Arithmetic Geometry And Number Theory

Series on Number Theoiy and Its Applications V o l . 1

ARITHMETIC GEOMETRY AND NUMBER THEORY

Editors Lin Weng & Iku Nakamura World Scientific

ARITHMETIC GEOMETRY AND NUMBER THEORY

Series on Number Theory and Its Applications

ISSN

1793-3161

Series Editor: Shigeru Kanemitsu (Kinki University, Japan) Editorial Board Members: V. N. Chubarikov (Moscow State University, Russian Federation) Christopher Deninger (Universitat Munster, Germany) Chaohua Jia (Chinese Academy of Sciences, PR China) H. Niederreiter (National University of Singapore, Singapore) M. Waldschmidt (Universite Pierre et Marie Curie, France) Advisory Board: K. Ramachandra (Tata Institute of Fundamental Research, India (retired)) A. Schinzel (Polish Academy of Sciences, Poland)

Vol. 1 Arithmetic Geometry and Number Theory edited by Lin Weng & Iku Nakamura

Series on Number Theory and Its Applications Vol. 1

ARITHMETIC GEOMETRY AND NUMBER THEORY

Editors

Lin Weng

Kyushu University, Japan

Iku Nakamura Hokkaido University, Japan

\[p World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING

• SHANGHAI

• HONG KONG • T A I P E I • C H E N N A I

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Foreword This series aims to bring together the very many applications of number theory in a fusion of diverse disciplines such as chemistry, physics and others. It aims to provide a comprehensive and thorough coverage of the whole spectrum of (state-of-the-art knowledge of) number theory and related fields, in the form of textbooks and review volumes. Presented as an organic whole, rather than as an assembly of disjointed subjects, the volumes in the series will include ample examples to illustrate the applications of number theory. The target audience will range from the undergraduate student who hopes to master number theory so as to apply it to his or her own research, to the professional scientist who wishes to keep abreast of the latest in the applications of number theory, to the curious academic who wants to know more about this fusion of old disciplines. Shigeru Kanemitsu Series Editor

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Preface Mathematics is a part of our culture. As such, the works presented here serve the purposes of developing branches involved, popularizing existing theories, and guiding our future explorations. Accordingly, the collection of this volume may be roughly divided into three categories. More precisely, first, Jiang's paper deals with local gamma factors t h a t appeared in the theory of automorphic representations; Obitsu-To-Weng's paper investigates the intrinsic relations between Weil-Petersson and Takhtajan-Zograf metrics on moduli spaces of punctured Riemann surfaces using Deligne pairings and an arithmetic Riemann-Roch isometry; Werner's paper explains her recent works with Deninger on vector bundles on curves over Cp; Yoshida's paper exposes his beautiful theory on CM periods; and Yu's paper studies the transcendence of special values for zetas over finite fields. All these well-prepared articles then bring us to the uppermost frontiers of the current researches in Arithmetic Geometry and Number Theory. Secondly, the lecture notes of Weng explains basic ideas and methods behind the fundamental yet famously difficult work of Langlands on the Eisenstein series and spectral decompositions. The reader will find these notes invaluable in understanding the original theory. Finally, Weng's paper of Geometric Arithmetic outlines a Program for understanding global arithmetic using algebraic a n d / o r analytic methods based on geometric considerations - the topics touched here are a continuation of Weil's approach on non-abelian Class Field Theory using stability and Tannakian category theory; new yet genuine non-abelian zetas and Ls which are closely related with the so-called Arthur's periods; and an intersection approach to the Riemann Hypothesis. While various important topics are selected, all papers share common themes such as the Eisenstein series, stability and zeta functions. Jiang's paper was presented at the Conference on L-Functions (February 18-23, 2006, Fukuoka). Partial contents of the papers of

vn

Preface

viii

Obitsu-To-Weng, Werner, Yoshida and Yu were delivered by W.-K. To, A. Werner, H. Yoshida and J. Yu, respectively, in the (series of) lectures at our Karatsu symposium on 'Arithmetic Geometry and Number Theory', held from March 21 to March 25, 2005, immediately after the huge Fukuoka earthquake of scale M7.0 (on March 20). The notes about Langlands' work is based on six lectures of Weng at the Mathematics Department, University of Toronto, between October and November, 2005. Finally, the Program paper, of which the first version was circulated around the turn of the millennium, is revised significantly for this publication and is indeed the driving force for the whole project 1 . The Editors

lr

rhis project is partially supported by JSPS.

Contents Foreword

v

Preface

vii

On Local 7-Factors D. H. Jiang

1

Deligne Pairings over M o d u l i Spaces of P u n c t u r e d R i e m a n n Surfaces K. Obitsu, W.-K. To and L. Weng Vector B u n d l e s on Curves over Cp A. Werner Absolute C M - p e r i o d s — C o m p l e x a n d p-Adic H. Yoshida

47 . . . .

Special Zeta Values in Positive Characteristic . . . . J. Yu A u t o m o r p h i c F o r m s , Eisenstein Series and Spectral Decompositions L. Weng Geometric Arithmetic: A Program L. Weng

IX

29

65

103

123 211

On Local 7-Factors Dihua JIANG

Contents 1 Introduction

1

2 Basic Properties of Local 7-Factors 2.1 Multiplicativity 2.2 Stability 2.3 Remarks

5 6 6 7

3 Local Converse Theorems 3.1 The case of GL„(F) 3.2 A conjectural LOT 3.3 The case of S02n+i(F)

7 7 9 12

4 Poles of Local 7-Factors 4.1 The case of G = S0 2 „+i 4.2 Other classical groups

15 17 20

1

Introduction

Let G be a reductive algebraic group defined over a p-adic local field F. We assume that F is a finite extension of Qp for simplicity. Let WF be the local Weil group of F and LG be the Langlands dual group of G, which is a semi-product of the complex dual group G v and the absolute Galois group r > = Gal(F/F). Consider continuous homomorphisms (j) from the Weil-Deligne group Wf x SL2(C) to the Langlands dual group L G, which is admissible in the sense of [B79]. The Gv-conjugacy class of such a homomorphism

