•; A\). This corresponds to the fact that the fundamental root system of A3 can be produced by adding a fundamental root to that of A2 in the Dynkin diagram. This strategy can be applied to various other cases. For instance, in [5], we deduced relation formulas for ft( • ;£ 2 ) from those of ft( • ;A2) and ft( • ;A\). Similarly, it is possible to deduce relation formulas for ft( •; B3) and ft( •; C3) from those of ft(•; A3), or alternatively, from ft( •; B2) and ft (•; A\). In [5], the process of the latter way was described in detail. These correspondences between functional relations for zeta-functions and the structure of root systems seems to be fascinating.

138

Y. Komori, K. Matsumoto & H. Tsumura

References [1] T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1976. [2] N. Bourbaki, Groupes et Algebres de Lie, Chapitres 4, 5 et 6, Hermann, Paris, 1968. [3] P. E. Gunnells and R. Sczech, Evaluation of Dedekind sums, Eisenstein cocycles, and special values of L-functions, Duke Math. J. 118 (2003), 229-260. [4] Y. Komori, K. Matsumoto and H. Tsumura, On Witten multiple zetafunctions associated with semisimple Lie algebras II, preprint, submitted for publication. [5] Y. Komori, K. Matsumoto and H. Tsumura, On Witten multiple zetafunctions associated with semisimple Lie algebras III, preprint, submitted for publication. [6] Y. Komori, K. Matsumoto and H. Tsumura, On Witten multiple zetafunctions associated with semisimple Lie algebras IV, in preparation. [7] K. Matsumoto, Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series, Nagoya Math J. 172 (2003), 59-102. [8] K. Matsumoto, On the analytic continuation of various multiple zeta-functions, in 'Number Theory for the Millennium II, Proc. Millennial Conference on Number Theory', M. A. Bennett et al (eds.), A K Peters, 2002, pp. 417-440. [9] K. Matsumoto, The analytic continuation and the asymptotic behaviour of certain multiple zeta-functions I, J. Number Theory 101 (2003), 223-243. [10] K. Matsumoto, On Mordell-Tornheim and other multiple zetafunctions, Proceedings of the Session in analytic number theory

Zeta Functions of Root Systems

139

and Diophantine equations (Bonn, January-June 2002), D. R. HeathBrown and B. Z. Moroz (eds.), Bonner Mathematische Schriften Nr. 360, Bonn 2003, n.25, 17pp. [11] K. Matsumoto, T. Nakamura, H. Ochiai and H. Tsumura, On valuerelations, functional relations and singularities of Mordell-Tornheim and related triple zeta-functions, preprint, submitted for publication. [12] K. Matsumoto and H. Tsumura, On Witten multiple zeta-functions associated with semisimple Lie algebras I, Ann. Inst. Fourier (Grenoble) 56 (2006), 1457-1504. [13] K. Matsumoto and H. Tsumura, Functional relations for various multiple zeta-functions, ('Analytic Number Theory', Kyoto, 2005), RIMS K = oky = uroku No. 1512 (2006), 179-190. [14] K. Matsumoto and H. Tsumura, A new method of producing functional relations among multiple zeta-functions, Tokyo Metropolitan University Preprint Series 2006: No.l. [15] L. J. Mordell, On the evaluation of some multiple series, J. London Math. Soc. 33 (1958), 368-371. [16] T. Nakamura, A functional relation for the Tornheim double zeta function, Acta Arith., to appear. [17] H. Samelson, Notes on Lie Algebras, Universitext, Springer-Verlag, 1990. [18] M. V. Subbarao and R. Sitaramachandrarao, On some infinite series of L. J. Mordell and their analogues, Pacific J. Math. 119 (1985), 245-255. [19] L. Tornheim, Harmonic double series, Amer. J. Math. 72 (1950), 303-314. [20] H. Tsumura, On some combinatorial relations for Tornheim's double series, Acta Arith. 105 (2002), 239-252. [21] H. Tsumura, Evaluation formulas for Tornheim's type of alternating double series, Math. Comp. 73 (2004), 251-258.

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Y. Komori, K. Matsumoto & H. Tsumura

[22] H. Tsumura, On Witten's type of zeta values attached to S 0(5), Arch. Math. (Basel) 84 (2004), 147-152. [23] H. Tsumura, On functional relations between the Mordell-Tornheim double zeta functions and the Riemann zeta function, Math. Proc. Cambridge Philos. Soc, to appear. [24] E. Witten, On quantum gauge theories in two dimensions, Comm. Math. Phys. 141 (1991), 153-209. [25] D. Zagier, Values of zeta functions and their applications, in Proc. First Congress of Math., Paris, vol.11, Progress in Math., 120, Birkhauser, 1994, 497-512.

Y. Komori Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602 Japan Email: [email protected] K. Matsumoto Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602 Japan Email: [email protected] H. Tsumura Department of Mathematics and Information Sciences, Tokyo Metropolitan University 1-1, Minami-Ohsawa, Hachioji, Tokyo 192-0397 Japan Email: [email protected]

Sums of Kloosterman Sums Revisited Y. MOTOHASHI

The aim of the present article is to show that the Spectral-Kloosterman and the Kloosterman-Spectral sum formulas are practically equivalent to each other. Previously, only the procedure for deriving the latter from the former was known, which is the inversion due to N.V. Kuznetsov [10][11]. We shall not only prove it in our own way but also establish a reverse procedure. Our motivation is detailed in the third and the final sections. Technically, our discussion is focused on two Bessel transforms which appear in those sum formulas. These transforms, especially their inversions, seem to be best understood as special instances of the Plancherel theorem on Lie groups as discussed by R.W. Bruggeman [2]. We shall, however, employ an alternative argument which appears to demand less background knowledge, yet it has a certain merit in applications of the sum formula. We shall restrict ourselves to the most basic instance offered by the full modular group, solely for the sake of simplicity. Notations are introduced where they are needed for the first time, and will continue to be effective thereafter.

1 Automorphic forms To begin with, we shall collect elements of the theory of r-automorphic representations of G, with G = PSL2(R),

r = PSL2(Z).

(1.1)

We write n[x] =

1 x , 1

a[y] = ^

l/># 141

,

k[0] =

cos 8 sin 6 • (1.2) - sin 6 cos 9

142

Y. Motohashi

Let N = {n[x] : x e R}, A = {a[y] : y > 0}, and K = {k[fl] : 0 € so that G = NAK be the Iwasawa decomposition of G. We read it as G a g = nak = n[*]a£y]k[0]. A Haar measure on G is defined by dg = (rry2)~ldxdyd9. The space L2(T\G) is composed of all left r-automorphic functions on G, vectors for short, which are square integrable over F\G against dg. Elements of G act unitarily on vectors from the right, and we have the orthogonal decomposition into invariant subspaces L2(r\G) = C • 1 © cL2(r\G) © eL2(r\G),

(1.3)

where CL2 is the cuspidal subspace, and eL2 is spanned by integrals of Eisenstein series. The cuspidal subspace splits into irreducible subspaces: c 2

L (r\G) = ®V;

n\v = (v£ - i ) • 1,

(1.4)

where Q. = y2 [p2. + d2 J - ydxdo is the Casimir operator. With (1.1), we can restrict our attention to two cases: either vy e (0, +00)/ or vy e N - \. According to the right action of K, the space V is decomposed into Kirreducible subspaces V=

© Vp,

dimVp
(1.5)

p~—00

If it is not trivial, Vp is spanned by a r-automorphic function

y.

.

.

-^=="^sgn(n)0p(a[|n|]g; Vy),

(1.6)

ntToo VN where
Au(pp{g;v)= I *J— 00

exp(-2m'Mv)^(wn[v]g;v)rfv,

w= k[^].

(1.7)

Sums of Kloosterman Sums Revisited

143

In particular, we have the expansion i


rrVy / , gv(n)Kvv(2n\n\y)exp(2mnx),

(1.8)

with ATV standing for the ^T-Bessel function of order v. This is a cusp-form of unit length on the hyperbolic upper half plane G/K. Let us consider a V in the discrete series; thus vy = q-\, with a ^ e N . We have, in place of (1.5), oo

either

V = © V„

—q

or

V=

© Vp,

(1.9)

with dim Vp = 1, corresponding to the holomorphic and the antiholomorphic discrete series, respectively. The involution g = nak i-» n _1 ak _1 maps one to the other. In the holomorphic case, we have a complete orthonormal system {ipp : p > q) in V such that

MB) = " W f r ^ f t ^iMa[»]g;w).

(i.io)

In particular, we have 22%'?+i 1 J \ ¥>«(g) = (~D , exp(2ig%* V ev(n)n*~* exp(2mn(* + *»). g

Vr(2
£i

(1.11) This infinite sum is a holomorphic cusp-form of weight 2q on G/A", the length of which is equal to 2_2%_<7-2 -\jT{2q). We shall not deal with irreducible subspaces of complementary series type, for there are no such subspaces under (1.1), as indicated above. We note relevantly that our argument below can be extended to general cofinite groups with cusps, without essential changes if the corresponding Kloosterman sums satisfy a Weil type bound; see a remark following (2.3) below. Let V be generic as in (1.4). It should be observed that the coefficients Qv{n) in either (1.6) or (1.10) are normalized in such a way that they do not depend on the weight. We recall the basic bound: For any m e N YJ v \vv\
\Qv(m)\2 « N2 + m5 +£

(1.12)

144

Y. Motohashi

as N tends to +00, where e is an arbitrary small positive constant, solely on which the implicit constant may depend. This is an extended version of [12, (2.3.2)] under our present normalization; those V in discrete series are to be treated with Petersson's sum formula (see (4.5) below).

2 Sum formulas We are going to display the two versions of the sum formula involving Kloosterman sums. We stress again that the aim of the present work is to discuss the relation of these two but not to develop any proof of the sum formula itself; see the remarks (l)-(3) in the final section. Our notation depends on (1.6) and (1.10) and is different from those employed in [1][2][10][11 ] [ 12], for instance. We shall skip over convergence issues, as they will be touched on in later sections. We define a Kloosterman sum attached to r by S(m,n;f!)=

V w mod t

exp I — (/nw + nw)J, '

ww = 1 mod I,

(2.1)

'

with arbitrary integers I > 0, and m, n. We put, for x> 0, Bh(x) = -±-. f —^—J2r(x)h(r)dr 2m J (0 ) cos(Trr) i f J-2r{x) ~ Jlrix) . . ,, . . ^ •^ - — —— rta.n(nr)h(r)dr, An J(o) sin(^-r) where Jv stands for the 7-Bessel function of order v; and (77) denotes the vertical line passing through the point 77. We require that h be even, regular, \h(r)\ <£ (|r| + l ) - 2 - 6 , provided |Re r\ < - + e,

(2.3)

where the implicit constant may depend on e. Note that this width of the vertical strip is relevant to the Weil bound for S{m,n;C); the sum over I in (2.4) below converges absolutely, since Bh(x) «: xi+£ for x close to 0, as is implied by an appropriate shift of the contour in the first integral of (2.2), provided (2.3) holds. Thus the width has to be altered according to the choice of underlying discrete subgroups; in general, the condition

145

Sums of Kloosterman Sums Revisited

|Re r\ < \ + E is appropriate, on which as well our argument works fine with minor changes. With this, the first sum formula that we are concerned with asserts that it holds that for any m, n e N

v

Qv(m)QV(n)h{vv) + — -— ——h(r)dr = 4m J ( O) (m«)r^(l + 2r)£(l - If)

i

/ 47r

- i

r

\

(2 4)

-

<5m,„-^ I /-tan(wr)A(r)£/r + J ] - 5 ( m , n ; ^ ) B A l y VwJ). Here E u indicates that V belong to the unitary principal series; £ the Riemann zeta-function, cra{n) = Y^d\n da, and 6m
qeN.

(2.5)

We put 1 °° IMx) = Bh(x) + — 2(-D 9 (2<7 - \)h (q - I) J2g-i(x),

(2.6)

q=\

A(h) = i I rt<m(nr)h(r)dr+Y(2q-l)h(q-\).

(2.7)

Then we have, in place of (2.4),

£^m)^(v,) + -L f A

°° i

^f^fn /4

\

,^dr (2-8)

with V running over all irreducible cuspidal subspaces enlisted in (1.4). The equivalence of (2.4) and (2.8), under (2.3) n (2.5), will be shown in Section 4.

146

Y. Motohashi

The maps L and A came later from the effort to understand Kuznetsov's inversion of (2.4), i.e., the second version of the sum formula (2.12)/(2.13) below in the framework of the spectral decomposition of L2(r\G), as developed in [2]; their origin can be traced back to [1] and [10][11]. They might appear irrelevant to (2.4) itself, for one may employ in fact any function on N - j as far as the decay condition is satisfied. For more discussion see the remark (1) in the final section. Next, let us introduce another Bessel transform K)-Jlr{x)

dx

K/(r) = In f0 J~2r(X\ 7w/(x)-, (2.9) sin(;ir) x Jo sin'— which is to be compared with the second line of (2.2); one may call K the adjoint integral transformation of B. Here we require that there exists a sufficiently large constant A > 0 such that for a < A fa\x)

« (* + 1 lx)~A,

x>0,

(2.10)

where the implied constant may depend on A. Also, we put 1 °° P / W = fix) - — £ ( - l ) * ( 2 * - W

(q - i) Jlq-l(x).

(2.11)

q=\

Then the second sum formula we are concerned with is Y> -jS(m,n; W ( y V ^ l + < W ^ J

rtm(nr)Kf(r)dr

^QvWevtWfivv) v +— f ^rimtorin) Ani J (0) (mnmi + 2r)f(l - 2r)

(2.12) JKn

r

'

where m , n e N are arbitrary. Corresponding to (2.8), we have the alternative expression: V -S{m, n; €)f ( ^ > f f l = YJ X v t=i '

- f;

(0)

4m Jio)

Qv(m)fMn)Kf(vv)

( ™ / f ( l + 2r)f(l - If)

(2.13) JyJ

•

Sums of Kloosterman Sums Revisited

147

The equivalence of (2.12) and (2.13) is analogous to that of (2.4) and (2.8), apart from the fact that we need to incorporate AK = 0,

(2.14)

as we shall show in Section 4 on the condition (2.10). The spectral decomposition (2.13) is a version of the Kloosterman-Spectral sum formula of Kuznetsov [11, Theorem 2 supplemented with (6.33)]. The condition (2.10) is introduced solely for the sake of simplicity; it is immaterial to discuss how large the constant A should be. In fact, (2.10) could be either relaxed or strengthened considerably. For instance, the mild condition [12, (2.4.6)] should work well and also one may start with an arbitrary / which is smooth and compactly supported in (0, +oo), only we need to reduce respective situations to (2.10) appealing to a usual approximation procedure. In the same token, the condition on h could be highly strengthened; (2.3) appears presently to be the least restrictive.

3 The inversion problem Historically, (2.4) was proved first and (2.12) second, by Kuznetsov [10]. It is stated in [11, three lines above (6.32)] that they are, nevertheless, essentially equivalent. To endorse this fundamental assertion, it is proved in [ibid, (6.32) and Appendix] that Tift

\

Pf(x) = -i

• C

r

r < x r°

—-j^x) J(0)cos(7rr)

J

-2r(y) - hr(y)

J0

r-— sin(^r)

-,

.dy

f(y)—dr; y

(3.1)

namely, BK = P,

(3.2)

on (2.10). In fact, a condition less restrictive than (2.10) is assumed there, but such an aspect is irrelevant to our present purpose as remarked already. We shall show at the end of the next section that the image K / is in the space (2.3) n (2.5), and (2.8) should hold with h = K/. On noting (2.14), this procedure yields (2.13), for LK - BK = 1 - P by definition, and (3.2) means that on (2.10) LK=1. (3.3)

148

Y. Motohashi

Provided (2.10), say, holds, the Kloosterman-Spectral sum formula follows from the Spectral-Kloosterman sum formula. However, it does not appear that the endorsement from the opposite direction, i.e., KL = 1 (3.4) on (2.3) n (2.5) has ever been asserted and proved either in [11] or anywhere else. This would practically complete the proof of the equivalence in question. We shall establish Theorem. The identities (3.3), and (3.4) hold. The sum formulas (2.8) and (2.13) are equivalent to each other; that is, they can be derived from each other. This assertion depends on the conditions (2.3) n (2.5) and (2.10) implicitly and in a somewhat peculiar way. In fact, the map L does not generally take h in the space (2.3) n (2.5) to the space (2.10), which is in a sharp contrast against the property of K under (2.10) mentioned above. Nonetheless, to the image Lh the map K is applicable, and (3.4) holds. This causes a complication in our derivation of (2.8) from (2.13). Thus we first modify both h and L or rather B so that we may apply (2.13) to the result; then a limiting procedure is employed to reach (2.8) with the aid of (3.4). In the course of discussion, we shall actually prove, solely on (2.3), KB = 1,

(3.5)

which is slightly deeper than (3.4), for it implies that Kuznetsov's equivalence assertion on (2.4) and (2.12) is indeed correct. Details are developed in Section 5. It should be stressed that we know already that the relations (3.3) and (3.4) are valid in L2-sense. To explain this, let us equip the set Y = Ri U N - 5} with the measure dr defined by

AW

1«'4 '

(36)

Then in [2, Propositions 14.2.6 and 14.2.8] Bruggeman showed that K and

149

Sums of Kloosterman Sums Revisited L extend to the unitary isomorphisms Ku : L2((0, +00), <j*) —• L\Y, dr), Lu : L2{Y, dr) —» L2((0, +00), ^ , with dxx = dx/jt; moreover, KULU = 1,

LUKU = 1.

(3.8)

It remains thus to prove (3.8) on spaces of test functions for which the resulting integrals converge absolutely. This is not merely a theoretical exercise. When we apply the sum formula, we need test functions that are well defined on the relevant sets. Relations in L2-sense such as (3.8) are useless for that purpose. We may then try to work in practical spaces like those introduced above and proceed weakly in L2(Y, dr) and L2 ((0, +00), d*), being aided with (3.8). However, this leads us inevitably to an estimation procedure that is shared to a certain extent by our lowbrow argument to prove the theorem. As a byproduct of our discussion, we shall be able to see how Bh deviates from the reasonable space (2.10), which appears to have applications to the circle of problems on Hecke L-functions that we dealt with recently in [14] and [15].

4 Proof (1) We shall begin with the equivalence of (2.4) and (2.8). It suffices to show that 2]

Qv(m)gv(n)h(vv)

v discrete series

1

°°

(_°°. 1

IA

\

S,

The left side converges absolutely because of (1.12) and the decay condition in (2.3). The double sum on the right converges also absolutely, as we

150

Y. Motohashi

have the Weil bound for S(m,n; () and

\Jq{x)\<mm[\,-^\,

(4.2)

uniformly in x > 0 and q e N, which follows from a combination of J(x)=-

I cos(q6-xsm8)d6, 7T Jo /vW = ^ = -(^Y (1 - J2)""* cos(^)^, Re v > - - . z ym(v+ j) Jo 2

(4.3) (4.4)

The convergence of the last sum in (4.1) is obvious, because of (2.5). In passing, we note that the convergence of (2.11) is confirmed analogously. With this, we invoke Petersson's sum formula under the normalization (1.10); thus the last sum over I is equal to ^{-\)q-X6m,n In

+ {-\f—^-(2q - 1)

V Qv(mj^(nj, ^

(4.5)

vV=q-\

where V are in the holomorphic discrete series (see [12, (2.2.9)]). Hence we get (4.1); as remarked above, we see that any value of h (q - jj works as far as (2.5) holds. We shall prove (2.14), which states that on (2.10)

J

rtan(^)K/(ryr = i V ( 2 9 - l ) K / ( ? - I ) .

(°)

(4.6)

U

We return to the definition (2.9). Shifting the contour to (|), we see that the left side is equal to iKfU)-4nf vz/

—r— ^ hr(x)f(x)—dr. J(s) cos(Trr) Jo *

(4.7)

This double integral converges absolutely; in fact (4.4) gives the uniform bound J2r(x) <s (x/|r|) 5/2 e" |r| . Exchanging the order of integration, and

151

Sums of Kloosterman Sums Revisited

computing the new inner integral with residue calculus, which is possible with x fixed as (4.4) implies, we find that (4.7) is equal to

r

dx

l

(U = - 4ni I Jo

)

(4.8)

Jo(x)f(x)dx,

where we have used the relation Jy-i(x) + Jv+\{x) = (V/JC)/V(JC).

(4.9)

Similarly we have that the right side of (4.6) is equal to Jo(x)f(x)dx + 4ni l i m ( - l ) e I 2->°° Jo

J2Q(x)f(x)dx.

(4.10)

The limit can be taken inside the integral because of (4.2). Hence, (4.6) follows from (4.8) and (4.10); and the proof of the equivalence of (2.12) and (2.13) ends. Next, we shall prove Kuznetsov's inversion formula (3.1) in a manner different from his that is developed in [11]. To this end, let us put D±

s=-y1[j]

+s{s-l)^y2,

?€(0,+oo).

(4.11)

We claim that the resolvent kernels of y~2D* give rise to (+): Kuznetsov inversion formula, (-): Kontorovitch-Lebedev inversion formula.

(4.12)

In fact, the minus case is treated in [12, Lemma 2.6], and is pertinent to the analogous sum formula involving Kloosterman sums of opposite sign, i.e., S(-m,n;£) with m, n e N. We are going to show that the argument there works for the plus case as well. Thus, we observe that the differential equation D*f(y) = 0 has two linearly independent solutions: y,_i(y) = yfyJs-i(y), nJy i 5

2 cos ns \

x 2

i

s

J

(4.13)

152

Y. Motohashi

Their Wronskian is equal to - 1 . Hence a resolvent kernel R+ ofy 2D+ is given by ^_i(j)^_i(v),

v>y.

To eliminate the separation of cases, we put 1 C r Cs(y,y) = sin(7rr) kr(y)kr(y)dr i J(0)r2_^_i)2 = niyfv I

(4.15)

-^Jr(v)kr(y)dr.

The power series expansion of J±r gives that the integrand is «: (|r|2 +1) - 1 , and the integral is absolutely convergent. Move the line of integration to Re r = 2£>, with a large Q e N. We have, for 2Q > Re s > \, v < y, Q 2-1 Cs(v,y) = n2R:(v,y) + In1 V ^ £ ^— -j 2 =i (2
(416)

•••^.

J(2Q)

Again by the series expansion of J±r, this integral is «: (v/y) 2 ^ /(?, which vanishes as Q —> oo; and we have *>,?) = 4 : f ^ J ( 0 ) r 2 _ (s oo

3 sin(^r)^(v)A:r(y)rfr 1 2 - 1

(417)

- 2 Vvy X —, rj ^-i(v)/ 2 9 -i(y),=i(2?-l)2-(5-i) The same holds for v > y as well, because of symmetry. Now, we have, for a smooth function / which is compactly supported in (0, +oo), v 2 /(v) = Z>X/(v)(4-18) Insert the last expression for the resolvent kernel, change the order of integration, and take the operator D+s inside the outer integral. In the resulting

153

Sums of Kloosterman Sums Revisited identity replace /(v), f(y) by /(v)v obtain f(v) = ~i

3/2

, f(y)y

:—r-Mv) J (0) cos(7ir) Jo

+ 2 Yciq

U

3/2

, respectively. Then we

r—— sin(Trr)

- l)/ 29 -i(v) Jo f

f(y)—dr v

(4.19)

f(y)J2q-i(y)—, y

which ends our proof of (3.1) essentially. What remains is to check if / can be as in (2.10); and it depends on the nature of convergence on the right side or that of the image K/. For this sake, let /* be the Mellin transform of / satisfying (2.10). We have, for reRi, K/(r) = -i [ f(s) f J(-|) Jo

J2rix) J

- ->r(x)x-^dxds; sin(Trr)

(4.20)

the necessary absolute convergence is endorsed by a well-known asymptotic expansion for 7-Bessel functions [16, eq. (1) on p. 199]. Thus we have, by [ibid, eq. (1) on p. 391], K/(r) = - f m

2-2scos(7rs)T(s + r)r(s-r)f(-2s)ds.

(4.21)

JG)

Analogously we have, for any q € N,

Kf(q-$)=

2~2s^j

2

-{f\-2s)ds.

(4.22)

Since (2.10) implies that f*{s) is regular for |Re s\ < \A, and
154

Y. Motohashi

5 Proof (2) We turn to the opposite direction of operations, that is, KL or rather KB. We shall try to extend [12, Lemma 2.10] that concerns the reverse of the minus case in (4.12). Let h satisfy (2.3) but not necessarily (2.5). By definition, we have, for v € Ri, KBfc(v) = -i

—— ——J2r(x)h(r)dr—. (5.1) Jo sin(Trv) J ( 0 ) cos(^-r) x The image Bh might not satisfy (2.10), but K is applicable to it. In fact, shifting the contour of the inner-integral to ( | j , we see that the resulting double integral is absolutely convergent; note that we need (4.4) for J2r(x) and the asymptotic expansion for J±2v(x) as x —> oo. With this observation, we consider then M(s,v) Jo

Sin(7TV)

^ J ( l ) COS(7IT)

)

When |Re.y| < | , this is absolutely and uniformly convergent; thus M(s, v) is regular there, and Af(0, v) = KB/z(v),

v 6 Ri.

(5.3)

Then, we assume that 0 < Re s < | . Exchanging the order of integration, we have M(s,v) = . . \ , f -^r-Mr) isin(7rv) J ( i ) cos(Trr)

r J0

J2r{x)(J-2v{x) -

J2v{x))x2s-Xdxdr.

(5.4) The inner integral can be computed explicitly using the Mellin-Parseval reasoning or rather invoking [16, eq. (2) on p. 403]; that is,

f

J2r(x)J2v(x)x2s

X

dx

0

2 2 '- 1 r(l - 2s)T(r + s + v) 'T(l - s - r + v)T(r - s + v + \)T(r - s - v + 1)'

(5 5)

155

Sums of Kloosterman Sums Revisited Inserting this into (5.4), we get M{s,v) _22s ~ ni

S

f rh(r)cos(n(r + s))T(r + s + v)T(r + s - v) J{\) cos(nr)T(r - s + v + l)T(r - s - v + I)

(5.6)

We move the contour to (-%)• Computing the residues at r = -s ± v and letting s tend to 0 in the result, we have

M(0,v) = 2h(v) + - f

4^kdr-

<5-7)

m J(-i) rL - vz

Moving further the contour to (j), we see that this integral is equal to -inhiy). Hence we find that M(0,v) = h(y),

veRi.

(5.8)

This and (5.3) give (3.5) under (2.3). In passing, we observe that analytic continuation of (5.5) implies Hi

(M(JX J2q-\(,X)hv(,X)—

(-I)*" 1

Jo x Since this is even in v, we have readily KBh(q-±)

cosfrv)

= —r

-J

(q-l)2-v2

In

= 0,

qeN.

«Q, •

(5.9)

(5.10)

Also, the equality KBh(y) = h(y) itself extends off the imaginary axis, to a certain extent. This is not relevant to our present purpose but to the situation where one has to take into account the possible existence of the contribution of complementary series representations; then the line of integration in the inner integral of (5.2) has to be adapted, provided the vertical strip in (2.3) is widened appropriately. It remains to derive (2.4) from (2.12) on the assumption (2.3). To this end, let yS > 0 be arbitrary and hp(r) = h(r) exp^Sr2); note that hp{r) «: exp(-/?|r|2). Also, let Q € N be arbitrary and put Q

Bhpix) = Bhpix) - J ] yqfihq-x{x\ q=\

(5.11)

156

Y. Motohashi

where yqfi are to satisfy Q


(5.12) (r)r2k+1 tsm(7rr)dr,

0
•^n Jro) (0)

which is obviously soluble. Later in the present section, it is proved that as x —» +oo

— I StyOc) «* oris,

a<4<2,

(5.13)

provided Q is large. We then put f(x) - o>a(x)Bhp(x), where u>a(x) = exp(-a/x), a > 0. On noting that the left side of (5.13) is «: x?~a as x —> 0 + and that g is arbitrary, this / satisfies (2.10) and we can apply (2.12) as it is supposed now to hold. We shall consider the situation as a -» 0 + and/? -* 0 + . The Kloosterman-sum side, i.e., 00

1

£

IA

\

-S (m, n; e)PtoaBhp ( y V ^ J

(5.14)

is easy to handle. In fact, we have seen that Bhp(x) decays appropriately at both ends of (0, +oo), whence for q e N

r

dx \Bhp(x)\—

(5.15)

is finite, which means in turn that the sum (5.14) converges uniformly for a > 0. Since (5.10) gives that P B = P B = B,

(5.16)

the value of (5.14) at a = 0 is equal to the Kloosterman-sum part of (2.4)^, the result of replacing h by hp in (2.4). On the other hand, we have, invoking the Mellin transform of a>a, KojaBhp(v) = -

f a-2sT(2s)Mp(s,v)ds,

m J(fi)

veRi,

(5.17)

157

Sums of Kloosterman Sums Revisited where i2i-l

M,(s, v) = M,(s, v) -

St" 1 ^

T(2s)cos(7Ts)

r{q+s + v-\)r[q r(q-S

+

+

s-v-j)

(5.18)

v+^)r(q-s-v+l1)'

It is expedient to observe here that Mp(s, v) is of polynomial order in |s|+|v|, as far as |Re s\ < | and v e Ri. In fact, returning to (5.6)^ with the same convention as above, we apply Stirling's formula and the trivial bound for trigonometrical functions. We may assume without loss of generality that r, s, v € Ri; then the argument of the exponentiated factor is -- (\r + s + v\ + \r + s - v| - \r - s + v| - \r - s - v|) + n(\r+s\-\r\-\s\)

(5 19)

= - n {max (|r + s\, |v|) - max (\r - s\, |v|)} + nQr +

s\-\r\-\s\)<0,

and the assertion follows. Thus we see immediately that the integrand of (5.17) decays exponentially in \s\, while being of polynomial order in |v|. With this, shifting the contour to (-e), we get the residue Mp(0, v) = hp(v), which gives rise to the terms of (2.4)^ other than the Kloosterman part. For a large v, the new integral is truncated to the segment [-e - |v|ei, -E + \v\Ei], and the integral (5.6)/? with s in this range to the segment f | - |v|ei', I + \v\Ei\. The error thus caused is clearly «: a2e exp(-|v| e ), and Stirling's formula implies that the remaining part is «; alE\v\~2~Ae, where implicit constants may depend on/?. Since we have (1.12), the relevant contribution vanishes as a —* 0 + . Hence we have derived (2.4)/} from (2.12) under the condition (2.3). Invoking Bh(x) «: xi+s with x close to 0 and taking limit as /? -» 0 + , we end the proof of our theorem, provided (5.13) is confirmed. Now, to prove (5.13) we adopt an argument employed in [14, Section 3]. In what follows, we let g stand for hp\ thus g is regular in |Re r\ < | + e and decays wildly there. We observe first that B*(x) = - - i - j f g(s)l^)~2S ds, 2 (2;r) J(_i) \2/

(5.20)

158

Y. Motohashi

where g(s) is understood to be an analytic continuation of f

s

TO + r)

(5)=

rg(r)

FT;—r^—r^dr'

Re

*>0-

(5-21)

J(o) H I - s + r) COS(7ZT)

In fact, we may shift the latter contour to Q j , getting a continuation to Re s > -\. We insert the result into the right side of (5.20); the double integral is absolutely convergent, and exchanging the order of integration we obtain the identity (5.20) via [16, eq. (7) on p. 192]. Assuming that Re s > 0 and Im J > 0, we move the last contour to £ which is the result of connecting, with straight lines, the points | - ooi, \-ci,-\ + ci, -j + ooi in this order, where c> 0 is arbitrary. An analytic continuation of g(s) to the right of £ is obtained. On noting that g is even and L symmetric, we have g(s) = — x

COS(TTS)

I T(s + r)T(s - r)g(r)rtan(nr)dr. Jc

(5.22)

Let us suppose that s = cr + ti is such that t tends to +oo while cr > -\ remains bounded. Then an appropriate deform of L and a truncation give rg(r) tan(7rr)r(s + r)Y{s - r)dr + O ie'1*). (5.23)

g(s) = — cos(ns) 7T

V

J-t*i

'

Here we have, by Stirling's formula, T(s + r)r(s -r) = - Imt^'^21'

exp(mo- - lit - nt)

Q l

~

x i + ^rW)

>

+

-1

o(((W + i)2r'f) ,

(5.24)

with any fixed Q e N, where the polynomial /?*(£) is of degree k in £; pk depends on cr, and the implicit constant on cr and Q. Thus we have g(s)

=i^-l+2i'e-2it C { Q~X \ X I rg(r) tan(^r) j 1 + £ t~kpk (r 2 ) I rfr + OO^" 2 ).

(5 25)

159

Sums of Kloosterman Sums Revisited By (5.12), we may write this as g(s) =27rt2CT-1+2V2'''

i+ r

2 +o(f2cr e)

i(-i)^4 S ^(^-w )l q=\

{

k=l

" '

(5.26)

)

and, by (5.24), (5.27) which ends the approximation of g(s) for the case Im s > 0; the case Im s < 0 is analogous, and the result is the same as (5.27). This holds with s large in any fixed vertical strip contained in the half plane Re r > -j. On the other hand, we have, by (5.11), (5.20), and [16, eq. (7) on p. 192],

i

r

L, , . v

T q+s

( -i)\(xv2sJ

(5 28)

-

We may shift the contour to Re s = \Q, because of (5.27); and, differentiating a times inside the integral, we obtain (5.13) and end the proof of the theorem.

6

Concluding remarks

(1) Kuznetsov's proof of (2.12)—(2.13) is in fact an ingenious transformation of a prototype of (2.4) which is a spectral decomposition in L2(r\G/K) of inner products of real analytic Poincare series of weight 0. Thus the appearance in (2.13) of the contribution of holomorphic cusp forms or the discrete series representations was a mysterious phenomenon, in the sense that his method did not provide any structural background for the fact, since L2(r\G/K) lacks holomorphic elements. However, as is developed in [2], if we employ instead Poincare series of a non-zero even integral weight

160

Y. Motohashi

coupled with the relevant L2-space, then holomorphic cusp forms play in fact a role. One may vary weights and try to view all, and then comes naturally to the space L2(r\G) as a proper working place. The maps L and A convey such a procedure of extending (2.4) to reach (2.13) via (2.8). (2) There exists, however, an alternative, direct approach to (2.13) due to J.W. Cogdell and I. Piatetski-Shapiro [7]. Their argument starts with a construction of a particular r-automorphic Poincare series on G whose Fourier expansion is closely related to the Kloosterman part of (2.13). Thus the discrete series representations come into play in a way as natural and indispensable as unitary principal series representations. Technically, the main subject of [7] was to compute projections of the Poincare series to irreducible subspaces of I?(T\G) in such a way that the resulting spectral decomposition is explicit with respect to the seed function which is related to / , that is, the task was to fix the kernel of the relevant integral transform of / . To this end, a use was made of the theory of the Kirillov model; and in particular the Bessel transform K got a clear-cut structural explanation as being an analytic expression of the action in the model of the Weyl element w (see (1.7)) and thus essentially the same as the Bessel function of representations in the sense of [8] and as the Mellin inversion of the T-factor of the local functional equation introduced in [9] (see also [13]). In a similar context, it is worth stressing that the theory has yielded a new approach [6] to the fourth moment of the Riemann zeta-function, which is far more transparent than the argument developed by the author in [12], and suggests further developments (see [15]). (3) However, there has remained a basic problem, that is, the derivation of (2.4) from (2.12) or (2.13); such is highly desirable, as (2.4) is equally fundamental in applications. Section 5 solves this and closes the discussion of [7] as far as the two sum formulas of Kloosterman sums are concerned; to achieve this was a motivation of the present work. In other words, both Bruggeman-Kuznetsov's and Cogdell-Piatetski-Shapiro's approaches come to the same conclusion, if we restrict ourselves to those arithmetic configurations represented by (1.1). (4) The formula (4.19) is a completion of the Webb-Kapteyn theory of Neumann expansions with /-Bessel functions of odd integral order. See [16, Section 16.4], where a sceptical opinion about the value of relevant works is expressed. This episode makes the inversion formula (4.19) just

Sums of Kloosterman Sums Revisited

161

more significant, for it is one of the most basic implements which have brought about recent profound changes in Analytic Number Theory. (5) Most probably our discussion extends to the situation with G = PSL2(C) and r being Bianchi groups. The extension of the sum formulas (2.8) and (2.13) in the case of the Picard group r = PSL2(Z[/]) is, in fact, fully discussed in [5] but the analogue of the latter formula is deduced from that of the former; thus the procedure is similar to Bruggeman-Kuznetsov's. It is worth extending the argument of [6]. A preliminary work for this purpose has already been done in [4] and [15]. Further extensions to the situation like that dealt with in [3] should be of considerable interest. Lastly, we mention that an inversion of the explicit formula for the fourth moment of the Riemann zeta-function might be discussed along the lines developed in Section 5. Acknowledgements. The present work incorporates expert comments provided generously by R.W. Bruggeman, which the author appreciates greatly. He is indebted also to the organizers of the conference to which this volume is attributed for the financial aid that made his participation possible.

References [1] R.W. Bruggeman. Fourier coefficients of cusp forms. Invent, math., 45 (1978), 1-18. [2] R.W. Bruggeman. Fourier Coefficients of Automorphic Forms. Springer LNM, vol. 865, Springer-Verlag, Berlin etc 1981. [3] R.W. Bruggeman and R.J. Miatello. Sum formula for SL2 over a totally real number field. To appear in Memoirs AMS. [4] R.W. Bruggeman and Y. Motohashi. A note on the mean value of the zeta and L-functions. XHI. Proc. Japan Acad., 78A (2002), 87-91. [5] R.W. Bruggeman and Y. Motohashi. Sum formula for Kloosterman sums and the fourth moment of the Dedekind zeta-function over the Gaussian number field. Functiones et Approximatio, 31 (2003), 7-76. [6] R.W. Bruggeman and Y. Motohashi. A new approach to the spectral

162

Y. Motohashi

theory of the fourth moment of the Riemann zeta-function. J. reine angew. Math., 579 (2005), 75-114. [7] J.W. Cogdell and I. Piatetski-Shapiro. The Arithmetic and Spectral Analysis of Poincare series. Perspectives in Math., vol. 13, Academic Press, San Diego 1990. [8] I.M. Gel'fand, M.I. Graev, and LI. Pyatetski-Shapiro. Representation Theory and Automorphic Functions. W.B. Saunders Company, Philadelphia 1969. [9] H. Jacquet and R.P. Langlands. Automorphic Forms on GL(2). Lecture Notes in Math., vol. 114, Springer-Verlag, Berlin etc. 1970. [10] N.V. Kuznetsov. Petersson hypothesis for forms of weight zero and Linnik hypothesis. Khabarovsk Complex Res. Inst. Acad. Sci. USSR. Preprint 1977. (Russian) [11] N.V. Kuznetsov. Petersson hypothesis for parabolic forms of weight zero and Linnik hypothesis. Sums of Kloosterman sums. Math. USSR-Sb., 39 (1981), 299-342. [12] Y. Motohashi. Spectral Theory of the Riemann Zeta-Function. Cambridge Tracts in Math., vol. 127, Cambridge Univ. Press, Cambridge 1997. [13] Y. Motohashi. A note on the mean value of the zeta and L-functions. XII. Proc. Japan Acad., 78A (2002), 36-41. [14] Y. Motohashi. A functional equation for the spectral fourth moment of the modular Hecke L-functions. Proc. MPIM-Bonn Special Activity on Analytic Number Theory, Bonn 2002, Bonner Math. Schrift., vol. 130 (2003), 19 pages. [15] Y. Motohashi. Mean values of zeta-functions via the representation theory. To appear in Proc. Symp. Pure Math., AMS, Workshop on Multiple Zetas, Bretton Woods 2005. [16] G.N. Watson. A Treatise on the Theory of Bessel Functions. Cambridge Univ. Press, Cambridge 1996.

Sums of Kloosterman Sums Revisited

Yoichi Motohashi Department of Mathematics, Nihon University, Surugadai, Tokyo 101-8308, Japan Email: [email protected]

163

The Lindelof Class of L-Functions K. MURTY

We define a class of functions that contains the Selberg class and which has a natural ring structure.

0

Introduction

About 15 years ago, Selberg [3] formulated a class of L-functions having properties enjoyed by L-functions that arise in arithmetic and geometry. He formulated two key conjectures about this class. A few years later, Ram Murty [2] showed that these conjectures of Selberg have far reaching implications. In particular, he showed that the holomorphy of Artin Lfunctions belonging to solvable Galois extensions of Q would follow from these conjectures. The Selberg class, while capturing many of the features of familiar Lfunctions, has several shortcomings. One of these is the lack of a ring structure. Ram Murty asked if one could formulate a larger class of Lfunctions that would have a ring structure. This note is a possible answer to this question. We define the Lindelof class which contains the Selberg class. This class has a natural ring structure, though we lose the key properties of Euler product and functional equation enjoyed by elements of the Selberg class.

1

The Selberg class

Selberg [3] defined the class S of L-functions F(s) satisfying the following properties: 165

166

K. Murty

(51) Dirichlet series: can be expressed as a Dirichlet series

n=\

in the region Re(s) > 1 satisfying the following properties: (52) Analytic continuation: for some non-negative integer m, {s- l)mF(s) extends to an entire function; (53) Functional equation: there exist non-negative real numbers Q and a,, complex numbers /i, with Reijii) > 0, and w satisfying |w| = 1 such that

(54) Euler product: there is an Euler product of the form

F(s) = Y\Fp(s) P

where oo

k=l

wiihbpK =

r y

0(pk9),6
(55) Ramanujan hypothesis: for any e > 0, a„(F) = 0(n€). The implied constants in the estimates of (S4) and (S5) depend on F. The parameter Q that appears above in (b) is usually referred to as the "conductor" of F and the degree dp of F is denned as dF = 2^or,-. i

It is an interesting conjecture in the Selberg theory that this degree is always an integer.

167

LindelofClass ofL-Functions

2

The Lindelof class

The Selberg class «S is closed under product but not under sums. In this section, our aim is to define a larger class that is closed under sums and products. This larger class will be defined in terms of growth conditions. Towards this end, we begin by defining the // function.

2.1

The function fiF(o~)

For F e S, let us define the function fiF as follows: /uF(cr) = inf{A>0 : \F(s)\ «: (|s| + 2)X for all s with Re(s) = cr}. Note that in this definition, the implied constant is independent of s. Since the Dirichlet series defining F(s) converges absolutely for Re(s) > 1 (by (S5)), we have (2.1)

nF(cr) = 0 for cr > 1.

By the functional equation (S3), we also get that (2.2)

juF(<7-) < -dF(\ - 2cr) for a < 0.

Indeed, by Stirling's formula

rfeU-sHA) Tiats+fi,)

(a,-(|f| + 2)ff(1-2
From the Phragmen LindelSf theorem, for c < cr < d, we have (2.3)

fiF(cr)
fiF(d)^^. a—c

Thus, for 0 < cr < 1, we have (2.4)

fiF(cr) <

l

-dF{l-
168 2.2

K. Murty Nonvanishing in a half plane

From (SI), it follows that F does not vanish in a half plane. Indeed, by (S5), the Dirichlet series defining F converges absolutely for cr > 1. Then, forcr > 1, we have n>2

and so for any cr > a > 1, this is

|1 Fv ( 5 ) l > 1 _ i _ e '

2 ~

'

n

n>2

if cr is sufficiently large, say cr > cr0. Hence, F(s) * 0 in this half plane. It is clear that it is not essential to assume that the Dirichlet series converges absolutely for cr > 1. From (SI), it converges for cr > 1 and therefore converges absolutely for cr > 2. Also, it is not essential that « F ( 1 ) = 1- We only need some non-zero Dirichlet coefficient and this exists if F ± 0. 2.3

The number of zeros

Next, let us denote by NF(T) the number of zeros of F(s) with 0 < cr < 1 and \t\ < T. From (S2), it follows that NF(T + 1) - NF(T) «: log T. Indeed, let CTQ be such that F(s) + 0 for Re(s) > CTQ. Set np(r) to be the number of zeros of F{s) in the circle of radius r with center CTQ + 1 + iT. Take R > e ^(crQ + l) 2 + 1. Then, by Jensen's theorem f ^ ^ d r = — f log\F(cr0 +l+iT + Reie)\dG - logF(cr0 +1 + iT). Jo r 2n Jo In (To + 1 - R < cr < (To + 1 + R, we have for some A = A(F), by (2.2) and (2.3), (2.5)

\(s - l)mF(s)\ « F (\s\ + 2)A.

Hence, on the circle above log \F(crQ + 1 + iT + Rei6)\ < A log T + O(l).

169

Lindelof Class of L-Functions Thus,

CR nF(r) r)

Jo

r

dr<& logT.

In particular (2.6) < nF(yj(o-0+l)2

NF(T+\)-NF(T)

+ l)<

\

^-^-dr

Jo

r

<£ log T.

Hence, for some c > 0, NF(T) < Define

^TlogT.

2nNF(T) 5F = h m s u p r ^ ^ ^ ^ r .

For F ecr, note that 6F = dF. 2.4

The class M

Let us define the class M to be the set of functions F(s) satisfying the following conditions: (Ml) Dirichlet series: For cr > 1, F(s) is given by the absolutely convergent Dirichlet series 2 aF{n)n~s. (M2) Analytic continuation: F can be continued as a function of order 1, analytic apart possibly for a pole at s = 1. (M3) Growth condition: The quantity fiF((r)/(l - Ixf) is bounded for cr < 0. (M4) Ramanujan hypothesis: \aF(n)\ 0. From the arguments of the previous paragraphs, we see that for an element F € M, the function fiF and the constant 6F are defined. For an F in M, let us define cF = limsup^Q

2fiF(cr)

l-2o-'

170

K. Murty

Note that cF > 0. Moreover, for 0 < cr < 1, we have 1*F(CT) < -cF(\

- cr).

Example 1. Consider F(s) = 1 - 2~s. For cr < 0, we have |(r|

( \*

log max(|cr|, 2) and in particular, F e M and cF = 0. Example 2. By similar reasoning, any Dirichlet polynomial F is in M and has Cf = 0.

Example 3. If F(s) = £(s), the Riemann zeta function, then CF = 1. Moreover, for any constant a, the function F(s) = a + £(s) is also in At and has Cf = 1.

The following is evident from the definition. Proposition 1. CFG = Cf + Co

and cF+G < max(cF,cG)From these, it is easy to see that M is a ring. We also have a relation between the invariants cF and 8F. Proposition 2. For FeM,ifcF

= 0 then 6F = 0.

Proof. In (2.5), we can take A = e. Inserting this into (2.6), we see that NF(T + l) - NF(T) = o(logT). Hence, NF(T) = o(T log T) and so 6F = 0. Now we are ready to prove the following result. Proposition 3. Suppose F e M. Then cF>\orcF

= 0.

Lindelof Class of L-Functions

171

The proof is modeled on earlier work of Bochner, Selberg and ConreyGhosh [1]. Proof. For cr > 1, we have YaF(n)e-nx td n=l

= 2m —

Jot

F(s)r(s)x-Sds.

By (M3) and convexity, F has polynomial growth in \t\ in vertical strips. Thus, moving the line of integration to the left and taking into account the possible pole at s = 1 of F(s), as well as the poles of F(s) at s = 0, - 1 , - 2 , •••, we find

(2.7)

P(log*) Y. r( »)
n=\

+

y

+

F{-n)i

Y^W

n=0

where P is a polynomial. By (M3), \F(-n)\

As

«

n i^(i+2n) +e

c-r-

1
the «-th term in the sum on the right of (2.7) is <3C

^(l^e^^^^lf^lj".

If Cf < 1, then the series converges absolutely for all values of x. Hence, the left hand side of (2.7) is analytic in C\{x < 0 : x € R}. But it is periodic (with period 2ni) and so it converges for all x. Then, the function

is entire and for any real x and any k > 0, we have aF(n)e-"x = -L J " f(x + iyy"ydy

« n~k

172

K. Murty

by applying integration by parts repeatedly. Taking x = l/n, we deduce that a.F{n)nk «: 1 for all k > 1. Thus, the series

Z, aFJn) n s

n=l

converges absolutely for all values of s. Hence, HF{C) = 0 for all cr. We note that the above argument also includes a proof of the following result. Proposition 4. We have CF = 0 if and only ifforallk>0, 2.5

ap(n)nk «: l.

The Lindelof class

Denote by Alo the subset of M with CF = 0. It is, in fact, a subring and this can be seen from Proposition 4. It is clear that if F, G e Alo then F -G e Alo- Moreover, if F, G e Mo then for any k > 0, we have aFG(n)nk = J ] a F ( r f ) / a G ( ^ ) ( ^ ) * « £ d\n

1 « n£

d\n

for any e > 0. It follows that for any k > 0, aFG(n)nk «c 1 and so FG e MoNow let Atoo denote the non-zero elements in Mo. It is a multiplicative subset. The Lindelof class is denned as the localization

£ = M^M. In particular, elements of £, are analytic for Re(s) > 1. However, for Re(s) < 1, they are meromorphic and may have infinitely many poles. We have inclusions S ^ M ^ £. Note that »S n Alo = {1}. Also, note that by construction, the elements of Moo are units in £.

173

Lindelof Class of L-Functions

For an element G e X, we define eg as follows. We can write G = F/u where F € M and u e Aioo- We set CG

- cF.

If we can write G = H/v for some other H e M and v e Atoo, and CH > Cf, then CFV-HU = c#. On the other hand, by the definition of localization, cpv-Hu - 0 and this is a contradiction. Note, in particular, that the only elements G of X with CQ = 0 are units. Conversely, if G is a unit of X then CQ = 0. 2.6

Lindelof primitive functions

An element F e X that is not a unit is Lindelof primitive if it cannot be factored as F\F2 with F\, Fi € X with cpx,c/r2 < cp and Fi, F2 non-units. Proposition 5. Every element in X con be factored as a product of a unit and Lindelof primitive elements. Proof. Consider an F e X that is not a unit. Then CF > 1. If cp = 1, then F is Lindelof primitive so there is nothing to prove. Assume that we have proved the result for all G € X with CQ < CF- If F is not Lindelof primitive, then it can be written as F1F2 with cpx,cp1 < cp and F\,F% nonunits. By the induction hypothesis, F\ and F2 can be written as a product of primitive elements. The result follows. The natural question to ask next is whether this factorization is unique, and if so what consequences it would have.

References [1] B. Conrey and A. Ghosh, On the Selberg class of Dirichlet series: small degrees, Duke Math. J., 72(1993), 673-693. [2] M. Ram Murty, Selberg's conjectures and Artin L-functions, Bull. Amer. Math. Soc, 31(1994), 1-15.

174

K.Muity

[3] A. Selberg, Old and new conjectures and results about a class of Dirichlet series, Proc. Amalfi Conf. on Analytic Number Theory, ed. E. Bombieri et.al., pp. 367-385. (See also Collected Papers, Vol. II, pp. 47-63, Springer-Verlag, 1991.)

V. Kumar Murty Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Canada Email: [email protected]

A Proof of the Riemann Hypothesis for the Weng Zeta Function of Rank 3 for the Rationals M. SUZUKI

0

Introduction

Let F be an algebraic number field. Recently, L. Weng introduced zeta functions of rank n associated to F as a generalization of the Iwasawa-Tate zeta integral from the Arakelov geometric point of view. In the article, we call them the Weng zeta function of rank n. In the case of rank one, the Weng zeta function coincide with the Dedekind zeta function of F. The background and precise definition of Weng zeta functions were published in [4, sec.B.4]. One remarkable fact for Weng zeta functions is that we can prove the Riemann hypothesis in the case of rank 2. It had been done by the author and J.C. Lagarias in [2] for the rational number field and was extended to the case of general number field F by Weng in [4, sec.C.4]. The proof of the Riemann hypothesis for a zeta function of rank 2 depends on the explicit expression for it. In this spring, Weng obtained an explicit expression for the zeta function of rank 3 in the case of F = Q. It is given in [5] this volume. By using his explicit formula, the author proved the Riemann hypothesis for the zeta function of rank 3 over the rational number field. In the article we give the proof of the Riemann hypohesis for it and the idea of the proof in self-contained fashion, as far as possible. The zeta function £FMS) °f r a n k 3 is obtained by an integral of the completed Epstein zeta function of rank 3 over a moduli space of semistable lattices of rank 3 over F. In detail, see [5, sec.5.3, chap.9] (in [5] our £F,I(S) is denoted by feO)). In the case F = Q, Theorem 4 in [5] 175

176

M. Suzuki

assert that the explicit expression for CQAS) is given by

fe'w=( (0.1)

3 ( 7 r i r 2p7=» >?(3s) ^.I.-Ijjfe,.,) -(l7"2(37^)'?<3s-2x

where £(s) = n~s/2T(s/2)^(s) and £(s) is the Riemann zeta-function. From the general theory of zeta functions of rank n, the function ^3(5) satisfies the functional equation ^3(5) = ^ 3 ( 1 - 5 ) . Hence the Riemann hypothesis for £F,3(S) is that all zeros of £F,3(S) lie on the line Re (s) = 1/2. Using (0.1) we can prove the Riemann hypothesis for £0.3(5). Theorem 1. The function £Q,3(.S) satisfies the Riemann hypothesis. That is, all zeros ofgqjis) lie on the line Re(s) = 1/2 which is the central line of the functional equation £Q,3(S) = £Q,3(1 — s). The article is organized as follows. In section 2, we explain the idea of the proof. In section 3, we state and prove Theorem 2 which is a easily version of Theorem 1. We describe the frame of the proof of Theorem 1 by proving Theorem 2 by using Lemma 1 and Lemma 2. In section 4, we give the proof of Theorem 1. It is proved by using Lemma 3 and Lemma 4 whose are modifications of Lemma 1 and Lemma 2. In section 5 we give the proof of Lemma 1 which is also used in the proof of Lemma 3. In section 6, we give the proof of Lemma 5 which is used in the proof of Lemma 4. In section 7, we comment on a possibility of applications of Theorem 1 and Theorem 2 to the original Riemann zeta function.

Acknowledgment. The author thanks professor Lin Weng for his helpful information on zeta functions of rank 3 and encouragement for the proof of the Riemann hypothesis.

111

Riemann Hypothesis for Rank 3 Zeta

1 The idea of the proof For simplicity, we denote by Zi(s) the function £Q,30):

Z3(s) = 7(2) • - i - • 7(3*) - 7(2) • ~ 7(3* - 2) 3s-3

(L1)

3^

4~-a3.-i)-5-~«3-i>

By using the well-known functional equation £(s) = £(1 - s), we can directly check that Zs(s) has the functional equation Z3O) = Z3(l - s). Since £(s) is holomorphic except for two simple poles at s = 0, 1 with residues - 1 and 1 respectively, the possible poles of Z3O) are s = 0, 1/3, 2/3, 1. We can easily check that the possible poles 5 = 1/3, 2/3 are removable and s = 0,1 are simple poles of Z${s). To investigate the zeros of Z3(s), we consider the entire function &(*) = 3s(3s - l)(3s - 2)(3s - 3)Z3(s) (1.2)

=^

• (3* - 2) • f(3s) + &±.(l-3s).

£Qs - 2)

+ f(3*-l) + | - j . « 3 5 - 2 ) + |-(l-s).f(3*), where £(s) is Riemann's xi-function g(s) = s(s - 1)£0). To describe the idea of the proof of Theorem 1, we recall the proof of the Riemann hypothesis for the zeta function £Q,2(S) of rank 2. The key of the proof was in the explicit expression ft(j) = 2s(2s - l)(2s - 2)7Q>2(*) = &2s) - £(2s - 1). The right-hand side has the properties (A) g(2s) and g(2s-1) are related by the functional equation £(2(1 -s)) = £(2*-l), (B) all zeros of g(2s) lie in the strip 0 < Re (s) < 1/2.

178

M. Suzuki

By using these properties we derived \g(2s - l)/1f(2s)\ < 1 for Re (s) > 1/2 and |f(2s)/£(2s - 1)| < 1 for Re (s) < 1/2. These inequality and (B) gave the Riemann hypothesis for £Q,2(S)- Here we note one more important property, (C) £(2s) has the functional equation £(2(± - s)) = €(2s). The role of (C) was hidden in the back of (B) in the proof of the Riemann hypothesis for £Q,2(S). However the property (C) is the key of the generalization to the proof of rank 3 case. The idea of our proof of Theorem 1 is to recast the right-hand side of (1.2) into the form fr(s) = Y(s) + F(l - s) so that all zeros of Y(s) are in some vertical strip on the left of the line Re(j) = 1/2. To obtain such expression, we notice properties (A) and (C) in rank 2 case. If we ignore polynomial factors, the right-hand side of (1.2) consists of £(3s), g(3s - 1) and £(3s - 2). Using functional equations (1.3)

#3(1 - s)) = £(3s - 2)

and (1.4)

f(3(l-s)-l) = f(3s-l),

we can easily obtain an analogue of (A) for &(s). However such expression has many possibilities. Here we notice the functional equations (1.5)

£(3( | - * ) ) = # 3 * - 1 )

and (1.6)

£(3(^-s) -2) = £(3s-l).

These extra functional equations allow us to recast the right-hand side of (1.2) as £3(5) = Y(s) + y(l - s) so that Y(s) satisfies an analogue of (B). In this process, the extra functional equations (1.5) and (1.6) play a role of (C). At the time, we obtain Theorem 1 by a way similar to the proof of rank 2 case. Now we put

(1.7)

h(s) = {|(f(2) - 1). - m) - I ))&3s).

179

Riemann Hypothesis for Rank 3 Zeta

This h(s) consists of all terms in the right-hand side of (1.2) containing £(3s). We have (1.8)

A(l - s) = {|(^(2) - 1)(1 -s)- m) - \)}SOs-

2).

This coincides to all terms in the right-hand side of (1.2) containing £(3s 2). Further (1.9)

A ( | - s) + h(s- i ) = (1 - |(f(2) - l ) ) £ ( 3 s - 1).

Hence if we take (1.10)

X(s) = h(s) + Jr((2/3) - s),

then we have (1.11)

£,(*) - X(J) + X(1 - 5) + ^ # 3 * - 1).

Here we used £(2) = 7r/3. Since £(3s - 1) = £(3(1 - s) - 1), this equality also shows the functional equation fr(s) = £j(l - s). By taking (1.12)

Y(s) = X(s) +

^-{(3s-\),

we obtain (1.13)

&(s) = Y(s) + Y{l-s).

As shown in below this expression satisfies properties analogous to (A) and (B).

2 The frame of the proof To describe the frame of our proof of Theorem 1, we give the proof of the following result which is similar to Theorem 1. Theorem 2. Let X(s) be the function defined by (1.10) with (1.7). If we take £3(5) = X(s) + X(l - s), then all zeros ofQ(s) lie on the line Re (s) = 1/2.

180

M. Suzuki

To prove Theorem 2, we prepare following two lemmas. Lemma 1. Let F{s) be an entire function of genus zero or one, that has the following properties. (i) F(s) is real on the real axis and satisfies afunctional equation (2.1)

F(s) = ±F(l - s),

for some choice of sign. (ii) There exists a > 0 such that all zeros ofF(s) lie in the vertical strip (2.2)

|Re(j)-l/2|
Then for any real c I>a, (2.3)

F(s + c) >1 F(s - c)

for

Re(*)>l/2,

F(s + c) < 1 for F(s - c)

Re(j)
and (2.4)

In particular, all zeros ofF{s + c) ± F(s - c) lie on the line Re (s) = 1/2. Proof. We prove the lemma in section 5.

•

Lemma 2. All zeros ofX(s) lie on the line Re (s) = 1/3. Remark. From definition (1.10), X(s) has the functional equation X(s) X(2/3 - s). The line Re (s) = 1/3 is the center of the functional equation. Proof. We put x\s)

=

X(s-l/6).

Then Lemma 2 is equivalent to the assertion that all zeros of X\s) lie on the line Re(s') = 1/2. Since

(2.5)

+ {(^)(f-»)-(f-f)}tf3'-l).

181

Riemann Hypothesis for Rank 3 Zeta we have

&(*) = { i ^ ) ( s ~ \) ~ (f " | ) } • «3(* - 1/6)) (2.6) x 1+

("arXl - (J - ^)) - (f - 1 )g(3(J -1/6) - 1 ) '

If we take F(s) = £(3s - 1), then F(s) = F(l - J) and all zeors of F(s) lie in the strip |Re(5) - 1/2| < 1/6. Applying Lemma 1 to F(s) with c = 1/6, we obtain (2.7)

1>

F(s-l/6) F(J+1/6)

£(3(s-l/6)-l) £(3(s-l/6))

for

Re(j)>l/2.

On the other hand, we find that

(¥)(§-(*-i))-(f-j) (^rX^-^-d-l)

(2.8)

<1

for

Re(s)>l/2

by an elementary calculation. Since

{ ( ^ ) ( ^ - | ) - ( f - |)}^(3(^- 1/6)) has no zeros in the right-half plane Re(s) > 1/2, equality (2.6) and (2.7), (2.8) implies X\s) has no zeros in the right-half plane Re (s) > 1/2. The functional equation X\s) = X\l - s) shows that X\s) also has no zeros in the left-half plane Re Re (s) < 1 /2. Thus we obtain the assertion of Lemma 2. • 2.1

Proof of Theorem 2

We have

4(s) = X(5)|l (l ++ X ( 1

^4

We show that (2.9)

X(l - 5) <1 X(s)

for

Re(s)>l/2.

182

M. Suzuki

This yields the non-existence of the zeros of Q(s) in the right-half plane Re(s) > 1/2, since X(s) has no zeros in the right-half plane Re (s) > 1/3 by Lemma 2. We put F(s) = X(s-1/6). The functional equation X(s) = X((2/3) - s) yields F(s) = F(l - s). By Lemma 2, all zeros of F(s) lie on the line Re (s) = 1 /2. These means that F(s) satisfies all conditions of Lemma 1 for any a > 0. Applying Lemma 1 to F(s) with c = 1/6, we obtain F{s-1/6) F(s+1/6)

<1

for

Re(s)>l/2.

Using X(s) = X((2/3) - 5) we have FQ - 1/6) _ X(s - 1/3) _ X(l - s) F(s+1/6) ~ X(s) " X(J) ' Thus we obtain (2.9). Because of the functional equation ^(s) = £J(1 - 5), we also obtain the non-existence of the zeros of Q(s) in the left-half plane Re (s) < 1 /2. Now we complete the proof of Theorem 2. • As the above, the proof of Theorem 2 consists of two steps. These two steps correspond to different two symmetries. Lemma 1 is used for each symmetry. In the first step corresponding to X(s), we used Lemma 1 for h(s)+h((2/3)-s) which yields the functional equation X(s) = X((2/3)-s). In the second step corresponding to Q(s), we used Lemma 1 for X(s) + X(l - s) which yields the functional equation tj\(s) = ^ ( 1 - s). This is the rough frame of the proof of Theorem 1.

3 Proof of Theorem 1 Let Y(s) be the function defined by (1.12). Recall equation (1.13); 6(5) = Y(s) + Y(l - s). Since Y(s) does not have a functional equation, to prove Theorem 1, we need the following modification of Lemma 1.

Riemann Hypothesis for Rank 3 Zeta

183

Lemma 3. Let F(s) be an entire function of genus zero or one. Suppose that (i) F(s) is real on the real axis, (ii) there exists CTQ < 1/2 such that all zeros of F(s) lie in the vertical strip 0-0
(3.1)

(Hi) there exists C > 0 such that (3.2)

N(T) < CT log T

as

T -> oo,

where N(T) is the number of zeros p ofF(s) satisfying o~a < Re (p) < 1/2 and 0 < Im(p) < T. Further F(\ - o~)/F(cr) > 0 for large o~ > 1/2 and (3.3)

F(l-o-) F(o-)

0

as

1/2 < cr —> oo.

Then we have (3.4)

F(l - s) < 1 for F(s)

Re(s)>l/2,

F(l - s) > 1 for F(s)

Re(s)
and (3.5)

In particular, all zeros ofF(s) ± F(\ - s) lie on the line Re (s) = 1/2. To apply Lemma 3 to Y(s), we prepare the following Lemma 4 which is a modification of Lemma 2. Lemma 4. All zeros ofY(s) lie in the strip 1/3 < Re (s) < 1/2.

184 3.1

M. Suzuki Proof of Theorem 1 under Lemmas 3 and 4

Now we give the proof of Theorem 1 by using Lemma 3 and Lemma 4. We prove these lemmas after the proof of the theorem. By definition of Y(s), it is an entire function which is real on the real axis. By Lemma 4, Y(s) satisfies (ii) of Lemma 3. Hence if (3.2) and (3.3) are shown for Y(s), then we obtain Theorem 1 by Lemma 3. At first we prove (3.2). It is well known that \f;(s)\ < exp(M|s| log \s\) as \s\ —> oo for some constant M > 0. Since Y(s) is a linear combination of g(s) up to degree one polynomials, we have \Y(s)\ < exp(M'|j|log|5|) as \s\ —» oo for some constant M' > 0. Recall Jensen's formula for entire function f(s) with /(0) + 0: r'ln(r)dr = -L J * log \f(Reie)\ d6 - log |/(0)|,

J

where n(r) is the number of zeros in \s\ < r. We apply Jensen's formula to Y(s). Then the right-hand side is estimated as «: RlogR for sufficiently large R. Since J^ r~xn{r)dr > n(R) log 2, it follows that n(R) = 0(7? log/?). Hence we obtain (3.2). From the definition of Y(s) we have y(l-cr)

£(3cr-l)

\ 2

Since f(s) = s(s - 1)TT-S/2T(S/2)£(S), £(3o--l) £(3cr)

=

4"

> v2

6"

^

we have

w _ 2vr((3o--l)/2)f(3o--l) V 3tr' T(3cr/2) ^(3cr) '

W

Using the Stirling formula and the Dirichlet series expansion of £(s), we have r

«3-1)/2 r(3o-/2)

)

=

g

- ^ " > i + ^ - ; ) and ^ 1 ) V^ 1 + O(o-!) C(3o-)

=

j

+

^

for large cr > 1. Hence we obtain 7(1 - cr)/Y(cr) > 0 for large <x > 1 and Y(l-
___1/2 = 0(
as cr -> +oo.

185

Riemann Hypothesis for Rank 3 Zeta

Thus (3.3) holds for Y(s). Now we complete the proof of Theorem 1 under Lemma 3 and Lemma 4. o 3.2

Proof of Lemma 3

We prove the lemma only if F(s) has genus one, since if F(s) has genus zero it is easily proved by a way similar to the case of genus one. The genus one assumption is equivalent to the assertion that the Hadamard product factorization F(s) = eA+BssmY\(l--)ep

(3.6)

(m e Z>0)

p

converges absolutely and uniformly on any compact subsets of C. This assumption is also equivalent to the bound £ p \p\~z < oo. Assumption (i) implies symmetriy of the zeros under p H-» p. It follows that the set of zeros p = B + iy, counted with multiplicity, can be partitioned into blocks B(p) comprising [p,p\ if y > 0 and {p} if )S £ 0 and y = 0. Each block is labeled with the unique zero in it having B < 1/2 and y > 0. Using assumption (ii), we show F(s) = s*"eA+B'*Y]

(3.7)

fit

1

--)

Bip) KpeB(p)

where the outer product on the right-hand side converges absolutely and uniformly on any compact subsets of C. This assertion holds because the block convergence factors exp(c(B(p))s) are given by c(B(p)) = 2B\p\~2 for y > 0. Assumption (ii) gives cr0 < B < 1/2. Hence

2 \c(B(p))\ < B(p)

J]

^l_1 + max{1' 2|crol}( Z

0*pe(o- 0 ,l/2)

lpl 2

~ ) < °°-

p

Thus the convergence factors can be pulled out of the product. Hence we have (3.7) with (3.8)

B' = B + J]c(B(p)). Bip)

186

M. Suzuki

To establish (3.4) we proceed block by block in (3.7), using the factorization F(\ - s) F(s)

(3.9)

^rn n

aB'(l-2Re(s))

11

B(p) \peB(p)

1-s p _ £ p

Using assumption (iii), we show (3.10)

B' > 0.

We will prove this later. If (3.10) is shown, we have (3.11)

e

1 — Sm <1

B'(l-2Re(s))

for

1 Re(^)>-.

In a single block we can clear denominators to obtain

n

1-1=1

>-J

peB(p)

s-\+p

•n

P

s-p

pzB(p)

We compare the term in the numerator with p against the term in the denominator with p' := 1 - p. We find that + P

(3.12)

<1

for

Re(s)>-.

s-p In fact, writing s = cr + it, we have s-l+p2

(o--l+B)2 2

(o--B)

s-p

+ +

(t-y)2 - V > 2: (t-y)

= 1-

(l-2y8)(2cr-l) (cr-y8)2 + ( r - r ) 2 -

This imphes (3.12) since B < 1/2 by assumption (ii). Thus we conclude for Re (s) > 1/2 that the absolute value of the product over terms in each block on the right in (3.9) is less than 1. Hence (3.4) holds by (3.11) and (3.12). Therefore it remains to show (3.10). By (3.3), we have (3.13)

R 3 log(

F(l-tr) j —» -oo F((T)

as

1/2 < cr —> +c

Riemann Hypothesis for Rank 3 Zeta

187

Using (3.3) and (3.7), we have F(l-cr) F<<(r)

_

2 _,_ „,2

B,{1_2(r),cr-1

\ cr ) _U„

p=/?eR

cr-B

11

(cr-^)2+r2

y>0

Thus (3.14)

M,F(l-cr), ^ ) = ^-^)-log(l-i)+ Z log(l-^|) p=/3eR

+ l'0g('p=P+iy

(l-2/?)(2
(cr-^

+ y2

)

y>0

Here we note that log(l -

(l-2/?)(2
for cr> 1/2

by assumption (ii). Suppose that B' < 0. Then (3.13) and (3.14) claim that

J>g(l-

(3.15)

(\-2B)(2cr-\)

p=P+iy y>0

2 j. (cr-Bf+y

„,2

> 2\B'\cr

for sufficiently large cr > 1/2, because the number of real zeros of F(s) is finite by assumption (ii). On the other hand, for large cr > 1/2, we have (l-2B)(2cr-l) I M

Z M / - (i-zp)^cr--m2 1

p=0+iy y>0

(o

+ y2

)

E Mi

p=P+iy y>0

(l-2a-(,X2o--l) 2 (cr- 1/2)2 ++ yr )

«(2o--l) V 5-^— p=p+iy Ay>0( ^ - l / 2 ) 2 + r The sum in the right-hand side can be written as the Stieltjes integral

r J 70

dN(t) (cr -1/2)22 +, tf:2 -

188

M. Suzuki

Using (3.2) we have r°° dN(t) JY0 (cr - 1/2) 2 + t2

C (log 0 dt log(o- + yo) L (" (22 <

Hence we obtain

I ^

(3.16)

(l-2yg)(2cr-l) (o--^)2+r2

«: log(cr + yo)

y>0

for sufficiently large cr > 1/2. This contradict (3.15). Thus (3.10) holds. Inequality (3.5) is proved by a way similar to the proof of (3.4). We complete the proof of Lemma 3. • 3.3

Proof of Lemma 4

Now we complete the proof of Theorem 1 by proving Lemma 4. We put (3.17)

,<„-<4V(Hx

Then X(s) and Y(s) are written as (3.18)

X(s) = p(s)f(3s) + p((2/3) - s)t(3s - 1)

and (3.19)

Y(s) = p(s)&3s) + {p(--s)

+

Z^-}&3s-l).

We find that (i) the only one zero of p(s) is

2?r-9

=* -6.40,

2 1 (ii) the only one zero of p( — - s) is 3

O

(iii) the only one zero of p( - - s) H 3

7T — 3 ^

=* 7.06, 1

— is =* 7.56. 4 2n - 6

189

Riemann Hypothesis for Rank 3 Zeta

At the first, we show that Y(s) has no zeros in the left-half plane Re (s) < 1/3. To prove the assertion, we divide Y(s) as lp(~s)+7^}f(3s-\){l+

Y(s) =

pis) p((2/3)-s) +

ti3s) > (7r-3)/4'£(3s-iy

Here we note that {p(\-s)+n-^)i;(3s-\) has no zeros in the left-half plane Re is) < 1 /3. Therefore if we show that (3.20)

pis) p((2/3)-s) + i7r-3)/4

ti3s) < 1 for f(3s-l)

Re (s) < -,

then Y(s) has no zeros in Re is) < 1/3. By a way similar to the proof of Lemma 2, #3*) < 1 for Re(s) < -. f(3*-l) On the other hand, from (i) and (iii), we have pis) pi(2/3) -s) + in- 3)/4 "

s-j2n9)/(3*r - 9) s - in - l)/(2*r - 6) '

Thus Pis) p«2/3)-s) +

in-3)/4

<1

for

Reis)
2 ^ 12(;r- 3)

0.583.

Hence we obtain (3.20). Next, we prove that Y(s) has no zeros in the right-half plane Re is) > 1/2. To prove the assertion, we divide Y(s) as Y(s) = p(s)Z(3s)-(\ +

p((2/3)-s)

+ (n-3)/4 Pis)

f(3s-l) &s. •) >

Here we note that p(s)^(3s) has no zeros in the right-half plane Re is) > 1/3. Therefore we show that (3.21)

Vis) = 1 +

PH2/3)-s)

+ in-3)/4 Pis)

£(3s-l) ZOs)

190

M. Suzuki

has no zeros in the right-half plane Re (s) > 1/2. By a way similar to the proof of Lemma 2, we obtain «3*-l)

< 1 for

Re(s) > -. 3

«3J)

On the other hand, we obtain / * ( 2 / 3 ) - j ) + (ff-3)/4 < 1 for P(s)

Re (s) > ??.

2

12(TT -

? , * 0.583 3)

by an elementary calculation. Thus V(s) + 0 for Re (s) >

7*-21

I2(n - 3) Hence to complete the proof of Lemma 4 we should prove that V(s) has no zeros in the strip 1 „ ., 7;r - 21 -
(3.22)

Let/? be the rectangle [l/2,(7^-21)/(12(^-3))]x[-15,15]. Clearly V(s) is holomorphic in R. From a numerical computation (for example by using MATHEMATICA), we obtain

_L f YL (s)ds = 0 2ni JdR V where dR is the boundary of R. This means V(s) £ 0 in R by the argument principle. Now we prove V(cr + it) + 0 for cr in (3.22) and |/| > 15 by showing that (3.23)

#3s-l)

p((2/3)-s) + (7T-3)/4

#3*)

P(s)

<1

holds in this region. In the proof of Lemma 2, we obtained (3.24)

£(3*-D

t(3s)

< 1 for

Re (s) > -

by using the method established in [2], which is also explained in section 5. In detail, (3.24) is obtained by showing that this inequality holds term-byterm in the (modified) Hadamard product on a zero-by-zero basis, where

191

Riemann Hypothesis for Rank 3 Zeta

the zero p = /? + iy of g(s) in the numerator is paired against the zero p' = 1 -/? + iy in the denominator. That is, we established (3.24) using the fact that 3s-l-p' (3.25) < 1 for R e ( s ) > - . 3* - p Hence, to establish (3.23), it suffices to show that for each s in (3.22) with |/| > 15 there exists a zero p of g(s) such that (3.26)

3s-I-p' 3s - p

/?((2/3) - s) + (*r - 3)/4 < 1. P(s)

Then (3.25) and (3.26) give (3.23). To prove the existence of a zero p of £(s) satisfying (3.26), we need the following lemma which will be proved in section 6. Lemma 5 (Lagarias). For any real \t\ > 14 there exists a zero p = J3 + iy of g(s) such that 0 < /? < 1/2 and \t-y\< 5. For each s = cr + it satisfying \t\ > 15, Lemma 5 gives a zero p =fi+ iy with 0 < /? < 1/2 and \3t-y\ < 5. Therefore, to prove (3.26), it is sufficient that (3 21) "

3o--2+/3 + it0 3or-p + it0

P(s) ' p((2/3) -s) + (n- 3)/4

for any 1/2 < o- < (In - 21)/(12TT - 36), \t\ >l5,0<J3<\/2 Denote by \i and v the zeros of p(s) and p((2/3)-s)+(n-3)/4, By squaring the inequality, (3.27) is equivalent to (3.28)

(3o--2+yS) 2 + , o

(3cr-py

+ t.

<

(
2

and \t0\ < 5. respectively.

+ t2

(o- --v >v)22 + t2

For fixed * and ft it suffices to prove (3.27) with to = 5, for it would then hold for |f0| < 5, using the fact that (3cr-/?)2 > (3cr-2+/3)2 for Re (s) > 1/3. Next, when to = 5, it suffices to verify the inequality for/3 = 1 / 2 since if it holds there then the left-hand side of (3.28) with to = 5 increases as/? > 0 increases, while the right side is fixed. To establish (3.27) it thus suffices to verify the inequality (3.29)

(3cr - h2 + 25 (3o- - i ) 2 + 25

(o--p)2 + t2 (a- - v)2 + t2'

192

M. Suzuki

The left-hand side of (3.29) decreases as cr increases in 1/2 < cr < (7TT 21)/(12TT - 36). On the other hand, for fixed \t\ > 15, the right-hand side of (3.29) increases as cr increases in 1/2 < cr < {In - 2\)l{\2n - 36). Hence to establish (3.29) it suffices to verify the inequality (3 30) K

}

25 26

<

( ( 1 / 2 ) - j i f + i' ((l/2)-v)2 + f2-

By an elementary calculation, inequality (3.30) is valid for \t\ > 3. Now we obtain (3.26) and complete the proof of Lemma 3. •

4

Proof of Lemma 1

In this section, we give a proof of Lemma 1 according to Lagarias-Suzuki [2]. The genus one assumption is equivalent to that the Hadamard product factorization (4.1)

F(s) = eA+Bssm [~](l - - )e? p

p

converges absolutely and uniformly on any compact subsets of C. This assumption is also equivalent to the bound £ p Ipl-2 < °°- Assumption (i) implies symmetries of the zeros under p H-» p andp i-» 1 - p . It follows that the set of zeros p - B+iy, counted with multiplicity, can be partitioned into blocks B(p) comprising {p, 1 - p,p, 1 - p ) if/? + 1/2, {p, 1 - p ) if/3 = 1/2 and y ^ O , and {p} if p = 1/2. Each block is labeled with the unique zero in it having/? < 1/2 and y > 0. Using assumption (ii), we show

(4.2)

F(s) = eA+B'*Y] f i t 1 - - ) B(fi)[peB(fi)

P

.

where the outer product on the right-hand side converges absolutely and uniformly on any compact subsets of C. This assertion holds because the block convergence factors exp(c(B(p))s) are given by \B\p\-2 + {\-B)\\-p\-2 c(B(p)) = {\p\-2 2

if B* 1/2, if B = 112 and y * 0, ifp = 1/2.

193

Riemann Hypothesis for Rank 3 Zeta Assumption (ii) gives -a < B - 1/2 < a. Hence 2

\c(B(p))\ < (1 + 2a)( £

B(p)

]p\~2 ) < oo.

P

Thus the convergence factors can be pulled out of the product. Hence we have (4.2) with B' = B + £ B(p ) c(B(p)). Using the functional equation (2.1) we infer that B' = 0 in (4.2), so that (4.3) B(p)\peB(p)

y

.

Indeed the change of variable s H 1 - J permutes the factors in each block B(p), with a possible sign change for p = 1/2, so it must be that eA+B's = ±eA+B'V-s\ which forces B' = 0. To establish (2.3) and (2.4) we proceed block by block in (4.3), using the factorization (4.4)

F(s + c) F(s - c)

=n n

s+c

1 _

P s c ~ P

1 _

B(p) \peB(fi)

In a single block we can clear denominators to obtain

n

p£B(p)

1 _ s+ c P

i-f

s + c- p

•n

S - C-

p

peB(p)

The main point is to compare the term in the numerator with p against the term in the denominator with p ' : = l - p = l - p . We show that (4.5)

s+c- p s-c-(l -p)

2

s+c- p s-c-(l-p)

2

>1

for

Re(s)> - ,

<1

for

Re(s)<-.

and (4.6)

If (4.5) is shown, then we may conclude for Re (s) > 1/2 that the absolute value of the product over terms in each block on the right in (4.4) exceeds 1,

194

M. Suzuki

and (2.3) follows. Similarly (4.6) implies that for Re (s) < 1/2 the product of terms over each block is smaller than 1, and (2.4) follows. Therefore it remains to show (4.5) and (4.6). Writing s = cr + it, we have

(a-+c-/3y + (t-yy + (t-y)2

s+ c•

s-c-(l-p) (cr-c-l+p)2 Now (4.5) reduces to the assertion that (4.7)

(cr + c-p)2

>(cr-c-l+P)2

for

Re(s)>i

To show this we note that Re (s) > 1/2 gives cr + c-/3>

- + a - / ? > 0,

whence (4.7) makes the two assertions cr + c-j3> (T-c-

1 + p,

cr + c - / ? > -(cr - c- 1 +/?). The second of these asserts that cr > 1/2. While the first asserts that 2c > 208 - 1/2). This holds since c > a > /3 - 1/2. Thus (4.7) holds, whence (4.5) holds. A similar argument is used to establish (4.6). It reduces to the assertion that (4.8)

(cr + c-p)2

<(cr-c-\+pf

for

Re (5) < - .

We have -{cr - c - \ +P) >- +

a-p>0,

so that (4.8) is equivalent to the two assertions -0- + C+ 1 -p> -cr + c+l-p>

-(cr + c-p), cr +

c-p.

The second of these is equivalent to cr < 1/2. While the first of these is equivalent to c + 1/2 -/? > 0. This holds by our choice of c. The conclusion that all zeros of F(s + c) ± F(s - c) lie on the line Re (s) = 1/2 follows, because two terms F(s+c) and F(s-c) have different absolute values off the line Re (s) = 1/2. D

Riemann Hypothesis for Rank 3 Zeta

195

5 Proof of Lemma 5 In this section, we give a proof of Lemma 5 according to Lagarias [1]. Throughout the section, we denote by y the imaginary part of the nontrivial zeros of £(s). The spacing between consecutive ordinate y goes to zero as y —» oo. Therefore the result of Lemma 5 holds for \t\ exceeding some bound and gives an explicit bound. First, we prove that the result of the lemma holds for |?| > 14. Since the zeros are symmetric around the real axis, it suffices to consider the case t > 14. We verify the lemma directly for 14 < t < 1687T + 5 < 525 by inspection of a table of zeros of ({s). In fact there is no gap of size 5 between any consecutive zeros of £(s) starting with 72 - 21.02, and the smallest zeros have ordinates 71 =* 14.13. For the remaining range we use numerical estimates of Turing [3]. Let N(T) be the number of zeros p with 0 < Im (p) < T and let nS (T) be the argument of £(1 / 2 + iT) obtained by analytic continuation along a horizontal line from 00 + iT. We have (5.1)

N(T)

= 2K(^)

+

\+S(T),

where K(T) = (4m')_1 log(r(l/4 + 7rrr)/r(l/4 - KVT)) - (l/4)rlog;r. Theorem 1 of Turing [3] gives (5.2) 1/ 1x 0.006 K(T) = - IV r l o g T - T - - 1 + e(r) with \S(T)\ < for T > 64. 2 2' r ForSi(f) = J^ S(u)du, Theorem 4 of Turing [3] assert that if ti >t\> 168TT then (5.3)

|5Kb) - St(fi)\ < 2.30 + 0.128log(^).

Now we prove Lemma 5 under the following Lemma 6 which is shown in the end of the section. Lemma 6. Suppose that there is no zero p of ((s) with 1687T < t\ < Im (p) < t2 and that S (T) has one sign over the interval [t\, £2]- Then (5.4)

t2-h<

10/3.

196

M. Suzuki

Suppose that t > 168^ + 5 and there is no zero on [t - 2, t + 2], that is, N(T) is constant on [t - 2, t + 2]. Because of Lemma 6, inside this interval S{T) must have a zero-crossing in each subinterval of length 10/3. Hence it must have a zero-crossing at some point t\ = t + x with \x\ < 4/3. Since N(T) is constant, S(T) varies like -2K(T/2TT) - N(tt) - 1. Therefore (5.2) implies that all other zero-crossing of S(T) in [t - 2,t + 2] are localized within a distance e = 0.006/(fi log(/i/2;r)) of this one. If there were no zero on [t - 5, i], then N(T) is constant there. Hence S(T) varies approximately linearly on the interval. If t + x falls in [t - 5, t], zero-crossings of S(T) are located within 0.001 of t + x, and otherwise it has no zerocrossings. Since |x| < 4/3, S(T) has single sign on [t - 5, t - 5 + 10/3]. This contradicts Lemma 6. Thus there is a zero with the ordinate y in [/ - 5, t - 2]. By a similar argument there is a zero with the ordinate y in [r + 2,f+ 5]. If £ > 1687T + 5 and there is a zero with the ordinate y'm[t-2,t + 2], this of course implies our assertion. We complete the proof of Lemma 5.