2

D. Jiang

is denoted by $(G/F). Let U(G/F) be the set of equivalence classes of irreducible admissible complex representations of G(F). The local Langlands conjecture for G over F asserts that for each local Langlands parameter <j> € $(G/F), there should be a finite subset 11(0), which is called the local L-packet attached to 0 such that the set {11(0) | 0 £ $(G/F)} is a partition of U(G/F), among other required properties ([B79]). The map 0 — i > 11(0) is called the local Langlands correspondence or the local Langlands reciprocity law for G over F. The main problem is of course how to construct the local Langlands reciprocity map 0 — i > 11(0). Prom the classification-theoretic point of view, the local Langlands conjecture provides a classification for irreducible admissible representations up to L-packet. It is interesting to characterize the local L-packets in general. The most wellknown approach to characterize local L-packets is in terms of stability of distribution characters following from the idea of Arthur trace formula approach to the discrete spectrum of automorphic forms. We refer to [MW03], [KV05], [DR05], [R05] and [V93] for further discussions. In this note, we discuss the roles of local factors attached to irreducible admissible representations of G(F). They yield information about the classification theory and the functorial structures of irreducible admissible representations of G(F). First, we recall the local Langlands conjecture for GL n over F, which is proved by Harris-Taylor [HT01] and by Henniart [H00]. Theorem 1.1 ([HT01], [H00], [H93]). There is a unique collection of bisections rec F : Il(GLn/F)

^

$(GLn/F)

for every n > 1 such that 1. for 7r € n ( G L i / F ) , recp(7r) = 7r o Art^ 1 , where Art^ is the local Artin reciprocity map from Fx to Wf; 2. for TTi € U(GLni/F)

and TT2 €

U(GLn2/F),

L(s, 7Ti x 7r2) = L(s, rec,F(7ri) ® r e c / r ^ ) )

On Local y-Factors

3

and e(s,7Ti x TT2,ip) = e(s,recF(wi)

(girecF^),^)

w/iere ^ is a given nontrivial character of F; 3. for IT e U(GLn/F)

and X € n ( G L i / F ) ,

recp(7r ( x o det)) = r e c p ^ ) <8> recp( x); ^. /or 7T € Tl(GLn/F)

with central character cu^ = x> det orecF(7r) = rec^Cx);

5. for re e U(GLn/F), contragredient.

Tecf{nv) = recir(7r)v, where V denotes the

We note that the existence of the local Langlands correspondence (the reciprocity map satisfies conditions (l)-(5) is proved in [HT01] and [H00]. The uniqueness of the such maps is proved in [H93]. We refer to [HT01] for historical remarks on the proof of the local Langlands conjecture for GL n (F). The local factors on the GL n (F) side is given [JPSS83] and the local factors on the WF X SL2(C) side is given in [T79]. One can define as in [JPSS83] the local 7-factors by 1.1)

7(S,TTI

x ir2,i>) = e S,TTI x ir2,i>) • -—-,

i-

f^.

L(S, 7Ti X 7T2J

On the WF X SL2(C) side, one defines the 7-factor in the same way [T79]. Note that for GLn(F), the local Z-packets always contains one member. This fact follows from [H93] and the Bernstein-Zelevinsky classification theory ([BZ77] and [Z80]). For general reductive groups local factors have been defined for many cases. When irreducible admissible representations -K of G(F) are generic, i.e. have nonzero Whittaker models, the Shahidi's theory of local coefficients defines the local L-, e-, and 7-factors. It is expected that the local factors defined by Shahidi should be essentially the same as the ones defined by the Rankin-Selberg method if they are available, although it has to be verified case by case. It

4

D. Jiang

should be mentioned t h a t for nongeneric representations, there are cases where the local factors can be defined by the Rankin-Selberg method ([GPSR97] and [LR05]), and also t h a t the work [FG99] has the potential to define the local factors for nongeneric representations, which can be viewed as the natural extension of Shahidi's work. Of course, one may define the local factors by means of the conjectured local Langlands conjecture for G over F, and this definition should be consistent with all other definitions. We recall the local Langlands functoriality principle. Let G and H be reductive algebraic groups defined over F. For an admissible homomorphism Lp ([B79]) from the Langlands dual group LH to the Langlands dual group LG, there should be a functorial transfer p from U{H/F) to n ( G / F ) , which takes L-packets of H(F) to L-packets of G(F), and satisfies the following conditions. L 1. For any local Langlands parameter
2. For any finite-dimensional complex representation r of LG and a € 11(0//), one has L(s, p(cr),r) = L(s, a, r o

L

p),

and e(s, p(a),r, ift) = e(s, a, r o Lp, %/>). It follows t h a t 7 ( s , p(a), r, tp) — 7(5, a, r o Lp, ip). From the formulation of the local Langlands conjecture for GLn(F) (Theorem 1.1), the functorial transfer should be characterized by the conditions similar to conditions ( l ) - ( 5 ) in Theorem 1.1, in particular, by L(s,a

x T ) = L(s,p(a)

x r)

and e ( s , a XT,I/J)

= e(s,p(a)

X T,if>),

On Local ^-Factors

5

for all irreducible supercuspidal representations r of GL n (F) for all integers n > 1. For the local 7-factors, one expects (1.2)

7 ( s , a x r , ^ ) = 7 ( s ,p(cr)

xr,ip).

Of course, if one assumes the validity of the local Langlands functoriality from reductive groups to the general linear group, then one may use (1.2) to define the twisted local 7-factors in general. We refer [K95], [K00], [BKOO] and [GK99] for some very interesting discussions in this aspect. For a given ir € H(G/F), we have two collections of local 7factors: one is (1.3) {-y(s, re x r, ip) I for all r G II(GL n /F), and for all n = 1,2,... }, the twisted local 7-factors of ir, and the other is (1.4)

{7(s,7r,r,V) I for all r},

the local 7-factors attached to all finite-dimensional complex representations r of LG. Although the exact definition of these collections of local 7-factors is still conjectural in general, it is clear that they are invariants attached to irreducible admissible representations n of G(F) up to equivalence. The basic questions are the following. 1. How do the collections of local 7-factors classify the irreducible admissible representations? (the Local Converse Theorem) 2. How do the explicit analytic properties of the local 7-factors determine the functorial structures of the irreducible admissible representations? (a local version of Langlands problem) We first recall some basic properties of local 7-factors, and then discuss these two basic problems in details, including some typical known examples in the following sections.