• Proof of Lemma 6. We talced = h+c withe > 10/3. From the assumption S(T) has single sign on [t\,t%\. Hence we have

r

/->c

S(h+u)du\=1

/"10/3

I \S(h+u)\du> I Jo Jo

\S(ti+u)\du.

Suppose that S(T) is positive. Since N(t) is constant on [ti, t{\, we obtain (5.5) 5(/ 1 + M ) = ^

1

) - 2 ^ ( ^ ) - l = 2 ^ ) - 2 ^ ( ^ ) + 5(f1)>0

for any 0 < u < 10/3. Thus (5.6)

S(h)>2K{^)-2K{^).

Combining (5.5) and (5.6) we obtain S{tx

+

u)>2K{^)-2K(!^)

for any 0 < u < 10/3. Hence we have io/3

X

t

t +u

\2K(^)-2K(-^)du.

197

Riemann Hypothesis for Rank 3 Zeta Thus (5.2) yields (5.8) l*ite) - 5 l ( n ) | > I

f10/V2 fi + " v , ( £ - ^-)(log

h ,\, „ r/0.006N ± - l)du - 3 . 5 ( — )

> 0.884 l o g ( ^ - ) - 0.886.

Suppose that 5(7) is negative. By (5.1) we have N(h) < 2K(h/(2n)) + 1. Since N(T) is constant on [t\, t2], we obtain

-S(fi + ii) = 2 * ( ^ ) - iV(/i) + 1 > 2 * ( ^ ) - 2 * ( £ )

for 0 < u < 10/3. Hence we have

X

t +U

10/3

t

du.

p(-^)-2^)

Thus (5.2) yields f10/3 (5.10)

I5i(ii)-5,(r,)| > Jo

u /

u

\

,0.006 \

- ( l o g £ - i>fc-3.5(—)

> 0.884 l o g ( ^ - ) - 0.886.

Now tx > 168TT gives logfo/(2TT)) > 4.4 and logft/(2TT)) > log(f2/(2;r)) 0.01. Hence (5.8) and (5.10) gives

(5.11)

|5 !fe) - 5 ^ O l > 0.884 log(^-) - 0.886

under the assumptions of Lemma 6. This contradicts (5.3). We complete the proof. •

198

M. Suzuki

6 A question Equality (1.11) can be written as

(6-1)

6(') = *5« + ^ £ ( 3 * - l ) ,

where £(s) = X(s) + X(l - s). This yields (6-2)

^

£(3*-1) = 6 « - £ § ( * ) .

As this, the Riemann xi function g(s) can be written as the sum of £3(5) and £}($). Equality (6.2) was first pointed out by Weng. We have already known that all zeros of &(s) and ^(s) lie on the line Re (s) = 1/2. Therefore, equality (6.2) may be a remarkable relation. We conclude the article by the following natural question: Can one say some nice things for the zeros of^(s) by using the properties ofgj(s) andQ(s) via relation (6.2)?

References [1] J.C. Lagarias, On a positivity property of the Riemann ^-function, Acta Arithmetica 89 (1999), No. 3, 217-234. [2] J.C. Lagarias and M. Suzuki, The Riemann hypothesis for certain integrals of Eisenstein series, J. Number Theory 118 (2006), no. 1, 98-122. [3] A.M. Turing, Some calculations of the Riemann zeta function, Proc. London Math. Soc. 3 (1953), 99-117. [In: Collected Works of A. M. Turing, Volume I (J. L.Britton, Ed.) North-Holland 1992, pp. 79-97. Notes, pp. 254-261.] [4] L. Weng, Geometric arithmetic: a program, Arithmetic Geometry and Number Theory (Series on Number Theory and Its Applications, vol. 1), pp. 211-400, 2006. [5] L. Weng, A geometric approach to L-Functions, this volume.

Riemann Hypothesis for Rank 3 Zeta

Masatoshi Suzuki Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan Email: [email protected]

199

Elliptic Curves Arising from the Spectral Zeta Function for Non-Commutative Harmonic Oscillators and ro(4)-Modular Forms K. KIMOTO1 & M. WAKAYAMA2

The Apery-like numbers hin) introduced in [6] are indispensable to a description of the special value £Q(2) of the spectral zeta function £Q(S) for the non-commutative harmonic oscillator Q. These numbers have remarkable modular form interpretation as the original Apery numbers possess. In fact, we show that the differential equation satisfied by the generating function W2(t) of hiri) is the Picard-Fuchs equation for the universal family of elliptic curves equipped with rational 4-torsion. The parameter t of this family can be considered as a modular function for the congruent subgroup T0(4). Further, we see that the function wz(t) is regarded as a ro(4)-modular form of weight 1 in the variable T by taking t as the classical Legendre modular function A(T).

0

Introduction

Let us consider the Apery numbers an and bn defined by the recurrence relation (0.1)

n2un - (Un2 -Un

+ 3)w„_i - (n - l)2w„_2 = 0

together with the initial conditions OQ = 1, a\ = 3 and bo = 0, b\ = 5. These numbers were introduced in 1978 by Roger Apery, who utilized 2

them to prove the irrationality of the special value £(2) = *g- of the Riemann zeta function £(s). The important point is that a parallel technique allows 'Partially supported by Grant-in-Aid for Scientific Research (B) No. 16740021. Partially supported by Grant-in-Aid for Scientific Research (B) No. 15340012.

2

201

202

K. Kimoto & M. Wakayama

us to provide a proof of the irrationality of £(3) (we refer to [11] for further information on his irrationality proof). Since then, several people have tried to understand deeply the nature of the Apery numbers and to generalize the theory to £(n) (n > 3). For instance, there are many works on congruence and/or supercongruence properties, algebro-geometric interpretations, modular properties, etc. For detailed information, we refer to [1], [2], [3] and their references. Among them, in this note, we focus particularly the study on the relation between Apery numbers and elliptic curves developed by Beukers [1]. Let us hence recall the result in [1] briefly. Consider the generating functions A(t) = £™=0 anf and B(t) = £™=0 bnt" of the Apery numbers an and bn. Then A(t) and B(t) satisfy the differential equation L2CA) = 0 and La{B) = - 5 respectively, where La denotes the Fuchsian differential operator L2 = tit1 + lit - 1 ) —1 + (3J2 + 22t - 1 ) - + {t + 3). at at The main result of [1] shows that the equation La(Y) = 0 is the PicardFuchs equation associated with the family of curves (0.2)

y2 = x1 + -(I2 + 6t + l)* 2 + -t(t + l)x + -t2.

We notice that any elliptic curve equipped with rational 5-torsion is birationally equivalent to (0.2) for some value oft [7]. By this fact, the function A(t) is interpreted as the period of the holomorphic 1-form 2 ^ £ . Moreover, A{t) is a r"i(5)-modular form of weight 1 in the variable T (Imr > 0), the ratio of the fundamental periods. In the present note, we deal with analogous objects of the Apery numbers, say, the "Apery-like numbers" Jzin) associated with the special value £Q(2) introduced in [6]. Here £Q(S) is the spectral zeta function of a certain differential operator called the non-commutative harmonic oscillator Q. We recall shortly basic properties of the spectral zeta function £Q(S) and how the numbers J2O1) arise in connection with the value £Q(2). Let Q = Qafi be the ordinary differential operator on L2(R) ® C 2 defined by

Elliptic Curves Arising from the NCHO and ro(4)-Moduiar Forms

203

where the parameters a,/3 e R>o satisfy ar/3 > 1. The system defined by the operator Q is called the non-commutative harmonic oscillator [10]. The operator Q is positive and self-adjoint, and has only a discrete spectrum; the eigenvalues are 0 < A\ < \2 < ••• < \n < ••• f +oo with uniformly bounded multiplicity. Then, the spectral zeta function £Q(S) of Q is defined as the Dirichlet series oo

n=\

in order to study the structure of the spectrum of Q. This series converges absolutely if Res > 1 [6]. In [5], it is shown that £Q(S) is continued to the whole complex plane as a meromorphic function which has a unique simple pole at 5 = 1, and trivial zeros at s = 0, - 2 , - 4 , - 6 , . . . like the Riemann zeta function £(s). We note that the operator Q is unitarily equivalent to a pair of usual quantum harmonic oscillators when a = /?. In particular, we see that £Q(s) = 2(2* - l)£(s) when a = /3 = V2. The numbers J2(n) in question are defined by <03)

''2(n) •= Jo J. T 3 7 ^ ( a - ^ . , , 2

J **•

which arise in the expression of the special value of (Q(S) at s = 2. Actually, we have

In [6], a recurrence formula for J2(n) is obtained as (0.5)

4n2J2(n) - (8n2 - 8n + 3)/ 2 (« - 1) + 4(« - l)2J2(n - 2) = 0.

Introducing the generating function w2(t) of /2(«) by CO

n=Q

one finds that the recurrence formula (0.5) is equivalent to the singly confluent Heun differential equation (0.6)

t(l - tfw2\t)

- (1 - 30(1 - t)w'2(t) + (t- \)w2(t) = 0.

204

K. Kimoto & M. Wakayama

We show in [8] that the rational numbers / 2 (n) := hW/hiO) satisfy the congruence relation Jjimp") = J2{mp"~l) (mod p") for any odd prime number p and positive integers m, n, which is quite similar to the one for the Apery numbers; amp»_i = ampn-\_x (mod p"). Here we note that the Apery-like numbers / 3 (n) are introduced similarly to describe the value £Q(3), and satisfy the recurrence formula (0.7) 4n2J3(n)-($n2-Sn

+ 3)Mn-l)

+ 4(n-l)2Un-2)

= ?" ( "~ *?,!. (2« - 1)!!

Remarkably, the only difference between the two recurrence equations (0.5) and (0.7) is the existence of the inhomogeneous term in (0.7). As a result, the generating function w 3 (0 for y3(n) satisfies almost the same differential equation as (0.6) but with inhomogeneous term (see (5.2)). Furthermore, one can obtain a congruence formula for the normalized sequence / 3 (n) := 73(n) - 73(0)/2(«) € Q too. See [8] for detail. Based on these similarities between the Apery numbers and our Aperylike numbers hin), / 3 (n), in the talk at the Conference on L-functions, the second author had a chance to state a rather vague conjecture; like the results in [1] (resp. [2]) for the Apery numbers, the differential equation for W2(t) (resp. w3(?)) would be understood as a Picard-Fuchs differential equation attached to a certain family of elliptic curves (resp. a certain family of K3 surfaces). At that time, he presented some numerical data for J2(n), from which Zagier immediately showed his interest and indicated that the numbers fan) would be coming from a certain modular form. In fact, on the next day, he kindly suggested us the precise formula from his study [14]. The aim of this note is, therefore, to establish an analogue of the results in [1] for the Apery-like numbers hin). Precisely, we show that the differential equation satisfied by the generating function w 2 (0 of 72(n) is the Picard-Fuchs equation for the universal family of elliptic curves equipped with rational 4-torsion. The parameter t for the family of such elliptic curves can be considered as a modular function for the congruent subgroup To(4). Moreover, we show that the function w2(r) is regarded as a ro(4)-modular form of weight 1 in the variable T by taking t as the classical Legendre modular function A(r). Our strategy is almost parallel to the one in [1].

Elliptic Curves Arising from the NCHO and r0(4)-Modu/ar Forms

1

205

Setting the stage

Changing the variables of integral, we have .-(^)/2 /(l - e -*)(i - e~ls) e-{t+s)

1

1

\

(i _ e-(m))2

44

r r Hi - Xx )(i ) ( l --yY4 4))V \ " d) dXdY Jo Jo I ( 1 - X W / l - xW

_

Thus we obtain the expression (1.1) w 2 (0 = V h(n)f £o

=4

=-— - dXdY Jo Jo (1 - ^Y2)2 - t{\ - X*)(l - F4)

of W2(t) as an integration of a rational function. This integral expression is calculated as

1. ( fl f1 l-X2Y2 4W2(t) ~ Jo Jo (1 - 0(1 - VY2)2 + KX2 - Y2)2 1 nX

dXdY 2(11-oUo - 0 Jo Jo Jo (1 - X Yi) + T(X2 - Y2) (>1 r \1 dXdY v 2 2 2 2 {l-X Y )-T{X -Y )> V

2

n

- J_ f1 f1

dXdY

" 1 - t Jo Jo (1 - X2Y2) + T{X2 - Y2)' where we put T = Jjzi ort = ^ - j - . Set (1.2)

QT(X, Y) = (1 - X2Y2) + T(X2 - Y2).

Then we define "WiiT) by ,-. „*

*., „™

1 —?

,.

0.3)

W2(r) =

_„2(,)

=

C dXdY

j

o

_ _ ,

where n denotes the domain [0,1] x [0,1]. The denominator QT(X, Y) of the integrand in 'W^CO defines an algebraic curve QT : QT(X, Y) = 0 in C 2 . From (0.6), it is easy to see that the function ^WjiT) satisfies the

206

K. Kimoto & M. Wakayama

differential equation JXWi) = 0. Here the differential operator £ is given by

(1.4)

£ = ( r 3 - D ^ j + (3T2 - 1) A + T.

In the sequel, we explain actually that the differential equation XCW2) = 0 has a geometric origin, that is, (1) the algebraic curve QT is birationally equivalent to a certain elliptic curve CT for all but finite values of T, and {Cr}7-e-vcYox §i y e s t n e family of elliptic curves having rational 4-torsion. (2) the differential equation XCW2) = 0 is the Picard-Fuchs equation corresponding to the family [CT)TRemark 1.1. In [9], the differential equation (0.6) is solved so that we have

n ^

(i.5)

fA w

^ =

M0)

/r (l

l

1

M

Tl2F\rru7-[)>

where 2^1 (a, b;c;z) denotes the Gaussian hypergeometric function. As a corollary, we obtain the hypergeometric expression

W ) = £ 2^(1,1;!; 7-) for 'WiiT). This yields the binomial expression for foiri) as ,-2

"

/ 1\2/

—T§<-><(7)(;).

Elliptic Curves Arising from the NCHO and To(4)-Modular Forms

2

207

Elliptic curves associated with J2O1)

Let us consider the series of birational transformations (2.1a) (2.1b)

(2.1c)

X=

1

xi+yi + l

i+*r

vfui +yi)+r

(1 + VT)y2 (i + 7 > 2 - ( i + Vf)y 2 -i' (i + Vf)(i + Vf(i - Vf)ya) y\ = (l + D((i + DJC2 - (l + Vf)y 2 -i)' JCI =

x2 = JC3,

y2

y3 *3

- i + 6 V f - 6 r +6 V f r - r 2 +12*

,„, JX (2. Id)

*3 = J3 =

—

i2Vf(i + r) 2

,

1 - 5T - 5T2 + 7 3 - 12(1 + T)x + 24y

247(1 + Vf)(l + T)2 The first transformation (2.1a) reduces the curve QT defined in (1.2) to the cubic curve 2(1 + 7)(1 - Vf)yi + (1 + 7)(1 - T)y\ - 2 Vf(1 - Vf )2*i + 2(1 - Vf )(1 - y/f + T + T Vf )xiyi + Vf ( V f - l ) ( r + Vf - 4);c* - 2 Vf (1 - Vf )jtfri - 2 Vf (1 - Vf )*? = 0, and the remaining three steps (2.1b), (2.1c), (2. Id) are the regular procedure to obtaining the standard form of an elliptic curve (see, e.g. [12]). By a tedious step-by-step calculation, we obtain the following key lemma. Lemma 2.1. By the transformation X = X(x,y) and Y = Y(x,y) given in (2.1), the curve Qj is birationally equivalent to the elliptic curve Cj : CT(X, y) = 0 defined by ,„„,„, (3 (2.2) CT(x,y)x = *3

r 4 + 147 2 + l

x+

r 6 - 3 3 r 4 - 3 3 7 2 + l\ , ) - y\

208

K. Kimoto & M. Wakayama

Furthermore, the equality dXdY _ dxdy QT{X,Y) 2CT(x,y)

(23)

holds.

a

Let us look at the resulting elliptic curve , : y = xr 2

CT

T 4 + 14T2 + 1

T6 - 33T4 - 33T2 + 1

x+ 48 864 = STld - 6T + T2 ~ 12 *X1 +6T + T2- 12JC)(1 + T2 + 6x). 864 The points of order 2 on CT are y

(1-6T

+ T2

\ (1+6T + T2

\ I l + T2

\

For any T e C, these three points together with the point of infinity (the identity element of the group Cr(C)) form a finite subgroup of C^(C) isomorphic to Z/2Z x Z/2Z. We also conclude that the curve CT is singular if and only if T = 0, ±1, oo since the discriminant of CT is equal to 3 ^ r 2 ( i - r2f. When T = V^Tw 6 V ^ Q , CT has only one rational 2-torsion ( i L ^ i , 0). In this case, there are just two rational points l + 5u2 H(1 + u2) 12 ^ 4 on CT of order 4, and hence the torsion part of Cr(Q) is isomorphic to Z/4Z by Mazur's theorem on the structure of torsion subgroups. In [7], it is shown that the universal family of elliptic curves with rational 4-torsion is given by Eb • y2 + xy - by = x3 - bx2,

A = b\\ + 16b) * 0.

In this form, the origin (0,0) is a generator of the 4-torsion. If we set ,

Vf(7 + 1) F 7' 2(Vr-i)4 then we see that Eb and CT are birationally equivalent, and the family {CT} is the universal family of the elliptic curves equipped with rational 4-torsion described by another form. b =

Elliptic Curves Arising from the NCHO and To(4)-Modular Forms

3

209

Geometric interpretation of the differential equation for W2(T)

We follow the discussion expanded in [1] to see that the differential equation (0.6) is the Picard-Fuchs equation associated with the family of elliptic curves {C7-}. We recall that the function "it^CO is a unique (up to constant) holomorphic solution of the differential equation X.i'Wi) = 0 (X is given in (1.4)) around 7 = 0. Fix a point TQ e C near the origin and take a closed curve 7 through TQ such that 7 does not contain any singularity of the form QX,^YY Then, following the idea in [1], we consider the analytic continuation of ^M^CH along 7, which is given by

where D c C^is a suitable domain such that 3D = du (the same boundary). Notice that IViiT) also satisfies the differential equation XCW2) = 0> a n d the integral

is over the closed and oriented surface (2-cycle) D-D. Namely, this integral can be written as the form h{T)

-i .

dXdY QT(X,Y)

for a certain 2-cycle T in C2 \ QT. By Lemma 2.1, the integral Ir(T) becomes 1 f dxdy Is(T) CT(x,y) 2

-u

for some S e H2(C \ CT,Z). By the same 'topological reduction' discussion as in [1], it follows that

>y

y

210

K. Kimoto & M. Wakayama

for a certain 1-cycle y e H^C2 \ C r ,Z). Thus the difference TV 2 (r) ^(T) is a period IS(T) of the holomorphic 1-form ^nid: . Hence rH/2(7') 2y (and <W2{T)) and the period Is (T) of :^nidx satisfy the same differential equation (see also Remark 3.1). It is well known that the integral . C dx I7(T) ::== m ni I — Jy y

2

(y € Hx ( C \ CT, Z))

satisfies a second order Fuchsian differential equation (the Picard-Fuchs differential equation for {CV}). The local exponents of Iy(T) at the singularities 0,1, - 1 , oo are respectively given by 0,0,0,1. Therefore, the differential equation for Iy(T) is of the form (3.1)

(T 3 - T)F"(T) + (3T2 - l)F\T) + (T + A)F(T) = 0

for a certain constant A. To determine this constant A, we take y as a closed path in the x-plane surrounding just two of the singularities (1±6f2+r ,0) of the 1-form ^ j 2 , which go to ^ as 7* —» 0, so that this integral Iy{T) gives the holomorphic solution of this differential equation (3.1) around the origin T - 0. Calculating the integral Iy(T) directly using the residue theorem, we notice that the Taylor expansion of Iy(T) for sufficient small T is given by

/r(7) = i + i r 2 + ^ r 4 + ....

(3.2)

From this expansion (3.2), it follows that A = 0. Hence, putting the discussions above together, we now conclude the Theorem 3.1. The differential equation £S^Wi) = 0 is the Picard-Fuchs equation associated with the family (3.3)

, Cr-.y ^ 2

,

T 4 + 14T2 + 1

T6 - 33T4 - 33T2 + 1

X+

of elliptic curves equipped with rational A-torsion.

•

Remark 3.1. Let -£(/ r ) = 0 be the second order linear differential equation for the period Iy(T) of the holomorphic 1-form ^ associated to the elliptic curves {CT). From the equation <W2(T) - W2(T) = \ly(T), we have

Elliptic Curves Arising from the NCHO and To(4)-Modular Forms

211

-CCW2) = JX^Wi)- Hence we see that the monodromy group of XCW2) is trivial so that it is a rational function. In our case, we know a priori that the function <W2(T) itself satisfy the second order linear differential equation of the form XCW2) = 0, and it follows that the operators X, and X are identical, up to a constant multiple.

4 Modular properties Let A(T) be the classical Legendre modular function

where Imr > 0, q = e™ and n{f) = q& n^Li(l -
A e 5L 2 (Z); c = 0(mod4)|. %

Further, it is known that the modular function field for To(4) is equal to C(/1(T)). Based on the study [14] (see also Remark 4.1), Zagier pointed out the following theorem from the list therein. Theorem 4.1. The equality (4.2)

W2U(T))

=

#o(r) 2

T?(2T)22

1 - A(T)

77(T) 1 2 77(4T) !

holds. Here #0(1") is the elliptic theta function {which is a TQ(4)-modular form of weight 5) (4.3)

# 0 (T) = ^ r

= V ( - I ) " / = 1 - 2q + 2q4 + • • • .

Notice that W2(A) is not holomorphic. We show that the theorem is deduced by the same mechanism as the discussion in [1].

212

K. Kimoto & M. Wakayama

Proof. Let CO(T) be a ro(4)-modular form of weight 1 and put CO(T) = Then

TCO(T).

lar + b\ _ CO

\CT + d)

for I

c CO(T) + d CO(T),

OJ(T)

CO

\CT + dj

+ b CO(T)

J € To(4). Hence, if we regard to and Co as a multivalued functions

in A, then the analytic continuation coy{A) and Co7(A) along a closed path y in the /l-plane corresponding to an element I (4.4)

coy(A) = a OJ(A) + b co(A),

J e To(4) is given by

toy(A) - c C)(A) + d co(A).

It is immediate that both co(A) and co co co'

(4.5)

co co co Co'

F' +

CO

CO

F =0

where the prime denotes the differential with respect to A. The coefficients in the differential equation are calculated as (4.6)

CO CO'

d)

CO'

CO2 dT

CO CO

co" Co"

Kl

i

dA V dr >

•t

co' co" Of co"

1/da>\2

, .d2to

(dA\3

Notice that the monodromy groups of these determinants are trivial due to (4.4), so that they are rational functions in A. Now we take CO(T) = f°^T) • Using well-known formulas among the values of elliptic functions and theta functions, we see that the equations

(4.7)

co co' co co' co' co" a/ Co"

1

1 2

niA(l-A) ' 1

CO

CO

Co Co'

1 (1 - 3/t)(l - A.) ~ni

A2{\ -A)4

'

*~\

7tiA2(l-A)4

holds (we postpone the calculation; see Lemma 4.2 below). Thus we find that co satisfies

(4.8)

,1(1 - A)zco"{A) - (1 - 3/l)(l - A)co'{A) + (A- -)co(A) = 0.

Comparing this differential equation with (0.6), we have proved the lemma. •

Elliptic Curves Arising from the NCHO and To(4)-Modular Forms

213

Lemma 4.2. The equations of determinants in (4.7) hold. Proof. We use the convention for elliptic and/or theta functions in [4]. Denote by co\, o>2 the fundamental periods so that r = ^ , and by #/(V,T) 0' = 0,1,2,3) the elliptic theta functions. We put #, = 0/(0, T) (j = 0,2,3) and 0\ = T£-(0,T). We also put e\ = p(
4m/lrflog#i &>? \3

rfr

dlog?M dr

and it follows that ~ e2) = —

# 0 = -TW l

n

m\

— j

j

dT

•

dT J

On the other hand, since A(T) = (|* j , we have rflog/l

ld\og&2

dlog#3\

4

Thus we obtain to to' _to2 dX to to' dr

The third de termina nt

K A( 1

xpdiogx

1

1

2 niA{\-A)

to to" is also calculated in the si to to"

the calculation is rather complicated. The second determinant _ _, to to ~^(y(f-V)

is readil

y

obt

ained by differentiate ; j j - } ^ by A.

n

Remark 4.1. Consider the differential equation (4.9)

((f3 + at2 + bt)F'(t))' + (t- A)F(t) = 0

with rational parameters a,b,A, which is due to Beukers. When a = 11, ft = - 1 and A = - 3 , this equation (4.9) is exactly the one for the generating function A(t) of the Apery numbers an. In [14], Zagier searches the triplets

214

K. Kimoto & M. Wakayama

(a, b, A) of integers within a certain domain such that (4.9) has an integral solution (i.e. solutions in Z[[*]]), and presents a list of 36 such solutions. It is shown that the twelve solutions of them have parametrizations in terms of modular functions. Zagier noticed and pointed out that our generating function W2(t) of Jjiri) is #19 in his list.

5

Closing remarks

In the final position of the note, we give several remarks for the future study. 5.1 In the discussions above, we see that the numbers Jjin) arising from the value £Q(2) acquire the ro(4)-modularity associated with the family of elliptic curves equipped with rational 4-torsion. However, at this moment, we have no intrinsic explanation from the level of non-commutative harmonic oscillators why such things hold. 5.2 The Apery-like numbers hin) are arising in the expression (0.4) for £Q(2) via the generating function g2(z)=> , J (~ 2 U2(n)z".

•S(v)

This function satisfies the differential equation D(g2) = 0 (see [8]). Here D is given by D = 8^(1 +z)24i+i 24z(l + z)(l + 2z)—L + 2(4 + 27z + 27z2)-^- + 3(1 + 2z). dz dz dz

Does the function gi{z) have a modular form interpretation like W2(t)l If this is true, then g2(z) should be a modular form of weight 2 from the result in [13]. We also note that g2(z) has the following explicit expressions by hypergeometric functions [9]:

82iz) =

vrW 3/?2(i'rhh 1; rb)=2Fl(i' h1; _ z 2 ) •

Elliptic Curves Arising from the NCHO and ro(4)-Moduiar Forms

215

5.3 The Apery numbers an and bn are corresponding to ((2), satisfying the same recurrence formula. Their generating functions satisfy the differential equations L2{A) = 0 and L2{B) = - 5 for the same operator L2 but one is homogeneous and the other is inhomogeneous. Denote by an and /?„ the Apery numbers corresponding to £(3). They satisfy the same recurrence formula n3un - (34n3 - 51«2 + 27n - 5)M„_I +(n- l)3w„_2 = 0 with initial conditions oro = 0, a\ = 6 and/?0 = l>/?i = 5 . Their generating functions A(t) = I~ =0 a n f" and B(f) = XZtifint" satisfy the differential equations L-}(A) - 0 and £3(8) = 5 for the same operator

L3 = / V - 34f + l ) 43r + 3/(2^ - 51f + 1)-^1 + (7f* - 112/ + 1)4 + (f - 5),

af at at but one is homogeneous and the other is inhomogeneous. Thus the situation is exactly the same for the Apery numbers corresponding to ({2). It was proved in [2] that the differential equation Z,3(A) = 0 is the PicardFuchs equation associated to a certain family of K3 surfaces. In our spectral case, the situation seems similar at a glance, but it is different. The Apery-like numbers J2(n) are corresponding to the special value £g(2), while (5.i)

,300 := 8 j Q j

ojQ

(

( 1

_

x W ) 2

)T T ^ 2

are corresponding to the special value £Q(3) (see [8]). Let wj(t) be the generating function for J-}(ri) defined by w-}(t) = J^=o h{n)f. The generating functions w2(t) and wj(t) satisfy the differential equations (5.2)

D(w2) = 0

and

U;

H H

D(w3) = - 2FX

respectively for the same operator

D = , ( 1 - ^ , | - ( i - 3 , ) a - „ | + ( ,-2,. Notice that the former equation is homogeneous, but the latter is inhomogeneous. This fact shows that the structure of 'pairing' in our theory is different from the case of the Apery numbers.

216

K. Kimoto & M. Wakayama

5.4 From the expressions (0.3) and (5.1) of the Apery-like numbers associated to £Q(2) and £g(3), it is quite natural to consider the numbers A(n) defined as Jo Jo"'Jo 1-,-('—« 1

(1-,-e—^

)*>*-*

Note that the numbers Jk(n) are all positive. Their generating functions are GO

wk(z):=YjJk(nV

n

/"•!

l

1 — Jl^ • • • J t 2

•••Jod-^-^-d'-^a-x*-^^1^-^

Jo Jo

Jo (1 _ •z){.\-x\---x\f+z{x\-xl---x^ <

- 3LIL ( f' f' ~ 1 -z VJo Jo

dx\dx2 • • -dxic

C dxidx2--dxk 2 Jo (l-x 14---4) + Z(x2l-4--x2k)

r1 r1 r1

dxxdx2-dxk

JoX'"jo(.l-x214---4)-Z(x2i-x22--4)j

\

where Z = J p y - It is not hard to see the formula w*(0) = (2* - 1 )£(&). Then, we can ask the question; is it true that the function Wk(z) (k > 3) satisfies a Picard-Fuchs type differential equation coming from certain family of algebraic varieties? We remark that we have the double integral expression for /*(«) y (n) =

<

roTT) Jo Jo T^W

[ ( 1 - ^ ) 2 J <««

as well as for w*(z) (-1)* 2 2 Jo i - z r ( * -- i 1) ) JJo 0 J o (i-x y2) + z(x -y2)

Elliptic Curves Arising from the NCHO and r0(4)-Modular Forms

217

Acknowledgement The second author would like to thank Lin Weng and Masanobu Kaneko, the organizers of the Conference on L-functions, for presenting him the precious occasion to give a talk on the subject. We are also very grateful to Don Zagier, who gave us an accurate suggestion and his strong interest on our Apery-like numbers, and kindly sent us his unpublished note [14].

References [1] Beukers, F.: Irrationality of n2, periods of an elliptic curve and Ti(5). Diophantine approximations and transcendental numbers (Luminy, 1982), 47-66, Progr. Math., 31, Birkhauser, Boston, Mass., 1983. [2] Beukers, F. and Peters, C. A. M.: A family of K3 surfaces and £(3). J. ReineAngew. Math. 351 (1984), 42-54. [3] Deutsch, E. and Sagan, B. E.: Congruences for Catalan and Motzkin numbers and related sequences. /. Number Theory 111 (2006), no. 1, 191-215. [4] Hurwitz, A. and Courant, R.: "Vorlesungen fiber allgemeine Funktionentheorie und elliptische Funktionen." Interscience Publishers, Inc., New York, (1944). [5] Ichinose, T. and Wakayama, M.: Zeta functions for the spectrum of the non-commutative harmonic oscillators. Commun. Math. Phys. 258 (2005), 697-739. [6] Ichinose, T. and Wakayama, M.: Special values of the spectral zeta function of the non-commutative harmonic oscillator and confluent Heun equations. Kyushu J. Math. 59 (2005), 39-100. [7] Kubert, D. S.: Universal bounds on the torsion of elliptic curves. Proc. London Math. Soc. 33 (1976), 193-237. [8] Kimoto, K. and Wakayama, M.: Apery-like numbers arising from special values of spectral zeta functions for non-commutative harmonic oscillators. Kyushu J. Math. 60 (2006), 383-404.

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[9] Ochiai, H.: A special value of the spectral zeta function of the non-commutative harmonic oscillators. To appear in Ramanujon J. (ArXiv:math.RT/0507111). [10] Parmeggiani, A. and Wakayama, M.: Oscillator representations and systems of ordinary differential equations. Proc. Natl. Acad. Sci. USA 98 (2001), 26-30. [11] Van der Poorten, A. J.: A proof that Euler missed... Apery's proof of the irrationality of f(3). Math. Intelligencer 1 (1978/1979), 195-203. [12] Silverman, J. H. and Tate, J.: "Rational points on elliptic curves." Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1992. [13] Zagier, D.: Modular forms and differential operators. Proc. Indian Acad. Sci. Math. Sci. 104 (1994), no 1, 57-75. [14] Zagier, D.: Integral solutions of Apery-like recurrence equations. Preprint.

Kazufumi KIMOTO Department of Mathematical Science, University of the Ryukyus, Senbaru, Nishihara, Okinawa 903-0231, Japan Email: [email protected] Maasato WAKAYAMA Faculty of Mathematics, Kyushu University, Hakozaki, Fukuoka 812-8518, Japan Email: [email protected]

A Geometric Approach to L-Functions L.WENG

In this paper, we initiate a geometric approach to the theory of L-functions. It consists of two parts: Chapters 1-4, used to be called 'Non-Abelian LFunctions for Number Fields', reveal a general theory, and Chapters 5-9 give a detailed account of ranks two and three zetas. The starting point is a stability for lattices in terms of intersection. This is followed by a new yet genuine geo-arithmetical cohomology, which satisfies Serre duality and Riemann-Roch. Based on them, then we introduce high rank zeta functions for number fields and establish their basic properties. At it turns out, high rank zetas are canonical: When rank is one, our zetas give the well-known Dedekind zetas; moreover, they can be meromorphic extended, satisfy functional equations, and admit only two singularities with the same residues given by volumes of certain compact moduli spaces of stable lattices. All this is done in Chapter 1. By singling out the volume part, using Mellin transform, we find that high rank zetas can also be understood as integrations of certain Epstein type zetas over the above mentioned compact moduli spaces. Note that, despite we call them zetas, Epstein type zetas are indeed a special kind of Eisenstein series. This then leads to our definition of non-abelian Lfunctions (as integrations of relative Eisenstein series associated to L2automorphic forms on some far-reaching generalizations of moduli spaces). As such, basic properties such as meromorphic continuations, functional equations and singularities of these non-abelian L-functions are also studied based on Langlands' fundamental theory on Eisenstein series associated L2-automorphic forms. This is given in Chapter 2. With our primitive objects introduced and basic properties being established, our next aim is to study them quantitatively. This then leads to the discussion in Chapter 3, in which we successfully relate our geoarithmetical truncation with Arthur's analytic truncation, a result already 219

220

L. Weng

appeared in Lafforgue's work on Langlands conjecture for curves defined over finite fields. Built on it, in Chapter 4, we develop an advance version of Rankin-Selberg & Zagier method, by generalizing a result of JacquetLapid-Rogowski. As a direct application of this, we then give a general formula for what we call high rank L-functions associated to cusp forms. Surely, the point here is that unlike in Arthur's theory of analytic truncations, parameters used in our natural geo-arithmetical truncation are just normal, or better, are not sufficiently regular. So additional cares have to be taken. After this settled, at the end of Chapter 4, we give an explicit formula for the volume of mentioned moduli space based on an earlier work of Henry Kim and the author. This then reveals an intrinsic relation between special values of Dedekind zetas and that of our zetas, a result rooted back to Siegel and Langlands. From Chapter 5, we concentrate our attention to the study of our zeta functions in ranks two and three. In particular, in Chapter 6, we show that a lattice of rank two is semi-stable if and only if the generalized Siegel type distances of the corresponding modular point in the upper half space model to all cusps are bigger or equal to 1. This is a beautiful fact. Based on this, in Chapter 7, we then give an explicit expression of rank two zetas in terms of Dedekind zetas, by further developing Rankin-Selberg & Zagier method in a classical fashion with the help of our study on Fourier expansions for Epstain type zeta functions involved. Meeting everyone's expectation, the fact exposed in Chapter 8 then is that all zeros of rank two zetas lie on the central line Re = 1/2, an analogue of the standard Riemann Hypothesis for Dedekind zetas, based on a method of Suzuki and Lagarias who first prove the result for the field of rationals. Finally, in Chapter 9, we give a detailed account of rank three zeta for the field of rationals. Our approach uses classical Eisenstein series such as Koecher zeta functions and Siegel's Eisenstein series and the powerful formula in Chapter 4 mentioned above. The up-shot is a precise formula for the associated rank three zeta. We end our presentation by the following recent result of Suzuki: All zeros of this rank three zeta lie on the central line Re = 1/2 as well. As such, then it appears that the time is ripe to openly make the following generalization of the Riemann Hypothesis: The RH. All zeros of zetas, rank one or higher, lie on the central line

Res = i.

A Geometric Approach to L- functions

221

Contents 1 High Rank Zetas for Number Fields

223

1.1 Iwasawa's ICM Talk on Dedekind Zetas 1.2 A Geo-Arithmetical Cohomology 1.3 Geometric Truncations 1.4 High Rank Zeta Functions for Number Fields 1.5 First Example

223 226 230 244 249

2 Non-Abelian L-Functions

254

2.1 Automorphic Forms and Eisenstein Series 2.2 Non-Abelian L-Functions

254 260

3 Geometric and Analytic Truncations

264

3.1 Geometric Truncation: Revised 3.2 Arthur's Analytic Truncation 3.3 Arthur's Periods 3.4 Geometric and Analytic Truncations

264 272 284 288

4 Rankin-Selberg & Zagier Method 4.1 Regularized Integration over Cones 4.2 JLR Periods of Automorphic Forms 4.3 Arthur's Periods and JLR's Periods 4.4 Bernstein Principle 4.5 Eisenstein Periods for Cusp Forms 4.6 Volumes of Truncated Fundamental Domains 4.7 When Parameters Are Small 4.8 Fundamental Relations Between Special Zeta Values

296

. .

296 300 303 304 305 306 310 312

5 High Rank Zetas and Eisenstein Series

315

5.1 High Rank Zetas for Number Fields 5.2 Space of OK -Lattices 5.3 High Rank Zetas and Epstein Zetas

315 316 319

222

L. Weng

6 Stability and Distance to Cusps 6.1 Upper Half Plane 6.2 Upper Half Space 6.3 Upper Half Space Model 6.4 Cusps 6.5 Stablizer Groups of Cusps 6.6 Fundamental Domain 6.7 Stability and Distances to Cusps 6.8 Moduli Space in Rank Two 7 Explicit Formulae for Rank Two Zetas 7.1 Epstein Zetas and Eisenstein Series 7.2 Fourier Expansions 7.3 Explicit Formula for Rank Two Zetas 8 Zeros of Rank Two Zetas

321 321 321 322 323 324 326 329 329 331 331 331 333 336

8.1 Zeros of Rank Two Zeta of Q 8.2 Zeros of Rank Two Zetas

336 341

9 A Rank Three Zeta and Its Zeros

342

9.1 Various Eisenstein Series 9.2 What Matters to Us 9.3 Relation with the Rank Three Zeta 9.4 Elementary Calculations 9.5 Rank Three Zeta References

344 352 354 355 362 365

1 High Rank Zetas for Number Fields 1.1 Iwasawa's ICM Talk on Dedekind Zetas We start with a discussion on Dedekind zeta functions. However, we will not adopt the classical approach, rather we would like to recall Iwasawa's interpretation. Let F be a number field. Denote by 5 the collection of all (inequivalent) normalized places of F. Set 5 M be the collection of all Archimedean places of F and S n n := S\S«,. Denote by I the idele group of F, N : I -» R>o and deg : I —» R the norm map and the degree map on ideles respectively. Also introduce the following subgroups of I:

1° ={a = (av) e I : deg(a) = 0}, F* ={a = (av) e 1° : av = a e F\{0], Vv e S], U ={a = ( a v ) e l ° : |av|v = lVveS}, Ifin ={a = (av) e I : av = lVv € Sooh loo ={a = (av) € I : av = lVv e 5fln). Set C/fin := U n Ifin. Then, with respect to the natural topology on I, we have (1) F* ^ I is discrete and I°/F* is compact. Write Pic(F) = I/F*; (2) U °-»I is compact; (3) [/nn «-» Ifin is both open and compact. Moreover, the morphism / : [a = (av)] \-* 1(a) := TYves^ ^v induces an isomorphism between Ifin/f/fin and the ideal group of F, where Pv denotes the maximal ideal of the ring of integers Ofv corresponding to the place v; and, N(a) = Fives l«vl^v:=[Fv:Qp] = N(I(a)yl with N(I(a)) the norm of the ideal 1(a); (4) I = Ifin x loo. Hence we may write an idele a as a = a nn •floowith «fin € Ifin and «oo £ loo respectively. In particular, if dfi(a) denotes the normalized Haar measure on I as say in Weil's Basic Number Theory, we have dfi(a) = dn(a%n) • d^aoo) corresponding to the decomposition I = Ign x !«,. 223

224

L. Weng

Set fl, e(«fin):=

if/(afl n )c0|?;

„

10,

otherwise,

e(<3oo) = exp ( - n ^

a2v - In ^

v:R

|a v | 2 )

v:C

and «(«fin • a*,) = e(afin) • e(aoo), for a nn € Ifin and #<» e la,, Denote by Af (the absolute values of) the discriminant of F, r\ and r2 the number of real and complex places in Sco as usual. With this preparation in notation, next we recall Iwasawa's interpretation of Dedekind zeta function for F in the form suitable for our later study. This goes as follows. For5€C,Re(5')> 1, fr(s) := Af(2;Tir^)[ I ((2;r)-T(s))'' 2

£

#(/)"'

0*IcOF

=4- J]

W

0±IaOF

x f

f]|^exp(-^2^-

JtveFv,veS«,

= A|-

„

v:R

N(iys-

J]

i

1

F

VOKt/fin)

VOl(t/fin)

X I

'

| 2

)n^

v

^(aoo)Jc(aoo)rfju(aoo)j

i A J. F

(A^(afin)A'r(a0o)) (e(afi n )e(a 0O ))(^/x(a fin )^(a O o))

_I—-A*. VOl([/fin)

v:C

S^

f tf(a„)*e(aoo#/i(aoo)

! x ( I iV(flfin) «(%)^(«iin)' I

1

2 7 r

v

'

fN(aYe(a)dfi(a). Jl

1 High Rank Zetas for Number Fields

225

Now denote by dn([a]) the induced Haar measure on the Picard group Pic(F) := I/F*. Note that fe(aoo), e(afintfoc) := {n 10,

if/(afin)c<9F, otherwise,

and that by (1) above, F* °-> I is discrete, (hence taking integration over F* means taking summation), we get (voltt/ita)-Aj)-fr(j) N(aa)se(aa)da)dix([a])

= f

( f

= f

( V e(aa)\ • N([a])sdn([a]) (by the product formula)

= f

tf([a])*d/i(M)

= f

•( V

iV([fl])V/i([fl]) • (

I/F

*

= f

efraenWaaJ) J]

e(aaoo))

aeF*,/(aa fin )cO f

N([a])XM) • (

J]

e(Qrfloo)).

For an idele L = (av), define the 0-th algebraic cohomology group of the idele L _1 by H°(F,L~l) := {or e F*,aa v c Ov, Vv e ,Sfin} = /(L _1 ); moreover for the associated idele class, introduce its associated geometric 0-th cohomology via

h\F,Vx) :=log(

YJ aeH°(F,i- 1 )\{0)

exp(-^^|gva|2-2^2|gva|2)). v:R

v:C

226

L. Weng

Then easily, we have (vol(C/ f l n )-A; 1 ) •£=•(*)

= LF) ^ ^Pic(F) ^F

-p ( - z I^-I2 -2* z i^-i2)

z aemFL-i)m

v : R

'

v : C

s

• N{L) dn(L) JPic(F) WF

WF

f

V

'

X Jpic(F) 'Pic(f)

_A°(F,L)

' i

e l ~ {e^myMLr hic(F) #Aut(L) JPk Here Wf denotes the number of units of F, and #Aut(L)(= Wf) denotes the number of automorphisms of L. With this said, it is quite clear how a high rank zeta for number field should be introduced. The key points are as follows: (1) We should develop a geo-ari cohomology for which Serre type duality and Riemann-Roch type results must hold; (2) We should use moduli spaces of high rank lattices, instead of the Picard, which works only for rank 1. However, by doing this, then a serious problem appears: In the case of rank 1, the Picard can be factorized as the product of Picard zero, which is compact, with the volume part. For higher ranks, even we have a similar decomposition, the analogue of Picard zero is no longer compact. Hence, convergence problem does arise. It is for this, borrowing a powerful idea from algebraic geometry, (2)' We should introduce a suitable stability in terms of intersection for hermitian bundles over number fields, or better, for lattices. 1.2 1.2.1

A Geo-Arithmetical Cohomology Cohomology Groups

For number field F, denote by A = Ap its ring of adeles, by GLr(A) the rank r general linear group over A, and write A := Afin © Aoo, GLr(

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1 High Rank Zetas for Number Fields

GLr(Afin) x GLr(ATO) according to finite and infinite partition. Also for simplicity, we often use v (resp. cr) to denote elements in Sfin (resp. Soo). For any g = (gfin : #«>) = (gv',g
r

(av; a„-) e A :

(ii.a) gv(f) e Orv, Vv;

(ii.b) (/; a
Then we have the following 9-diagram with exact columns and rows: 0 Ar(g) n Fr(g)

0 ~

^

Fr(g) Fr(g)/Ar(g)nFr(g)

Ar(g)

0 -»

A r (g)/A r (g)nF r (g)

-»

Ar/Fr(g)

-»

Ar/Ar(g) + Fr(g)

J,

>->

Ar

>-» A7A r (g)

J,

•^

•*?

4*

0

0

0.

Guided by Chevelley's adelic cohomology theory for divisors over algebraic curves, (see e.g, [S]), we introduce the following Definition. For any g e GLr(A), define its 0-th and 1-st geo-arithmetical cohomology groups by H°(AF,g) := Ar(g) n F\g), and H\AF,g)

:= Ar/Ar(g) + Fr(g).

Proposition 1. (Serre Duality = Pontryagin Duality) As locally compact groups, Hl(AF,g)^H°(AF'^F~<8>g-1). Here KF denotes an idelic dualizing element of F, and " the Pontryagin dual. In particular, H° is discrete and Hl is compact.

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Remark. For v e 5fin, denote by dv the local different of Fv, the vcompletion of F, and by Ov the valuation ring with nv a local parameter. Then dv =: n°xUdv) • Ov. We call KF := (d°rdv(5v); 1) e I F := GLi(A) an idelic dualizing element of F. Proof. As usual, fix a basic character ;f on A b y x ='• Ofv \xV) where := Av ° T r J with Av : Qv -> Q v /Z v --> Q/Z «-» R/Z, and x* •= A*, o T r ^ with /*«, : R -> R/Z. Then the pairing (x,y) i-» e2ni^x-y) induces natural isomorphisms Ar =* A r (as locally compact groups) and (F r ) x =* Fr (as discrete subgroups). With this, a direct local calculation, see e.g, [T], shows that (A r (g)) x = Ar(KF ® g _1 ) and (F r (g)) x =* Fr(KF ® g _1 ). This completes the proof since

Xv

(Arfc) n Fr(g)f

= (A'(s)) x + (F r (g)) x .

• 1.2.2

Geo-Arithmetical Counts

Motivated by the Pontryagin duality and that the dimensions of a vector space and its dual are the same, one basic principle we adopt in counting locally compact groups is the following: Counting Axiom. If #ga counts a certain class of locally compact groups G, then #ga(G) = #ga(G). Practically, our counts of arithmetic cohomology groups are based on Fourier inverse formula, or more accurately, the Plancherel formula in Fourier analysis over locally compact groups. (See e.g, [Fo].) While any reasonable test function on A r would do, as a continuation of a more traditional mathematics and also for simplicity, we set / := Y\v fv • Ho- fo-, where (i) / v is the characteristic function of Orv\ (ii.a) fo-ixa-) := e'"^ / 2 if <xis real; and (ii.b) fo-ixa) '•= e-*1*0"! if cr is complex. Moreover, we take the following normalization for the Haar measure dx, which we call standard, on A: locally for v, dx is the measure for which Ov gets the measure N(dv)~l/2;

1 High Rank Zetas for Number Fields

229

while for o~ real (resp. complex), dx is the ordinary Lebesgue measure (resp. twice the ordinary Lebesgue measure). Definition. (1) The geo-arithmetical counts of the 0-th and the 1-st cohomology groups for g e GLr(A) are # g a(tt°(A f , g)) :=#JH°(AF, g); f, dx) := f

\f(x)\2dx;

JtfO(A f ,g)

#ga(//1(AF,g)):=#ga(//1(AF)g);/,^):= f

|/(£)|2^.

J//l(A F ,g)

Here dx denotes (the restriction of) the standard Haar measure on A, dg (the induced quotient measure from) the dual measure (with respect to x)> and f the corresponding Fourier transform off; (2) The 0-th and the 1-st geo-arithmetical cohomologies ofg e GLr(A) are fc°(AF)g):=log(#ga(tf°(AF)g))); h\AF,g):=log(#g!i(H\AF,g)j). Remark. Even groups H°(AF, g) and Hx (AF, g) do depend on g e GLr(A), from definition, it is easy to see that their counts h°(AF,g) and hl(AF,g) depend only on the class of g e GL r (F)\GL r (A). 1.2.3

Serre Duality and Riemann-Roch

For arithmetic cohomologies just introduced, we have the following Theorem. For any g e GL r (F)\GL r (A), (1) (Serre Duality) h\AF,g) = h°(AF,KF®g-1); (2) (Riemann-Roch Theorem) h°(AF,g) - hl(AF,g) = deg(g) - r- • log |AF|. Proof. By the choice of our / in the definition, (1) is a direct consequence of the topological Serre duality, i.e, Prop. 1 in 1.2.1, and the Plancherel Formula; while (2) is a direct consequence of the Serre duality just proved and Tate's Riemann-Roch theorem ([T, Thm. 4.2.1] and/or [Neu, XIV, §6]), i.e, the Poisson summation formula, by the fact that (H°(AF, g)) = H°(AF,KF®g-1). a

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L. Weng Often, we also write ehi(-AF-g) := H),a(F, g), i = 1,2.

Remarks. (1) Our work here is motivated by the works of Weil [W2], Tate [T], Iwasawa [Iwl,2], van der Geer-Schoof [GS], and Li [Li], as well as the works of Lang [Ll,2], Arakelov [L3], Szpiro [L3], Parshin [Pa], Moreno [Mo], Neukirch [Ne], Deninger [De 1,2], Connes [Co], and Borisov [Bor]. Also, it would be extremely interesting if one could relate the work here with that of Connes [Co] and Deninger [Del,2,3]. (2) One may apply the discussion here to wider classes of (multiplicative) characters and test functions so as to obtain a more general class of Lfunctions as done in Tate's Thesis. (3) Instead of working on SF, the set of equivalence classes of valuations of F, or better, normalized valuations over F, one may also work on the set of all valuations of F, which in particular is parametrized by S r x R to get a much more refined geo-arithmetical cohomology theory. To understand the above theory, we suggest the reader to use language of lattices (whose details will be recall in the following section): Denote by OF the ring of integers of F. Then for an (^F-lattice A of rank r, its geoarithmetical cohomology groups are defined to be locally compact groups H°(F, A) := A

and

Hl(F, A) := (R n x

C2)r/A.

Consequently, then (i) H°(F, A) is discrete while Hx (F, A) is compact; (ii) The Serre duality coincides with the Pontryagin duality, say in the case F = Q, ((Rri

x C 2 ) 7 A ) V * A;

(iii) the 0-th cohomology is then simply given by h\F, A) := log ( £ exp ( - n V^I £ ||x|L, - In V = T £ ||x||T)), xeA

o-.R

T:C

a definition due to van der Geer and Schoof [GS]. 1.3 1.3.1

Geometric Truncations Projective O/f-modules

Let K be an algebraic number field, i.e., a finite extension of the field of rationals Q. Denote by OK the ring of integers of K. Then by definition, an

1 High Rank Zetas for Number Fields

231

OK -module M is called projective if there exists an (9/c-module N such that M © N is a free OK -module. For projective OK -modules, it is well-known that (i) If M is a projective (9#-module and is a part of a short exact sequence (of Otf-modules) 0 -» M0 -» Mi -* M -» 0, then Mi ^ Mo © M. (ii) All fractional 0#-ideals are projective; and (iii) Rank 1 projective (9/f-submodules in K are simply fractional OKideals. Thus, by finiteness of the ideal class group of K, up to equivalence relation (defined by isomorphisms as (^-modules), there are only finitely many rank 1 projective (9^-modules in K. Therefore, we may choose integral OK -ideals a,-, i = \,...,h with h = h(K), the class number of K, such that (a) Any rank 1 projective (^-module is isomorphic to one of the a,; while (b) None of the a, and a, are isomorphic to each other if i + j , i, j =

\,...,h. We will fix a choice of a,-, i = \,...,h satisfying (a) and (b) above for the rest of this paper, and sometimes use a as a running symbol for them. With this discussion of the rank 1 projective <9tf-modules, now let us consider high rank OK -modules. Clearly for a fractional ideal a,

P„:=/%:=0k _1 ® a is a rank r projective O^-module. The nice thing is that such types of projective O^-modules, up to isomorphism, give all rank r projective OKmodules. Indeed, we have the following structural Proposition. (1) For fractional ideals a and b, Pra =* Pr$ if and only if a^b; (2) For a rank r projective OK-module P, there exists a fractional ideal a such that P =s Pa. The proof of this proposition is based on an induction on the rank and the following basic relation about fractional ideals: For two fractional ideals o and b, as <9/r-modules, a © b =* OK © ab. (See e.g., [FT].) So we omit the details.

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Now use the natural inclusion of fractional ideals in K to embed Pna into Kr. View an element in Kr as a column vector. In such a way, any OK -module morphism A : Pna —* Pnb may be written down as an element in A e MrXr(K) so that the image A(x) of x under A becomes simply the matrix multiplication A • x. In particular, one checks easily the following Lemma. IfA € GIXX,OK) defines an OK-isomorphism A : Pra -» Pr;f>> then b =* (detA) • a. In particular, in the case when both a and b are integral OK-ideals, (i) detA e UK, the group of units ofK; (ii) a = b; and (iii) A € Autor(P,;a).

1.3.2

(9/r-Lattices: First Level

We start our first level discussion on lattices in the next three sections, mainly follow [Gr] for the expositions. Let cr be an Archimedean place of K, and Ka be the cr-completion of K. It is well-known that the R-algebra Ka is either equal to R, or equal to C. Accordingly, we call cr (to be) real or complex, write sometimes in terms of cr : R or cr : C accordingly. If Va- is a finite dimensional ^-vector space, then an inner product on Va- is a positive definite bilinear form Va- x Va- -> Ka- which is symmetric if cr is real and is hermitian if cr is complex. When equipped with an inner product, Va- is called a metrized space. By definition, an OK-lattice A consists of (1) a projective O/r-module P = P(A) of finite rank; and (2) an inner product on the vector space Va :- P ®oK Ka- for each of the Archimedean place cr of K. Set V = P®z R. Then V = Ylaes^ Va- Indeed, this is a direct consequence of the fact that as a Z-module, an <9#-ideal is of rank n = r\ + 2r2 where n = [K : Q], r\ denotes the number of real places and rj. denotes the number of complex places (in Soo)-

233

1 High RankZetas for Number Fields 1.3.3

Space of (^-Lattices: First Level

Let P be a rank r projective OK -module. Denote by GL(P) := AutoA.(P). Let A := A(P) be the space of (0/s;-)lattices A whose underlying OKmodule is P. For cr € Soo, let Aa be the space of inner products on VV; if a basis is chosen for Va as a real or a complex vector space according to whether cr is real or complex, Aa may be realized as an open_set of a realor complex vector space. (See 5.2 below for details.) We have A = rio-es^ ^oand this provides us a natural topology on A. As above, consider GL{P) to act on P from the left. With the chosen basis, elements of A will be thought of as column vectors, and matrices of linear maps will be written on the left. Given A 6 A and u,w e Va-, let (u, W)A,O- or (u, w)PA(cr) denote the value of the inner product on the vectors u and w associated to the lattice_A. As such, if A e GL(P), we may define a new lattice A • A in A by the following formula (U,W)AK,O-

•= (A~l

• u,A~l

• W)K,O-

This defines an action of GL(P) on A. Clearly, then the map v H-» Av gives an isometry A = A • A of the lattices. (By an isometry here, we mean an isomorphism of OK -modules for the underlying Oj^-modules subjecting the condition that the isomorphism also keeps the inner product unchanged.) Conversely, suppose that A : Ai = A2 is an isometry of Oj^-lattices, each of which is in A. Then, A defines an element, also denoted by A, of GL(P). Clearly A2 = A • A\. (Here we follow a nice notation initiated by Miyaoka; an isomorphism of corresponding algebraic structures is only denoted by =s while an isometry is denoted by =.) Therefore, the orbit set GL(P)\A(P) can be regarded as the set of isometry classes of OK -lattices whose underlying <9jt-modules are isomorphic to P. _ We end this section by introducing an operation among the lattices in A. If T is a positive real number, then from A, we can produce a new O^-lattice called A[T] by multiplying each of the inner products on A, or better, on Ao- for cr e 5 00, by T2. Let then A = A(P) be the quotient of A by the equivalence relation A ~ A[T]. As such, A admits a natural topological

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L. Weng

structures well. Furthermore, as it becomes clear later, the construction of A from A plays a key role when we want to get the compactness statement for our moduli spaces. (Indeed, the [^-construction naturally fixes a specific volume for a certain family of lattices, while does not really change the 'essential' structures of lattices involved. Consequently, by reduction theory, semi-stable lattices of a fixed volume form a compact family.) 1.3.4

Semi-Stable Lattices: First Level

Let A be an O^-lattice with underlying O^-module P. Then any submodule P\ c P can be made into an (^-lattice by restricting the inner product on each Va- to the subspace V\>a- := P\ «S>K Ka. Call the resulting (^-lattice Ai := A n P\ and write Ai c A. If moreover, P/P\ is projective, we say that Ai is a sublattice of A. The orthogonal projections na : Va- -> Vfa to the orthogonal complement Vf- of V\,o- in Va- provide isomorphisms (P/P\) <8>oK K& - Vf-^, which can be used to make P/P\ into an 0^-lattice. We call this resulting lattice the quotient lattice of A by Ai, and denote it by A/Ai. There is a procedure called restriction ofscalars which makes an ORlattice into a standard Z-lattice. Recall that V = A <8>z K. = Flo-es*, VaDefine an inner product on the real vector space V by (U, Vf)oo := 2_/«
e

<"
Denote Resj^/QA the Z-lattice obtained by equipped P, regarding as a Zmodule, with this inner product (at the unique infinite place oo of Q). We let rk(A) denote the OK -module rank of P (or of A) and let dim(A) denote the rank of P as Z-module. Clearly, dim(A) = rk(A) • dim(0jf) = rk(A) • [F : Q]. We define the Lebesgue volume of A, denoted by VolLeb(A), to be the (co)volume of the lattice Resjf/QA inside its inner product space V. As such, this volume may be computed as det oo. For examples, (a) If dimA = 0, then VolLet>(A) = 1;

1 High Rank Zetas for Number Fields

235

(b) If dimA = 1, then VolLeb(A) is the length of a generator of A; (c) If dimA = 2, then VolLeb(A) is the area of a fundamental parallelopiped, and so on. Clearly, if P' is a submodule of finite index in P, then VolLeb(A') = [P : P'JVolLebCA),

where A' = A n P is the lattice induced from P'. Examples: 1) Take P = OK and for each place or, let {1} be an orthonormal basis of Va = Ka, i.e., equipped V„- = R or C with the standard Lebesgue measure. This makes OK into an ^-lattice OK = (OK, 1) in a natural way. It is a well-known fact, see e.g., [LI], that

VolLebpI) = 2-" • VA7, where A/r denotes the absolute value of the discriminant of K. More generally, take P = a an fractional idea of K and equip the same inner product as above on Vp-. Then a becomes an O^-lattice a = (a, 1) in a natural way with rk(a) = 1. It is a well-known fact, see e.g., [Ne], that VolLeb(a) = 2~n • (N(a) • VAT), where N(a) denote the norm of a. Due to the factor 2~n, we also define the canonical volume of A, denoted by Volca„(A) or simply by Vol(A), to be 2r2rk(A)VolLeb(A). (This canonical volume is theoretically correct. In fact, in Arakelov theory, where the minus of the log of the canonical vomule is defined to be the Arakelov-Euler characteristic x(A) of A. That is, *(A):=-log(Vol(A)). For details, see the justification given in 1.3.5 below.) So in particular, Vol(a) = N(a) • y[K^, with

Vol(0^) - VA7 as its special case.

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L. Weng

2) The [T] -construction changes volumes of lattices in the following way Vol(A[r]) = r dim(A) • Vol(A). Now we are ready to introduce our first key definition. Definition. An OK lattice A is called semi-stable (resp. stable) if for any proper sublattice Ai of A, Vol(Ai)rk(A) > (resp. >)Vol(A)Vol(Al). Clearly the last inequality is equivalent to VolLeb(A1)rk(A) > VolLeb(A)Vol(A>). So it does not matter which volume, the canonical one or the Lebesgue one, we use. Remark. Despite the fact that we introduce the stability for lattices independently, many others, notably Stuhler, introduced the stability earlier. (See e.g. [Gr], [St].) 1.3.5

Minkowski Metric versus Canonical Metric

This section is specially introduced for the reader who wants to see clearly the relation between Lebesgue and canonical volumes. We mainly follow [Neu] for the presentation. For the first reading, one can skip it, except perhaps the Arakelov Riemann-Roch. Minkowski's basic idea on using Geometry of Numbers to study algebraic number field K/Q of degree n is to interpret its numbers as points in an n-dimensional space. For this, let us consider the canonical map ;': K -» Kc := [~[ C,

a H-» j(a) := (TO)

T

induced from the n complex embeddings T : K —> C. (Here, if cr : K —* Ka- = R is a real Archimedean place, we use the natural embedding R. <^-> C = R + /R so as to get a natural map T : K -» C ending with C.) The C-vector space #c is equipped with the hermitian scalar product (X,y) = YjXryr-

237

1 High Rank Zetas for Number Fields

In the sequel, we always view Kc as the hermitian space with respect to this standard metric. Let G(C|R) be the Galois group generated by complex conjugation F : z i-> z. Then F acts on both the factors C of the product l\T C and on the index set of T'S at the same time; that is to say, a H o for a e C, while T i-> f for each embedding r : K —> C with associated complex conjugate f : K -» C. Altogether, this defines an involution F : Kc -> Kc, which, in terms of points z = (zT) e Kc, is given by (Fz)T = If- Clearly, the scalar product <•, •) is equivariant under F. That is to say, (Fx, Fy) = (x,y). We now concentrate on the R-vector space *rR:=/£:=[f[c]+ T

consisting of the G(C|R)-invariant, i.e., f-invariant, points of KQ- Easily, the F-invariant points of Kc are exactly these points (zT) such that Zf = zTIn particular, then we have KR^Rn via the map (x^,...

XC" 2 .

,*o-r] ;Za>1+i>Zo>1+1, • • • >Zo-ri+r2,Zo-ri+r2) |—

* \XCTl > • • • 5 XCTr{ > ZcTry + 1 > • • • '

^0-ri+r2)-

Since 7(a) = TO" for a e K, we have F(j(a)) = F(a) for all ae K. This then further induces a map j : K —> KR. The restriction of hermitian scalar product (•, •) from KQ to £ R gives a scalar product <•, •> : KR x KK -> R on the R-vector space £ R . We call the Euclidean vector space K& = [ ] 1 T C ] + the Minkowski space, its scalar product <•,•) the canonical metric, and the association Haar measure the canonical measure. Moreover, the map j : K -* £ R identifies the vector space £ R with the tensor product K ®Q R. That is, we have a natural identification if®QlR^ .KR,

a® x i-» 0'(a)) • JC.

Likewise, K ®Q C =* ^c- With this said, then the inclusion KR C # C corresponds exactly to the canonical map K<8>Q R —» A"®Q C induced from the natural inclusion R <^-» C. And simply, F corresponds to a ® z i-» a ® z.

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On the other hand, the Lebesgue volume is associated with the following alternative explicit descripion of the Minkowski space KR. When embeddings r : K —» C are real, their images land already in R. As for complexes, i.e., these which are not real, by writing them in pairs, we may list (all real embeddings as cr\,..., crn : K —» R, and) all complex (conjugate) embeddings as T\, f \,...,Tn, fn : K -» C. (Here [K : Q] = n - r\ + 2r2.) Choose from each pair r 7 , f7 a certain fixed complex embedding Tj. With this done, then let cr vary over the family of real embeddings and T over the family of chosen complex embeddings. Write p as a (common) running symbol for cr, and [TJ, f ; }. Since F leaves the cr invariant, but exchanges T and f, we have KR = j(zP) e f ] C : v e R , Z f = z ? e C ) . p

As a direct consequence, we then obtain a natural isomorphism

V

where for the reals Xa- - Zo- while for the complexes xT = Re (zT), xf = Im (zT). (So in particular for complex T, fe.Zf) = (*T + iyT,xT ~ iyT) i-+(xT,yT),

(^+y?) + (^+^)^2U?+y?).) This isomorphism transforms the canonical metric (•, •) on JSR into the scalar product (x,y) = EP^P-^PJP where A^p = 1 resp. Np = 2 if p is real, resp. complex. The scalar product (x, y) = EpNp*pyp transforms the canonical measure from KR to a measure on R ri+2r2 . It obviously differes from the standard Lebesgue measure by V0lcan(X) = 2'2V0lLeb(/(X)). Minkowski himself worked with the Lebesgue measure on R ri+2r2 , and afterwards, most works, e.g., research papers and textbooks, follow suit. As we said before, we will write Volcan simply as Vol. Also in the calculations below, we use both canonical and Lebesgue measures without any clear indication on which one we really use for our convenience.

1 High Rank Zetas for Number Fields

239

As said too, the canonical measure has an advantage theoretically. For example, we have the following Arakelov-Riemann-Roch Formula: For an O^-lattice A of rank r, - log (Vol(A)) = deg(A) - T- log A*. For a proof, see e.g., [L2]. (For the reader who does not know the definition of the Arakelov degree, he or she may simply take this relation as the definition.) 1.3.6

Canonical Filtration

Based on stability, we may introduce a more general geometric truncation for space of lattices. Let us start with the following well-known Lemma. ForafixedO-lattice A, jVol(Ai) : A j C A c R>o is discrete and bounded from below. Proof. It is based on the follows: 1) If a lattice Ai induced from a submodule of the lattice A has the minimal volume among all lattices induced from submodules of A of the same rank, then Ai is a sublattice; 2) Taking wedge product, we may further assume that the rank is 1. It is then clear. • As a direct consequence, we have the following Proposition. Let A be an O-lattice. Then (1) (Canonical Filtration) There exists a unique filtration of proper sublattices 0 = A0 c Ai c • • • c As = A such that (i) for all i = 1,..., s, A,7A;_i is semi-stable; and (ii) for all j=

l,...,s-l, (Vol(A;+i/A;-)J

> (VoliAj/Aj-i))

;

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L. Weng

(2) (Jordan-Holder Filtration) If moreover, A is semi-stable, then there exists a filtration of proper sublattices, 0 = A' +1 c A' c • • • c A0 = A such that (i) for allk = 0,. ..,t, A*/A*+1 is stable; and (ii) for all I =

\,...,t, ( V o K A ' / A - ) ) * ^ 1 ^ = (Vol ( A'-VA')) rk(A ' /A ' +1) .

Moremore, up to isometry, the graded lattice Gr(A) := ©jj._0A*7A*+1 is uniquely determined by A. Proof Existence is clear and the uniqueness of Jordan-Holder graded lattice is fairly standard. Let us look at the uniqueness of canonical filtration, an analogue of the well-known Harder-Narasimhan filtration for vector bundles: Existence of two such filtrations will lead to a contradiction by applying the fact that for any two sublattices Ai, A2 of A, Vol(Ai/(Ai n A2)) > Vol((Ai + A 2 )/A 2 ). • 1.3.7

Compactness

For a lattice A e A with the associated canonical filtration 0 = Ao c Ai c • • • c Aj = A define the associated canonical polygone p A : [0, r] —> R. by the following conditions: (1) p A (0) = pA(r) = 0; (2) p A is affine over the closed interval [rkA,-, rkA,+i]; and (3) 7>A(rkA,-) = deg(A,) - rk(A,) • %m.

1 High Rank Zetas for Number Fields

241

Clearly, the canonical polygon is well-defined on the space A. _ To go further, let us introduce an operation among the lattices in A. If T is a positive real number, then from A, we can produce a new (9j^-lattice called A[T] by multiplying each of the inner products on A, or better, on AQ- for a- e 5oo, by T 2 . Obviously, the [T]-construction changes volumes of lattices in the following way Vol(A[T]) = rrk(AH*:Q] . Vol(A). That is to say, the |T]-construction naturally fixes a specific volume for a certain family of lattices, while does not really change the 'essential' structures of lattices involved. Accordingly, let A = A(P) be the quotient topological space of A modulo the equivalence relation A ~ A[T]. Since canonical polygon is invariant under the scaling operation and hence descends to this quotient space A. With all this, we are ready to state the following fundamental Theorem. For any fixed convex polygon p : [0, r] —* R, the subset |[A] 6 A : p A < p\ is compact in A. Proof. This is based on the follows: (1) Dirichlet's Unit Theorem; and (2) Minkowski's Reduction Theory: Due to the fact that volumes of lattices involved are fixed, semi-stability condition implies that the first Minkowski successive minimums of these lattices admit a natural lower bound away from 0 (depending only on r). Hence by the standard reduction theory, see e.g., Borel [Bol,2], the subset |[A] € A : /?A < p\ is compact. • 1.3.8

Geometric Truncation: Adelic Version

Let X = Spec (2/r. If £ is a vector sheaf of rank r over X, i.e, a locally free Of-sheaf of rank r, denote by EF the fiber of E at the generic point Spec(F) <^-> Spec OF of X (Ep is an F-vector space of dimension r), and for each v e Sf := Sfin, set E0v :- H°(SpecOFv,E) a free <9v-module of rank r. In particular, we have a canonical isomorphism: can v : Fv 0v E0v =* Fv ®p EF.

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Now if E is a vector sheaf of rank r over X equipped with a basis ap • Fr =* £> of its generic fiber and a basis aov '• Orv ^ Eov for any v e Sf, the elements gv := (F v ®f QTF)-1 O canv o (F v ®
* (E.)F,

and with an isomorphism of filtrations of free <9v-modules a , A : ((0) c O J ' c - C (%+n+~+r,=r) ^

( £

^

for every v e Sf. Clearly, this bijection induces a bijection between the set of isomorphism classes of filtrations of vector sheaves of rank ( n , r\ + r2,..., r\ + r2 + • • • + rs = r) over X and the double coset space Pi(F)\ P/(A/)/P/(G>). The natural embedding P/(A/) <^-» P/(A) (resp. canonical projections P/(A/) -> M/(A/) -> GLrj(Af) for j = 1,..., s, where M7 denotes the standard Levi of P/,) admits a modular interpretation (E»;a,tF : (a*,ov)v€S/) •-* (Es;aSiF : (as,ov)veS/) (resp. (Z?*;a.,F : (a*,ov)vesf) ^ (gr/^*);gr ; -(a*,f),gr/ar* I a)ves / ),

243

1 High Rank Zetas for Number Fields

where gr,(£.) := EJ/EJ-U gr/ar.,/0 : FrJ =* gr/£*)f and gr;(or»>0v) : Ovrj =* grj(Et)ov, v e S / are induced by a»,f and or,^,, respectively.) Moreover, any g = (g/;goo) e GL r (A/) x GLr(A<x,) = GLr(A) gives first a rank r vector sheaf £ g on Spec (9f, which via the embedding £> ^-» (Rri x C2)r gives a discrete subgroup, a free rank r G>-module. In particular, goo = (go-) then induces a natural metric on Eg by twisting the standard one on (R n x C2)r via the linear transformation induced from goo. As a direct consequence, see e.g, [L3], deg(£ g ,p g ) = -log(AT(detg)) with N : GL\(AF) = IF -> R >0 the standard norm map of the idelic group ofF. With this, for g = (g/; goo) € GLr(A) and a parabolic subgroup Q of GLr, denote by EfQ the filtration of the vector sheaf Egf induced by the parabolic subgroup Q. Then we have a filtration of hermitian vector sheaves (E8'Q,pfQ) with the hermitian metrics p8-Q on E8j'Q obtained via the restrictions of pgoo. Now introduce an associated polygon p i : [0, r] -» R by the following conditions: (i)p*(0) = p*(r) = 0; (ii) p i is affine on the interval [rkE8^, rkE8'G]; and (iii) for all indices i,

p* (rk£f G ) = deg(£f;G,pf;G) - ik£f fl •

^l£l.

Then, by Prop. 1.3.6, there is a unique convex polygon p*, the canonical polygon associated with g, which bounds all p8Q from above for all parabolic subgroups Q for GLr. Moreover there exists a parabolic subgroup Qg such that p8 = p8. Hence we have the following fundamental Main Lemma. For any fixed polygon p : [0, r] —> R and any d e R, the subset [g e GLr(F)\GLr(A): degg = d,p* < p) is compact.

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Proof. This is a restatement of the classical reduction theory, i.e., Thm. 1.3.7. More precisely, it consists of two parts: In terms of 0/r-lattices, by fixing the degree, we get a fixed volume for the free (9f-lattice of rank r corresponding to (Eg,pg), by the Arakelov Riemann-Roch formula. Thus the condition p^ < p gives an upper bound for the volumes of all the sublattices of (Eg,pg) and a lower bound of the Minkowski successive minimums. That is to say, at the lattice level, the corresponding space is compact. On the other hand, a general theory says that fibers of the natural map from GLr(F)\GLr(A)/GLr(OF) to isometry classes of (^-lattices are all compact: This is in fact generally true for all reductive groups, a result due to Borel [Bol,2]. In fact, our current case for which only the group GLr is involved is rather obvious: As to be seen in 5.2, essentially, the fibers are the compact subgroup S 0 r (R) n x S Ur(C)r2. a Similarly yet more generally, for a fixed parabolic subgroup P of GLr and g 6 GLr(A), there is a unique maximal element p8p among all p8Q, where Q runs over all parabolic subgroups of GLr which are contained in P. And we have Main Lemma'. For any fixed polygon p : [0, r] -* R, d e R and any standard parabolic subgroup P ofGLr, the subset [g € GLr(F)\GLr(A):

degg = d,pgp

-p)

is compact. 1.4 1.4.1

High Rank Zeta Functions for Number Fields Definition

For a number field F with (absolute value of) discriminant A/r, denote by M.F,r the moduli space of semi-stable 0/r-lattices of rank r. Then naturally MF,K :=Ur€R+ MFAT] where A1f>|T] := MFA^S, T) denotes the volume T part of the moduli. For an adele element g e GLr(A), denote its associated (isometric class of) OF -lattice of rank r by U(g). Clearly, II factors through GLr(F)\GLr(A). Denote by MAF,r c GL r (F)\GL r (A) the inverse image of M.F,r- It is well known that fibers of Et : M.AF,r —> MFS

1 High RankZetas for Number Fields

245

are all compact. Hence so is MAFAT] '•- n 1(MFAT]\ Consequently, we obtain natural measures dp. on MFJ and M.AF,rU"\ induced from that on GLr(A). In particular, being compact, MAF,r[A£] is °f finite volume. For any A e MAF>r, set h'(F, A) := h'(Ap, g) where g is an element of GLr(A) whose associated <9f-lattice coincides with A. This is well-defined since for any a e GL r (F), ti(AF,a- g) = ti(AF,g) by definition. Indeed, as said earlier, h°(F, A) = log ( J ] exp ( - 7T V^T J ] Hx"- " <x:R

X€A

27r

^

Z

I|X|IT

))"

T:C

Definition. The (completed) rank r zeta function £JFAS) °f F *s defined by frAs) := {4)S

{eh°iF'A) ~ l ) ( ^ ) d e g ( A ) • dp, Refc) > 1.

f JAeMFr

Following 1.1, i.e., Iwasawa's interpretation of Dedekind zeta functions, we see that £;F,\(S) is essentially the completed Dedekind zeta function gp(s) for F. 1.4.2 Basic Properties Just as Dedekind zeta functions, high rank zetas are canonical as well. Theorem. (1) Rank r zeta function €FA'):={4)'

f

(^° ( / r ' A ) -l)(O d e g ( A ) -^

JheMFr

converges absolutely and uniformly when Re(^) > 1 + 8 for any 6 > 0. Moreover, £;FAS) admits a unique meromorphic continuation to the whole complex s-plane; (2) The extended £FAS) satisfies the functional equation €FAS) = ZFA1 - *); (3) The extended £FAS) has two singularities only, both are simple poles, at s - 0,1 with Res,=i fr>(*) = Res J=0 &>(.*) = Vol(AtF,r[A*]).

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L. Weng

Remarks. The global non-abelian invariant Volf MFA^F^) for F can be calculated in terms of special values gf (2), %p(3),..., £F(>") of the Dedekind zeta. For details, see 4.8. Proof. Motivated by Weil's treatment of functional equation in [W2], we first give a formal discussion to show the key points of functional equation, but under a more general context. For the number field F with discriminant Af, denote by MAFS the moduli space of semi-stable adelic bundles of rank r, that is, where N MAF,r :=UAf£R+ MAFAN] MAF,r[N] := MAFA1°B )- A S topological spaces, MAFS - MAFA\^F\^] X R.+- Hence we obtain a natural measure d/j. on MAF,r from the Tamagawa measures on MAF ,r[N] and %L onR+. Recall that for E e MAF,r, H^E) := H^(F,g) = eh'(AF'8) where g 6 GLr(A) is such that E = [g]. Since for any a € GLr(F), H'g!i(F, a- g) = H^iF, g) by definition. H^iF, E) is well-defined for i = 0, 1. With respect to fixed real constants A,B,C,a and/?, introduce the formal integration

ZF,rA,B,C;afi(s)

'• =

( < ( F , E)A • N(E)Bs+c - N{E)as+B)d»{E)-

(IAH-T)' f

Then formally, Zpr^B,C\afi{s) = I(s) - II(s) + III(s), where

/(*):= (|AFr"r f j£eM A/r ,r,W(£')<|AF|2

(H^(F, E)A • N(E)Bs+c - N(E)as+B)d/x(E); as+B II(s):=(\AF\-?y f dti(E); r N(E) JEeMAF„N(E)>\AF\'Z

III(s) := (|A F |-T)* f ,

r

H°sa(F, E)A • N(E)Bs+cdfi(E).

j£ eM A F , r ,W(£)>|Af|J

By Theorem 1.2.3, i.e., the Serre duality and Riemann-Roch theorem, we

247

1 High Rank Zetas for Number Fields have III(s) = (IA/H-T) x f

r

H°UF, E)A •

N(E)B^-^^Cd^E\

JEeMAFsME)<\AFP by the fact that N(Ei®E%) = A ^ O ^ ^ - J V C / ^ ) - ™ * ^ . Hence, formally, A + 2C = Ks) + I(-S - — — ) - 7/(5) + IV(S),

ZFMC;afi{s) where

rff

/V(s)

A+2C

:= ( | A F | - T )

»

x f NiEr^-^^d^E). r JEeMAF,rME)<\AF\i Moreover, by definition, -II(s) = - f , N(E)as+Bd/i(E) JE£MAF,rME)>&} =- f

,d*E).

JM AE .. r [|AF|5] fas+B

= - WF(r)

("<»*% Jl

as+fl l

*

= WF(r)

1 as+fi'

provided that as +/3 < 0. Similarly,

iv(S)= f

r

^A
= WF(r) provided that a(-s - ^^-) Therefore, formally,

fta(-s-^)+/3-

dmJO

*

1 a(-s-^)+J3 +j3>0. A + 1C

ZF,rA,B,C;aJ3(s)

= I(s) + I(-S

+W WFF(r) (r) •• (— + as+J3

)

a(-s-^)+p

248

L. Weng

As a direct consequence, we have £F,r\A,B,C;afi\S) = ^F,r\A,B,C;afi{~s

£

)•

D

This is the functional equation. Next let us justify the arguments above, we consider two types of convergence problems. Type 1. Convergence for II(s) and IV(s), where II(s)={\AF\-Ty

as+B

f

rN(E)

dn{E);

JEeMAFirME)>\AF\l rfl

-

A+2C

IV(s)= (\AF\~T) * B X JEeM*rrME)<\AF\i >EeMAF,rME)<\AF\

From above calculation, whenRe(or-s+B) < 0 andRe(a-(-s-^±g£)+8) 0, being holomorphic functions, 11(5) = -WF(r) • —-^—:, IV(s) = WE(r) • — Type 2. Convergence for I(s) and I(-s - ^^-) I(s)

>

\

where

=(IA F I-*)' r

lEeMAF,rME)<\AF\i JEs

(flJa(F, E)A • N(E)Bs+c -

N{E)as+p)dp(E).

By the discussion above about II(s), unless B = a, I(s) and I(-s ^ j p ) , and hence ZFr^B,C;ajs(s) cannot be meromorphically extended as a meromorphic function to the whole s-plane. (See also the discussion below.) Thus, we introduce the following Compatibility Conditions: a = B and/3 = C. With this, by a change of variables, we may afterwards assume that B = 1 and C = 0 without loss of generality so as to obtain the following

249

1 High Rank Zetas for Number Fields integration: ZF,rA(s)

:

= Z/v-;A ( -l,0;-l,o(s)

:=(|A F |i)'- f

(/^a(F,£)A-l)-iV(£)-VM£).

j£eA
(Note that then Z/rr;i(.s) = &>(*)•) Furthermore, for any g e GLr(A), note that H°(AF, g) is discrete, so from the definition, the associated measure is simply the standard counting measure. Thus by writing down precisely, H°sa(F,g) = 1+

J]

exp(~"J]\g
aeH°(SpecOF,£(g))\{0}



=:l+H's&(F,g). In this expression, the first term is simply the constant function 1 on the moduli space, while each term in the second decays exponentially. Hence, by a standard argument about convergence of an integration of theta series for higher rank lattices, see. e.g, [Ne], and the fact that 1 in the first term H^a(F, E)A cancels with 1 appeared as the second term in the combination H^a(F, E)A - 1, we conclude that II(s) and II(-s - A) are all holomorphic functions, provided A > 0. • 1.5

First Example

1.5.1 Epstein Zeta and High Rank Zeta Accordingly, the rank r zeta function ^q,r(s) of Q is given by

&,,(*) = f

(eft°«-A) - l) • ( e -) deg(A) dtfA),

Re(,) > 1,

where h°(Q, A) := log I £ exp ( - n\x\2) I, And we have

deg(A) = - log Vol(Rr/A).

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L. Weng

Theorem', (i) £Q,I(S) coincides with the (completed) Riemann-zeta; (ii) ^q,r(s) can be meromorphically extended to the whole plane; (iii) £Q,,-(S) satisfies the functional equation £JQ,r(s) = fQ,r(l - s); (iv) ^Q,r(*) has two singularities at s = 0, 1 only, all simple poles with the same residues Vol (A1Q,,-[1]J. Decompose according to their volumes, Mq>r = there is a natural morphism MQ^[T] —> AtQ,r[l], A quently,

and r ? • A. Conse-

U^OAIQ^IY], H

(e"0«-A>-i)-(Odeg(A)^(A)

€oA*)= f J<JT>oMQ,r[T]

Jo

l JAVm \

'

But h°(Q, Tlr • A) = log (-ZxeA exp ( - n\x\2 • 7"?)) and

r

T

B

\Bl

we have ^r(s)=rx-n-isT(r-s)-

f

2

2

\x\~rs

V

JMQAU UAMO)

J

Accordingly, introduce the completed Epstein zeta function for A by E(A;S):=n-sns)-

\x\~2s.

£ *eA\{0)

We then arrive at Proposition. (Eisenstein Zeta and High Rank Zeta) r f ^Q,rW = 2

JMQAD

r E(A,-s)d/ii(A). l

251

1 High Rank Zetas for Number Fields 1.5.2

Rankin-Selberg & Zagier Method

Consider the action of SL(2, Z) on the upper half plane (H. Then a standard 'fundamental domain' is given by D = {z = x + iy 6 *H : \x\ < \,y > 0, x2 + y2 > 1}. Recall also the completed standard Eisenstein series E(z;s):=7r-Sr(s)-

YJ _

y> \mz + n\2s

(m,n)eZ2\{(0,0)) '

Naturally, we are led to consider the integral fD E{z, s)-^. However, this integration diverges. Indeed, near the only cusp y = oo, by the ChowlaSelberg formula, E{z, s) has the Fourier expansion YJ^n(y,s)eZninx

E(z;s)=

n=—oo

with

an{y S)

'

(£(2s)ys + t(2-2s)yl-s, \2MH
ifn = 0; if n * 0,

where £,(s) is the completed Riemann zeta function, crs(n) := Y^d\n ds, and Ks(y) •= \ £° e-y{t+^l2ts% is the K-Bessel function. Moreover, \Ks(y)\ < e-y/2KKe(s)(2), if y > 4,

and

Ks = K-s.