2

Basic Properties of Local 7-Factors

We recall briefly some basic properties of the local 7-factors. Among them are mainly the multiplicativity and stability of the local 7factors.

6

2.1

D. Jiang

Multiplicativity

For the twisted local 7-factors, one expects the multiplicativity holds. More precisely, it can be stated as follows. For an irreducible admissible representation 7r of G(F), there is a supercuspidal datum (M(F),cr) where P = MN is a parabolic subgroup of G defined over F and a is an irreducible supercuspidal representation of M(F) such that 7r is isomorphic to an irreducible subrepresentation of the G( F\

induced representation I n d p ^ {a). Then one expects (2.1)

7(7r x r, s, ip) = 7(0- x r, s, tp)

for all irreducible supercuspidal representations r of GLj (F) with I — 1,2,.... Further, if M = GL r x H, then one may write a = r r ® a', and expect (2.2)

7(0- x r, s, ip) — 7(r r x r, s, •*/>) • 7(0-' x r, s, ip).

Properties (2.1) and (2.2) are called the Multiplicativity of the local 7-factors. F. Shahidi proved in [Sh90b] the multiplicativity for irreducible generic representations n of all F-quasisplit reductive algebraic groups G(F) by using his theory on the local coefficients. For G = GL(n), it is proved by Soudry by the Rankin-Selberg method ([S00]). One may expect that the work ([GPSR97] and [FG99]) has implication in this aspect for irreducible admissible representations, which may not be generic.

2.2

Stability

Another significant property of twisted local 7-factors is the Stability, which can be stated as follows. For irreducible admissible representations 7Ti and 7T2 of G(F), there exists a highly ramified character X of Fx such that 7(71-1 x x, s, V>) = 7(^2 x x, s, ip). It was proved by Jacquet-Shalika ([JS85]) for the group GL m x GL n . For irreducible generic representations of classical groups, it was

On Local ^-Factors

7

proved in [CPS98] and [CKPSS04]. For irreducible generic representations of general F-quasisplit groups, the approach is taken in [CPSS05]. For F-split classical groups of either symplectic or orthogonal type, the stability of local 7-factors has be proved for general irreducible admissible representations via the doubling method ([RS05]). 2.3

Remarks

It is important to mention that E. Lapid and S. Rallis ([LR05]) determines the sign of the local e-factors via the doubling method. They introduce the ten properties, called the Ten Commandments of the local 7-factors, which determines the local 7-factors uniquely. We would also like to mention the explicit calculations of the local 7-factors for irreducible supercuspidal representations via the local Rankin-Selberg method (see [jKOO] for example).

3

Local Converse Theorems

The local converse theorem is to find the smallest subcollection of twisted local 7-factors 7(s, n x T,tp) which classifies the irreducible admissible representation TT up to equivalence. However, this is usually not the case in general. From the local Langlands conjecture, one may expect a certain subcollection of local 7-factors classifies the irreducible representation TX up to L-packet. On the other hand, if the irreducible admissible representations under consideration have additional structures, then one may still expect that a certain subcollection of local 7-factors classifies the irreducible representation n up to equivalence.

3.1

T h e case of GL„(F)

Let 7r be an irreducible admissible representation of GLn(F). Then there is a partition n — [ T ^ i rij (rij > 0) and an irreducible supercuspidal representation 71 ® • • • ® Tr

8

D. Jiang

of GL n i (F) x • • • x GLnr(F) such that the representation -K can be realized as a subrepresentation of the (normalized) induced representation (3.1)

I(n, ...,rr)

= I n d £ M F ) l ( F ) ( r i ® • • • ® rr).

By the multiplicativity of the local 7-factors ([Sh90b] and [S00]), we have r

(3.2)

7(s,

7T x r, i/>) = Yl 7(s, Tj x r, ^)

for all irreducible admissible representations r of GL;(F) for all/ > 1. It reduces the problem for the case of general irreducible admissible representations to the case when the irreducible admissible representations are supercuspidal. It should be remarked that even if the irreducible supercuspidal representations can be determined by the twisted local 7-factors up to equivalence, it is the best one can expect that in general the twisted local 7-factors determines the irreducible admissible representations up to the equivalence of supercuspidal data. We first consider the case of irreducible supercuspidal representations of GLn(F). The first local converse theorem (LCT) for GL n (F) is proved by G. Henniart in [H93], which can be stated as follows. Theorem 3.1 (LCT(n,n-l) [H93]). Letn\,%2 be irreducible supercuspidal representations of GLn(F) with the same central character. If the twisted local 7-factors are the same, i.e. 7 ( S , 7Tl X T, ifi) = 7 ( S , 7T2 X T, lj))

for all irreducible supercuspidal representations r ofGLi(F) 1,2,..., n — 1, then 7Ti and KI are equivalent.

with I =

It follows that an irreducible supercuspidal representation ir can be determined up to equivalence by the subcollection of twisted local 7-factors {7(s, 7r x T,tp) I for all r as in the theorem}.

On Local 7-Facfcors

9

The remaining problem is to reduce the 'size' of the subcollection of twisted local 7-factors, that is, to prove LCT(n,r) for r < n — 1. In this direction, we have Theorem 3.2 (LCT(n,n-2) [C96], [CPS99]). Letiri,ir2 be irreducible supercuspidal representations ofGLn(F). If the twisted local 7-factors are the same, i.e. 7 ( S , 7Ti XT,lp)

= 7 ( 5 , 7T2 X T, V>)

for all irreducible supercuspidal representations r ofGLi(F) 1,2,..., n — 2, then TT\ and -K2 are equivalent.

with I =

This theorem is proved in [C96] by a purely local argument, and prove in [CPS99] as a consequence of the global converse theorem for automorphic forms. It is well known to expect Conjecture 3.3 (Jacquet). Let 7TI,7T2 be irreducible supercuspidal representations ofGLn(F). If the twisted local 7-factors are the same, i.e. 7 ( 5 , 7Ti X T, V>) = 7( S > *"2 X T, 1p)

for all irreducible supercuspidal representations T ofGLi(F) 1,2,..., [§], then -K\ and 7T2 are equivalent.

with I =

There are not strong evidence to support this conjecture, which is known for n = 2, 3,4 for example. On the other hand, one may expect an even stronger version of this conjecture from the conjectural global converse theorem in [CPS94]. In order to prove a better local converse theorem, it is expected to use the explicit construction of irreducible supercuspidal representations of GL„(F) and reduce to the case over finite fields. On the other hand, it is also important to consider the local converse theorem for general reductive groups.