So an±o(y, s) decay exponentially, and the problematic term comes from ooCy. s), which is of slow growth. Therefore, to make the original integration meaningful, we need to cutoff the slow growth part. There are two ways to do so: one is geometric and hence rather direct and simple; while the other is analytic, and hence rather technical and traditional, dated back to Rankin-Selberg. (a) Geometric Truncation Draw a horizontal line y = T > 1 and set DT = {z = x + iy € D : y < T),

DT = {z = x + iy e D : y >T).

252

Then D = DjUD7.

L. Weng Introduce a well-defined integration

y2

JDT

(b) Analytic Truncation Define a truncated Eisenstein series Ej(z; s) by

\E(z,s)-a0(y;s),

ify>T.

Introduce a well-defined integration

4™(s):=

dxdy [ET(Z;S)-

f

JD

With this, from the Rankin-Selberg method, one checks that we have the following: Proposition. (Analytic Truncation = Geometric Truncation in Rank 2)

S-

1

5

J

For a proof, please see Ch. 7. Each of the above two integrations has its own merit: for the geometric one, we keep the Eisenstein series unchanged, while for the analytic one, we keep the original fundamental domain of "K under SL(2, Z) as it is. Note that the nice point about the fundamental domain is that it admits a modular interpretation. Thus it would be very nice if we could keep the Eisenstein series unchanged, while offer some integration domains which appear naturally in certain moduli problems. Guided by this, in Ch. 2, we will introduce non-abelian L-functions using integrations of Eisenstein series over generalized moduli spaces. (c) Arithmetic Truncation Now we explain why the above discussion and Rankin-Selberg method have anything to do with our high rank zeta functions. For this, we introduce yet another truncation, the stability one.

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1 High Rank Zetas for Number Fields

So back to the moduli space of rank 2 lattices of volume 1 over Q. Then classical reduction theory gives a natural map from this moduli space to the fundamental domain D above: For any lattice A, fix xi 6 A such that its length gives the first Minkowski minimum A\ of A. Then via rotation, we may assume that xi = (A\, 0). Further, from the reduction theory j-A may be viewed as the lattice of the volume A^2 = yo which is generated by (1,0) and u) = xo + iyo £ D. That is to say, the points in DT constructed above are in one-to-one corresponding to the rank two lattices of volume one whose first Minkowski minimum A~[2 < T, i.e, A\ > T~i. Set AC 5 2 ° S [1] be the moduli space of rank 2 lattices A of volume 1 over Q all of whose sublattices Ai of rank 1 have degrees < \ log T. As a direct consequence, we have the following Proposition. (Geometric Truncation = Arithmetic Truncation) There is a natural one-to-one, onto morphism

In particular, M|° 2 [1] = M Q>2 [1]-£>1. Consequently, we have the following Example in Rank 2.

£Q,2(S)

=

f(2s) 5-1

f(2*-l)

2 Non-Abelian L-Functions 2.1

Automorphic Forms and Eisenstein Series

To facilitate our ensuing discussion, we make the following preparation. For details, see e.g., [MW] and [We7]. Fix a connected reductive group G defined over F, denote by ZQ its center. Fix a minimal parabolic subgroup PQ of G. Then Po = MQUQ, where as usual we fix once and for all the Levi MQ and the unipotent radical UQ. A parabolic subgroup P is G is called standard if P D PQ- For such groups write P = MU with Mo c M the standard Levi and U the unipotent radical. Denote by Rat(Af) the group of rational characters of M, i.e, the morphism M —» Gm where Gm denotes the multiplicative group. Set a*M : = Rat(M) ® z C,

oM : = Homz(Rat(M), C),

and Rea^ := Rat(M) ® z R,

ReaM := Homz(Rat(M), R).

For any x e Rat(M), we obtain a (real) character \%\ : M(A) —> R* defined by m = (mv) i-> m^ := fives \mv$v with | • |v the v-absolute values. Set then M(A)1 := r\eRat(M)Kerl^|, which is a normal subgroup of M(A). Set XM be the group of complex characters which are trivial on M(A)1. Denote by HM '•= logM : M(A) -» OLM the map such that SX € Rat(M) c o^, <x, logM(m)> := log(mM). Clearly, M(A)1 = Ker(logM);

logM(M(A)/M(A)1) - ReoM.

Hence in particular there is a natural isomorphism K : a*M =* XM- Set ReXM := K(Rea*M),

IHIXM

:= K(i • Rea*M).

Moreover define our working space XM to be the subgroup of XM consisting of complex characters of M(A)/Af(A)1 which are trivial on ZG(A)Fix a maximal compact subgroup K such that for all standard parabolic subgroups P = MU as above, P(A) n l = (M(A) n K ) • (t/(A) nK). Hence we get the Langlands decomposition G(A) = Af (A) • f/(A) • K. Denote by mP : G(A) -» M(A)/M(A)1 the map g = m • n • k \-> M(A)1 • m where g € G(A), m e M(A), n € t/(A) and A: € K. 254

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2 Non-Abelian L-Functions

Fix Haar measures on Afo(A), Uo(A), K respectively such that (1) the induced measure on M(F) is the counting measure and the volume of the induced measure on M(F)\M(A) 1 is 1. (Recall that it is a fundamental fact that M(F)\M(A) 1 is of finite volume.) (2) the induced measure on Uo(F) is the counting measure and the volume of U(F)\U0(A) is 1. (Recall that being unipotent radical, U(F)\U0(A) is compact.) (3) the volume of K is 1. Such measures then also induce Haar measures via logM to the spaces CLM0, a*M , etc. Furthermore, if we denote by po the half of the sum of the positive roots of the maximal split torus To of the central ZMQ of Mo, then ft-*

I Jilfo(A)-£/0(A)-K

f(mnk)dkdnm~2podm

defined for continuous functions with compact supports on G(A) defines a Haar measure dg on G(A). This in turn gives measures on M(A), U(A) and hence on CLM, &*M, P(A), etc, for all parabolic subgroups P. In particular, one checks that the following compatibility condition holds I

f{mnk) dk dn m~

Po

dm

JM0(A)U0(A)K

= I

f(mnk) dk dn m~

pp

dm

JM(A)U(A)K

for all continuous functions / with compact supports on G(A), where pp denotes one half of the sum of all positive roots of the maximal split torus Tp of the central ZM of M. For later use, denote also by Ap the set of positive roots determined by (P, Tp) and Ao = A/>0. Fix an isomorphism To - G^. Embed R+ by the map t \-* (l;f). Then we obtain a natural injection (R*+)R ^-> To(A) which splits. Denote by AM0(A) the unique connected subgroup of To(A) which projects onto (R* ) R . More generally, for a standard parabolic subgroup P = MU, set n AM(A) '•= -AMO(A) %M(A) where as used above Z» denotes the center of the group *. Clearly, M(A) = AM(A) • M(A)1. For later use, set also ^ L A ) := [a e AM(A) • logG a = 0}. Then AMW = AGm ©^M(A)Note that K and U{F)\U{A) are all compact, and M(F)\M(A) 1 is of finite volume. With the Langlands decomposition G(A) = C/(A)M(A)K in

256

L. Weng

mind, the reduction theory for G(F)\G(A) or more generally P(F)\G(A) is reduced to that for AM(A) since ZG(F) n ZG(A)\ZG(A) n G(A)1 is compact as well. (The reader should know that one of the chief purposes to have a nice reduction theory is to make sure that the integrations of moderate growth functions are controlled. Thus with M(F)\M(A)1 of finite volume, a good control is not a big problem. This then leads us to the affine part A.) As such, for to € Mo(A) set AM0(A)('O)

:= [a e AMo(A) • a" > ffla e A0}.

Then, for a fixed compact subset a> c Po(A), we have the corresponding Siegel set S(CJ; t0) := {p • a • k : p e OJ, a e AMo(A)(to), k e K}. In particular, for big enough a> and small enough to, i.e, t^ is very close to 0 for all or 6 Ao, the classical reduction theory may be restated as G(A) = G(F) • S(a>; to). More generally set A

M0(A)('o) := {a 6 AMo(A) : aa > igVor e A£},

and Sp{io; to) := {p • a • k : p € CJ,a € ApMo{K){to),k e K}. Then similarly as above for big enough a> and small enough to, G(A) = P(F)-Sp(a>; to). (Here Ap denotes the set of positive roots for (PoPiM, To).) Fix an embedding ig '• G «-» SLn sending g to (gif). Introducing a height function on G(A) by setting \\g\\ := fives sup{|g,yiv : V/,;'}. It is well known that up to 0(1), height functions are unique. This implies that the following growth conditions do not depend on the height function we choose. A function / : G(A) —» C is said to have moderate growth if there exist c, r € R such that |/(g)| < c • \\g\\r for all g e G(A). Similarly, for a standard parabolic subgroup P = MU, a function / : U(A)M(F)\G(A) -> C is said to have moderate growth if there exist c, r 6 R, A e ReX^o such that for any a e A^(A)> k e K, m € M(A)1 n Sp(co; to), \f(amk)\ < c • \\a\\r • mPo{m)\

257

2 Non-Abelian L-Functions

Also a function / : G(A) —» C is said to be smooth if for any g = gf goo € G(Af) x G(Aoo), there exist open neighborhoods V* of g* in G(A) and a C°°-function / ' : V*. -» C such that f(g'f • g^) = / ' ( g ^ ) for all gj € V, and g'n, € Vco. By contrast, a function / : S (o»; fo) -> C is said to be rapidly decreasing if there exists r > 0 and for all A e RCXMQ there exists c > 0 such that for a e AM(A),g € G(A)1 nS(w; f0), |0(ag)| < c • ||a|| • mPo(g)x. And a function / : G(F)\G(A) —» C is said to be rapidly decreasing if f\s(w,t0) is so By definition, a function 0 : U(A)M(F)\G(A) -» C is called automorphic if (i) 0 has moderate growth; (ii)

(k\ • * -^2) parametrized by (k\, Z^) € K x K is finite dimensional; and (iv) is 3-finite, i.e, the C-span of all S(X)(f> parametrized by all X e 3 is finite dimensional. Here 3 denotes the center of the universal enveloping algebra u := U(LieG(Aco)) of the Lie algebra of G(ATO) and 6(X) denotes the derivative of 0 along X. For such a function 0, set fa '• M(F)\M(A) —> C by m i-> m~pp(t>{mk) for all k € K. Then one checks that fa is an automorphic form in the usual sense. Set A(U(A)M(F)\G(A)) be the space of automorphic forms on U(A)M(F)\G(A). For a measurable locally L1 -function / : U(F)\G(A) —» C, define its constant term along with the standard parabolic subgroup P = UM to be fP : C/(A)\G(A) -> C given by g H-> fU(F)\G{A) f(ng)dn. Then an automorphic form e A(U(A)M(F)\G(A)) is called a cusp form if for any standard parabolic subgroup P' properly contained in P, (pp> = 0. Denote by A0([/(A)M(F)\G(A)) the space of cusp forms on U(A)M(F)\G(A). One checks easily that (i) all cusp forms are rapidly decreasing; and hence (ii) there is a natural pairing (•,•) : Ao(U(A)M(F)\G(A))xA(U(A)M(F)\G(A)) defined by ( 0 , 0 ) := / z „ ( A ) t / ( A ) M ( F n G ( A ) Hg)4>(g) dg.

-» C

258

L. Weng Moreover, for a (complex) character £ : Zu{K) —* C* of ZM(A) set A(U(A)M(F)\G(A))€

:= {0 € A(U(A)M(F)\G(A))

:


:=

J]

A(U(A)M(F)\G(A))f

feHom(ZM(A),C*)

and Ao(U(A)M(F)\G(A))z

:=

£

A0(U(A)M(F)\G(A))f.

feHom(ZM(A).C*)

One checks that the natural morphism C[ReaM] ® A(U(A)M(F)\G(A))Z

-> A([/(A)M(F)\G(A))

defined by (Q, ) i-» (g H> 2(logM(m/>(g))) • 0(g) is an isomorphism, using the special structure of A^(A)-finite functions and the Fourier analysis over the compact space AM(A)\ZM(A>- Consequently, we also obtain a natural isomorphism C[ReaM] ® Ao(U(A)M(F)\G(A))z

-+

A0(U(A)M(F)\G(A))f.

(See 4.2 below for a more concrete and useful form.) Set also rio(M(A))f be isomorphism classes of irreducible representations of M(A) occurring in the space Ao(M(F)\M(A))^, and n 0 (M(A) := U^Hom(ZA,(A),c«)n0(M(A))f. (More precisely, we should use Af(A/) x (M(A)nK, Lie(M(Aoo))®RC)) instead of M(A).) For any n e n 0 (M(A)) f set Ao(M(F)\M(A))„ to be the isotypic component of type n of AQ(M(F)\M(A)\, i.e, the set of cusp forms of

259

2 Non-AbeJian L-Functions

M(A) generating a semi-simple isotypic M(A/)x(Af(A)nK, Lie(M(Aoo))®R C))-module of type TT. Set A0(f/(A)M(F)\G(A))^ := { 6 A0(£/(A)M(F)\G(A)) : ^ € A0(M(F)\M(A))n,

Vk e K}.

Clearly A 0 (t/(A)M(F)\G(A)) f = e^no(M(A))^o(f/(A)M(F)\G(A))^. More generally, let V c A(Af(F)\M(A)) be an irreducible M(A/) x(Af(A) n K, Lie(Af(Aoo)) ®R C))-module with no the induced representation of M(A/) x (M(A) n K,Lie(M(Aoo)) ®R C)). Then we call n0 an automorphic representation of M(A). Denote A(M(F)\M(A))^0 the isotypic subquotient module of type no of A(M(F)\M(A)). One checks that V ® HomM(A/)x(M(A)nK,Lie(M(A„))®RC))(V; A ( M ( F ) \ M ( A ) ) )

-A(M(F)\M(A))„ 0 . Set A(t/(A)M(F)\G(A))„0 := { € A(C/(A)M(F)\G(A)): <j>k e A(M(F)\M(A))„0, Vk e K}. Moreover if A(M(F)\M(A))no c A0(M(F)\M(A)), we call n0 a cuspidal representation. Two automorphic representations n and no of M(A) are said to be equivalent if there exists A e X^ such that n ^ no ® A. This, in practice, means that A(M(F)\Af(A))^ = ,1 • A(M(F)\M(A))no. That is, for any „0(m). Consequently, A(£/(A)M(F)\G(A))^ = (A ° mP) • A(f/(A)M(F)\G(A)V0. Denote by ^P := [no] the equivalence class of no. Then ty is an X^ -principal homogeneous space, hence admits a natural complex structure. Usually we call (M, ty) a cuspidal datum of G if no is cuspidal. Also for n e ty set ReTT := Re;^ = \x„\ e ReXjtf, where Xn is the central character of n, and Inur := n <8> (-Re?r).

260

L. Weng

Now fix an irreducible automorphic representation n of M(A) and an automorphic form (p e A{U{A)M(F)\G{A))n, define the associated Eisenstein series E( C by

E(4>,7r)(g) :=

J]

0(t%).

<5eP(F)\G(F)

Then one checks that there is an open cone C c ReX^ such that if Re7r e C, E(A • 0, Va e A^}. As a direct consequence, then E(<j),n) e A(G(F)\G(A)). That is, it is an automorphic form. 2.2

Non-Abelian L-Functions

2.2.1

Definition

Being automorphic forms, Eisenstein series are of moderate growth. Consequently, they are not integrable over G(F)\G(A) 1 . On the other hand, Eisenstein series are also smooth and hence integrable over compact subsets of G(F)\G(A) 1 . So it is very natural for us to search for compact domains which are intrinsically defined. As such, let us now return to the group G = GLr. Then form 1.3.8, we obtain compact moduli spaces M g [ A | ] := {g € GLr(F)\GLr(A):

degg = 0, p* < p)

for a fixed convex polygon p : [0, r] -* R. For example, A ^ t l ] = AiQ,r[l], (the adelic inverse image of) the moduli space of rank r semistable Z-lattices of volume 1. More generally, for a standard parabolic subgroup P of GLr, we introduce moduli spaces MP£p[A}r] := {g € P(F)\GLr(A) :tegg = 0,pgp p-<

r

-p).

By 1.3.8, these moduli spaces MF'~p[Aj.] are all compact. As usual, take the minimal parabolic subgroup Fo to be the one consisting of upper triangular matrices corresponding to the partition ( ! , . . . , ! )

261

2 Non-Abelian L-Functions

with Mo consisting of diagonal matrices. Then P = Pi = UJMI corresponds to a certain partition / = (n,...,r\p\) of r with Mi the standard Levi and Ui the unipotent radical. (See e.g., 3.2.5.) Now for a fixed irreducible automorphic representation n of M/(A), choose

r

E{(p,n){g)dg,

ReTreC.

JM%[A1]

More generally, for a standard parabolic subgroup Pj = UJMJ D Pi (so that the partition 7 is a refinement of /), we obtain a relative Eisenstein series Ej(cf>,n)(g) :=

YJ 6eP,(F)\Pj(F)

WS>>

Vg e

PJ(F)\G(A).

There is an open cone C\ in ReXfy such that if Re;r 6 C\, then EJ(
Definition. Define the non-abelian L-function by LpfW, n) : - f Pp < , F/(0, n)(g) dg, JM Ff[tf\

Rene

C\.

Remark. Here when defining non-abelian L-functions we assume that

262

L. Weng

only serves the purpose of giving the constructions and results in a very neat form. We end this subsection by pointing out that the discussion for rank r Lfunctions Lppr holds for general non-abelian L-functions LFfp as well. So from now on, we will leave the discussion on general LF^~P to the reader, while concentrate ourselves to the special case LFpr. 2.2.2

Meromorphic Extension and Functional Equations

With the same notation as above, set ^P = [n]. For w 6 W the Weyl group of G = GLr, fix once and for all representative w e G(F) of w. Set M' :wMw~x and denote the associated parabolic subgroup by P' = U'M'. W acts naturally on automorphic representations, from which we obtain an equivalence classes w^3 of automorphic representations of M'(A). As usual, define the associated intertwining operator M(yv, n) by (M(w,n)4>)(g) := f

1

l

(p(w-

n'g)dn', Vg e G(A).

J U'(.F)nwU(F)w- \U'(A)

One checks that if (Re7r, a v ) » 0, Vor e A£, (i) for a fixed automorphic depends only on the double coset M'(F)wM(F). So M(W,TT) is well-defined for weW; (ii) the above integral converges absolutely and uniformly for g varying in a compact subset of G(A); (iii) M{w,n)

,ri)forRen eC is well-defined and admits a unique meromorphic continuation to the whole space ^3; (II) (Functional Equation) As meromorphic Junctions on ^8, LFPr(, wn),

VweW.

Proof. This is a direct consequence of the fundamental results of Langlands on Eisenstein series and spectrum decompositions and explains why only L2-automorphic forms are used in the definition of our Ls. (See e.g, [Arl], [Lai], [MW] and/or [We7]. ) •

2 Non-Abelian L-Functions 2.2.3

263

Holomorphicity and Singularities

Let n e ty and a e AM. Define the function h:ty-*Cbyn®Ai-> (A, av), VA 6 X^j =* a^. Here as usual, av denotes the coroot associated to or. Set H :- {n' e ^B : h(n') = 0} and call it a root hyperplane. Clearly the function h is determined by H, hence we also denote h by hn- Note also that root hyperplanes depend on the base point n we choose. Let D be a set of root hyperplanes. Then (i) the singularities of a meromorphic function / on ^8 is said to be carried out by D if for all n e ^8, there exist nn : D —» Z>o zero almost everywhere such that n' i-» (Il//e£)/*ff(7r')">r(//)) • /(^') is holomorphic at TT'; (ii) the singularities of / are said to be without multiplicity at n if n„ e {0,1}; (iii) D is said to be locally finite, if for any compact subset C c $ , {H 6 D : H n C * 0} is finite. Basic Facts of Non-Abelian L-Functions. (III) (Holomorphicity) (i) When Ren eC, L^fty,n) is holomorphic; (ii) Lj?r(, n) is holomorphic at n where Ken = 0; (IV) (Singularities) Assume further that <\> is a cusp form. Then (i) There is a locally finite set of root hyperplanes D such that the singularities ofL~fr(

0,VaeA^; (iii) There are only finitely many of singular hyperplanes ofLj.p((p, n) which intersect [n e ^ : 0, Va e AM}Proof As above, this is a direct consequence of the fundamental results of Langlands on Eisenstein series and spectrum decompositions. (See e.g, [Arl], [Lai], [MW], and/or [We7]). a

3

Geometric and Analytic Truncations

3.1 3.1.1

Geometric Truncation: Revised Slopes, Canonical Filtrations

Following Lafforgue [Laf], we call an abelian category ft together with two additive morphisms rk : ft -> N,

deg : ft -» R

a category with slope structure. In particular, for non-zero A e ft, (1) define the slope of A by ^(A) := ^ g ? ; (2) If 0 = A0 c A\ c • • • c A/ = A is a filtration of A in ft with rk(A0) < rk(Ai) < • • • < rk(A/), define the associated polygon to be the function [0, rkA] -» R such that (i) its values at 0 and rk(A) are 0; (ii) it is affine on the intervals [rk(A;_i),rk(A,-)] with slope ^(A,/A,_i) -

tfA); (3) If a is a collection of subobjects of A in ft, then o is said to be nice if (i) a is stable under intersection and finite summation; (ii) o is Noetherian, i.e., every increasing chain of elements in a has a maximal element in a; (iii) if Aj e a then A\ * 0 if and only if rk(Ai) + 0; and (iv) for A\, A2 e a with rk(Ai) = rk(A2). Then A\ c A2 is proper implies thatdeg(Ai)<deg(A 2 ); (4) For any nice a, set fi+(A) :=sap[n(Ai): Aj e a,rk(A0 > l}, p~(A) :=inf{/i(AMi): A\ e Q,rk(Ai) < rk(A)). Then we say (A, 0) is semi-stable if fi+(A) = p{A) = pT{A). Moreover if rk(A) = 0, set also p.+(A) = -00 and n~(A) = +00. Proposition 1. ([Laf]) Let ft be a category with slope structure, A an object in ft and 0 a nice family of subobjects of A in ft. Then 264

3 Geometric and Analytic Truncations

265

(1) (Canonical Filtration) A admits a unique filtration 0 = Ao C A\ C • • • c A/ = A with elements in a such that (i) Ai, 0 < i < k are maximal in a; (ii) Ai/Ai-i are semi-stable; and (Hi) p(M/A0) > /i(A2/Xi > > n(Ak/Ak-iy, (2) (Boundness) All polygons of filiations of A with elements in a are bounded from above by ~p, where ~p :=~p is the associated polygon for the canonical filtration in (1); _ (3) For any A\ e a, rk(AO > 1 implies p(Ax) < p(A) + Pifft1'))); (4) The polygon ~p~ is convex with maximal slope p+(A) - n(A) and minimal slope n~(A) - n(A); (5) If (A', a') is another pair, and u: A —» A' is a homomorphism such that Ker(w) € o and \m(u) e a'. Then pT(A) > p.+(A') implies that u = 0. This results from a Harder-Narasimhan type filtration consideration. A detailed proof may be found at pp. 87-88 in [Laf]. As an example, we have the following Proposition 2. Let F be a number field. Then (1) the abelian category of hermitian vector sheaves on Spec OF together with the natural rank and the Arakelov degree is a category with slopes; (2) For any hermitian vector sheaf (E,p), a consisting ofpairs (E\,pi) with Ei sub vector sheaves ofE andp\ the restrictions ofp, forms a nice family. Proof. This is a simple reformulation of Prop. 1.3.6. Indeed, (1) is obvious, while (2) is a direct consequence of the following standard facts: (i) For a fixed (E,p), ideg(Ei,pi) : (E\,p\) e a} is discrete subset of R; and (ii) for any two sublattices Ai, A2 of A, Vol(Ai/(Ai n A2)) > Vol((Ai + A 2 )/A 2 ).

• Thus we get canonical filtrations of Harder-Narasimhan type for hermitian vector sheaves over Spec OF- Recall that hermitian vector sheaves over Spec OF are Of-lattices in (R n xC 2 ) r = r k ( £ ) : Say, corresponding (9/r-lattices are induced from their H° via the natural embedding Fr <^-> (R n x C 2 ) r where r\ (resp. r 2 ) denotes the real (resp. complex) embeddings of F.

266 3.1.2

L. Weng A Micro-Global Relation

In algebraic geometry, or better in Geometric Invariant Theory, a fundamental principle, the Micro-Global Principle, claims that if a point is not GIT stable then there exists a parabolic subgroup which destroys the corresponding stability. Here even we do not have a proper definition of GIT stability for lattices, in terms of intersection stability, an analogue of the Micro-Global Principle holds. Let A = Ag be a rank r lattice associated to g € GLr(A) and P a parabolic subgroup. Denote the sublattices filtration associated to P by 0 = Ao c Ai c A2 c • • • c A|p| = A. Assume that P corresponds to the partition / = {d\, di,..., dn=-\p\). Consequently, we have rk(A,) = rt := d\ + di + • • • + di,

for i= 1,2,..., |P|.

Let p, q : [0, r] -» R be two polygons such that p(0) = q(0) = p(r) q(r) = 0. Then following Lafforgue, we say q is bigger than p with respect to P and denote it by q >/> p, if q{ri) - p(n) > 0 for all i = 1,..., |P| - 1. Introduce also the characteristic function l(p* < p) by

lo,

otherwise.

Here as before, JP denotes the canonical polygon for the lattice corresponding to g. Recall that for a parabolic subgroup P, pp denotes the polygon induced by P for (the lattice corresponding to) the element g e G(A). Fundamental Relation. For a fixed convex polygon p : [0, r] —* R such that p(0) = p(r) = 0, we have

KF)=

S

(-i)17,1-1

P: standard parabolic

2

I(P?>PP).

6eP(F)\G(F)

Remarks. (1) This result and its proof below are motivated by a similar result of Lafforgue for vector bundles over function fields.

267

3 Geometric and Analytic Truncations

(2) The right hand side may be naturally decomposite into two parts according to whether P = G or not. In such away, the right hand side becomes

1G-

(-l) 1 ^ 1 -.

£ P: proper standard parabolic

This then exposes two aspects of our geometric truncation: First of all, if a lattice is not stable, then there will be parabolic subgroups which take the responsibility; Secondly, each parabolic subgroup has its fix role - Essentially, they should be counted only once each time. In other words, if more are substracted, then we need to add one fewer back to make sure the whole process is not overdone. Proof. (Following Lafforgue) Note that all parabolic subgroups of G may be obtained from taking 5-conjugates of standard parabolic subgroups with S € P(F)\G(F), and that ~p* = ~pgsps-X- Therefore, it suffices to prove the following Fundamental Relation'.

KP8
{-\f^\{pp>PP).

P: parabolic

To establish this equality, we consider two cases depending on whether (a) p^ < p; or (b) J?8 is not bounded from above by p. Assume that (a) holds. Then the LHS is simply 1. On the other hand, for a proper parabolic subgroup Q 4- G, by definition, PQ < p. Hence 1(/A >Q p) = 0. Consequently, the right hand side degenerates to the one consisting of a single term involving G only, which, by definition, it is simply 1 as well. We are done. Next, consider (b), for which p g is not bounded from above by p. This —g,p is a bit complicated. To start with, we set A% to be the unique filtration of A8 such that the associated polygon ~pPp is maximal among p8Q where Q runs over all parabolic subgroups contained in P. That is ~pp := max {/A : Q c P, parabolic subgroup].

268

L. Weng —8

Denote the associated parabolic subgroup by QP and call it P-canonical. Clearly, G-canonical is simply canonical. Moreover, since JP £ p, the LHS = 0. So we should prove that the RHS = 0 as well. For this, regroup the parabolic subgroups P appeared in the RHS according to the refined canonical parabolic subgroup Q8P. Then clearly, it suffices to establish the following Proposition. With the same notation as above, if ~pg ^ p, then for any fixed parabolic subgroup Q,

2 (-iri(ppP>PP)=o. P&P=Q

Proof. We break this into the following 3 steps. Step A. Fix a non-negative real number p.. For the lattice A = A8 and a fixed parabolic subgroup P of G, set (1) MQ to be the unique parabolic subgroup of G such that if^A, is the induced filtration of A8, then

fA!} = JA5:/,(A5/A5_1)>MA?+1/A3+4 where A» is the canonical filtration associated with Ag; —g

—g,P

(2) ^Qp to be the unique parabolic subgroup of G such that if ^A„ is the induced filtration of A8, then

i"A5'P] - « ) eP

where A • is the filtration of Ag induced from the parabolic sub—g,p group P, and A, is the P-canonical filtration associated with A8. In particular, from this construction, we have: (i) For —gP A, , so-called P-canonical filtration, the Harder-Narasimhan con—g,P i—g,P

ditions hold, i.e., A ; j^j-\ are semi-stable, and -rg,P tTS

rrsf rr8
.(Af/A-,)^;,/^).

3 Geometric and Analytic Truncations

269

except for those j appeared as that associated to P itself; (ii) While the canonical filtration A, of A8 satisfying ^ ( A J / A J . , ) > /i(X5+1/A*),

V; = 1,..., Kf I - 1,

A,} is the one obtained from this canonical one by selecting only the part where much stronger conditions

are satisfied. Consequently,

Q8 c "£f; (iii) The filtration PAf' } contains not only that part of the P-canonical filtration where stronger conditions

are satisfied, it contains also the full filtration of Ag associated to P. So, Q8P c "g£,

while

»~Q% c P.

Step B. For g e G(A), set Gl /(GO =jrkAf ,fl2 KQi) ={ikAj

MQi)

=(rkAf'e'

WQi)

=(rkAfe' G2

KP(Q2) =(rkA^

i=

l,2,...,\Qi\-l};

; = l,2,...,ie2|-l);

M[^i

/A/_i j > fi[AM /Az

)+H)\

(4 2 -p)(rkA^)>0).

Lemma. With the same notation as above, the following correspondence [PeP:Q8P

= Qu "Qp = Q2, PgP>p P]

- * ( l : I c I(Q2), J^Qx) c 7(J22), (/(fii) - •*>(&)) U (/(&) - MQi)) c S c ATp(j22)}

270

L. Weng

defined by P H+ /(/>) := (rkAf P : i = 1 , 2 , . . . , | P | - 1 } is a well-defined bijection. Proof of the Lemma. (I) Well-defined. For this, let us look at the conditions for P in {PeP: QP = QU tlQp = Q2,PgP>Pp}. —8

(i) Qp - Q\ gives the filtration (01 = Ag'Ql

c A8'Ql c Ag'Ql

c • • • c Ag'Ql

c A8,Ql

= A8

with p8Q be the maximal among all [p8Q : Q c p). Hence, (a) (2i c P; and (b) Except possibly at indices corresponding to the partion coming from P,

/4A*

/ V l j > A*(At+1 /A t j ,

by the maximal property, or more clearly, the Harder-Narasimhan property. Therefore, we have /(Gi) - MQi) c 1(F). (ii) ^Qp = Qi gives the filtration (0) = A8,Ql

c Ag'G2 c A*'22 c • • • c A8,Ql

c Ag'G2 = A8

which by definition coincides with {A8/

:j=l,2,...,|P|-l)

U {Af'^1 : ft[Ak

JAk_x ) > n[Ak+l JAk

) + n j.

By taking the rank for each lattices, we get the relation KQi) - I(P) U Jfi(Q\)- Therefore, (a) Q2cPor equivalently I(P) c 7(<22); and (b) KQi) - / „ ( & ) c /(/>). (iii) psp >p p is, by definition, equivalent to the following group of inequalities

(/ p -p)(rkAf)>0.

3 Geometric and Analytic Truncations

271

Consequently, I(P) c KP(Q2) since I(P) c I(Q2). Therefore, P i-> /(f) is well-defined. (II) Injectivity. This is clear since with a fixed partition /, there exists only one unique standard parabolic subgroup Pj such that I(Pi) = A-1. (III) Surjectivity. For a subset S of [l, 2 , . . . , A such that

I c/(fi2), ^(Gi) c /(&), (/(Gi) - /o(Gi)) U (/(fiz) - /„(&)) c S c KP(Q2), clearly, tiiere exists a parabolic subgroup P = P%. So we need to check whether QP = Qu * ~QP = Qi and p£ >/» p. _ 8 It is clear that p p >P p since E c KP(Q2). On the other hand, QP = Gi» ^Qp = Qi are direct consequences of the definition of P-canonical filtration and the conditions that (/(Gi) - MQi)) c I and (/(G2) - ^ ( G i ) ) c S as explained in the proof above of the well-defined statement. This completes the proof of the Lemma. Step C. With the Lemma, we have

(-lfll(pp>Pp)

£

=\

( 1)#(
2

-

s:(/(eiWo(ei))u(/(G2)-^(Gi))csc/rp(e2) =

jo, if (/(GO - /o(Gi)) u (/(G2) - ^(Gi)) * ^(G2); \ i , if (/(Gi) - MQi)) u (/(G2) - //i(Gi)) = ^ ( 6 2 ) .

Surely, we want to show that the value 1 cannot be taken for g such that ~pp is not bounded by p. To this end, as the final push to establish our Fundamental Relation, set// = 0 and Q\ = Q2 = Q in the above discussion so that

2

(-l)ml(pP >P p)

P<2P=QU»QP=QI

=

J]

(-lf\\{pp>pp)

P{QP=Q=°QP

= J] P-X>SP=Q

(-lfll(pp>Pp)

272

L. Weng

is the summation in the Proposition. Then, by definition and the convexity property of p, (/(0i) - MQi)) U (/(&) - Jp{Qi)) = KP{Q2) implies that KQi) = JoiQi), or better Ksp{Q2) = 0. Consequently, this then shows that JP < p, a contradiction. This completes the proof of the Proposition and hence the Fundamental Relation. •

3.2 3.2.1

Arthur's Analytic Truncation Parabolic Subgroups

Let F be a number field with A = Ap the ring of adeles. Let G be a connected reductive group defined over F. Recall that a subgroup P of G is called parabolic if G/P is a complete algebraic variety. Fix a minimal F-parabolic subgroup Fo of G with its unipotent radical No = Np0 and fix an F-Levi subgroup Mo = Mp0 of Fo so as to have a Levi decomposition PQ = MoNo- An F-parabolic subgroup P is called standard if it contains Fo. For such parabolic subgroups F, there exists a unique Levi subgroup M = Mp containing Mo which we call the standard Levi subgroup of F. Let N = Np be the unipotent radical. Let us agree to use the term parabolic subgroups and Levi subgroups to denote standard F-parabolic subgroups and standard Levi subgroups respectively, unless otherwise is stated. Let F be a parabolic subgroup of G. Write Tp for the maximal split torus in the center of Mp and T'p for the maximal quotient split torus of Mp. Set a> := X»(7» ® R and denote its real dimension by d(P), where X»(T) is the lattice of 1-parameter subgroups in the torus T. Then it is known that 5/> = X*(T'p) ® R as well. The two descriptions of 5/> show that if Q c F is a parabolic subgroup, then there is a canonical injection ap ^-» 5g and a natural surjection Q.Q -» ap. We thus obtain a canonical decomposition O.Q = 5^ © ap for a certain subspace 5^ of 5g. In particular, ac is a summand of 5 = ap for all F. Set ap := ap/aG and a^ := O.Q/OG. Then we have ag = apQ © ap and ap is canonically identified as a subspace of ag. Set % := ap0 and a£ = a£ then we also have ao = a£ © ap for all F.

273

3 Geometric and Analytic Truncations 3.2.2 Logarithmic Map

For a real vector space V, write V* its dual space over R. Then dually we have the spaces aj, a*p, (OQ J and hence the decompositions a

5 = ( a $T®( a Gf ea P-

So o|, = X(MP) <S> R with X(MP) the group HomF(MP,GL(lj) i.e., collection of characters on Mp. It is known that a*p - X(Ap) <8> R where Ap denotes the split component of the center of Mp. Clearly, if Q c P, then MQ C Mp while Ap c AQ. Thus via restriction, the above two expressions of a*p also naturally induce an injection a*p ^-> a*Q and a surjection aQ -» ap, compactible with the decomposition a*Q = (a^J © ap. Every x = Z -Wi in a/>c : = af> ® ^ determines a morphism P(A) -» C* by /? i-> p* := n Wi(p)\s'- Consequently, we have a natural logarithmic map Hp : P(A) -» ap defined by <#/>(/>),*> = ^ .

YY 6 a*p.

The kernel of ///> is denoted by P(A)1 and we set MP(A)1 := P(A)1 n Mp(A). Let also A+ be the set of a e Ap(A) such that (1) av = 1 for all finite places v of F; and (2) x(ao-) is a positive number independent of infinite places cr of F for all AT e X ( M P ) .

ThenM(A)=A + -M(A) 1 . 3.2.3

Roots, Coroots, Weights and Coweights

We now introduce standard bases for above spaces and their duals. Let Ao and Ao be the subsets of simple roots and simple weights in OQ respectively. (Recall that elements of Ao are non-negative linear combinations of elements in Ao.) Write AQ (resp. AQ) for the basis of ao dual to Ao (resp. Ao). Being the dual of the collection of simple weights (resp. of simple roots), AQ (resp. AQ ) is the set of coroots (resp. coweights). For every P, let A^> c aj be the set of non-trivial restrictions of elements of AQ to ap. Denote the dual basis of Ap by A„. For each a e Ap, let

274

L. Weng

a v be the projection of By to ap, where B is the root in Ao whose restriction to ap is a. Set A^ := |arv : a e Ap}, and define the dual basis of A^ by Ap. More generally, if Q c P, write APQ to denote the subset a e AQ appearing in the action of TQ in the unipotent radical of Q n MP, (Indeed, MP n Q is a parabolic subgroup of Mp with nilpotent radical NQ := NQ n Mp. Thus AQ is simply the set of roots of the parabolic subgroup (Mp n Q,AQ). And one checks that the map P i-» A^ gives a natural bijection between parabolic subgroups P containing Q and subsets of AQ.) Then ap is the subspace of OQ annihilated by A£. Denote by (A V )Q the dual of AQ. Let (AJJ)V := ja v : a e AJ) and denote by AJ the dual of (A£) v . 3.2.4

Positive Cone and Positive Chamber

Let Q c P be two parabolic subgroups of G. We extend the linear functiona l in AQ and A~ to elements of the dual space o* by means of the canonical projection from Oo to a^ given by the decomposition ao = a~ © a^ © ap. Let TQ be the characteristic function of the positive chamber {Heao:(a,H)>OVaeApQ} = a® © {# e apQ : (a,H) > 0 for all a e AQ] © aP and let TQ be the characteristic function of the positive cone [H 6 0o:(tiT,//>>OVtzTeAj} = a® © {// e apQ : (m,H) > 0 for all m e A^} © ap. Note that elements in AQ are non-negative linear combinations of elements in AQ, SO we have I*

3.2.5

>T

P

Example with 5 L r

Set G = SLr(R). Let P 0 = B be the Borel subgroup of G consisting of upper triangular matrices, and Mo be the Levi component consisting of diagonal matrices. One checks easily that the unipotent radical No consists

275

3 Geometric and Analytic Truncations

of upper triangular matrices whose diagonal entries are all equal to 1 and that (the positive part Ao+ of) the split component AQ consists of diagonal matrices diag(ai,a2,...,ar) with a\,a2,...,ar € R>o and II)=i a ' = *• Hence Oo = [H := (HUH2, ...,Hr) eW : £- = i Hi = OJ and the natural logarithmic map Hp0 : PQ -» ao is given by g i-» Hp0(a(g)J = loga\ + log«2 + • • • + loga r , where we have used the Iwasawa decomposition g = n-m- a(g) • k with n € No, m e MQ, a(g) = diag(<2i, a2, ...,ar)e A0+ and keK = SO(r). Set ei e aj such that e,(/T) = Hi, then Ao = [ei - e 2 , e 2 - e j , . . . , e r - \ - e r y We identify Ao with the set A := {1,2,..., r-1} by the map a, := e,-e,+i *-> i,i = l , 2 , . . . , r - 1. It is well known that there is a natural bijection between partitions of r and parabolic subgroups of G. More precisely, the partition I := (d\,d2,...,dn)of r(sothatr = d\+d2 + --- + dn andd,- € Z>o)corresponds to the parabolic subgroup Pj consisting of blocked upper triangular matrices whose diagonal entries are blocked matrices of sizes d\,d2,...,dn. Then clearly, the unipotent radical Nj consists of blocked upper triangular matrices whose blocked diagonal entries are identity matrices of sizes d\, d2,..., d„, the Levi component Mj consists of blocked diagonal matrices whose blocked diagonal entries are of sizes d\,d2,...,dr, and the corresponding A/+ consits of matrices of the form diagfai / 0 for i = 1,2,..., n, and n"=i ai = 1. As such, a/ is a subspace of do defined by a7 := [(HuHz,...

,Hr) e do : Hi = H2 = • • • = Hn,

Hn+i = Hn+2 = • • • = Hn, . . . , Hrn_1 + i = Hrn_l+2

= ••• =

Hrn\

where we set r,- = d\ + d2 + • • • + d{ for i = 1,2,..., n. Moreover, with the help of Iwasawa decomposition, the corresponding logarithmic map may be simply seen to be the map Hi : A/+ -> a/ denned by diag{a\Idl, a2Idl,...,

anIdn)

.-» ( ( l o g f l l ) W l ) , 0 o g f l 2 ) ( * ) , • • • , (logfln)***).

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L. Weng

Obviously, the natural bijection between parabolic subgroups P of G and subsets of Ao or better of A is given explicitly as follows: The parabolic subgroup P = Pj corresponds to the subset I{P) of A = | l , 2 , . . . , r - 1 j such that A - I{P) = {ru r2,..., rn}. That is, 7(7>) = { l , 2 , . . . , r 1 - l ) | J ( r 1 + l , r 1 + 2 , . . . , r 2 - l }

U " • U I'"-1 + lf r"~i+2' • • •'r" ~ AFor simplicity, from now on, we will use / to indicate both the partition and its corresponding subset I(Pi). For example, the subset 7 = 0 corresponds to the partition = ( 1 , 1 , . . . , 1) of r = 1 + 1 + • • • + 1 (since A - 0 = {1,2,..., r-1}) whose corresponding parabolic subgroup is simply Po the minimal parabolic subgroup. Consider now 0 c 7 c / . From A - / c A - 7 c A, we get 7Jo = Pq, c Pi c Pj. Set A- J = {s\,s2,..., sm} corresponding to the partition r = fi+h + --- + fm- Then f\ = d\ + d2 + • • • + dkl, f2 = 4,+i + dki+2 + ••• + dk2,...,fm = dkm_l+i + dkm_l+2 + --- + dkm and aj is a subspace of a0 defined by aj : = { < > , 7 7 2 ^ , . . . , / # " > ) 6 «o). Clearly aj is in a/ since a/ is a subspace of ao denned by a/:=((A?'U?\...,A^)eoo} and we have the map aj

->

0/

Moreover, with the help of Iwasawa decomposition, the corresponding logarithmic map may be simply viewed as the map Hj : Aj+ -» a.j denned by di&g(biIfl,b2If2,...,bmIfm) »

((log bi)&\

(log b2)^\

...,

(log

bm)^).

To go further, for i e A, set _ .

mi := wf =

i

^k-ak r

k=\

.

r-\

+r

^(r-k)-ak k=i+l

277

3 Geometric and Analytic Truncations

be the corresponding fundamental weight, mi is the Q-linear form on oq, = ao given by l

mi(H) =Hi+H2+-+Hi-

-(H\

+ H2 + --- + Hr)

= Hi+H2 + --- + Hi for all H = \H\,H2,...,

Hrj e oo, so that A/ = {trr,- : / e A - /} since A/ = (or,1

: i e A - /}.

la;

I

Note that aj = (A/)R and elements of A/ are non-degenerate positive linear combination of elements in A/, so both A/ and A/ generate c^. Consequently, we have a* = (mt : i e A - / ) R => ( ^ : j e A - j ) R = a}. Moreover, by definition, the space aj is the subspace of aj annihilated by a}. Therefore, aj = [H 6 a7 : a(tf) = 0 Var e a}) = JH e a/: GT,(#) = 0 V; e A - /}

= {(,...,^>)ea0: rfi#i + d2H2 + ••• + dklHkl = 0, dki+iHki+\ + dkl+2Hk1+2 + ••• + dk2Hk2 = 0,... j . Note that clearly dim aj + dim ay = dim aj and a^ n ay = 0, so we have the natural decomposition aj = aj® ay. Furthermore, note that n

I = {J{rj-\

+

hrj-i+2,...,rj-l}

i=i

={l, 2 , . . . , r - l} - {n, r2,..., r„_i}.

278

L. Weng

Set, for k = 1,2,-•• ,dj- 1, d-k

k

k

dj l

~

motivated by the construction of fundamental weights above. Then if, as usual, we define the coroot aY e oo by 1 (<*,v)* = ' - 1 0

if k=i, ifk=i+l, otherwise,

then jaY : i e /j is a basis of the subspace a^ of am and easily one checks that TiT,-(ap = Sij for all i,j e I. Consequently, jroj

;

: i € /} is its dual

0

basis in (a^)*. In particular, we also get A/ = [VJ{ : k e / - / = (A - /) - (A - /)). We end this preperation with the following remarks about the notation. With respect to the decomposition

we have that the space (ajH is generated by A^ = (or/: / € / - 0 = /} c A0 the root system, similarly (a'A is generated by A£ = [ak : k e I - 0 = /) c A0 while (aj) is generated by A^ which is not a subset of A0 but we have A, c A/ consists of the restriction of a* to aj with k e J -1 - (A -1) - (A - J). That is Aj = {ak\aJ :k€j-I = (A~I)-(AJ)}. In particular, if / = A we have <4 = ao, af = a/,

and

A^ = A0, Af = A/.

279

3 Geometric and Analytic Truncations 3.2.6

Partial Truncation and First Estimations

Denote T£ and T^ simply by Tp and T>. Basic Estimation. ([Ar2]) Suppose that we are given a parabolic subgroup P, and a Euclidean norm \\ • \\ on ap. Then there are constants c and N such that for all x € G(A)1 and X € ap,

Y;

-TP(H(SX) -

X) < c(\\x\\emf.

6eP(F)\G(F)

Moreover, the sum is finite. As a direct consequence, we have the following Corollary. ([Ar2]) Suppose that T e OQ and N > 0. Then there exist constants c' and N' such that for any function (f> on P(F)\G(A)\ and x, y e G(A)1,

£

\tf6x)\-Tp(H(6x)-H(y)-X)

6eP(F)\G(F)

is bounded from above by c'\\xf'-\\yf

• sup (|0(M)| • \\u\rN). ueG(A)1

3.2.7

Langlands' Combinatorial Lemma

In this subsection, following Arthur, we recall Langlands' combinatorial lemma. If Pi c P 2 , following Arthur [Ar2], set

oi(H):=o-Pp]:= £

{-\f^A^r\(H)

• f,{H),

for H € do. Then we have Lemma 1. If Pi c Pz, cr* is a characteristic function of the subset of H e ai such that (i)a(#)>0foralla€A2; (ii) cr{H) < 0 for all cr € Ai\A*; and

280

L. Weng

(iii) m(H) > 0 for all w e A2. As a special case, with Pi = P2, we get the following important consequence: Langlands' Combinatorial Lemma. IfQcP then for all H e %,

are parabolic subgroups,

(-l)dim(A*M,,)T£(tf)f£(tf) =6QP;

2 R-.QcRcP

YJ (-lf^^iHtfiH)

=SQP.

R-.QcRcP

We next give an application of this combinatorial lemma, which will be used in 3.2.8. Suppose that Q c P are parabolic subgroups. Fix a vector A 6 <1Q. Let 4(A)

:= (_1)#(-A-A(^)
and let
He QO,

be the characteristic function of the set r„ l"

£

w{H)>0, °° wm * 0, :

ifA(av)<0u .Pl v VQf£A ifA(a )>0' G|-

Lemma 2. With the same notation as above,

Z

B

p

p

R-.QcRcP 3.2.8

fo,

if A(or v )<0, 3 a e A£

I1'

Ulnerwl!

>e

Langlands-Arthur's Partition: Reduction Theory

Our aim is to derive Langlands-Arthur's partition of G(F)\G(A) into disjoint subsets, one for each (standard) parabolic subgroup. To start with, suppose that to is a compact subset of Aro(A)Mo(A)1 and that To 6 - o j . For any parabolic subgroup P\, introduce the associated Siegel set s''1 (To, co) as the collection of pak,

p e o), a

281

3 Geometric and Analytic Truncations

where a(H0(a) - To) is positive for each a e AQ. Then from classical reduction theory, we conclude that for sufficiently big co and sufficiently small T0, G(A) = Pi(F)sPl(T0, to). Suppose now that Pi is given. Let 5Pl(To,T,co) be the set of x in s Pl (To, co) such that VT(HO(X) - T) < 0 for each w e KP'. Let FPx (x, T) := Fx(x, T) be the characteristic function of the set of x e G(A) such that 5x belongs to sPl(Jo, T, co) for some 6 e P\(F). As such, Fl(x,T) is left Ai(R)°M(A)Mi(F)-invariant, and can be regarded as the characteristic function of the projection of the space sPi(T0,T,io) onto the space Ai(R)°Afi(A)Mi(F)\G(A), a compact subset of A i (R)°JVi (A)Ml (F)\G(A). For example, F(x, T) := FG(x, T) admits the following more direct description which will play a key role in our study of Arthur's periods: If Pi c P2 are (standard) parabolic subgroups, we write A~ :- Ap for Ap^A) 0 , the identity component of Ap,(R), and AZ2:=A™itP2:=APinMP2(A)\ Then the logarithmic map Hpl maps A™2 isomorphically onto cij, the orthogonal complement of Q2 in ai. If T0 and T are points in ao, set A™2(T0, T) to be the set [a € A~2 : a(H\{a) -T)>0,

ae A2X; m(H\(a) -T)
A?j,

where A^ := Ap,nM2 and Aj := AP]nM2. In particular, for T0 such that -T0 is suitably regular, F(x, T) is the characteristic function of the compact subset ofG(F)\G(A)1 obtained by projecting N0(A) •

MQ(A) 1

• Ap°oG(To,T) • K

ontoG(F)\G(A) 1 . All in all, by Lemma 2 of 3.2.7, we arrive at the following Arthur's Partition. ([Ar2]) Fix P and let T be any suitably point in T0+a+. Then

J]

J]

Pi:P0cPicPSePi(F)\G(F)

F\6X)-TP(H0(SX)-T)=1

VxeG(A).

282

L. Weng

3.2.9

Arthur's Analytic Truncation

Recall that an element T e aj is called sufficiently regular, if for any a e A0, a(T) » 0. Definition. Fix a suitably regular point T 6 a^. If

)(X) to be the function (AT4>)(X)

:= £(-i)dim(A/Z) />

£

M&c) •

TP(H(.6X)

- r),

6eP(F)\G(F)

where

0H*) := I

(f>(nx)dn

JN(F)\N(A)

denotes the constant term of

)(x) = ^(-l) d i m ( A / Z ) <M*) • TP(H(X) - T), where the sum is over p

all, both standard and non-standard, parabolic subgroups; (b) If

283

3 Geometric and Analytic Truncations

This is because by definition, all constant terms along proper P : P ±G are zero. Moreover, it is a direct consequence of the Basic Estimation in 3.2.6 for partial truncation, (see e.g. the Corollary there), we have (c) If(f> is of moderate growth in the sense that there exist some constants C, N such that |0(x)| < c\\x\\N for all x e G(A), then so is A r 0 . Fundamental properties of Arthur's analytic truncation may be summarized as follows: Theorem. ([Ar2,3]) For sufficiently regular T in ao, (l)Let(p: G(F)\G(A) -^Cbea locally L1 function. Then ATAT (g) for almost all g.If is also locally bounded, then the above is true for all (2) Let 0i, 02 be two locally Lx functions on G(F)\G(A). Suppose that 0i is of moderate growth and 02 is rapidly decreasing. Then f

ATl(g)-4>2(g)dg= f

JZG(A)G(F)\G(A)

Mg)-ATcf>2(g)dg;

JZO(A)G(F)\G(A)

(3) Let Kf be an open compact subgroup ofG(Af), and r, r' are two positive real numbers. Then there exists a finite subset lX( : i = 1,2,..., N\ C < U, the universal enveloping algebra of g^, such that the following is satisfied: Let be a smooth function on G(F)\G(A), right invariant under Kf and let a € AG(A), g 6 G(A)1 D S. Then N T

\*
where S is a Siegel domain with respect to G(F)\G(A); and (4) The function A T 1 is a characteristic function of a compact subset of G(F)\G(A) J . The key to the proof is that nilpotent groups have simple structure. Say, for (1-3), while a bit complicated, it is based on the discussion in 3.2.6-8. As for (4), which has been considered to be only secondary so far, please see 3.3.2 below.

284

3.3 3.3.1

L. Weng Arthur's Periods Arthur's Periods

Fix a sufficiently regular T e Oo and let 0 be an automorphic form of G. Then by Theorem 3.2.9, A r 0 is rapidly decreasing, and hence integrable. In particular, the integration AT^g)dg

A(cf>;T):= f JG(F)\G(A)

makes sense. We claim that A{
AT
f JZc(A)G(F)\G(A)

A r (A r 0)(g) dn{g).

= f JZo(A)G(F)\G(A)

Moreover, by the self-adjoint property, for the constant function 1 on G(A),

I

ug)-AT(ATyg)dti(g)

Zc(A)G(F)\G(A)

(ATiyg) • (Ar0)(g) dub)

= r JZG( A ) G(F)\G(A)

A r ( A T l W ) •
= f JZG( A ) G(F)\G(A) r

T

since A 0 and A 1 are rapidly decreasing. Therefore, using A T o A r = A r again, we arrive at A(0;T)= f

A7" 1(g) • 0(g) dn(g).

(*)

JZG( A ) G(F)\G(A)

To go further, let us give a much more detailed study of Authur's analytic truncation for the constant function 1. Introduce the truncated subset 1(7) := (ZG(A)G(F)\G(A))r of the space G(F)\G(A) ! by S(T) := (ZG(A)G(F)\G(A))7, := [g e Z G ( A ) G ( F ) \ G ( A ) : ATl(g) = l).

285

3 Geometric and Analytic Truncations

We claim that E(T) or the same (z G( A)G(F)\G(A)) r , is compact. In fact, much stronger result is correct. Namely, we have the following Basic Fact. ([Ar4]) For sufficiently regular T e a+, A r l(*) = F(x, T). That is to say, A r l is the characteristic function of the compact subset S(T) of G(F)\G(A) 1 obtained by projecting N0(A) • Mo(A)1 • A~ G(T0, T) • K onto G(F)\G(A)\ Thus, by the equation (*) and the Basic Fact above, AT(f>(g)dp(g)

Jz, Zc(A)G(F)\G(A)

ATl(g)-4>(g)dfi(g)

= f JzG(A)G(F)\G(A)

= r


JJ.(T)

That is to say, we have obtained the following very beautiful relation: Arthur's Periods. For a sufficiently regular l e a o , and an automorphic form (p on G(F)\G(A),

f JUT)


AT(g)dn(g). 1

The importance of this fundamental relation can hardly be over-estimated. Say, when taking (f> to be Eisenstein series associated with an L2-automorphic forms, (i) (Analytic Evaluation for RHS) The right hand side can be evaluated using certain analytic methods. For example, in the case for Eisenstein series associated to cusp forms, as done in the next chapter, the RHS can be evaluated in terms of integrations of cusp forms twisted by intertwining operators, which normally are close related with L-functions; (ii) (Geometric Evaluation for LHS) The left hand, by taking the residue (for suitable Eisenstein series), can be evaluated using geometric methods totally independently. A good example is given in [KW], as to be explained in 4.6. See also 4.7-8, where this is used to obtain certain new yet fundamental relations among special values of zeta functions.

286

L. Weng

3.3.2

A r l Is a Characteristic Function

Proof of the Basic Fact is based on Arthur's partition for G(F)\G(A) and an inversion formula, which itself is a direct consequence of Langlands' Combinatorial lemma. Recall that for T e do, level P-Arthur's analytic truncation of

^P4>(g) := Z

2

(-D«-^>

fl:/fc/>

**<*> • T£(*<*> - 4

SeR(F)\P(F)

As such A r stands simply for A r , G . Thus, together with Langlands' combinatorial Lemma, we have the following: (a)Ar^):=^-1^(/eW(G) R

Z

M8g)TR(H(6g)-T);

and

6€R(F)\G(F)

(b) (Inversion Formula) For a G(F)-invariant function
#*> = Z

Ar,/

Z

P

V^) • TP(H(<5S) - T).

SeP(F)\G(F)

(c) Assume that T 6 Cp and

(m) is rapidly decreasing on M(F)\M(A) 1 . In particular, the integration JM(F)\M(A\i AT,p
Y, P

FP Sx T

Z

( ' )"

T

P{HP(6X)

-T) = l

SeP(F)\G(F)

where r/> is the characteristic function of [H € do : a(H) > 0, a e AP] and Fp[nmk, T) = FMp{m, T),

n e NP(A), m e MP(A), k e K.

On the other hand, by the inversion formula, applying to the constant function 1, we have £ P

£ 6eP(F)\G(F)

(AT'Pl)(6x) • rp(HP(8x) - T) = 1,

287

3 Geometric and Analytic Truncations

where AT'P is the partial analytic truncation operator defined above. With this, the desired result is immediately obtained by induction. As a direct consequence of this proof, we conclude also the following Proposition 1. ([Ar4]) For sufficiently regular T, the characteristic Junction F@(g, T) coincides with the partial truncation along Pfor the constant Junction 1. That is to say, Fp(g,T) = AP>T1 =

sp

£ R-.PocRcP

£

?R(H(6x)-T).

6eR(F)\P(F)

Note that the right hand side makes sense even when T is not sufficiently regular. This then leads to the definition of Fp(g, T) for general

Fp(g,T) = AP-TI=

2

sp

R-.PncRcP

2

rR(Hm-T).

6eR(F)\P(F)

When preparing our Arthur period paper, we note that such a function is compactly supported. (See the lemma below.) As such, a natural question is whether Fp(g, T) is a characteristic function for a natural compact subset. As it appears, the answer is well-known among those working with trace formula: When I addressed it to Arthur in 2005, he immediately gave me his paper [Ar4], which leads the discussion above for sufficiently regular T. For general T, we have the following Proposition 2. ([Ar5], see also [Ho]) As a function of g e G(A)1, Fp(g, T) is bounded and compactly supported modulo U(A)M(F), uniformly for T varying in a compcat subset. Proof The proof is standard. Following Arthur, let

TQp(H,X):= YJ

44(Hrf;(H-X).

R-.PcRcQ

Basic properties of Tp(H,X) are given in Lemma 2.1 of [Ar5], from which, in particular, we know that Tj,{H,X) is compactly supported in if € aj,, uniformly for X varying in a compact subset. On the other hand, by the Langlands Combinatorial Lemma, easily, Fp(g,T + X)=

J]

£

FR(Sg,T)rp(H(6g)-T,X).

R:P0cR
As such, the final result comes from Prop. 1: for sufficiently regular T, Fp(g, T) is a characteristic function of a compact subset. •

288 3.4 3.4.1

L. Weng Geometric and Analytic Truncations A Micro Bridge

For simplicity, we in this subsection work only with the field of rationals Q and use mixed languages of adeles and lattices. Also, without loss of generality, we assume that Z-lattices are of volume one. Accordingly, set G = SLr. For a rank r lattice A of volume one, denote its sublattices filtration associated to a parabolic subgroup P by 0 = A 0 c Ai c A2 c • • • c A|/.| = A. Assume that P corresponds to the partition / = (d\, d2,..., d\p\). Recall that a polygon p : [0,r] -* R is called normalized if p(0) = p{r) = 0. For a (normalized) polygon p : [0, r] —» R, define the associated (real) character T = T{p) of MQ by the condition that

am = [p(i) - p(i - 1)] - [pii + 1) - pit)] for all i = 1,2,... ,r - 1. Lemma. With the same notation as above, T(p) = (p(l) - p(0),p(2) - p ( l ) , . . . , p(i) - p{i - 1), ...,p(r-l)-p(r-2),p(r)-p(r-l)). Proof. Let T = T(J>) = (hip), t2ip),...,

Up)) = (h,t2,...,

tr) 6 do.

Note thato/CT) = f,- - f,+i. Hence U - tM =[pii) - pii - 1)] - [pii + 1) - pii)] =Apii) - Apii + 1),

i = 1,2

r- 1

where Apii) := pii) - pii - 1). Consequently, t\ - ti+\ = Apii) - Apii +1),

i = 1,2,..., r - 1.

289

3 Geometric and Analytic Truncations In particular, h-tr

= Ap(l) - Ap(r) = p(\) - p(r - 1),

and tM =h+ Ap(i + 1) - p{\),

i = 1,2,..., r - 1,

since Ap(l) = p(\) - p(0) = p(\) and A(r) = p(r) - p(r - 1) = -/?(r - 1). But r

r-\

0=YJti = YJ{h-p(l) + &P(i+D) j=l

i=0

=(r-D(h-p(l))

+ {p(0)-p(rj)

=(r-l)(h-p(D). Therefore t\ = p{\) and hence t{+\ = Ap(i +1).

D

Now take g = g(A) e G(A). Denote its lattice by A8, and its induced filtration from P by 0 = A f c A f c - . - c A J p f = A*. Clearly, the polygon p^ = p^* : [0, r] —> R is characterized by (1) /7p(0) = /4(r) = 0; (2) /?p is affine on [r,-, r,-+i], i = 1,2,..., |P| - 1; and (3) Pp(n) = deg(Af) - n • *&?!, i = 1,2,..., |P| - 1. Note that the volume of A is assumed to be one, therefore (3) is equivalent to (3)' pgP(n) = deg(Af p ), i = 1,2,..., \P\ - 1. The advantage of partially using adelic language is that the values of Pp may be written down precisely. Indeed, using Langlands decomposition g = n • m • a(g) • k with n e Np(A),m 6 Mp(A)l,a € A+ and k e K := Up SL(Oqp) xSO(r). Write a = a(g) - di&g{a\Idi,a2Id2

a\P\Idm)

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where r = d\ + d2 + • • • + d\p\ is the partition corresponding to P. Then it is a standard fact that

deg(Af') = " log ( f ] fl?)

=

- Z ^ l o ^ aJ>

' = 1 • • • • • 1^1-

Set now l(/?p >p /?) to be the characteristic function of the subset of g's such that p8p >p p. Micro Bridge. For a fixed convex normalized polygon p : [0, r] —* R, and g € S Lr(A), with respect to any parabolic subgroup P, we have tp(-Ho(g)-T(p))

=

l(pP>Pp).

Proof. First let us consider the right hand side. By definition, Upp >PP)

= 1

if and only if

ppin) - pin) > o,

pp(r2)

-p(r2)>o,

Pp(r\p\-\) - p(r\p\-i) > 0. That is, deg(AfP) - p{n) > 0,

deg(Af) - p(r2) > 0,

deg(A^_j)-/7(r| P |_ 1 )>0. In this way, we have established the following Subiemma 1. With the same notation as above, Upp >PP) = 1 if and only if

(-rfilogai)-p(n) >0, ( - d\ log ax - d2 log a2) - p(r2) > 0,

( - d\ logai -d2loga2

d\P\-i loga|/>|_i) -p(r\ P \-i) > 0.

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3 Geometric and Analytic Truncations

Next let us consider the left hand side, that is, the function Tp(-Ho(g) T(p)). When does it take the value one? Recall that for P = P/, 77. is the characteristic function of the following subset of % = O.Q © a/ defined to be a j ® j f f e a , : m&H) > 0 Vi € A - /}. That is to say, if H e % belongs to a£0[H e a7 : m&H) > 0 i =

rur2,...,r\P\-i),

then T > ( # ) = 1. But H e a7 means H = (Hfx),H{*2\...,H^) Therefore, the conditions that H € a/

such that

and JJQ dM = 0.

vJi(H) > 0 for all i = r\, r2,..., r\p\-\

is equivalent to the conditions that \P\

H = (H^\ Ef\...,

flgW)

where

j

dtHt = 0

;=i

and d\ H\ > 0, diHi + d2H2 > 0 , . . . , d\H\ + d2H2 + ••• + d\p\-iH\p\-i > 0. On the other hand, for H = (H\ ,H2,..., Hr\ e ao, the projections of H to a/ (resp. to a!Q) is given by /,Hi+H2 + --- + Hn^dl) fHn+l+Hn+2 + --- + Hr2s ' r#/i f l -i+i

'1

+

^nfi-i+ 2 + • • • + Hw \(d\p<)\

^

) J e a/

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(resp. Hx + + tiH22 + + •• •• •• + +H H\ tirn, d\ '

I

Hi+H2+-+Hri Hi+H2

d\ + --- + Hn

#r|/>|_i+l + -^n/1-1+2 + • • • + Hm • • • • >

H

r \ P \ - i + \ -

,

\p\ H

+2

m-\

d

^/•i-i =\H\,H2,..

.,HrJ

_ (,Hl+H2

lv ••'[

\P\ ^npi-i+2 + • • • + Hm >

+1 +

+ --- + Hn,(dl)

*

'

(Hn+1

+ Hn+2 + • • • + Hr2^(d2)

'^

J2 )eao->

)

d{p]

^ '

Therefore, Trp(-H0(g)-T(p))

=l

if and only if the coordinates H of -Ho(g) - T(p) =: H = (Hn satisfy the conditions that Hi + H2 + • • • + Hri > 0, (Hi+H2

+ --- + Hn) + (Hn+i + Hri+2 + --- +

(Hl+H2

+ --- + Hn) + (Hn+l + Hn+2 + --- + Hn) + ••• + (^n/i-i+i

+

Hr2)>0,

^n/i-i+2 + • • • + # w - i ) > °-

Recall now that, by definition, -H0(g) - T{p) = (-t\-\ogbx,

-t2 -\ogb2,...,-tr-

log br)

where aig) = diag(fei, b2, ...,br)

= diag( fl < dl) , af\...,

a$f)

293

3 Geometric and Analytic Truncations and Tip)=(tut2,...,tr) = (pil), pil) - pi\), ...,pir-\)-pir-

1), -pir - 1)).

As such, this latest group of inequalities, after replacing Hi's with the precise coordinates of -H0ig) - Tip), are then changed to the following group conditions: ( - log bx - pil)) + ( - log b2 - pil) + pi\)) + ••• + ( - logb n - pin) + pin - l)) > 0 (( - logfcj - pil)) + (-logb2-

pil) + Pil))

+ ••• + ( - log bn - pin) + Pin - 1)))

+ ( ( - iogbn+\ - pin + l) + Pin)) + ( - logfcn+2 - Pin + l) + pin + l)) + ••• + ( - log bn - pin) + pin - l))) > 0

( ( - \ogbx - pil)) + ( - iogb2 - pil) + pil)) + ••• + ( - log bn - pin) + pin - i))) + ( ( - iogfcri+i - pin + l) + pin)) + ( - logbn+2 - pin + 2) + pin + i)) + ••• + ( - log bn - pin) + pin - l)))

+ ((- l o g ^ j + i - /?(>|p|_2 + 1) + pir\p\-i)) + ( - l o g ^PI-2+2 - Pir\p\-2 + 2) + pir\p\-2 + 1)) + ••• + ( - log fcn/>H - pir\P\-{) + pir\Pl-i - 1)) > 0. This, after the obvious simplification with Bx := -logbi,B2

= -logb2

Br - -\ogbr,

rQ := 0,

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L. Weng

becomes the group of conditions Bi + B2 + --- + Bri -p(n)

+ p(0)>0,

(B1+B2 + --- + Bn) + (fl n+1 + Bn+2 + --- + Br2)- p(r2) > 0,

(Bi+B2 + --- + Bri) + (Bn+1 + Bn+2 + --- + Bn) + BnPl_2+2

+ ••• + Bm_^

- p(r\P\-i)

> 0.

Thus note that (bub2, • • • A ) = a(g) = (a
aj™0),

we arrive at the following Sublemma 2. With the same notation as above, TP(-H0(g)-T(p))

=l

if and only if - J i l o g a i -pin)

>0,

( - d\ log ai - d2 log a2) - pin) > 0

( - dx loga! - d2 loga 2

d\p\-i loga|/>|_i) - p(r|p|_i) > 0.

By comparing Sublemmas 1 and 2, we see that fp(-Hoig)-Tip))

=

l(pgp>Pp). D

3.4.2

Global Bridge Between Geometric and Analytic Truncations

With the micro bridge above, now we are ready to expose a beautiful intrinsic relation between algebraic and analytic truncations.

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3 Geometric and Analytic Truncations

Global Bridge. For a fixed normalized convex polygon p : [0, r] —> R, let T{p) := (Kl), P(2) - p{\),...,

p{i) - p(i - 1),.. •, p(r - 1) - p(r - 2), -p(r - 1))

be the associated vector in do. IfT(p) is sufficiently positive, then Kp8
(AT^l)(g).

Proof. This is a direct consequence of the Fundamental Relation in 3.1.2, the definition of Arthur's analytic truncation and the Micro Bridge in 3.4.1.

•

4 Rankin-Selberg & Zagier Method Global bridge between geometric and analytic truncations in 3.4.2 shows that at least for polygons p whose associated T(p) e ao are sufficiently positive, the corresponding non-abelian L-functions may be viewed as a special kinds of Arthur's periods. Therefore, to understand our Ls, it is quite helpful to understand the structure of Arthur's periods. However, to write down them precisely, Arthur's periods for automorphic forms seems to be too general. It is for this purpose that we then narrow our class by considering Eisenstein periods: By definition, Eisenstein periods are integrations of Eisenstein series over truncated compact domains. Simply put, the aim of this chapter is to show that a special kind of Eisenstein periods and hence also certain classes of Ls can be evaluated via an advanced version of Rankin-Selberg & Zagier method. 4.1

Regularized Integration over Cones

To begin with, we here give a review of Jacquet-Lapid-Rogawski's beautiful yet elementary treatment of integrations over cones. Let V be a real r dimensional vector space. Let V* be the space of complex linear forms on V. Denote by S(V*) the symmetric algebra of V*, which may be regarded as the space of polynomial functions on V as well. By definition, an exponential polynomial function on V is a function of the form r ;=i

where the At are distinct elements of V* and the P,(JC) are non-zero elements of S(V*). One checks easily that (1) The Xi are uniquely determined and called the exponents of f. By a cone in V we shall mean a closed subset of the form C:= [xeV:

(pi,x)>0Vi]

where {p,} is a basis of Re(V*), the space of real linear forms on V. Let {ej\ be the dual basis of V. We shall say that A e V* is negative (resp. 296

297

4 Rankin-Selberg & Zagier Method

non-degenerate) with respect to C if Re {A, ej) < 0 (resp. (A, ej) * 0) for each;'= l , . . . , r . (2) The function f(s) = Ej_i e^^P&x) is integrable over C if and only if A{ are negative with respect to Cfor all i. To define regularized integrals over C, we study the integral

7c(/U):=

ff(x)eMdx.

It converges absolutely for A in the open set [A e V* : Re(At - A,ej) < 0 VI < i < n, 1 < j < r), and can be analytically continued as follows: In the case when Re(A) > 0 and P is a polynomial of one variable,

f Jo

m>0

We then can use the right hand side to obtain the continuation. To deal with more general case, fix an index k and let Ck be the intersection of C with the hyperplane Vjt := \x : {^k,x) = 0j. Then, we can write x = ink, x)ek + y with y e Vjt. Thus, for a suitable choice of Haar measures,

i=im>o[(A-Ai,ek))

Jc

*

This formula then gives the analytic continuation of Icif', ty to the tube domain defined by Re(/l, - A, e*) > 0 for j' ± k, 1 < i < n with singularities on hyperplanes //*,,- := i,A : (A,ek) = }, 1 < « < «. Moreover, since this is true for all k = l,2,...,r, the function iAf;A) has analytic continuation to V* by Hartogs' lemma with (only) hyperplane singularities along Hit, 1 < k, < r, 1 < i < r. In particular, from such an induction, one checks (3) Suppose that f is absolutely integrable over C. ThenIc(f\A) morphic at 0 and Icif', Oj = fc f(x) dx; and

is holo-

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L. Weng

(4) The function lAf; AJ is holomorphic at 0 if and only if for all i, A{ are non-degenerate with respect to C, i.e., (At, e^) =£ Ofor all pairs (i, k). Denote the characteristic function of a set Y by TY. For A e V* such that At - A are negative with respect to C for all i = 1,2,..., n, set F(A; C, T) := f f(x) • rc(x - T) • e~{x'x)dx Pi(x)TC(x = J]e-<^T)Ic{Pi(*

T)e~(x-Xi'x)dx + T)e<M;A).

The integrals are absolutely convergent and FyA; C, TJ extends to a meromorphic function on V*. With this, following [JLR], we call the function f(x) • ^(x - T) #-integrable if FLA; C, Tj is holomorphic at A = 0. In this case, set

r-

f(x) • TC(X - T)dx := F(0; C, T).

By (4), the #-integral exists if and only if each exponent Aj is non-degenerate with respect to C. To go further, suppose that V = W\ © W2 is a decomposition of V as a direct sum, and let C, be a cone in Wj. Write T = T\ + T2 and x = x\ + X2 relative to this decomposition. If the exponents Aj of / are non-degenerate with respect to C = C\ +C2, then the function

f

r A , 1(W\ - Ti)d\Vl W2 >-» I f(w\ + W2)T 1 JWi

is defined and is an exponential polynomial. And it follows by analytic continuation that (5) J

f(x)TC(x-T)dx

= \ [ I Jw2 \ JWi

f(wi+W2)TCl,(wi-Tl)dwi)-T^2(w2-T2)dw2; J

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4 Rankin-Selberg & Zagier Method

and as a function ofT, it is an exponential polynomial with the same exponents as f. More generally, let g(x) be a compactly supported function on Wi and consider functions of the form g{w\ - h)Tc(w2 - T2), which we call functions of type (C). It is clear that the integral e~Mdx

F(A) := f f(x) • g(Wl - h)TC(w2 - T2) •

converges absolutely for an open set of A whose restriction to W2 is negative with respect to C2- Furthermore, F(A) has a meromorphic continuation to V*. Following JLR, the function f(x)g(wi - h)Tc(w2 - T2) is called #integrable if F(A) is holomorphic at A = 0. If so we define J

/(*) • g(wx - h)TC(.w2 - T2)dx := F(0).

Note that as a function of W2, L f(w\ +W2)g{w\ -T\)dw\ polynomial on W2, hence I f(x) g(wi -1\ )TC(W2

Jw Jw, )W22 \ \JWi

is an exponential

-T2)dx

f{w\ + W2)g{w\ - T\)dw\ )T£(vi>2 - T2)dw2.

We end this discussion by the following explicit formulas: (6) J e^\c(x-

Jv

T)dx = {-\)rNo\(e v ue2,...

,er) •

* '

. , where

' 115=1 <^.«y>

C := I £y=i a ;' e ; : aj > 0} and Vol(ei,e2,-•• ,er) is the volume of the associated parallelepiped I 2;=i ajej '• aj

r

cone C c R , J

eM(i

_ T c ( ; c _Tj)dx

=

e

[0,1]). M

e dt =

_ r

eAT

A

and hence for any

x x) C

e<

' T (x - T) dx

since 1 - 1 ° is the characteristic function of the cone - C .

300 4.2

L. Weng JLR Periods of Automorphic Forms

Now let us follow [JLR] to apply the above discussion on integration over cones to periods of automorphic forms. Let Tk(X) be a function of type (C) on ap that depends continuously on k € K, i.e., we assume that there is a decomposition ap = W\® W2 such that T/t has the form gk(\Vl - T\)TC2k{W2 ~ T2)

where the compactly supported function gk varies continuously in the L1norm and linear inequalities defining the cone Czk vary continuously. Also with the application to automorphic forms in mind, let / be a function on the space M(F)N(A)\G(A) of the form s

aj{Hp{a),k)e{^pp'Hp(a))

f{namk) = J ^ / ™ . * ) • ;=i

for n 6 N(A), a e Ap, m € M(A)1 and k e K, where, for all j , (a) (pj(m, k) is absolutely integrable on M(F)\M(A) 1 x K; and (b) Aj e a*p and aj(X, k) is a continuous family of polynomials on ap such that, for all keK, aj{X, k)e(AJ-X)Tk(X) is #-integrable. Remark. This is really nothing but the micro structure for automorphic forms stated more formally at 2.1. The point is also related to the reduction theory: with M(F)\M(A) 1 of finite volume, (pj behaves very well. So the problem when say taking integration is about the A-part, which is affine the exponential polynomial function can be controlled nicely provided the exponents are negative. As such, following [JLR], define the ^-integral

f

f{g)-rk(Hp{g))dg

P(F)\G(A)

Y

f(

f

J-}JK\JM(F)\M(A)1

j(mk)dm)[ f I

\Jap

aj(X,k)Tk(X)dx)dk. J

Remark. The reader should notice that in this definition there is no integration involving JN(F)XN(A) dn : In practice, / is supposed to be the constant term along P, so is iV(A)-invariant.

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4 Rankin-Selberg & Zagier Method

Clearly, the set £/>(/) of distinct exponents [Aj] of / is uniquely determined by / , but the functions 4>j and a, are not. So we need to show that the #-integral is independent of the choices of these functions. This goes as follows: The function f(exmk)dm

Xv-> I 1 JM(F)\M(A) /M(F)\M(A)>

is an exponential polynomial on dp and the above #-integral is equal to f

M

f(exmk)dm)e-
f

JK Jap \ JM(F)\M(A)1

I

This shows in particular that the original #-integral is independent of the decomposition of / as used in the definition. Moreover, if each of the exponents Aj is negative with respect to Cik for all k e K, then the ordinary integral

f

m-Tk(HP(g))dg

JP(F)\G(A)1

is absolutely convergent and equals to the #-integral by Basic Property (3) of integration over cones listed in 4.1. That is to say, we have the following Lemma. (1) The above ^-integral fp{F)KGWf(g) • Tk(HP(gj)dg is welldefined; and (2) If each of the exponents Aj of f is negative with respect to C2k far all k € K, then the ordinary integral

f

)P{F)\G(A)11 JP(F)\G(A)

f(g)-Tk(Hp(g))dg v

'

is absolutely convergent and equals to the ^-integral. Fix a sufficiently regular element T € at. Then the above construction applies to Arthur's analytic truncation A^'PxP(g) and the characteristic function TP(HP(g) - T) where Y € JHP(G) := A((/(A)M(F)\G(A)), the space of level P = MU automorphic forms. Indeed, with ^(namk) = ^T Qj(HP(a)) • if/j{amk) 7=1

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for n e N(A),a e Ap,m e M(A)1 and k e K, where £?/ are polynomials and \f/j € 3ip(G) satisfies

Mag) =

e^VW^jig)

TtP for some Aj e aj, and all a e Ap, fM p. lxK A ij/j(mk)dm dk is wellT,p defined, since the function m \-> A if/j(mk) is rapidly decreasing on the space M(F)\M(A) 1 x K. Now Tp is the characteristic function of the cone spanned by the coweights A^, so

1

AT'Py(g)-Tp(Hp(g)-T)dg

JPU P(F)\G(A)

exists if and only if (AJ,VJV)*0

VweAji

and

Aj e SpQV).

The same is true for Ar-/'xI'(g)-T/>(#/>(,?-x)-F) for x e G(A). (For x e G(A), let K(g) € K be an element such that gK(g)'1 e P0(A), then HP{gx) = HP(g) + HP(K(g)x) and hence TP(X - T - HP(K(g)xj) is the characteristic function of a cone depending continuously on g.) Set then IP/W):=

AT'PV(g)-TP(HP(g)-T)dg.

f

For any automorphic form 0 6 !h{G), write £/>(0) for the set of exponents &p{ * 0

VGTV

6 A^, A e &P{
G).

If e J?l(G)*, then I^T((f>p) exists for all P; and define JLR's regularized period by

f

p):=ITG(ct>).

In the follows, let us expose some of the basic properties of JLR's regularized periods. For this, we make the following preparation. As above, for x € G(Af), let p(x) denote right translation by x: p(x)(f>(g) - (gx).

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4 Rankin-Selberg & Zagier Method

The space 31(G) is stable under right translation by G(A/). Furthermore, p(x)

P(namk) = J^ Qj(HP(a)) • ej(mk). j

Since amkx = n*aa'mm'K(kx) for some n* e N(A), we have (ppinamk • x) e^+ppMa)+^a'))
= J ] Qj(HP(a) + HP(a')) •

This shows that Gp(p(x)(p) - &p(
f

JG(F)\G(A) 1

4.3

(g)dg=

f

JG(F)\G(A)1

then $ e 3\(G\ and

<[>(g)dg.

Arthur's Periods and JLR's Periods

For / e 3lP(G) with

f(namk) = ^tjimk)

•

aj{H(a))e{x^pp'H(a))

;=i

where n e N(A), a e AP, m e M(A)1, k e K, and for all j , aj(X) is a polynomial, and (f>j(g) is an automorphic form in 3\p(G) such that
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(j)j{g) for aeAp, following [JLR], define

f

P(F)\G(A)1 I

m-TP(H(g)-T)dg (pj(mk) dm dk

^ri

JKJM K JM(F)\M(A)! X:

( |

aj(X)e{xi'X)TP{X - T) dx\.

This is well-defined provided that the following two conditions are satisfied: _ v V (i*) (n, tz7 ) * 0 V£ c P, SJ e (A v )£, /i € S e (0); and (ii*) <^, orv) * 0 Vor e A/., A € 6 P (0). Let ${{G)** be the space of

e ^(G)**,

X

G(F)\G(A)

= Y(-1)^ G > f 4.4

M*) • rp(//(g) - r)
Bernstein Principle

With above, to understand our L-functions at least for T » 0, i.e., for T € QJ sufficiently positive, we need to study Eisenstein periods. For this, the following result plays a key role. Bernstein's Principle. ([JLR]) Let P = MU be a proper parabolic subgroup and let o- be an irreducible cuspidal representation in the space L 2 (M(F)\M(A) ! ). Let E(g,(f>,A) be an Eisenstein series associated to a cusp form 4> e ^/>(G), A)) are defined.

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4 Rankin-Selberg & Zagier Method

We believe that this principle is the main reason behind special yet very important relations among multiple and ordinary zeta functions in literature. 4.5

Eisenstein Periods for Cusp Forms

As a direct consequence of Bernstein principle, with the explicit formula for constant terms of Eisenstein series associated to cusp forms, we have the following closed formula for Eisenstein periods associated with cusp forms: Theorem. ([JLR]) Fix a sufficiently positive T e aj. Let P = MU be a parabolic subgroup and cr an irreducible cuspidal representation in L 2 (M(/ r )\M(A) 1 ). Let E(g,(f>,A) be an Eisenstein series associated to

, A) dg is equal to (1) 0 if P ^ Po is not minimal; and (2) Vol(( Z a e A o aaa- : aa e [0,1)}) • £ n ^ Z - P ^ w<=W

X I

(M{w,A)
JM0(F)\MO(A)1XK

ifp = pQ = MQUQ is minimal. Consequently, from the very definition, we have the following Theorem'. Fix a convex polygon p such that T(p) e CLQ is sufficiently positive. Then for a P-level cusp form , Ljfitp, X) is equal to (1)0 if P is not minimal; (2) Vol({ Z« £ A„ <*? : aa e [0,1)}) • £ n ^ g U weW X

I

)M0l(F)\M0n(AyxK JMr (A)]xK

(M(w,A)(f>)(mk)dmdk,

ifP-PoMoNo is minimal. We remind the reader that this result is merely the beginning of a quantitive study about our L-functions. Say, (1) When the polygon p satisfies T(p) » 0, we still need to understand LFf{(p, A) where P' D Pisa, parabolic subgroup of G;

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(2) We have to weak the restriction that T(p) » 0; and (3) We need also to study our non-abelian L-functions for principal lattices, i.e., that for general reductive groups. In fact all this proves to be very interesting and quite fruitful: In (1), Artin L-functions naturally appears; to (2), the above discussion is valid as long as T{p) e aj U {0}; and for (3), no essentially difficulties appear despite its generality.

4.6

Volumes of Truncated Fundamental Domains

As an application, we here recall a recent result of H. Kim and the author on the volumes of Arthur's truncated domains [KW].