3.2

A conjectural L C T

For a general reductive algebraic group G defined over F, the collection of twisted local 7-factors 7(3, IT X r, ip) is expected to determine the irreducible supercuspidal representation IT up to the local

D. Jiang

10

L-packet. Note that all irreducible supercuspidal representations of GL n (F) are generic, i.e. have nonzero Whittaker models. It is natural to consider the local converse theorem for irreducible generic supercuspidal representations of G(F) in general. We recall the notion of Whittaker models for F-quasisplit reductive algebraic group G(F). Fix an .F-Borel subgroup B = TU. Let <E>(G, T) be the root system with the positive roots $ + determined by U and A be the set of the simple roots. Choose an F-split {Xa}, where Xa is a basis vector in the one-dimensional F-root space of a. Then we have (3.3)

U/[U,U]^®aeAF-Xa.

Let ip be a character of U(F). Then tj) factorizes through the quotient U(F)/[U(F),U(F)}, which is isomorphic as abelian groups to ©aeA-F • Xa. A character tf> of U(F) is called generic if ip is nontrivial at each of the simple root a, via the isomorphism above. By the Pontriagin duality, such characters of U(F) is parametrized by r-tuples a = (a1,...,ar) e (Fx)r, where r is the F-rank of G, i.e. the number of simple roots in A. An irreducible admissible representation (n, V^) of G(F) is called generic or ^-generic if the following space H o m ^ V ^ ) * HomG(F)(K,Ind^(^)) is nonzero. For any nonzero functional 1$ 6 Homj/(.F)(V^,^), under the above isomorphism, there is G(F)-equivariant homomorphism veVn^Wf(g)

= ^(7r(g)(v)).

The subspace {Wt{g) \ v G K-} is called the ^-Whittaker model associated to 7r. By the uniqueness of local Whittaker models ([Shl74]), the functional £^ is unique up to scalar multiple. For t G T(F), we define toip(u) = ^>(t -1 iti). If ip is generic, then t o ip is generic for all t G T(F). Also it is clear that

I n d ^ W - IndgJUt o
On Local 'y-Factors

11

for all t G T(F). It follows that if an irreducible admissible representation 7r of G(F) is ^-generic, then IT is also t o ^-generic for all t G T(F). In other words, the genericity of irreducible admissible representations of G(F) depends on the T(F)-orbit of the generic characters. It is an easy exercise to show ([K02]) that Proposition 3.4. The set of T(F)-orbits the generic characters of U(F) is in one-to-one correspondence with LT^IV, ZQ), where Yp is the absolute Galois group of F and ZQ is the center of G. For an irreducible admissible representation TT of G(F), we define (3.4)

•^"(7r) = {'0 1 n is ip-generic},

and call it the set of generic characters attached to ir. It is clear that the set F(ir) is T(F)-stable, and the T(F)-orbits 7 " ( T T ) / T ( F ) determines the genericity of IT. The following conjecture of Shahidi ([Sh90a]) is fundamental to the harmonic analysis of p-adic groups and representations. Conjecture 3.5 (Shahidi [Sh90a]). Every tempered local L-packet contains a generic member. In general some nontempered local L-packets may also contains generic members ([JS04]). We call a local L-packet with generic members a generic local L-packet. Shahidi's conjecture is known to be true for the case of GL„(F) ([BZ77], [Z80]), for the case of SL n (F) ([LS86]), for the case of S02n+i(F) ([JS04], [M98]), and for the case of U2,i(-F1) ([GRS97]). More recently, it is proved to be true for cuspidal local L-packets of F-quasisplit groups ([DR05]). For reductive algebraic groups at archemidean local fields, it follows from [L89]. Some relevant discussions can be found in [MT02], and global applications can be found in [Ar89] and [KS99]. The author proposes the following refinement of the Shahidi conjecture. Conjecture 3.6 (Refinement of Shahidi's Conjecture). In a generic local L-packet 11(0), for any generic members 7ri,7T2 € 11(0), the sets JF(7Ti) and -T 7 ^) are disjoint and the union of T(F)-orbits

12

D. Jiang

of generic characters over the subset Tl9((f)) ofTL(cf>) consisting of all generic members in Tl(4>), i.e.

is in one-to-one correspondence with

H1{TF,ZC)-

It is not difficult to check that Conjecture 3.6 holds if one knows the complete structure of local L-packets. This should be the case for the cases where Shahidi's conjecture holds. However, Conjecture 3.6 may be verified before one knows completely the structure of local Z-packets. We will explain in the next section that Conjecture 3.6 holds for G = SC>2n+i by means of the local converse theorem. Based on Conjecture 3.6, we formulate the following general version of the local converse theorem. Conjecture 3.7 (LCT). For any irreducible admissible generic representations 7ri and 7T2 of G(F), if the following two conditions hold 1. the intersection of J-(ir\) and J-{TT2J is not empty, and 2. the twisted local')-factors are equal, i.e. 7 ( S , 7Tl X T, 1p) = 7 ( s , 7T2 X T, if;)

for all irreducible supercuspidal representations r of GL;(F) with I = 1,2,..., [£], where r is F-rank of G, then 7i"i = 7T2.

We remark that any theorem of this nature should be called a local converse theorem. The number of twists up to the half of the F-rank of G is an imitation of Jacquet's conjecture for GL n . We have no strong evidence about this claim. In the next section we discuss the author's joint work with David Soudry for S02 n +i3.3

T h e case of

S02n+i(F)

We review briefly here the joint work with Soudry ([JS03]) on LCT for S0 2 n+i-

On Local 'j-Factors

13

Theorem 3.8 (LCT for S0 2 n +i [JS03]). Let TT and ir' be irreducible admissible generic representations of S02n+i(k). If the twisted local gamma factors ^{-n x r, s, ip) and y(ir' x r, s, ip) are the same, i.e. J(TT X T, S, Ip) = 7(71"' X T, S, Ip)

for all irreducible supercuspidal representations r o/GL;(fc) with I = 1,2,..., 2n — 1, #ien i/ie representations TT and ir1 are equivalent. We remark that this theorem was proved in [JS03] by using Theorem 2.1, from the work [H93]. It is clear that we can improve this theorem to LCT(n,n-2) by using the work of [C96] or [CPS99]. We did not do this in [JS03] because it was good enough for applications in that paper. We remark that the LCT for generic representations of U(2,1) and for GSp(4) was established by E. M. Baruch in [B95] and [B97]. In order to point out the essence of the local converse theorem for general reductive groups, we would like to recall some important applications of the local converse theorem for S02n+i to the theory of automorphic forms. Let A; be a number field and A = A& be the ring of adeles of k, First we obtain the injectivity of the weak Langlands functorial lifting established in [CKPSS01] (which is proved for example in [JS03] and [JS04] to strong Langlands functoriality). Theorem 3.9 (Theorem 5.2 [JS03]). Let IP m (S0 2 r i +i/A) be the set of all equivalence classes of irreducible generic cuspidal automorphic representations of S02n+i(A) and II a (GL2n/A) be the set of all equivalence classes of irreducible automorphic representations of GL2n (A). Then the Langlands functorial lifting from II 9 c a (S02n+i/A) to II a (GL2n/A) is an injective map. The second global application is to determine the generic cuspidal data for an irreducible cuspidal automorphic representations. It is a well-known theorem that every irreducible cuspidal automorphic representation of GL„(A) is generic (i.e. having nonzero WhittakerFourier coefficients). This follows from the Whittaker-Fourier expansion of cuspidal automorphic forms of GL n (A). In general, we

14

D. Jiang

consider a cuspidal datum (P, a), where P is a parabolic subgroup of GL n and a is an irreducible cuspidal automorphic representation of M(A) with P = MN being the Levi decomposition. Jacquet and Shalika proved the following theorem. Theorem 3.10 (Theorem 4.4 [JS81]). Let (P;a) and (Q;T) be two pairs of cuspidal data o/GL(n). / / the two induced representations I n d p ^ | (a) and Ind^/^ (r) share the same irreducible unramified local constituents at almost all places, then the two pairs of cuspidal data are associate. By the Langlands functorial lifting from SO271+1 to GLi2n and the local converse theorem for S02n+i> we prove in [JS05] an analogue of Jacquet-Shalika's Theorem for SO271+1 with generic cuspidal data. For the trivial parabolic subgroup P = SO27H-1, this was proved in [JS03] (and also in [GRS01]). Theorem 3.11 (Theorem 3.2 [JS05]). Let {P;a) and (Q;r) be two pairs of generic cuspidal data of S02n+i(A). / / the two induced representations Ind p( ^? (a) and Ind Q /^? (r) share the same irreducible unramified local constituent at almost all places, then (P; a) and (Q; r) are associate. It has the following consequences which are important to the understanding of structure of the discrete spectrum of S02n+i(A). Theorem 3.12 (Corollary 3.3 [JS05]). With notations as above, we have (1) Irreducible generic cuspidal automorphic representations TV of the group SO"2n+i(A) cannot be a CAP with respect to a generic, proper, cuspidal datum {P,a), i.eir cannot be nearly equivalent to any irreducible constituent of I n d p ^ ? + (c). (2) If two pairs of generic cuspidal data (P; a) and (Q; T), are nearly associate, i.e. their local components are associate at almost all local places, then they are globally associate. (3) The generic cuspidal datum (P;
On Local ^-Factors

15

We refer to [JS05] for more detailed discussions on their relation with the Arthur conjecture.

4

Poles of Local 7-Factors

In his recent paper ([L04]), R. Langlands gave detailed discussion on a conjecture relating the order of the pole at s = 1 of automorphic Lfunctions to his functoriality principle. This conjecture can be stated as follows. Let A; be a number field and A be the ring of adeles of k. For any reductive algebraic group G denned over k, LG denotes its Langlands dual group ([B79]). Let n be an irreducible cuspidal automorphic representation of G(A). Langlands denned automorphic Lfunctions L(s, IT, p) for all finite-dimensional complex representation p of LG ([B79]). It is a theorem of Langlands that the automorphic L-function L(s, ir, p) converges absolutely for the real part of s large. It is a basic conjecture of Langlands that every automorphic L-function L(s, TT, p) should have meromorphic continuation to the whole complex plane C and satisfy a functional equation relating s to 1 — s. This basic conjecture has in fact been verified in many cases through the spectral theory of automorphic forms. See [Bmp04] and [GS88] for some detailed account on this aspect. Problem 4.1 (Langlands [L04]). For a given irreducible cuspidal automorphic representation TT of G(A), there exists an algebraic subgroup Ji-K of LG such that for every finite-dimensional complex representation p ofLG, the order of the pole at s = 1, denoted by m-K{p), of the automorphic L-function L(s, n, p) is equal to the multiplicity, denoted by mjiir{p), of the trivial representation of 7in occurring in the representation p ofLG when restricted to the subgroup Ti^. That is, the following identity (4.1)

mw(p) = mnw(p)

holds for all finite-dimensional complex representations p of LG. In [L04], Langlands discussed in length some relations of this conjecture to many basic problems in arithmetic and number theory

16

D. Jiang

and suggested a trace formula approach to certain important cases. Some detailed discussions for a special case, along the main idea of [L04] has been carried out by A. Venkatesh in his thesis [V04]. For G = S02n+i, the relation between the order of the pole at s = 1 of the automorphic L-functions attached to irreducible unitary generic cuspidal automorphic representation IT and the fundamental representations of the complex dual group Sp 2n (C) and the endoscopy structure of IT has been discussed in detail in [J05], along the line of the Langlands Problem. The main point here is to develop the local theory for the Langlands problem and the local analogy of [J05]. For simplicity of discussions below, it is assumed that all reductive algebraic groups considered below are F-split, and the Langlands dual group LG is taken to be the complex dual group G V (C), without action of the absolute Galois group. Let 7r be an irreducible supercuspidal representation of G(F) and p be a finite-dimensional complex representation of the complex dual group G V (C). One may define the local 7-factor attached to (-zr, p, %p) (for a fixed nontrivial additive character of F) to be (4.2)