4.6.1

The Results

Let then G be a split, semi-simple group of rank r. Recall that for sufficiently regular r e a j , Arthur's analytic truncation A r l coincides with the characteristic function of a compact subset I(T) := (G(F)\G(Ay J c G(F)\G(A) 1 . (See e.g., 3.3.1.) Our result calculates the volume of E(r). In particular, letting T —> oo, we recover Langlands' famous formula on the volumes of fundamental domains G(F)\G(A) 1 . To state the result, for any w e W, let Jw := {a e AQ : wa e AQ] and Pw be the standard parabolic subgroup corresponding to Jw. Let rw := {a e Ao : wa < 0}, and, for i>\, set m :=#{<* > 0 : = /} - #{a > 0,: = / - 1}, n,> :=#[a > 0, wa < 0 : (p, orv> = i] - #{or > 0, wa < 0 : (p, av> = i - 1}. As usual, denote by A? the discriminant of F, r-i the number of complex embeddings of F, and £F(1) the residue of the completed Dedekind zeta function £F(S) at 5 = 1. Theorem. ([KW]) For a sufficiently positive T e a+, the volume of the

4 Rankin-Selberg & Zagier Method

307

truncated fundamental domain (G(F) \G( A)1J is given by Vol({ J ] aa<*v : aa e [0,1)}) • W b f

["[frOr*

FlaeAo-w^ (1 - <Wp, Or v »

w^Vo

Note that Z(oo) = G(F)\G(A) 1 , and in the above formula, as T -* oo, only the term corresponding to w = 1 survives. In this way, we then recover Langlands' formula on volumes of fundamental domains; Corollary. ([La2]) IfG is split and semi-stable, then the volume of a fundamental domain %for G(F)\G(A) 1 is given by Vol(5) =Vol(j YJ «*<*V : o« 6 [0> D}) oreAo

i>i

4.6.2

The Proof

Apply Theorem 4.5 to the special case where G is a split, semi-simple group and

where, by Gindikin-Karpelevich formula [La4],

Consequently, we have f

ATE(g,l,A)dg

JG(F)\G(A)'

Z w€n

e(W\-PJ)

— — —M(w,A) FLeAo( w * ~P>a >

n

(*) dm.

JM(F)\M(A)i

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L. Weng

since

{wX+

P'H(-m)) = 1 for m e M(A)1. (Here v := Vol(( £ a e A o a a a v :

e

a a e [0, l)Jj appears because the measure on ao is defined using co-weights Izja : a e Ao}. Indeed, v is equal to the order of the fundamental group of 1 if a $. Ao, a > 0. Hence Res,i=p£(g, I, A) = Jim [~[ (U,a v > - l) • M(w0,A) aeAo

where wo is the longest Weyl group element. Since


""" * " Q & w . «»> + i:

we have

ReS/l=p£(g, l.A) - fr(l)r • ["JfrOT'. i>i

where r denotes the rank of G. Now by 3.3.1, ATE(g,hA)dg=

f

f

JG(F)\G(A)'

E(g,\,A)dg. Jw)

Hence Res^ p ( f JG(F)\G(A)'

Ar£(g,l,A)^)= f

ResA=pE(g,\,A)dg

Js(r)

^ 0

= Vol(Z(T)) • ResA=pE(g, I,A) = fra/ f l ^ " ' • Vol(Z(r)). «>i

On the other hand, by Theorem 4.5, the residue at A = p of the left hand side of (*) is

Vo,( W \r(A)').jta(v£,«-«M M n < £ $ p r >

309

4 Rankin-Selberg & Zagier Method

Here following Tate [T], Vol(r(F)\T(A) 1 ) = (Vol(F*\lJr))" = frW with £>(D := Res,=ifr(j). (See e.g., [T].) Note also that (wA, arv> = (A, (w -1 ar) v ), and for a given w € W, the sets {or € Ao : war < 0} and {a e Ao : war € Ao} are disjoint. Thus, if we let Wo be the set of all w e W such that {a € Ao : war < 0} U {a e Ao : war e Ao} = Ao, and rw := #{a e Ao : war < 0}, then we have lim( V e^-^M(w,A)

Rav>-1 U / T * v\ i )

•2

«wp,ar v >-l)

vfeWo

Therefore, Vol(l(D) = v • (27r)'2rA7 [~[ ^(i)-" 1 ' i>i

To go further, let us recall that: there is a one-to-one and onto correspondence between the elements of Wo and the standard parabolic subgroups ofG. Indeed, given w € Wo, let Jw = {a e A : wa e A] and denote by Pw the standard parabolic subgroup corresponding to Jw. Then P := Pw is the standard parabolic subgroup of G associated to w. Conversely, any standard parabolic subgroup P of G is associated to a unique subset / c A. Set then Dj := [w e W/Wj : wJ > 0}. It is known that the set Dj gives distinguished coset representatives of W/Wj. Let wj be the element of the maximal length in Dj. Then wj e W0 and in fact Wo = {wj : J c A}. We claim that if w e Wo, <wp,arv) < 0 for all a e A - wJw: if ar € A - wJw, then w/(ar) = -J3 for some/? 6 A. But by [Ca], if w/ is the element of the maximal length in W, and wij is the element of the maximal length in Wj, then wj = wiwij. So w-1(ar) = -wijw(J3) since w _1 = wijwwi. If now/? € Jw, then -wijw(fi) € Jw and w~l(a) e /„,. But w~l(a) £ Jw, since

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a <£ wJw. Contradiction. Consequently, /? g Jw and wijw > 0. Therefore w~x(a) < 0. As such, (<wp,av)-l) = (-l)rk^

Y\

Y\

(l-(wp,av>).

This then completes the proof of the Theorem. The result may also be written as follows, oriented by standard parabolic subgroups; Theorem'. ([KW]) For a sufficiently regular T, Vol(SXr)) = Vol({ £

a

^

• a<* e [0,1))) • {2n)nrdF"5

aeAo

Ua€Ao-WjA^ ~

/cAo

(Wjp,av))

Remark. The formula, as it stands, suggests that the volumes of cuspidal regions corresponding to parabolic subgroups can be written in terms of special zeta values as well. Proof. By the Theorem, this is a direct consequence of the fact that for w e Wo, rw + 2 «,> = 0, and - £ "i = r, the rank of G. Indeed, - 2 «,• is the number of positive roots such that (p, av) = 1. It is exactly r. Similarly, - Z ni,w is the number of positive roots such that wa < 0, and (p, a v ) = 1. It is exactly rw = rankiV So rw + £](-«,• + n,>) = r for each w e Wo. o 4.7

When Parameters Are Small

By definition,

A^CS) := £(-l)<W-*G>

J]

P

SeP(F)\G(F)

^8) MH^

~ T)-

And, as indicated in 3.3.1, to have the formula

r JX(.T)


AT
we need the following basic properties of the truncation operator A r .

4 Rankin-Selberg & Zagier Method

311

Theorem. (Arthur) For T sufficiently large, (1) A r o A r = AT; (2) AT is self-adjoint; (3) A r 0 is rapidly decreasing, if is automorphic on G(F)\G(A); (4) A r l is a characteristic function of a compact subset Z(T) ofG(F)\G(A)] However for geometric truncation using stability, parameters T may not be very regular. So we need to check whether properties (1) ~ (4) still hold for such Ts. Let us look at (1) first. For T sufficiently large, this is proved by Author based on the following Lemma. ([Ar2]) Fix Pi = M\U\. Thenforcpe C(G(F)\G(A)1), I

AT (f>(u\g) du\ = 0

JUi(F)\Ui(A)

unless vj(H{g) - T) < Ofor each m € A/>,. By examining the proof of Arthur, we see that for S Ln, if T is taken to be in the positive chamber or T = 0, the whole proof still works. Indeed, only the part at p. 91 of Arthur's proof [Ar2] should be checked. But then this is quite obvious. (For general reductive groups, the extra element 0 should be replaced by a certain naturally defined element. For details see [Ar5].) Next, let us turn to (2) and (3). Clearly, formal calculations in Arthur's proof work for any T. So the point here is really about the convergence of all integrals involved. Put this in a more general term, convergence for the integrals appeared are really the central part of Arthur's truncation theory. Sure, one more key ingredient needed here is the combinatorial technique used. Luckily, this elementary yet highly non-trivial combinatorial part can be formally applied even when T is not that large. All in all, the up-shot is the so-called Arthur-Langlands partition for the fundamental domain listed at 3.2.8. It is here that we use the condition T being sufficiently regular to make sure that the function Fp(g, T) is a characteristic function of a certain compact subset. While it is a bit involved, however, by checking the proof on convergence very carefully, we can conclude that what one really needs in proving (2) and (3) for smaller T is that, as a function of g, Fp(g, T) is bounded and compactly supported modulo U(A)M(F) uniformly for T

312

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varying in a compact subset. But this is then exactly what Prop. 2 of 3.3.2 provides. Finally, Let us turn our attention to (4). This is a very hard one: In our treatment, this is proved using the Global Bridge between geometric truncation and analytic truncation established in 3.4.2 and a micro-global principle called Fundamental Relation established in 3.1.2. As such, we have proved the following Theorem. Let P = MN be a parabolic subgroup ofG = SL„ and let (M, er) be a cuspidal datum. Let E(
JG(F)\G(A)1

ATE(g,
(i) is equal to zero ifP is not a minimal parabolic; (ii) is equal to

Vol({ YJ «<*<*V : 0 < a a < l}) aeAo e<,wA-p,i)

x

n

/ , Ti 1~> U^ ^ , YlaeAoiwA -p,av>

JM(F)\M(A)lxK

M(w,A)
if P is minimal. Consequently, we also have the following Theorem. For any T in the positive chamber a^orT = 0, Vol(Z(T)) = Vol({ J ] aaay : aa e [0,1))) •

(Wd^

aeAo

./cAo

4.8

YlaeAo-wjj(l-(wjp,(XV))

Fundamental Relations Between Special Zeta Values

By Corollary 4.7, with a detailed calculation, we then obtain the following result on the volumes of moduli space AIF^OAFI^), which reveals certain fundamental relations among special values of classical Dedekind and our zeta functions.

4 Rankin-Selberg & Zagier Method

313

Theorem. Let£F,r(l) := ResJ=i£F,rO) = Vol(Mf>(|A/r|5)). Level 1.

fr,i(D

= fr(l);

Level 2.

fr>2(l)

= fr(2)fr(D -

Level 3.

fed)

=

fr(l)fr(l);

fr(3)fr(2)fr(l) -^F(2)^(1)2-^F(2)^(1)2+^(1)3;

Level 4.

fr,4(l)

=

fr(4)fr(3)fr(2)fr(l)

- ^F(3)^(2)^F(1)2 - ^H2)2fr(D2 - ^ F ( 3 ) ^ F ( 2 ) ^ ( D 2 + ^(2)2fr(l)2 + ^F(2)^F(D2 + ^ F ( 2 ) ^ ( D 3 0

9

6

Level r. The r-level relation can be described as follows. (1) %F,r(\) is a degree r homogenous polynomial in %p(f)s, for i = 1,2,... , r; (2) there is a one-to-one and onto correspondence between the terms of this polynomial and partitions of r, or better, standard parabolic subgroups. More precisely, in the case r = r\ + r2 + • • • + r^, the corresponding term consists of three parts, that is, sign • coefficient • product of zeta values; (i) the sign is given by

(-l)N~l;

(ii) The coefficient is

; and (n + r2)(/-2 + r 3 ) • • • (rN-\ + rN) (iii) the product ofzeta values is given by: Ifri > 2, the product ^f(r,)^f(r,— 1) • • -£F(2) does appear. Moreover, these subfactors together with certain powers of %F{\) then gives all the product of zeta values (under the constrain that the final product is of degree r). It is well known that following a result of Siegel, in order to read the special value £/K«), we have to go to rank n lattices. The above relations may be viewed as a refined version of this basic result. Roughly put, our result says that

314

L. Weng

(i) among all rank ^-lattices, only stable lattices are essential; (ii) other rank n lattices can be built up from stable lattices of lower ranks; (iii) corresponding modular points of lattices in (ii) are sitting in cuspidal regions; and (iv) volumes of cupidal regions can be expressed in terms of £Hm)'s, m < n. We end this discussion by pointing out that our result above is closely related with the result of Harder and Narasimhan ([HN]) on counting stable bundles over curves on finite fields: Their method, an arithmetic geometry one, is algebraic while ours, a geometric arithmetic one, is analytic.

5 High Rank Zetas and Eisenstein Series 5.1

High Rank Zetas for Number Fields

Let K be an algebraic number field (of finite degree n) with A# the absolute value of its discriminant. Denote by OK the ring of integers. For a fixed positive integer r e N, denote by Atf.r the moduli space of semi-stable OKlattices of rank r with dp the associated (Tamagawa type) measure (induced from that on GL). Recall that for each A e Af/r,r, the associated 0-th geoarithmetical cohomology h°(K, A) is h°(K, A) := log ( YJ exp ( - * £ xeA

ll^ll^ - 2* £ \\xa\\2Pcr))

a-.R

cr:C

where x = (x0-)cr€sij<) and (pa-)cresm denote the cr-component of the metric p = PA determined by the lattice A with 5«, a collection Archimedean places of K; and that the rank r zeta function %KAS) °fK1S t n e integration

f^™- 1 ) • M"l08V°1(A) <WA), Re W > 1.

€*A* •= f or the same, Ms)

:= (A* )' • f

(e"°^)

• (e-fSW

dp(A), Re (,) > 1.

JAeMK,r

Theorem. (1) (Meromorphic Continuation) £KAS) is well-defined when Re(s) > 1 and admits a meromorphic continuation, denoted also by %KAS)> to the whole complex s-plane; (2) (Functional Equation) £*,,.( 1 - s) = &>(.$); (3) (Singularities & Residues) £KAS) has two singularities, all simple poles, at s = 0,1, with the same residues Vol(Aljt>r([A|])J. r

Here, as before, Atjf>r([A|]) denotes the moduli space of rank r semistable Ojf-lattices whose volumes are fixed to be A£. The proof of this theorem is fairly standard now since we have a good arithmetic cohomology which satisfies the Serre duality and the Riemann-Roch formula. For details, see e.g., 1.4.2. 315

316 5.2

L. Weng Space of 0/r-Lattices

Recall that an OK -lattice A consists of a underlying projective (^-module P and a metric structure on the space V = A ®z R = rio-eSeo Va-- Moreover, in assuming that the 0#-rank of P is r, we can identify P with one of the (r— 1 ^

A" := ^r;o, := 0K © o„ where a,, i = 1,... ,h, are chosen integral OKideals so that |[ai], [a2], . . . , [a/,]} = CL(^T) the class group of K. With this said, following Minkowski, we obtain a natural embedding for/5: P := tf£_1) © A M A*r) *-> (Rri x C r2 ) r 3 (R'")''1 x (C")1"2, which is simply the space V = A ®z R above. As a direct consequence, our lattice A then is determined by a metric structure on V = Yla-es^ V or the same, on (R r ) n x {C)n. It is well known that all these metrices are parametrized by the space (GL(r,R)/<9(r))n X (GL(r, C)/t/(r)) r \ Next let us shift our discussion from GL to SL. For this, we start with a local discussion on OK -lattice structures: For complex places T, by fixing a branch of the n-th root, we get GL(r, C)/f/(r) = (SL(r, C)/S U(r)) x R*+; while for real places cr, we have GL(r, R)/0(r) s (sUjr, R)/50(r)) X R^. Consequently, metric structures on V = Uo-es^ V
1

x

(CT 2 are

((5L(r, R)/50(r)) n X (SL(r, C)/s t/(r)) r2 ) X (R;) ri+r2 Furthermore, when we work with (^-lattice strucures A = A(P) on P, from above parametrized space of metric structures on V = Yl
X (S L{r, C)/S U(r)f} x (R;) ri+r2

317

5 High Rank Zetas and Eisenstein Series

Hence we need (a) to study the structure of the group AutoK(0^~l) © a) in terms of SL and units; and (b) to see how this group acts on the space of metric structures ((SL(r,R)/SO(r))ri

X (SL(r,C)/SU(r))n)

X (R*+1+/-2 )ri<

View Auto*(^ _ 1 ) ©a) as a subgroup of GUr, K). Then, A\itoK(0^~l)® a) = {A<=GL(r,K)tl

0K \a * •••

detA € UK a

a 0K)

+

Introduce groups GL (r,R) := {g e GL(r,R) : detg > o} and 0 + (r) := {A e 0 0 , R) : detg > o}, and also the subgroup A u t ^ ( 0 ^ _ 1 ) © a) of AutoK(P^~ } © a) consisting of these elements whose local determinants at real places are all positive. Then, by checking directly, we obtain a natural identification of quotient spaces between AutoK(0(K-l) © a)\((GL(r,R)/0(r)) ri x (GL(r,C)/U(r))r2) and Aut

OK(°K~l) © a)\((GL + (r,R)/0 + (r)) ri X (GL(r,C)/U(r))n).

Recall that for a unit e € UK, (a) diag(e,..., e) € AutoK(0%.~l) © a); and (b) detdiag(e,..., e) = sr e UrF := {er: e e t/jdSimilar to the subgroup GL+, introduce a subgroup UK of UK by setting £/£ := {E<EUK:s(r> 0, V
Aut^O^

© a). Thus, choose elements Ml,. . . , M ^ K ) eUK

318

L. Weng

such that | [ « i ] , . . . , [«^(r,A:)]} gives a complete representatives of the finite quotient group UK/(uK n C/^), where fi(r, K) denotes the cardinality of the group UK/(UK n UrK). Set also SL(0K~l) © a) := SL(r, /Q n GL(0K~l) © a). Then we have the following Lemma. ([We6,8]) There exist Ai,... ,A^r>K) in GL + ((9^ _1) © a) such that (i) det Ai = ut for i = 1,... ,ju(r, A"); and (ii) \A\,... tA^icu is a completed representative of the quotient resulting from A u t + ^ O j r 0 © a) modulo SL{0^~X) © a) x (ur£ • diag(l,..., 1)). That is to say, for automorphism groups, (a) AutJ (<9^ © a) is naturally identified with the disjoint union U ^ f }A( • (sL(0K~l)

© a) x (urK+ • diag(l

1)));

and, consequently, (b) The (^-lattice structures A(P) on the projective OK -module P = 0£~ ©a are parametrized by the disjoint union

VJ^Ai^SUd^

© a)\((SL(r,R)/SO(r)f

x (5L(r,C)/5f/(r)) r2 )) x (\UrK n £/+|\(R;) n+r2 )).

Therefore, to understand the space of OK -lattice structures, beyond the spaces SL(r, R)/S 0(r) and SL{r, C)/S U(r), we further need to study (i) the quotient space \UrK n ^|\(R.+) , which is more or less standard (see e.g. [Ne] or [We6,8]); and more importantly, (ii) the quotient space SL(0K~l) © a)\((SL(r,R)/SO(r)f

X

(SL(r,C)/SU(r)f2\

319

5 High Rank Zetas and Eisenstein Series 5.3

High Rank Zetas and Epstein Zetas

Choose integral Ox-ideals <x\ = OK, 02, • • •»°-h such that the ideal class group CL{K) is given by {[ai],..., [a/,]}, and that any rank r projective Otf-module P is isomorphic to Pai for a certain i, 1 < i < h. Here, Pa := Pr,a '•= 0^ - 1 ) ©o for a fractional (9^-ideal a. Introduce the partial high rank zeta function gK,r,a(s) associated to a by setting ^ ; « ( . ) := f

(

^

- 1) • (*"')" l0SV ° 1(A) ^(A),Re(,) > 1

where A1/f,r(a) denotes the part of the moduli space of semi-stable OKlattices whose corresponding modular points lie in (SL(0{K~1) © a)\((SL(r,R)/SO(r))ri

x

(SL(r,Q/SU(r)f2)"j

x(\UrKnUi\\ 1, define the completed Epstein zeta Junction EK,r;a(s) by

*[("«• 4)''

E

^-

With this, from the discussion above, by applying the Mellin transform and the formula

(whenever both sides make sense,) we obtain the following Proposition. ([We6,8]) (1) (Decomposition) The rank r zetafuntion ofK admits a natural decomposition h 1=1

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L. Weng

(2) (High Rank Zeta = Integration of Epstein Zeta) The partial rank r zeta function £;K,na(s) of K associated to a is given by an integration of a completed Epstein type zeta function: fr,r;«(*) = ( J / 1 " 2 • f r EK,nMdp., 1 JMFs;a[N(ayA%]

Re(j)>l.

6 Stability and Distance to Cusps Chapters 6, 7, 8 are dealing with rank two zeta functions. 6.1

Upper Half Plane

On the upper half plane
A:=y

2

(dT

+

d2 dy-22} dy

The group SL(2, R) naturally acts on *H via Mz:=

az + b cz + d'

VM

•n

eSL(2,R), ze
The stablizer of i = (0,1) € *H is equal to S 0(2) := {A e 0(2) : det A = 1}. Since this action is transitive, we can identify the quotient 5 L(2, R)/S 0(2) with 'H with the quotient map induced from SL(2, R) —»(H, g n g • i.
Naturally, the

!

above action of SL(2,R) also extends to P (R) via a

b\

ax + by cx + dy

c dj 6.2

Upper Half Space

The 3-dimensional hyperbolic space may be written as H :=Cx]0,oo[= j( z ,r) : z = x + iy e C,r e R*+] ={(x,y,r) : x,y eR,r eR*+}. 321

322

L. Weng

We will think of H as a subset of Hamilton's quaternions with 1, i, j , k the standard R-basis. Write points P in H as P = (z, r) = (x, y,r) = z + rj

where z = x + iy, j = (0,0,1).

We equip H with the hyperbolic metric coming from the line element ds := dx2+dy'+dr2 with hyperbolic volume form d/i := **&p*L. Consequently, the hyperbolic Laplace-Beltrami operator associated to the hyperbolic metric ds2 is simply 2

W

+

dy* + dr2>

r

8/

The natural action of SL(2,C) on H and on its boundary P ! (C) may be described as follows: We represent an element of P 1 (C) by x,y € C with (x,y) + (0,0). Then the action of the matrix M SL(2, C) on P ! (C) is defined to be a b c d

where

-n

ax + by cx + dy

Moreover, if we represent points P e H as quaternions whose fourth component equals zero, then the action of M on H is defined to be P »-• MP := {aP + b){cP + d)~\ where the inverse on the right is taken in the skew field of quaternions. Furthermore, with this action, the stabilizer of j = (0,0,1) e H in 5L(2,C) is equal to SU(2) := {A e f/(2) : detA = 1}. Since the action of 5L(2,C) on H is transitive, we obtain also a natural identification H » S L(2, C)/S U(2) via the quotient map induced from S L(2, C) -» H, J H 8-J6.3

Upper Half Space Model

Identifying *K with S 1(2, R)/5 0(2) and H with S L(2, C)/5 U(2), using the discussion in 5.2, we conclude that % T O

• Ajf] - (SL(0K © a)\(
323

6 Stability and Distance to Cusps

where we use ss to denote the subset consisting of points corresponding to rank two semi-stable <9#-lattices in S UQK © o)\l(s L(2, R)/S 0{2)f

x (s 1(2, C)/5 U(l)f

).

More precisely, if the metric on OK © a is given by g = (go-)creS„o with go- 6 SLi2,Ko), then the corresponding points on the right hand side is g(ImJ) with ImJ := ( i ( r i \ / r 2 ) ) , i.e., the point given by (go-ToOo-es^, where To- = io-:= (0,1) if cr is real and T„- = ja'•= (0,0,1) if cr is complex.

6.4 Cusps As such, the working site becomes the space SL(OK®O)\('H'"1 XW2). Here the action of SL{0K © o) is via the action of SL(2, K) on 'J-C1 x HP"2. More precisely, K2 admits natural embeddings K2 <^-» (Rr> x C 2 ) =* (R 2 ) ri x (C2)r2 so that OK®CL naturally embeds into (R 2 ) n x(C 2 ) r2 as a rank two 0Klattice. As such, SL(0K © a) acts on the image of 0K © a in (R 2 ) n x (C 2 ) r2 as automorphisms. Our task here is to understand the cusps of this action of SUOK © a) on 9C1 x W2. For this, we go as follows. First, the space (Hn xW2 admits a natural boundary Rri x C 2 , in which the field K is imbedded via Archmidean places in Soo'- K ^-* R n x C 2 . Consequently, P 1 ^ ) ^> P 1 (R) r| x P1(C)r2 with

: = M H (oo(ri>, oo^>).

As usual, via fractional linear transformations, SL(2, R) acts on P^R), and 5L(2, C) acts on Pl(C), hence so does SL(2, K) on pl(^) ^ P ^ R / i

X P'(C)

r2

.

Being a discrete subgroup of 5L(2,R) n x 5L(2,C) r2 , for the action of l SL(OK © a) on F (K), we call the corresponding orbits (of SL{0K © a) 1 on P ^ ) ) the cusps. Very often we also call representatives cusps. With this, we have the following fundamental result rooted back to MaajS. Cusp and Ideal Class Correspondence. (See e.g., [We6,8]) There is a natural bijection between the ideal class group CL{K) of K and the cusps

324

L. Weng

CT ofT = SL(0K © a) acting on
[0Ka + afi].

CT -» CL(K),

Easily, one checks that the inverse map II ! is given as follows: For b, choose ab,fib e K such that OK • ab + a • J3b = b; then II - 1 ([b]) is simply the class of the point ab in 5 L(2,0 © a)\P ! (K). Moreover, there always K fib exists M

6.5

€ 5L(2, K) such that Mr i • oo

Stablizer Groups of Cusps

Recall that under the Cusp-Ideal Class Correspondence, there are exactly h inequivalent cusps m, i = 1,2,..., h. Moreover, if we write the cusp at for suitable a,, ft e K, then the associated ideal class n m fii is exactly the one for the fractional ideal Ojcor, + afii =: b,-. Denote the stablizer group of rj in S L{OK © o) by Yv. Lemma. ([We6,8]) The associated 'lattice' for the cusp n is given by ab -2 . Moreover, A~%A =

S e t r ; := U \l

Z

0 u

_A : ue UK,ze ab 2

Z

AA~1 : z e ab~2\, Then

Note that also componentwisely,

_x \z = ^

what we really get is the following decomposition

r„ = r; X [/|

= u2z. So, in practice,

325

6 Stability and Distance to Cusps with

With this, we are ready to construct a fundamental domain for the action of T^ c SL(OK © a) on 9in x W2. This is based on a construction of a fundamental domain for the action of T^, on
, e SL(2, K) (always exists!), we have

a ; and

m

ii) The isotropy group of r\ in A - 1 5 L(OK © a)A is generated by translations -2 T H-» T + z with z € ob and by dilations T H M T where u runs through the group U\. (Here, we use A, a, /?, b as running symbols for A,, or;, /?,-, b,- := OKOCI + qft.) Consider then the map *W> x W2 -^ Rr^n defined by (Zl,. . . ,Zr,;Pl, • • • ,Pr 2 ) >-» ( I m ( z i ) , . . . , I m ( z n ) ; J(Pl),..-,

J(Pn))

where if z = x + iy £ *H resp. P = z + v/ € H, we set Im (z) = y resp. J(P) = r. It induces a map (A- 1 • r , • A)\(W

x H'2) - » t / ^ R ^ 2 ,

which exhibits (A -1 • T^ • A)\(o with fiber the n = n + 2ri dimensional torus (W1 x C2)lab~2. Having factored out the action of the translations, we only have to construct a fundamental domain for the action of \j\ on R>0+r2. For this, we look first at the action of \]\ on the norm-one hypersurface S := (y e R ^ 2 : N(y) = l}. By taking logarithms, it is transformed bijectively into a trace-zero hyperplane which is isomorphic to the space R r ' +/ ' 2_1 S

!2f RH+,2-1

yi->

:=

| ( f l l > _. ^an+n)

(logy!,..., log y ri+r2 ),

e Rrl+n

:

^

a i

= 0 J,

326

L. Weng

where the action of UK on S is carried out over an action on Rri+''2"1 by the translations a, i-> a, + loge (,) . By Dirichlet's Unit Theorem, the logarithm transforms U\ into a lattice in R n+r2 ~ 1 . Accordingly, the exponential map transforms a fundamental domain, e.g., a fundamental parallelepiped, for this action back into a fundamental domain S ^ for the action of U\ on S. The cone over S ^ , that is, R>o • S ^ c R>0+r2, is then a fundamental domain for the action of U\ on R>^ n . Denote by T a fundamental domain for the action of the translations by elements of ab~2 on R ri x C 2 , and set ReZ(zi,...,z r ,;Pi,...,/ > r 2 ) := (Re(zO,... ,RQ(Zr,);Z{Px),...

,Z(Pnj)

with Re (z) := x resp. Z(P) := z if z = x + iy e 'H resp. P = z + vj € H, then what we have just said proves the following Theorem. ([We6,8]) A fundamental domain for the action o/A - 1 r v A on •Kri X Hr2 is given by E := (T E W ' x Hr2 : ReZ(r) € T, ImJ (T) e R >0 • S ^ } . For later use, we also set T^ '•= A~x • E. 6.6

Fundamental Domain

Guided by Siegel's discussion on totally real fields [Sie] and the discussion above, we are now ready to construct fundamental domains for 5 L{OK © a)\(«Hri xH r 2 ). As the first step, we generalize Siegel's 'distance to cusps'. For this, recall that for a cusp TJ =

e Fl(K), by the Cusp-Ideal Class Corre-

spondence, we obtain a natural ideal class associated to the fractional ideal b := OK -a + a-fi. Moreover, by assuming that a,fi are all contained in OK, as we may, we know that the corresponding stablizer group Tv is given by A"1 • r , • A = \y = (|j J.] 6 T : u € UK,Z € ab" 2 },

327

6 Stability and Distance to Cusps

where A e SL(2, K) satisfying Aoo = 77 which may be further chosen in the form A = r

"* J € 5 L(2, K) so that 0K/3* + a -1 a* = b" 1 .

Now for T = (zi,..., zri; P\, • • •, Pr2) 6
j=i

Thenforally = (^

J 6 [c d)

5L(2,K), v

(

JV(ImJ(T))

(Note that here only the second row of y appears.) Moreover, define the a reciprocal distance p(n, T)from the point r e *Hr> x W1 to the cusp r\ =

fi

in P\K) by Mi7.T):=^(a- 1 -(O j r a + (V8)2) lm(.zi)---Im(.zn)-J(P1)2---J(Pr2)2 (0 2

' nil, \wPzi+<* )i n J , IK-/*^+« w )ii 2 1

^(ImJ(T)) 2

N(ab~ )

\\N(-pT + a)\\2

Lemma 1. ([We6,8]) (i) p is well-defined; (ii) p. is invariant under the action of SUP K © &). That is to say, p(yn, yr) = p(n, T),

SyeS UQK © <»)•

(iii) There exists a positive constant C depending only on K and a such that ifp(n,T) > C andp(rj',T) > C for re (Hri x HP and 77, 77' e P 1 ^ ) , then T} = T]'.

(iv) There exists a positive real number T := T(K) depending only on K such that for r € fHn X HT2, there exists a cusp 77 such that pit], T) > T.

328

L. Weng Now for the cusp 77 = a e P (K), define the 'sphere of influence' of

77 by F , := [T e /i(rf ,T), V17' € P 1 ^ ) } . Lemma 2. ([We6,8]) 77ie action ofSL(OK © a) *'" the interior F° o/F,, reduces to that of the isotropy group Y^ oft], i.e., ifr and JT both belong to F®, then yr = r. Consequently, we arrive at the following way to decompose the orbit space into h pieces glued in some way along pants of their boundary. Theorem. (We]) Let iv : rv\Fv <^-> SL(0K © tO^Tf1 x W2) denote the natural map. Then SL{0K © o)\(0 • S ^ ) a fundamental domain for the action of A^TJJA^ on TY7"1 x W2. The intersections of E with irjiFjj) are connected. Consequently, we have the following Theorem'. ([We6,8]) (1) A, := A"'E n F , is a fundamental domain for the action ofY^ on Fv;

329

6 Stability and Distance to Cusps

(2) There exist ai, ...,ah e SUQK® a) such that\Jhi=xa{Dm) is connected and hence a fundamental domain for SIXPK ® <*)• That is to say, a fundamental domain may be given as Sy U T\{Y\) U • • • U rh(Yh) with 5 Y bounded, Ttfi) = At • TiiYd and fiiYd := jr e r • S ^ } . Moreover, all TiiYiYs are disjoint from each other when Yj are sufficiently large. 6.7

Stability and Distances to Cusps

Define now the distance ofr to the cusp n by d(r],TA) := —

> 1.

Then we are ready to state the following fundamental result, which exposes a beautiful intrinsic relation between stability and the distance to cusps. Theorem. ([We6,8]) The lattice A is semi-stable if and only if the distances of corresponding point T\ € (Hri X W2 to all cusps are all bigger or equal to 1. Our proof uses an algebraic result of Tsukasa Hayashi [Ha]. One can never overestimate the importance of this relation. Being stable, lattices should be away from cusps. More generally, while the stability condition is defined in terms of sublattices, the relation above transforms these volumes inequalities in terms of distances to cusps. In a more theoretical term for higher rank lattices, the essence of this fact is that, sublattices and cusps, as two different aspects of parabolic subgroups, are naturally corresponding to each other: the stability conditions for various sublattices are naturally related with generalized distances to all types of cusps. 6.8

Moduli Space in Rank Two

For a rank two O^-lattice A, denote by TA e {Hn x W2 the corresponding point. Then, from above, A is semi-stable if and only if for all cusps n, d(n,T\) := , * . are bigger than or equal to 1. This then leads to the

330

L. Weng

consideration of the following truncation of the fundamental domain D of r 2 SUPK © a)\( x IF ): For T > 1, denote by £>r := jr e £> : ^ , T A ) > T _1 , Vcusp 77}. The space DT may be precisely described in terms of D and certain neighborhood of cusps. To explain this, we first establish the following Lemma. For a cusp 77, denote by XV(T) := (T € *Hri x HP : < % T ) < T~l). Then for T > 1, mm,2(r)#0

0171=172.

With this, we are ready to state the following Theorem. ([We6,8]) There is a natural identification between (a) the moduli space of rank two semi-stable Ox-lattices of volume N(a)Atf with underlying projective module OK © a; and (b) the truncated compact domain T>\ consisting of points in the fundamental domain D whose distances to all cusps are bigger than 1. In other words, the truncated compact domain D\ is obtained from the fundamental domain D of SL(0K © a)\[
T > 1.

7 Explicit Formulas for Rank Two Zetas 7.1

Epstein Zetas and Eisenstein Series

For a fixed integer r > 1 and a fractional ideal a of a number field K, define the Epstein type zeta function Era\(s) associated to an 0^-lattice A with ' '

(r— W

underlying projective module Pa = 0K

© a to be

%,aAs)-.={n-rMj)f(2J:--Y(rs)f

x (w)4)' •xeO

<

J J"

1)

Z

®o/£/+ r ,x#(0

A

0)

where UKr := \er : e € UK, sr e £/+} = UK n UrK. For example, note that in the case r = 2, f/£ 2 = U\, we have £2,a;A(*) :=(*-*r(5))''1 {2n-2sY(2s)f

x (tf(a)Ajr)* •

1

2

\\x\\2s'

xeOKm/Ul,x*(0,0) " "A

Set also, for Re (s) > 1, JE2.«(T, J)

(n-sns))r\2n-2sr(2S))r2

:=

CvMic/0„a>n/f7„ r r u w m m ^\\X V JC •• TT + + y\ V (x,y)eO K®a/UK,(x,y)*(0,0)

/

Lemma. ([We6,8]) For a rank two OR-lattice A = (OK © a, PA), denote by ffte corresponding point in the moduli space SL(OK © a)\rHri x IF 2 ).

TA

Tften E2,a-\(s) = E2A(JK, 7.2

S).

Fourier Expansions

For simplicity, introduce the standard Eisenstein series by setting

£2 ,„
(^*»y,

z

(x,y)eOK®a/UK,{x,y)*(0,0)

Ml* ' T

331

+

V

/

Re(s)>1.

332

L. Weng

Then the completed one becomes E2,aCr, s) = (n-sr(s))n (2n-lsY{2s)f As before, for the cusp rj = a a"K _1

P.

• (N(a)AK)S • £2>a(r, s).

(*)

, choose a (normalized) matrix A =

e SL(2, F) such that and that if b = 0Ka + a/3, then 0KP>* + aa* =

b . Clearly, Aoo = rj, and moreover,

A lr =

~ ^ {(o ? i : w 6 0 b " 2

Since £2,0(1", s), and hence £2,0(1", s), is SUOK © a)-invariant, £2,0(1", s) is T^ c SL(OK © a)-invariant. Therefore £2J„(.AT, •$•) is ab-2-invariant, that is, E2A{AT, S) is invariant under parallel transforms by elements of ab - 2 . Consequently, we have the Fourier expansion £ 2>a (Ar,s)=

^(lmJ(T),5)- e 2 -<-'- ReZ(r) >,

YJ 2 v

a/e(ab- )

where (ab~2)v denotes the dual lattice of ab -2 . Thus, if we use Q to denote a fundamental parallolgram of ab - 2 in Rn x C"2, then a^(ImJ(r), s) := voi(.a

)

^ . (C;rf)€(OA ea)/4/f/x.i(C;d)#(00)

f ( T T T ^ F • e~2M"w') r K • n ***• j fi \iicr+
i.>

y

Theorem. ([We6,8]) We have the following Fourier expansion £2>a(Ar, s) =aior • N(ab+ lbl i 2s) Vly2s • N(ImS(r)Y _ J _2s. ( w i ) n 2 VoKab- ) *-i N(c) * * 1

(

c ca) + d)

" W v (S^T) "
333

7 Explicit Formulas for Rank Two Zetas

Vol(ab"2)

N(c)2s

^

V

'

\T{s)l

I* *)

I Jr X [~[ ST,_.(2*/Uv) • ( ^ r a V cr:R

7.3

*•

5

)" ' I I *k-l(2*l<"Vr>-

'

T:C

Explicit Formula for Rank Two Zetas

The original Rankin-Selberg method gives a way to express the Mellin transfrom of the constant term in the Fourier expansion of an automorphic function as the scalar product of the automorphic function with an Eisenstein series if the automorphic function is very small when approach to the cusp. In a paper of Zagier [Z], this method is extended to a much broad type of automorphic functions. Here, we use a generalization of the Rankin-Selberg & Zagier method to give an explicit expression for rank two zeta functions of number fields in terms of Dedekind zeta functions. In the sequel, T is assumed to be a positive real number > 1. Our aim is to compute the integration

E

£>T

Here, as in 6.8, DT is the compact part obtained from the fundamental domain D for S L(OK@a)\('J-(ri xH r2 ) by cutting off the cusp neighborhoods defined by the conditions that the distance to cusps is less than J - 1 . (Recall that, as such, D\ is simply the part corresponding to semi-stable lattices.) For doing so, first formulate the integration

JOE

(A*: E2,a(T, s) W ( T )

where a-:R

r.C

with

its2

d2

\

A

2iif

a1

s2\

a

334

L. Weng

(For the time being, by an abuse of notation, we use A# to denote the hyperbolic Laplace operator for the space {Hn xIF 2 , not the absolute value of the discriminant of K which accordingly is changed to DR.) Note that K(yl) = sis - 1) • yU

and

Ar(/?*) = 2si2s - 2) • ?TS

by the SL-invariance of the metrics, we conclude that Ajf (?2,«(T, s)) = (n • (s{s

- 1)) + r2 • (2si2s - 2))) • £2,a(r, *).

Hence I II

£2,a(T, 5) d//(ta«) =

'

_ * III

AKE2,aiT, s)

dflir).

On the other hand, using Stokes' Formula, we have J I I AKE2,aiT> s)dnir) JJJDT

(T 5) Aifi |(T)

~HT ^.« ' -( )*

= J [ ) J ((A^£2,a(r, s)) • 1 - E2,aiT, s) • (Ajfl)J =

(T

(dEuJT,s) 7

JJdD(J) \ rr JJdD(T)

OV

£/MT)

~ dl\ 1 - E^air, s) • — W dv)

dE2,ajT,s) dv

where ^ is the outer normal derivative, dp. is the volume element of the boundary dDj. Theorem. ([We6,8]) Up to a constant factor depending only on K, fff

SUr. .VWt) - & ^ ( A , • i f ' - ^^>(AK

• T)-.

7 Explicit Formulas for Rank Two Zetas

335

Consequently, we have the following Theorem'. ([We6,8]) The rank two zeta function £K,2(S) ofK is given by ZK(2S) ZK,2(S) =

fr(2s-D

:

s- 1

p

, , ^ .

Re (s) > 1.

s

8 8.1

Zeros of Rank Two Zetas Zeros of Rank Two Zeta of Q

Following what was happened in history, let me first start with Suzuki's weak result [Su] and then give Lagarias' unconditional result. Meanwhile, for an independent parallel work, please go to Ki's beautiful paper [Ki]. Theorem. If the Riemann Hypothesis for the Riemann zeta function holds, then all zeros O/£Q,2(S) lie on the critical line Re (s) = \. This is a very clever observation, rooted back to Titchimashi's book on Riemann Zeta Functions. 8.1.1

Product Formula for Entire Function of Order 1

Let f(z) be an entire function of order one on C, that is, (i) f(z) is analytic over C; (ii)/(z) = 0(exp(|z| 1+e )), V e > 0 . Let n(R) denote the number of zeros of f{z) inside CR, the circle of radius R centered at the origin. Then (1) n(R) = 0(Ra), Vor> 1; (2) E Pn ./(p„)=oKr a converges. In particular, ( l - ^ ) e x p ( ^ ) = l + 0 ( ( ^ ) 2 ) as n -* 00. As a direct consequnce, (3) P(z) := rip„ (l ~~ jr)" e x P \T) converges and ^ is an entire function of order 1 without zeros, hence should be in the form exp(A + Bz) for certain constants A, B. That is to say, we have the following Hadamard Product Theorem. Let f(z) be an entire function of order one on C, then there exist constants A, B such that

/B-**-n('-|)-«p(f> H

p

H

Example. (See e.g. [Ed]) We have

p

336

H

H

337

8 Zeros of Rank Two Zetas

with A = - l o g 2, B = - | - 1 + j log An, where y = lim,,-,^! + \ + ••• + i - log wj denotes the Euler constant. 8.1.2

Proof

Let F(z) = - Z ( i + 2fe)

with

Z(s) = s(l - s)£(s).

Proposition. (Suzuki) (1) F(z + | ) - F(z - | ) = iz(l + 4z2)£Q)2(5 + iz). (2) Assume the RH, then all zeros ofF(z + \) - F(z - %) are real. In particular, then the RH implies that ^q,ii\ + Zl') admits only real zeros. Proof. (1) Simple calculation. Indeed, F(z + i ) = - Z ( i + 2i(i + z)) = - Z ( i - i + 2iz) = -Z(2iz). So f ( z - l ) = -Z(l+2«) 4>

and

= (1 + 2iz)(-2iz)f(l + 2iz) - 2iz(l - 2iz)£(2iz)

'Z vk

(± z)-l (i ++ iiz)~

i + iz

'

= Jz(l+4z z )f Q , 2 (- + /z). (2) Clearly, F(z) is an entire function of order 1, so there are constants A, B such that

m = ^-- n 0-f)-«p(f> Note that essentially, p are zeros of the completed Riemann zeta but transformed from z to 2 + 2iz. Hence,frytfie/iff, all p are real.