7(S,T,P,^)

:= e(s,*,p,rj,) x

L(1

~ *' ^ ' / ^

where e(s, IT, p, ip) is the local e-factor attached to (TT, p, tp). This definition is based on the assumption of the local Langlands conjecture for G(F). Problem 4.2 (Local Version of the Langlands Problem). Let G be a reductive algebraic group defined over F. For an irreducible unitary supercuspidal representation IT of G(F), there exists an algebraic subgroup H^ o/G v (C) such that if p is a finite-dimensional complex representation o/G v (C), then the multiplicity m-n^p) of the trivial representation of Ti^ occurring in the restriction of p to Hn is the order mn(p) of the pole of the local 7-factor 7(s, n, p, ip) at s = 1, i. e. the following identity mv{p) = mn„(p) holds for all finite-dimensional complex representations of GV(C).

On Local

j-Factors

17

We remark that the key point here is to define the group "H* for a given ir and to study the relation between the structure of Hx and the endoscopy structure of ir. We refer to [J05] for discussion of H^ in terms of the observable groups in the classical invariant theory. In the following we discuss the local analogy of [J05]. 4.1

T h e c a s e of G = S 0 2 n + i

Let G = S0 2 n +i be the F-split odd special orthogonal group. Then its complex dual group is Sp 2n (C). The fundamental representations of Sp 2n (C) are the finite-dimensional irreducible complex representations associated to the fundamental weights. They can be constructed by the following split exact sequence (4.3)

0 - • Vgn) - • A a (C 2n ) -+ A a ~ 2 (C 2n ) -» 0,

where A a (C 2 n ) denotes the a-th exterior power of C2™, the contraction map from A a (C 2n ) onto A a " 2 (C 2 ") is as defined in Page 236, [GW98], and its kernel is denoted by V^n). By Theorem 5.1.8 in [GW98], V^n) is the space of the irreducible representation pa of Sp 2n (C) with the a-th fundamental weight. Let t be the natural embedding of Sp 2 „(C) into GL 2n (C). Let A2 be the exterior square representation of GL 2n (C) on the vector space A 2 (C 2 n ), which has dimension 2n 2 — n. The composition A2 o i of A2 with t is a complex representation of Sp 2n (C). By (2.1) and by complete reducibility of representations of Sp 2n (C), we obtain (4.4)

A2 o i = P2 © l S p 2 n

where p2 is the second fundamental complex representation of the group Sp 2n (C), which is irreducible and has dimension 2n 2 — n — 1, and lsp 2n is the trivial representation of Sp 2n (C). Let r = T(TT) be the image of -K under the Langlands functorial transfer from S02n+i(-F1) to GL 2 n (F) for irreducible admissible generic representations, which was established in [JS03] and [JS04]. One may expert the following identities hold, (4.5)

L(s,TT,pa) =

L(s,T(*),*-a-2)

18

D. Jiang

and (4-6)

7(8, 7T, pa, Ip) =

7(s,r(7r),A a ,
For the second exterior power representation pi, the above identities have been verified by G. Henniart ([H03]). The following theorem relates the endoscopy structure of a to the order of pole at s = 1 of the second fundamental local 7-factors. Theorem 4.3. Let n be an irreducible generic supercuspidal representation of S02n+i{F) and pi be the second fundamental complex representation of Sp 2n (C). Then the second fundamental local 7factors 7(5, IT, P2, VO are meromorphic functions over C and have the following properties. 1. The second fundamental local ^-factor j(s, IT, P2, ip) has a pole of order r — 1 at s = 1 if and only if there exists a partition n = 53f=i rij with nj > 0 such that IT is a Langlands functorial lifting from an irreducible generic supercuspidal representation -K\ • • • TXr Of

SCWi^x-.-xSCWi^)2. The partition [ni • • • nr] is uniquely determined by the irreducible generic supercuspidal representation TT. More precisely, the set of positive integers {m,n2,...,nr} consists of all positive integers m such that there exists an irreducible supercuspidal representation r o/GL m (F) such that the tensor product local 7-factor 7(5, n x r,ip) has a pole at s = 1. 3. The set {7Ti, 7T2,..., 7rr} of irreducible generic supercuspidal representations of S02m+i(F) is completely determined by the irreducible generic supercuspidal representation IT, up to equivalence, namely, it is the set of irreducible generic supercuspidal

On Local ^-Factors

19

representations n' (up to equivalence) of SO2i+i(F) such that the tensor product local ^-factor 7 ( s , 7T X T(7T'),V>)

has a pole at s = 1, where T(TT') is the local Langlands transfer of IT' to GL2j (F) and is irreducible and supercuspidal. 4- We have m^p-i) = w # v

(P2), where

[ni'-nr] tf[n1...nr]=Sp2n1(C)x---xSP2nr(C).

The proof of this theorem follows essentially from [JS03] and [JS04], and the arguments in [J05]. The relation between the theorem and the endoscopy can be briefly discussed as below. The theory of twisted endoscopy can be found in [KS99]. For simplicity, we first recall from [Ar04] and [Ar05] the basic structure of all standard elliptic endoscopy groups of S 0 2 n + i . Let n = n\ + n 0. Take a semisimple element ° "\ 'Til,712

hn2

e S P 2 n (C).

-W Then the centralizer of s ni) „ 2 in Sp 2n (C) is given by H

[num] = Cent S p 2 n ( c ) ( S r i l , n 2 ) = Sp 2 n i (C) x Sp 2n2 (C).