338

for

L. Weng Moreover, since F(z) = - Z ( | + 2iz) with Z(s) = s(\ - s)g(s), we have xeR, ~F(x) = - Z ( l / 2 + 2IJC) = -z(\/2

+ 2ix) = -ziX - 2ix)

which by the functional equation is simply -Z(l - (^ - 2ix)) = - Z ( ^ + 2ix) = F(x). That is to say, for x e R , F(x) takes only real values. Hence, constants A and B are both real. Now let zo = *o + iyo be a zero of F(z + l-) - F(z -'-) = iz{\ + 4z2) • f Q ) 2 (- + iz). Then zo = 0 and/or zo is a zero of £(3,2(5 + iz) since ^,2(2 + '^) admits simple poles at z = ±\i. In any case, F(zo + ^) = F(zo - ^). By taking absolute values on both sides,

^*.j..n{1.2ii).^(2ii)| Z0~J\

1Z0--

=l^<»-i».n(1_^).exp(^)l. Since B eR and p„ € R (which is obtained by the RH as said above), we hence get (x0 - pn)2 + (yo - \f 1 l 2 2 n=\ Uo -pn) + (yo + \) '

•n

Thus if yo > 0, then the right hand side is < 1, while if yo < 0, then the right hand side is > 1. Contradiction. This leads then yo = 0. • With this in mind, note that in the proof above, the RH was used to ensure thatp are real, which have the effect that then in the calculation for

339

8 Zeros of Rank Two Zetas

the exponential factor expf^-pj, the ratio

-ft*)

gives us the exact

value 1. That is to say, this factor of ratio of exp's does not contribute. However, one does not need such an argument from the very beginning to eliminate the factors exp (^-^V In fact, this is the improvement of Lagarias, who gets his own unconditional result totally independently, as a part of his understanding of de Branges's work. The trick is very simple: Use the functional equation. So instead of working on individual p„ in the product, we may equally use the functional equation to pairp and 1 - p for the zeros of the completed Riemann zeta function, or even to group p, 1 - p , p and 1 - p together. Consequently, the exponential factor appeared inside the infinite product may be totally omitted. That is to say, from the very beginning, we may simply assume that the Hadamard product involved takes the form

m-f".

fl (i-i) p:F(p)=0

H

where f ] ' means thatp's are paired or grouped as above. Form here, it is an easy exercise to deduce the following result of (Suzuki and) Lagarias. TheoremQ. All zeros O/£Q,2(S) lie on the line Re (s) = j .

Proof. Alternatively, as above, we have

Since B e R, so if we can take care of the factors exp (^jp) and exp ( ^ p ) in a nice way, we are done. For this, as said above, let us group p, p, 1 p, 1 - p together, we see that J + I = ^ and ^ + ^ = ^ ^ are all reals, hence, the same prove as above works. •

340 8.1.3

L. Weng A Simple Generalization

The above method works for the functions £Q2(S) a s w e ^ ' P rov ided that T > 1. Indeed, first, recall that we have the precise relation Ws>-

s-l

1

1 s

•

Consequently, F(z + '-) • H " * - F(z - l-) • H + , z = iz{\ + 4z 2 )£j i2 (^ + zi). Therefore, using the same proof, we arrive at the relation

=

|^ fa -i). n(l _5_i ) . exp( 2^i ) |.|rH|.

That is to say,

^ p r Q c o - P ^ + Cyo-l)2 T-y° ~l=\(xo-pn)2 + (yo + \)2' T*>' Or equivalently,

U(*0-Pn) 2 + 0'0+i) 2 ' Thus with 7 > 1, we have (i) if yo > 0, the left hand side is > 1, while the right hand side is < 1, contradiction; while (ii) if yo < 0, the left hand side is < 1, while the right hand side is > 1, contradiction. That is to say, we obtain the following Theorem— For T > 1, all zeros of^L 2(s) lie on the critical line Re (s) = \. Recall that 5Q.2W -

_ j •I

s

•

341

8 Zeros of Rank Two Zetas Clearly

while limr->+0o £Q 2(S) does not really make any sense. This says that even when we have a family of natural functions whose zeros all lie on the critical line, in general, we cannot take limit for this family to preserve this property, no matter how careful we are. 8.2

Zeros of Rank Two Zetas

Finally, we are ready to state the following Theorem. All zeros of rank two zeta functions for number fields are on the critical line Re (s) = | . Proof This is a direct consequence of the following two facts: First, as stated in Thm 7.2, we know that, by the Rankin-Selberg & Zagier method, up to a constant factor depending only on K, &(2s) €K,2(S) =

Z

&(2s-l) •

s- 1 s Secondly, s(s - 1) • £#(•?) is a l s o a n entire function of order one [LI]. Consequently, the proof in the previous section, more precisely, that of 8.1.2, on the zeros works here as well by a simple change from the Riemann £ for the field of rationals Q to the Dedekind £#• for the number field K. •

9 A Rank Three Zeta and Its Zeros In this chapter, we evaluate rank 3 zeta functions, using Langlands' theory of Eisenstein series, Arthur's theory of analytic truncations, and an advance version of Rankin-Selberg & Zagier method established in 4.7. The starting point is a general relation between Arthur's analytic truncation and the geometric truncation. Simply put, geometric truncation using stability is the same as Arthur's analytic truncation (taking effect on constant function 1) for suitable choices of parameters. But then, an essential difficulty appears: In the theory of Arthur, parameters involved are all supposed to be very large, i.e., the truncation is taken to be very near cusps. With such sufficiently large parameters, because the cut-off part is far away, basic properties that the truncation operator is indeed an orthogonal projection and that the truncation for constant function 1 becomes a characteristic function do hold. On the other hand, in our geometric truncation, say, the one for semi-stable lattices, parameters are relatively small. And, it is by no means clear that the above mentioned properties hold for such parameters. Fortunately, geometric truncation using stability is intrinsic and hence has to be understood at each and hence all levels of parabolic subgroups. Consequently, the corresponding analytic truncation for 1 is still a characteristic function and the associated truncation operator is an orthogonal projection too. With this cleared, for the calculation of rank 3 zeta, it suffices to find out the so-called constant terms of certain Eisenstein series with the help of a residue argument, providing that all the integrals appeared are convergent. However, this latest assumption turns out to be a very difficult one: The convergence argument is really the heart of Arthur's analytic truncation theory, in which, as mentioned above, the parameters are sufficiently large, not just 'normal' in our current situation. Luckily, this can be solved in due course as well, thanks to the advanced studies for trace formula. The notable points are 1) for large parameters, by the original argument of Arthur, all integrations appeared are convergent; and 2) the integrands, varying over difference of regions obtained between large parameters and for smaller parameters are compactly supported. (We here should mention that while 1) is quite hard, 2) is quite obvious. But still it is this easy new input that solves our problem instantly.) With the convergence problem being fixed, we are now ready to come 342

9 A Rank Three Zeta and Its Zeros

343

to the point of constant terms of Eisenstein series. As stated in Ch. 1, Eisenstein series used in high rank zetas are Epstein type zeta functions. Being a special kind of Eisenstein series, they come from suitable L2functions on maximal parabolic subgroups. Hence, according to general theory of Langlands on Eisenstein series, they can be obtained as residues of Eisenstein series associated to cusp forms on lower rank parabolic subgroups. As such, the problem is changed to the one of writing down concretely such a residue argument. In the case at hand, by a classical result, we know that the cusp form can be taken to be the constant function 1 on minimal parabolic subgroup. (Recall that being abelian, as a minimal parabolic of SL, the L2-function 1 is indeed cuspidal on the Levi.) To be more precise, we will use classical Eisenstein series such as Koecher zeta functions and Selberg's Eisenstein series associated to parabolic subgroups ofSL„. With this important matter being settled again, to go further we need to recall the theory of constant terms of Eisenstein series associated to cusp forms. According to Langlands, they are closed related and are indeed given by the so-called intertwining operators, which, following Langlands as well, can be naturally given by the so-called Gindikin-Karpelevich formula. So a calculation about the action of Weyl group on the roots system must be worked out in details. After that, by applying our generalization in 4.1 of a result of Jacquet-Lapid-Rogowski, an advance version of RankinSelberg & Zagier method, final expression of rank 3 zeta can be obtained by taking residues along suitable singular hyperplanes. Let us now briefly review the related works on constant terms of Eisenstein series for SLj. In fact, a right expression first appeared in Venkov's basic example of trace formulas for SL3. However Venkov only briefly states the final result and gives no details. (He justifies this by merely saying that what he gets comes from Langlands' general theory.) A few years later, in an effort of giving all Fourier coefficients of the Eisenstein series for SL(3,Z), Vinagradov-Takhtajan obtained the same result, from which in particular, they proved the functional equation for the Eisenstein series. Their derivation is completely elementary and is independent on anything except Epstein zeta result for SLq.. Sure, the story does not stop here. Immediately after obtaining the precise formula for the rank 3 zeta, I sent it to Masatoshi Suzuki, and asked

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L. Weng

him to check whether the Riemann Hypothesis works for this rank 3 zeta as well. Suzuki first did some numerical calculations from which he was convinced that all zeros do lie on the central line. So he started working more carefully. Now we have the following theorem of him: All zeros of rank 3 zeta for the field of rationals lie on the central line Res = \. As such then it appears that the time is ripe to openly make the following generalization of the Riemann Hypothesis: The RH. All zeros of zetas, rank one or higher, lie on the central line Re s = \. 9.1 9.1.1

Various Eisenstein Series Eisenstein Series Associated to 1

Let G be a reductive group defined over a number field F. Fix a minimal parabolic subgroup PQ and its Levi subgroup MQ, both are supposed to be defined over F as well. Take a standard parabolic subgroup M with Levi decomposition P = MU where M is the corresponding standard Levi subgroup and U is the unipotent radical. Then for an automorphic form ij/ e A(U(A)M(F)\G(A)) of /Mevel, by definition, the associated Eisenstein series is given by the convergence series Ety, A, g) :=

mP(Sg)A+" • ^Sg)

J] 6eP(F)\G(F)

where A e ty satisfying ReA e C a certain positive cone. (For unknown notions, please refer to 2.1.) In particular, if i)/ is a cusp form, then C may be chosen to be the positive Weyl chamber of *$. For cusp forms, by a result of Langlands, the constant term of E with respect to a parabolic subgroup P' = M' U' is given by Ep.ty,X\g):=

J]

EM'IM(M(w,A)ilr,wA)(g)

weW(M,M')

where W(M,M') := [w e w\w~\a)

>0

Va € T+(T0, M'), wMw~x •=-> M'standard Levi},

345

9 A Rank Three Zeta and Its Zeros

and EM'/M denotes the relative Eisenstein series. As a special case, if M' is conjuagate to M, then Ep.ty,A;g):=

£

(M(w, A) ifr)(g).

weW(M)

For our purpose, we take G to be SL3, P = P0 = MQJJQ the minimal parabolic subgroup consisting of upper triangular matrices with MQ diagonal matrices in PQ (SO it is a torus, in particular, is commutative) and UQ the upper triangular matrices with all entries on the diagonal being 1. As such, then the constant function 1 on Mo is cuspidal and its associated Eisenstein series is simply E(l,AXg)=

m

Po(68)A+po,

J]

VReAeCo

5eP0(F)\SLi(F)

and weW(Mo)

while £>(1, X)(g) = 0 since vvMovv""1 f+ M' if M' =* M0 (recall that while Po is minimal, its Levi MQ is maximal).

9.1.2

Koecher Zeta Functions

Defined the so-called Koecher zeta function by

Z m , n _ m (X,s):=

YJ m

\

AeZ *"\GL(.m,Z),tkA=m

X

s W '

n

Re(s) > -

where X[A] := A' • X • A, and | | denotes the determinant. It is convergent hence well-defined, and satisfies many interesting properties. (For a discussion related to classical Eisenstein series, a useful reference is [Te]. In

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L. Weng

fact we here adopt her notation as well.) In particular, when m = 1, we get Zhn-X(X,s):=

\X[A]\~S

YJ AeZ1)<"/GZ.(l,Z),rkA=l

=

FA]

Z

AeZ l x "/±l,rk/l=l

\X[A] AeZn\0A=-A

'

J-y veZ"\0 '

'

which is indeed the standard Epstein zeta functions. For example,

veZ\0 '

9.1.3

'

A Special Eisenstein Series

Suppose that ^ is an automorphic form on a lower rank parabolic subgroups, 1 < m < n, Res > jf, Ye fn, the space of positive nx n real matrices. Define then the associated Eisenstein series £m,„_m(^, s\Y) := E(ffr,s\Y)by

E(flf, s\Y) =

J]

Y[Ai]

• iKY[Ai]°).

A=(Ai,*)ern/P(m,n-m)

Here if Y e 9n we write Y = t* W with t = \Y\>0 and W =: Y° e SPn := {Y e 9n\ \Y\ = 1}; Ai e ZnXm, Tn = GL(n, Z), P(m, n-m):= the subgroup /*i 0 \ consisting of elements of the form where *i (resp. *2) is a matrix \U

*2/

of size m (resp. n). 9.1.4

Relation Between Z and £

Lemma 1. The lattice version and the group version of Eisenstein series are related by the following relations Zm,n-m(Y, S) = Zmfi(I, S) •

347

9 A Rank Three Zeta and Its Zeros and m-\

zm,0(x,s) = \xr-Y\a2s-j). 7=0

In particular, ZU-1(Y,S)

^\Yr-a2s)-Ehn-l(l,s\Y)

= \Y\-'-a2s)-

J]

\YIMT'.

A=(Ai,*)er„/P(l,n-l),AieZ" ><1

Proof. We need the following Lemma 2. The quotient Znxm rk m/Ym, which means the n X m, rank m integral matrices in a complete set of matrices inequivalent under right multiplication by matrices in Tm = GL(m,Z), for 1 < m < n, can be expressed by matrices A = BC where B e Znxm, (B, *) £ GL(n, Z)/P(m, nm), C e Zmxmrkm/GL(m, Z). For a proof of this lemma, see e.g., [Tr]. Indeed, then by definition, ^m,n-m\X,

S)

£

\X[A]\~S

nxm

AeZ

ikm/GL(m,Z)

^

IC'B'XBCf*

BCeZnxm,(B*)eGZ.(n,Z)/7J(m,n-m),CeZmx'"rkm/GL(m>Z)

by the lemma since \AB\ = \A\ • \B\, while

zm,0(/, s) • £w,„_m(i, S\x) =

J]

CeZ mxm

x

IC' • c r J

/GL(m,Z),rkC=m

YJ

\B'XB\-\

(B,*)eGL(n,Z)//>(m,n-m)

n With this, for high rank zetas, we shift our attention to E\ttl-.\.

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L. Weng

Remark. In fact what we really use for rank 3 zeta function is £1,2- For this, to see the relation between Zi,2 and E\^ stated above, we may also use the following more direct Lemma 2'. Let (x,y,z) e Z3\{0} such that gcd (x,y,z) = 1, then 'an au a^ (i) There exists a matrix A e SL(3,Z) with the form A = a2\ an an ; \x y zt (ii) The above matrix A is unique up to a left multiplication of a matrix in P(l,2)nSL(3,Z). Proof First let us prove (ii). If A and A' are as in (i). Then consider the product A' • A~l and pay attention to the last row of the resulting matrix. Since the last rows of A and A' are the same, namely (x, y,z), the last row of the matrix A' • A~l equals to (00 1). This proves (ii). f a\\ a\i ai3^ Next let us prove (i). The matrix A = ai\ an 023 € 5L(3,Z) we vx y z, 'a\\ an an want to construct will take the form A 0 «22 «23 , that is, the entry .x y z . an = 0. Then the condition that A € SL(3, Z) means that an an

an -a an n

H-

•\a23y-a22z

Let gcd (y, z) = S. There exist a, b e Z such that ay + bz = 8 and gcd(a,b) = 1. Set now 023 = a,022 = -b. Then the above equation becomes an an -a\\ -{ay + bz)= 1, -b or equivalently, an -b

an - an • 5 = 1. a

Now note that gcd(x,6) = gcd (x,y,z) = 1, there exist X, Y e Z such that x • X + 6 • Y = 1. Set an = ~Y- Then the problem becomes that whether there exist 012, a n £ Z such that X = an an . But this is clear, since -b a

349

9 A Rank Three Zeta and Its Zeros

gcd (a, b) = 1. Indeed, there exist a,fteZ such that a • a+/3 • b = \. As such, set a\2 := a • X, an := /3 • X will do the job. D 9.1.5

Selberg Eisenstein Series

Let s = O i , . . . , sn) e C" and Y e fn. Define the so-called Selberg Eisenstein series by

E{n)(s\Y):= J]

P-VM)

ver„/P(„)

where for a matrix A e Pn, ps(A) is the power function defined by ps(A) := rn=i \Aj\Sj with Aj e Pj, the j x j upper left hand corner in Y, j = l,...,n. (Note that when n = 2 and 7 € 5^2, the power function can be identified with the function on the upper half plane defined by ps(x + iy) - ys for y>o.) As usual, let P = P(n\,ni,...,nq) be the parabolic subgroup of SL„ consisting of blocked upper triangular matrices of sizes n\, «2. • • •. nq corresponding to the partition n-n\ +«2H—+nq. Then for Selberg Eisenstein series, we have the following Lemma 3. For each i = 1 , 2 , . . . , « - 1,

Eo,)(s\Y) =

2

v=(vi*)=(v 2 *)=-=(v„)er„// j ;,vieZ" x l,v2eZ' ,x2

v„€Z"x"

n

E{2)(si\Ti).

[~]

\Y[VJ]\-'J

j=hj*i

where P* := P(\,..., 1,2,1,..., 1) with 2 the i-th entry, and Tt is defined as follows: First write F[v,+1] = P [ V ^ l ]

°j 'j

1

f

, i = 2,..., n - 1

and Y[v2] = R\. We see that all /?,• e P2: then introduce T, e P2 by setting TX=RX= Y[v2] and Tt := \Y[vi-i]\ • Rifar i = 2 , . . . , n - 1. (In particular, \TX\ = \Y[v2]\, T,- = |y[v,-_i]| • |F[v,-+1]| for i = 2 T„_i = |F[v„_2]| • \Y\.) For a proof, see e.g., [Te].

n - 2 and

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L. Weng

Example. For n = 3 with / = 1,2, we obtain the following formulae

£(3)(s|r)'=

£ v=(v2*)=V3er3/P(2>l),V2eZ3><2)V3€Z3x3

Ev)(.s1\Ti)-\Y[v2]\-'2-\Y[v3]rSi

E

v=(vi*)=V3er3/P(l,2),v1eZ3x|>V3€Z3><3

£(2)(J2ir2)-invi]ri-iy[v3]r'3

where T, = Y[v2],T2 = \Y[vi]\ • * 2 and F[v] = J 1 ^ 1

^ J

0

72

Consequently, E0)(SUS2,0\Y) £(2)(^ir 2 ) • y[vi]- 51

^ ver 3 /P( 1,2),v=(vi *), vi eZ3.>

£

£(2)feinvi]-/?2)-nvi]-^

ver 3 /P(l,2),v=(vi*),vieZ 3 . 1

£(2)fei/?2)-nvi]-ii+^.

E 3 1

ver 3 /P(l ,2),v=(vi *),vi eZ '

More generally, Siegel introduced the following Eisenstein series for the parabolic subgroup P = P{n\,n2, ...,nq): i

:=

Efw)

n MA^~si

z

Aer„/P,A=(Aj*),AjeZnxNJ,j=\,...,q

J=l

where Nj := n\ + n2 + • • • + tij. For example, Ep*(sx, s$\Y) \Y[Ai]\-Sl-\Y[A3]\-S3.

E Aer 3 /P(l ,2),A=(A i *)=A3 ,A i eZ

3x l

,A 3 eZ

3x3

In particular, Ep>(s, 0\Y)

E Aer 3 /P(l,2),/i=(Ai*)AieZ 3x l

imi]r = |^-zi,2(r,*). ^

9 A Rank Three Zeta and Its Zeros 9.1.6

351

Taking Residues

It is also known that E(n)(s\Y)\Si=0 =Epr(si,..., s,-i, sM,..., Res,,.=i£(„)(s|y) =— • EP.(s\,...,

sn\Y),

s*_v s*M,..., s*n\Y)

with s*- - Sj for j ±i± 1, s*±l = s,±i + \. (See e.g. [Te].) In particular, E0){s\Y)\Sl=o=Ep.2{susi\n Res,2=i£(3)(s|y) =— • EP.2(Sl + -, s3 + -\Y). Or put it in a better form, Ei3)(s,t,0\Y)\,=o=Ep2(s,0\Y), Res, =1 £ (3) (5 - j,t,-l-\Y)

=^ • Ep^is,0|F).

To go further, note that, by definition, £(3)(*l, *2, s3\Y) =

f[

P-s(Y[y]) = \Y\-* • E{3)(su

s2,0\Y)

yer 3 /p (3)

since Illy]!--53 = Y/Yyt^ - \y\2 • \Y\~S\ Consequently, to understand all £(3)(*i> S2, sj\Y), we only need to understand E@)(s, t, 0\Y). 9.1.7

E2>i: An Example

As an example, we have in particular, E2,3-2(EU, s\Y) =

Y[CXYU • |r[C 2 ]r + M / 2

J] (C,-,*)=Cer3/P(3),CieZ3x'-

where Eu(z) := Zyer2/P(i,i)y"('Y^' a n d ^(3) = -P(l,l,l) is the minimal subgroup PQ. Indeed, the right hand side is equal to

irtCiir • \Y[C2]\-S+U/2

£ Cer 3 /P(l,l,l),C=(Ci*)=(C2*),CieZ

C=(C2*)eT3/P(2,l)

3xl

,C 2 eZ

3x2

(b*)eY2/P(l,l)

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L. Weng

since r 3 / P ( l , 1,1) = r 3 /P(2,1) x P(2,1)/P(1,1,1) and

iWD/l'CU.D-f™

±°

)•

Also by definition

J]

E(EU, s\Y) :=

|r[Ai]r • £MOTAi]°)

A=(Ai*)er3//>(2,l)

=Eo)(u,s--,0\Y). (Note that in £„, *° is used, consequently, in E@), the variable becomes ^ - | not simply s). 9.2 9.2.1

What Matters to Us The Function ps(Y)

In this section, for SL3, we give a precise relation between classical and current expressions of Eisenstein series. The key then is that for SL3, any element g can be written as

(V« -k °) (a 0 8=

0 - ^-L = 0" •

lo

0 1 ;

0 a ,0 0

0 W l 0 x\ 0 0 1 / a - 2 , 10 0 1)

where a > 0, M > 0, v, x, t e R, so

§' =

Consequently,

with

'I 0 0' / a 0 0 1 0 0 a ,ac t 1, 1,0 0

'yfu 0 x 0 1 a' ) [o

0 1

0^ 0

0

ij

353

9 A Rank Three Zeta and Its Zeros

In particular, 1^1 = a2u, \Y2\ = a\u • £& - v2) = or4. This then leads to, fory \- a6, psiX) := IKil" • \Y2\S2 = ( a V ( a 4 ) i 2

9.2.2

Root System for 5 L3

In R3, set e\ := (l,0,0),e 2 = (0,1,0), e3 = (0,0,1) be the standard orthonormal basis. Then with a\ := e\ - e2,a2 := e2 - e^, we get ||ari||2 = ||ar2||2 = 2. And from - 1 = (ai,a2) - \\a\\\ • \\a2\\ • cos0 = 2cos0, we get 6 = |;r. AS usual, introduce the so-called co-root by orv := T^T • a. Then 2 atf = j ^ 1

= ai

'

a

2

=

2 2a2

=

ai

'

Set also AQ := {a^a^} and denote its dual basis by Ao := {uj\,m2} (so that t3T,(orY) = dij). Note that (aa\ + ba2)(u\,u2, M3) = a(u\ - u2) + b(u2 - W3) = au\ +(b- a)u2 - bu^ from vj\{a\) = 1, m\(a2) = 0, we get a - {b - a) = l,(b - a) + b = 0, that is, a = ^,b = 5. Or better, ^

2 = -ax

+

1 2 1 1 . -a2 = ( 3 , - 3 . - 3 ) .

Similarly fromCT-2(ori)= O . ^ f e ) = 1, we get a-(fe-a) = 0,(Z?-a) + & = 1, that is, a - | , b = | . Or better, 1 2 . 1 1 2 . to* = 3-a, + - a * = ( - , - , - - ) . Consequently, \\vjxf = \ + \ + I = | and ||tjr2||2 = 5 + 5 + 5 = 5. Also by definition, a\ ± m2, that is, a\ and TZT^ are orthogonal to each other. So we may use them, or better use {3GT2 = a\ + 2a2,a\) as a new coordinate system so that every point X on the roots plane can be written as A = s • (3GT2) + t • a\ = 5 • (a\ + 2a2) + t • a\ = (s + i)a\ + 2sa2.

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L. Weng

In particular, we have gW.%))

:=

g*

2

0 - ^ 0

^ 0

0

Q:"2

" (^nJB *

= M*+f • or6* • u~s = u' • (a6)s = u' • ys. Consequently, «.<*»<>«> = M y = P{suS2)(Y) when sx = t, ^ p - = better si = t,S2 = ^y*. In particular, when A = s • (3?zr2) +1 • a\, E(l, A)(g) = E0)(t + l-, ^

s

or

+ l-, 0\g'g)

sincep 0 = a\ + a2, E(l, A)(g) = E2,i(Et+i, s + \\Y), and Eo)(suS2,~\Y)

= E0)(t,^^-,~\Y)

=

EO)(t,^^,0\Y).

Remark. It is the function £(1, A - po)(g) = E°(Y; s, t) that is used by Venkov ([Ve]). This explains the above change of variables s i-> s + \, 11-> t+\. 9.3

Relation with the Rank Three Zeta

Define the completed Eisenstein series associated to E°(Y; s, t) by E°(Y; s,t) := E(l,A-p0)(g)

:= %{2t)ZOs - t)£(3s + t-l)-

E°(Y;s,t).

Then we have the following very neat Functional Equations. With the same notations as above, (i)E°(Y;s,l-t) = E0(Y;s,t); (ii) E°(y; i ( l - s + t), \{-\ + 3s + t)) = E"°(Y; S, t); (iii) E°(Y; \{2-sf), i(2 - 3s + t)) = E^(Y; s, t); (iv) £°(y; i ( l - s + t), \0 -3s- t)) = E°(Y; s, t); (v) £°(y; i(2 - s - t), i(3s - 0) = &(Y; s, t). Proof. Replacing E by E, this is a precise realization of Langlands' result on functional equations for the Eisenstein series involved: The totality should be 6, the cardinal number of the Weyl group. Here we omit the trivial one corresponding to the identity. For details, please also refer the discussion in the next section. •

355

9 A JRank Three Zeta and Its Zeros Moreover, by 3.3, 5.3 and 9.1-2, we have, up to a constant factor, fr,3(*):=

=J

f

£Q3(A,j)rf/i(A)

Res¥+>=1£(3)(r + \, ^

Res3i_,=i J

+ i , 017(A)) dfi(A(Y))

£(3)(r + - , - ^ - + - , 0|F(A)) d/i(A(y))

Wf,3

= Res3 J /A(G(F)\G(A)) A(G(F)\G(A»

This latest integration can be evaluated, since by Theorem 4.7, A(»=^ =1 >i?(M)(g)^(g)

f

JA(G(F)\G(A)) M(w, A)

• " Z n r i a A o < W ' ^ - p oZ-L^MMF>\MM). ,a > V

vveVv

e

This then leads to the calculation of next section. Remark. While it is the completed Eisenstein series that enters into the final discussion of our rank 3 zeta, but to follow notations used by most of current researchers, we will use E°(Y;s + \,t + ^) = E(l,A)(g), in other words, without clearing the denominators for the £'s appeared in the constant term along with the minimal parabolic subgroup P(l, 1,1), to do the calculation. 9.4 9.4.1

Elementary Calculations Action of Weyl Group on Root System

This is rather elementary, but for inexperienced reader, we give all the details. For any root a, we have the associated reflection , v v (2a, x) sa : x i-» x - (a , x) • a = x • a. (a, a) In particular, if x = ka, then ka i-» ka - 2— ! —- • a = ka - 2ka = -ka. (a, a)

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L. Weng

So, a H-» -a while for any x such x ± a, i.e., (x,or) = 0, we have x i-» x - 0 • a = x remains unchanged. It is well-known that the Weyl group W of SL3 is isomorphic to 53, consists of 6 elements and is generated by Ti = sai : a\ i-» -a\,ct2

•-» «2 ~ <«l,a2>»i = #2 - (-1) • <*\ = ori + <*2

and T2 = sa2 : a ] h-» a j -
: a\

i-> -ori

i-» - ( a i

+ QT2)> «2 •-» (*1 + ^2) ^

(<*l + &2) - «2 = #i>

Ti o T2 : Qfl H* a\ + QT2 l-> (-Q^l) + CC\ + QT2 = <*2» <*2 l- * —#2 •""* _(<*1 + Q^) and n

°(T2°TI)

: orj i-» - ( « i +QT2) l-> ^ l

—

(<*i +0^2) = -ar2,#2 >-* oc\ y-* —a\.

That is to say, we have i) the identity map Id : ori

H-»

a i , QT2 i-> (X2;

ii) the reflection with respect to W2 T\ : ai i-> -ai,ct2

t-* a\ +0:2;

iii) the reflection with respect to m\ T2 '• a\ H-> a\ + (X2,Ot2 *-* —OC2\

iv) the rotation with the center at origin and the angle |TT T2 o T\ : a\ i-» -(a\ + 0:2),0:2 >-» ori; v) the rotation with the center at origin and the angle \n 1"! ° T2 : a\ Y-> Qf2, QT2 •-» ~(»1 + «2);

357

9 A Rank Three Zeta and Its Zeros vi) the reflection with respect to the line L g . Ti o T2 o T\ - T2 o Ti o T2 : OC\ l-» -QT2. <*2 •"* - <*1 •

As such, then if A = s • (3^2) + ta\ = (s + t)a\ + Isai, then g1* = u'ys, while the action of other elements in the Weyl group is given by T\A =TI((S

+ i)a\ + 2sa2j

= (s + t)(-ai)

+ 2s(a\ + 02) = (s - f)oc\ + 2soi2

hence

gTxX = « - y ; T2A = T2\(s + i)a\ + 2sa2J = (s + t)(ai + ct2) + 2s(-a\)

= (s + t)ai + (t - s)a2,

hence T-, J

gT2A

rA-t ill

= US+ly

—!zl

3»+(

t-s

i « T = | ( 2 ) i ! ;

T2 o T\A = T2 ° T\[(S + t)a\ + 2sa2)

= (s + t)[-(a\

+ ai)\ + 2sai = (s - t)a\ -(s + t)a2,

hence gn
=

u-ry-^r;

T\ O T2A = Tl O T2\(.S + t)a\ + 2SCC2) = (s + i)ct2 + 2s[-{a\

+ QT2)] = -2sa\ + (-s + f)<*2,

hence 1-. r,r~ i

—0 c

-"+'

gn°r2A _ u-2sy-T-u-T

s l

-3s-t

~

_

-s+t

M-T_;y^-.

Tl o T2 O T\A = T\ o T2 o Tl((5 + 0«1 + 25»2) = (s + t)(-a2)

+ 2s(-a\)

= -2sa\ - (s + t)a2,

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L. Weng

a change from the previous one via s H-> s, 11-» -t, hence

In particular, with po := - V a = -{ai + a2 + (a\ + a2j) = ax + a2, we have by taking s = t = \ in the above discussion, gP° _ u\y\t g r 2 o Tl po

_

M

^-^

gTlP0 = u-kykt

g r,oT 2 p 0

_ u-ly0t

gT2P0

_

||ly0>

gnoncrtpo

_

M

-iy-if

while gwPo-po j s given by gP°-p° = u°y0,

9.4.2

gTiPo-P° = « y ,

gTzPo-Po _ a - i y i ,

Intertwining Operator

By the Gindikin-Karpelerich formua, the action of the intertwining operator M(w, A) on constant function 1 is given by

M(w,A) = [~J n

v

n£F((A,a

a>0,wa<0

irvN

'

) + iy '

'

So we need to calculate the corresponding product. First recall the following table a T\a T2a T2 o T\a T\ o r2a T\OT20

T\a

a\ a2 a\ + a2 oc\ + a2 a2 -an a\ + a2 -a2 a\ <X\ -(ax + a2) -a2 a2 -(ax + ai) -a\ -a2 -a\ -(ax + a2)

9 A Rank Three Zeta and Its Zeros

359

from which we see that the following cases must be taken into the product appeared in the above formula: or T\a T2QT T2°T\a T\ o T2CC

* -a\ * -(ari+a^) *

T\OT2°T\(X

* * -QT2 * - ( « 1 + Ct2)

-QT2

—<Xl

* * * -&1 —<X\ —(Ofl + CK2).

Consequently, M(ld,A) = l; M T

^) = 3 T T T ; — v — 7 T = -r~^r—7T = cF(t + ~ )J ; fr(Rari>+l) fr(2r+l) 2 fr«^,g2» _ tFQs-t) frOs-t+l) ^F«^,«2> 1) frOs-t+1) 2 F{{A,a2) + \) fr«,t,gl» ^F«^,ori+or2» M(j2 °T\,X) = £ F ( a , a i > + 1) f j r « ^ , a i + a2) + 1) fr(2Q fr(3j + f) ( i'

fr(2f+l)'fr(3* .

M{T\

M (

T l

O

1,

+ r+l)

,3s + f

1

&•( + 1) frOs-t) £FQs + 0 ZFOS-t+l)' frQs + t + 1) ,3s-t 1 3s + f 1 = C H — 2 ~ + j ) • C/r(—2~~ + 2);

T2, A) =

o , o

T l

, ^

^ < ^ » fr(W,«i>+l)

fr(2f)

• fr«*°»» fr«A,a2> + D &K(/l, <*1 + <*2» fr(3*

^F«/l,Qri+a2> + l ) - 0 frOs + t)

fF(2t + 1) ' £FOS -t+l)' fr(3s + t+l) . 1. ,3s-t 1 ,3s + t 1

2

360

L. Weng

since (A, ori) = 2(s + t)-2s = It, (A,a2) = -(s + t) + 4s = 3s-t, a2) = 3s + t. Here cF{t) := %fff used in [Ve].

{A,ai +

Remark. If we make the variable changes tt-*t+^,st-*s+^ Venkov to our current situation, we see that 3s -t 3s-t+\-\ h-» - — 2 2 3s + t - \ 3s + t - \ + \ + \ H»

=

3s -t 2 3S+

=

2 2 So our result coincides with that of Venkov [Ve]. 9.4.3

Calculation of YlaeA0(wA-

from

1 2 1

1- -

t

+ -.

2

2

po,av)

We also need to calculate rioreAo^^ ~ Po>aV)- That is, (wA - po, a*) • (wA - po, <*2 > or the same (wA - po, ai) • (wA - po, a2). This is given as follows: (A-p0,ai)-(A-pQ,a2) = ((s + t - l)ori + (2s- l)a2»^i) •((s + t-\)ax+

(2s -

\)a2,a2)

= (2(s + t - 1) - (2s - 1)) • ( - (s + t - 1) + 2(2s - 1)) =

(2t-\)(3s-t-\);

(T\A-p0,a\)(r\A-po,a2) = ((s-tl)ai + (2s - \)a2,a\) •((s-t-

\)a\ + (2s - \)a2, a2)

= (2(5 - t - 1) - (2s - 1)) • ( - (s -1 - 1) + 2(25 - 1)) = (-2t - l)(3s +1 - 1);

361

9 A Rank Three Zeta and Its Zeros - po, a\) • foA - po, or2) = <(i + f- \)a\ +(t- s- l)a2,a\) • ((s + t- l)ori +(t- s(T2A ,

l)a2,02)

= (2(5 + t - 1) - (t - s - 1)) • ( - (s + t - 1) + 2(t - 5 - 1)) = (3s + f - l ) ( - 3 s + f - l ) ;



= ( ( 5 - f- l)ai - (s + t + l)a2,ai) • ((s-t= (2(s -t-l) =

l)ai + (s + t + 1)^2,a2>

+ (s + t-l))-(-(s-t-l)-2(s

+ t+l))

(3s-t-l)(-3s-t-l);

<Ti o T2A - po, <*i > • (TI o T2A -po,a2) = ((-2s- l)ori + (-s + t- \)a2,ax) • ((-2s - l)ai +(-s + t- l)a2, <*2> = (2(-2s -l)-(-s

+ t- 1)) • ((2 5 + 1) + 2(-s + t - 1))

= (-3s-f-l)(2f-l);

(Ti OT2OTi\-po,a\)-(T\

OT2OTi/l-po,Q:2)

= <(-25 - l)ai - (s + t + l)or2, a\) • ((-2s - l)ai -(s + t+ l)a2, a2) = (2(-2s - 1) + (s + t + 1)) • ((2s + 1) - 2(s + t + 1))

= (-3s + f - l ) ( - 2 f - l ) .

362 9.5

L. Weng Rank Three Zeta

By putting all the above calculation together, we have then obtain the following JA(G(Q)\G(A))

= 3- f

da-\ L 2

(A)11

(Q)\Aa0(A) JAAn0(Q)\A

( ^

(2f + 1)(3J + f - 1) M

s

2

iy 2

U'-2f -

l)(3s - t - \ )

f(2f+l)

2

g(3s - t)

(3s + t-\)(3s-t+l) 3s-f

j+r

1

,^

_„

v

« T - ly—:r-i

£(2f)

£(3s + 0

(3*-f-l)(3s + f+l)

£(2f+l)

£(3s + r + l )

-3J-<

1

-j+(

u~ 2y—-i (3s + t+l)(2t-l) u^-hy^-i. +

£(3s-f+l) 1

1

,_

g(3s-t) t-Qs-t+l) g(2f)

£(3s + t) £(3s + t+l) £(3s - Q

g(3 J + t) i

(3s - t + l)(2t + 1) "f(2f+l) ' f ( 3 s - r + l ) " ^(3s + r+ 1)J'

Thus in particular, by taking u - y = 1, we get A ( " =1 ' y=1) £(M)(g)4u(s)

Res 3 w=i f JA(G(F)\G(A))

= ReS3 =,

- [(2/-l)(3,-f-l)]

r es 3 ,- ( _i[ ( 3 5 + ^_ _R

1 1)(3

^_f+ly

€&s-t) i ^3s_t+ly\

r 1 f(2f) g(3j + r) -, es 3 j -,_i[ ( 3 5 _ f _ 1 ) ( 3 ^ + r + JJ • ^ + j ^ • ^ + r + i)J r 1 fflj-Q _£(3s_+0_ 1 eS3j 1 "'- K35 + f + l ) ( 2 f - l ) ' ((3s-t+l) ' Z(3s + t+l)i

+ Res . 3j - f =i[ (3s-t+l)(2t+l) #20 f(2r+l)

f(3s-r) £(3s-?+l)

£(3* + r) f(3s + f + l ) J

363

9 A Rank Three Zeta and Its Zeros

Moreover, note that then t = 3s - 1 and 2t - 1 = 6s - 3, 3s + t - 1 = 6s - 2, 3s - t + 1 = 2, 3s + t + 1 = 65, 2f + 1 = 6s - 1 (surely we list these trivial relations for the reason to get the next lines straightforward) we conclude that A^=l^E(l,A)(g)dM(g)

Res3,-,=i f JA(G(F)\G(A»

__L

1 _ I J _ _ j _ £(6s - 2) £(6s - 1)

~6*-3

+

65 - 2 ' 2 ' ((2) 6s ' £(6s - 1) ' 1 1 £(6s - 1 ) 65(65 - 3) ' #2) ' £(65) 1 £(65 - 2) 1 £(65 - 1) 2-(6J-1)'£(6S-1)"#2)"

1 1 1 1 "(65-3) 2(6^-2) £(2) 6s 1 1 £(65 - 1 ) 65(65 - 3) ' £(2) ' £(65) 1 1 £(65 - 2) + 2 - ( 6 5 - l ) ' # 2 ) ' £(65) '

£(65)

£(6*)

£(65-2) £(6s)

This, by Remark 9.2.2, then completes the proof of the following Theorem 4. The rank 3 zeta function for the number field F = Q is given by ZF,3{S)

=ZF(2) • —!— • fr(35) - £(2) • ^ • fr(3* - 2) 35-3 35 + \ • T^-i • &K35 - 1) - \ • 1 • £F(3s - 1) 3 35-3 3 35 +

\ • T ^ T - ^ ( 3 5 - 2 ) - i • - J — -^(35). 2 35- 1 2 35- 2

Moreover, we also have the following recent result of M. Suzuki. Theorem 5 (Suzuki[Su]). All zeros 0/^,3(5) lie on the central line Re 5 = I 2-

364

L. Weng

We end this paper with the following generalization of the Riemann Hypothesis: The RH. All zeros of zetas, rank one or higher, lie on the central line Res=±.

Acknowledgments: I would like to thank Arthur, Deninger, Elstrodt, Fesenko, Freitag, Hida, Henry Kim, Lagarias, I. Nakamura, Takayuki Oda, Takhtajan, Ueno, Zagier and Zink for their interests; Special thanks also due to M. Suzuki for sharing with me his discovery that rank two and rank three zetas for the field of rationals satisfy the Riemann Hypothesis. I would like to dedicate this paper to A. Fujiki for his 60 years birthday. This project is partially supported by JSPS.

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L. Weng

370

LinWENG Institute for Fundamental Research, The L- Academy and Graduate School of Mathematics, Kyushu University, Fukuoka 812-8581, Japan Email: [email protected]

THE CONFERENCE ON

Z-FUNCTIONS This invaluable volume collects papers written by many of the world's top experts on /.-functions. It not only covers a wide range of topics from algebraic and analytic number theories, automorphic forms, to geometry and mathematical physics, but also treats the theory as a whole.

The contributions reflect the latest, most advanced and most important aspects of L-functions. In particular, it contains Hida's lecture notes at the conference and at the Eigenvariety semester in Harvard University and Weng's detailed account of his works on high rank zeta functions and non-abelian /.-functions.

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