The standard elliptic endoscopy group associated to the partition n = n\ + n 2 is H[ni,n2] = S 0 2 n i + i X S0 2 „ 2 + i, and the groups ff[ n i n 2 ] exhaust all standard elliptic endoscopy groups of S02n+i, in the sense of [KS99]. In general, an endoscopy transfer of admissible representations from an endoscopy group H of G to G takes a local Arthur packet of admissible representations of H(F) to a local Arthur packet of admissible representations of G(F), which is characterized by the stability

20

D. Jiang

of certain distributions from the geometric side of the Arthur trace formula ([Ar05]). Since the admissible representations of H(A) and G(A) considered in this paper are generic and tempered, following the Arthur conjecture on the structure of the local Arthur packets, the admissible representations we are considering in this paper should be the distinguished representatives of the corresponding local Arthur packets. This must also take the Shahidi conjecture on the genericity of tempered local .L-packets into account, which is the case in Arthur's formulation of his conjecture. Then by the relation between the local Arthur packets (^4-packets) and the local Langlands packets (L-packets), the endoscopy transfer should take the distinguished member of a local Arthur packet to its image of the Langlands functorial lifting from H to G. In other words, the endoscopy transfer from H to G for the distinguished members of local Arthur packets should be the same as the Langlands functorial lifting from H to G. In the above theorem, the Langlands functorial lifting can be viewed as the local endoscopy transfer from Hini^ nj to S02n+iAs in the global case considered in [J05], we can discuss the order of the pole at s = 1 of the higher fundamental local 7-factors explicitly and its precise relation with the structure of the endoscopy group LT[ni! n r ] to S02 n +i, i-e. complete determination of the set {ni,n 2 ,

...,nr}.

Since the argument for the local case is about the same as that for the global case, we omit the details here.

4.2

Other classical groups

We also remark that there exist analogy of the above discussions for other classical groups. The functoriality from the classical groups to the general linear groups for generic representations are known through more recent work of [AS], [CKPSS04], [KK05], and [S05]. On the other hand, the analogy of the local theory for SC>2n+i in [JS03] and [JS04] is still work in progress.

On Local ^-Factors

21

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[H03]

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L-functions

On Local 7-Factors

25

[hKOO]

Kim, H. Langlands-Shahidi method and poles of automorphic L-functions. II. Israel J. Math. 117 (2000), 261284.

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Kim, H.; Krishnamurthy, M., Stable base change lift from unitary groups to GLAT. Intern. Math. Research Papers, No. 1, 2005.

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Kim, H.; Shahidi, F., On simplicity of poles of automorphic L-functions. J. Ramanujan Math. Soc. 19(2004) 267-280.

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Kim, J. Gamma factors of certain supercuspidal representations. Math. Ann., 317(2000)(4), 751-781.

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Lapid, E.; Rallis, S. On the local factors of representations of classical groups. Automorphic Representations, L-Functions and Applications: Progress and Prospects. Ohio State University Mathematical Research Institute Publications 11. (OSU 11) 2005.

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Reeder, M. Some supercuspidal L-packets of positive depth, preprint 2005.

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On Local ^-Factors

27

[Sh02]

Shahidi, F. Local coefficients as Mellin transforms of Bessel functions; towards a general stability. IMRN. 39(2002), 2075-2119.

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28

D. Jiang Dihua JIANG School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA Email: [email protected]

Deligne Pairings over Moduli Spaces of Punctured Riemann Surfaces K. OBITSU, W.-K. TO and L. W E N G

1. W P Metrics and TZ Metrics (1.1) Teichmuller Spaces and Moduli Spaces For g > 0 and N > 0, we denote by Tg^ the Teichmuller space of Riemann surfaces of type ( 0. The Teichmuller space Tg^ is naturally a complex manifold of dimension 3g - 3 + N. The moduli space M9
30

K. Obitsu, W.-K. To & L. Weng

a Riemann surface of genus ga and with na punctures, i.e., Ca is of type (ga, na). As such, the stable condition is equivalent to that 2ga — 2 + na > 0 for each a, that is to say, each Ca also admits the complete hyperbolic metric of constant sectional curvature —1. It is easy to see that 5ZQ=I {^9a — 3 + na) + m = Zg — Z + N. With respect to the disjoint union MQ = \Jra=lCa, one easily sees that tf7l ^mTg,N is a product of lower dimensional Teichmuller spaces given by "7i,--,7m-'g,Af — -^Si.ni

x

-Lg2,ri2

x

' ' ' *

-Lgr,nr

with each Ca G T gai „ Q , a - 1,2,..., r. (1.2) Weil-Petersson M e t r i c and Takhtajan-Zograf M e t r i c For any M° G Tg>jv, M° admits the complete hyperbolic metric of constant sectional curvature —1. By the uniformization theorem, M° can be represented as a quotient H\T of the upper half plane H :— {z £ C : Imz > 0} by the natural action of Fuchsian group T C PSL(2,R) of the first kind. T is generated by 2g hyperbolic transformations A\, B\,..., Ag, Bg and N parabolic transformations S\,..., 5JV satisfying the relation A i B i A ^ B r 1 • • • AgBga^Bg1

• 5 i 5 2 • • • SN = Id.

Let z\,..., ZN € R U {oo} be the fixed points of the parabolic transformations Si,..., S;v respectively, which are also called cusps, such that they correspond to the punctures Pi,-. ,PJV of M under the projection H* := H U {zi,..., zN} —> W*\r ~ M accordingly. For each i = 1,2,..., N, it is well known that Si generates an infinite cyclic subgroup of T, and we can select CTJ € PSL(2,R) so that (Ti(oo) = Zi and o~ Picn is the transformation z ^> z -{- \ on Ji. For s G C with Res > 1, the Eisenstein series Ei(z,s) attached to the cusp Zi is given by Ei(z,s):=

Im J2 (^rV)s, 7G <s«>\r

zeH.

It is uniformly convergent on compact subsets of Ti, and invariant under T. Thus Ei(z, s) descends to a function on M.

Deligne Pairings over Moduli Spaces

31

To describe the tangent and cotangent spaces at a point M of T 9) JV, we first denote by Q(M) the space of holomorphic quadratic differentials (f> — 4>{z) dz2 on M with finite L 1 norm, i.e., fM \cf>\ < oo. Also, we denote by B(M) the space of L°° measurable Beltrami differentials p = p(z) dz/dz on M (i.e., ||/i||oo := ess.sup 2eM |ju(z)| < oo). Let HB(M) be the subspace of B(M) which can be represented as p<(> for some (j> G Q(M). Here p — p{z) dz dz denotes the hyperbolic metric on M. Elements of HB(M) are called harmonic Beltrami differentials. There is a natural Kodaira-Serre pairing (, ) : B(M) x Q(M) -> C given by (p, (f>) -

I /j,(z)(z)dzdz,

where p e B(M),

(peQ(M).

JM

Let Q(M)L c B(M) be the annihilator of Q{M) under this pairing. Then one has the decomposition B(M) = HB(M) © Q(M)1, and natural isomorphisms TMT9)N ~ B(M)/Q(M)± ~ HB(M) and T^TgtN ~ Q(M) with the duality between TMTg,N and T^T9tN given by the pairing (•, •) above. The Weil-Petersson metric g W P and the Takhtajan-Zograf metric TZ g on Tgtw (the latter being introduced in [TZ1,2]) are defined as follows: for X € Tg^ and p, v E HB(M), one has . WP 3

( ^ , i / ) = / pup,

and

TZ

g (p,v)

JV

= / V ^ ( - , 2 ) -/zi/p.

In particular, flTZ(M, i/) - ^ t i 1). Moreover, we know that the Takhtajan-Zograf metric is also Kahlerian ([TZ1,2]) and non-complete ([Ob 1,2]). The metrics gwp and gTZ (but not each individual g^ unless N = 1) are invariant under ModgtN and thus they descend to Kahler metrics on (the smooth points of) Mg
32

K. Obitsu, W.-K. To & L. Weng

2. Line Bundles over Moduli Spaces (2.1) Deligne Pairing Deligne pairing, a refined version of intersection, plays a key role in understanding the Weil-Petersson and Takhtajan-Zograf metrics. To start with, we use a simple example to explain the essential point of such pairings. Example. Let C\ and C2 be two prime divisors, i.e., curves, on a surface S. Assume that they intersect transversally at three points PUP2 and P 3 . Then Cx • C2 = # C i n C 2 = ^{P1,P2,P3} = 3. Moreover, for any meromorphic function / on S, (C\ +div(/) J • C2 = 3 = C\ -C2, where div(/) is defined as the zeros minus the poles. Thus if set {d + div(/) : V/} =: 0 s ( C i ) , then Os(Ci) • Os{C2) = 3 is well-defined. One may try to refine the intersection with C\ • C2 : = -Pi + P2 + Pz- But for our purpose, consider a relative picture 7r : S —> B for a fibration ir over the curve B. Assume then that C\ and C2 are horizontal, i.e., ir(d) = B and denote the images of Pi, P2 and -P3 by Qi, Q2 and Q3 respectively. (Assume that Qi ^ Qj.) Then, viewing from B, we get (C\ • C2)B = Qi + Q2 + Q3, or better f(C?s(Ci) > Os(Ci)> B

= OB(Q1 + Q2 + Q3),

\

=

<1CI,1C 2 )B

1

Qi+Q2+Q3i

where 1 denotes the defining section. In summary, if the relative dimension is 1, then for any two line bundles L\ and L2 (together with two sections si, s2 whose divisors intersect transversally) over the total space, we get a line bundle (Li, L2) (together with a section (si, s2)) on the base B in a canonical way. More generally, if TT : X —> Y is 'nice', of relative dimension n, then for any (n + 1) line bundles Lo, L i , . . . , Ln on X, we get a unique line bundle (Lo, L±,..., Ln) on Y, the so-called Deligne tuple associated to Lo, L\,..., Ln with respect to IT ([De]). (2.2) Universal Curves Pretend that we are using the V-manifold language (so that what follows can be justified). We have a universal

Deligne Pairings over Moduli Spaces

33

curve 7r: C9}N -> M9,N, together with N sections P;'s of 7r. Hence, for x = [ ( M ; P i , . . . , Pjy)] € JW5|JV, Tr" 1 ^) = M and P4(x) = P{. In fact, CS>JV = -Mg,N+ii and essentially 7r is the map of dropping the last puncture. In particular, the fiber of TT at [(M; P i , . . . , PN)] € M-9tN is the compact Riemann surface M together with punctures Pi,..., PN- Hence, by gluing KM and P j , . . . , P ^ along M9tN, and extend to M9,N, then on C9)./v, we get (i) i^T,-, the relative canonical line bundles associated to TT; and (ii) P i , . . . , P/v; sections of n viewed by abuse of terminology as line bundles. (2.3) Primitive Line Bundles Using Deligne pairing formalism, we now introduce some primitive line bundles on M.9,N (Weng [Wei,2]). (i) The Weil-Petersson line bundle A W p := (Kn(¥1 + ••• + PJV), i ^ P i + • • • + PJV)>; (ii) The Takhtajan-Zograf line bundle A T Z : = ( # * , P i + --- + PAT); (iii) The m-th Mumford (type) line bundle, for m > 1, Am := \(mKv

+ (m - 1)(Pi + • • • + PAT)) .

Essentially, Xm\[(M;P1,...,PN)] = det^H°^M,mKn

+ (m-l)(Pi

+- • •+

PN) ) ) , the determinant of the space Tm of cusp forms of weight 2m. Moreover, for m < 0, Am can be defined by using GrothendieckMumford determinant formalism and Serre duality. (See [We 1,2].)

3. Fundamental Relations on M9,N' Algebraic Story Among line bundles Am, A W P , ATZ and Abdy, there are the following fundamental relations. ( F R I) (Deligne [De], Mumford [Mu] N=0; Weng [Wel,2] N > 0) XT

* A ^ m 2 - 6 ™ + 1 > ® A?,- 1 ® A b d y ;

K. Obitsu, W.-K. To & L. Weng

34

( F R II) (Weng [Wei,2]) 2

A®N

^

A ®(2
2

.

( F R III) (i) (Xiao [X] & Cornalba-Harris [CH]) On M9 = Mg,Q,

\8 +

-)\i>Abdy;

( F R III) (ii) (Weng [Wel,2]) On Mg,N, 2 N

\ i

N>1, A

^ A

R e m a r k s . (1) L/M9tN > 0, by definition, if \/B curves in M9,N, deg(L|s) > 0; ' (2) When N = 0, Moriwaki [Mo] has a sharp version for FR III(i); When N = 1, Harris-Morrison (resp. Hain) obtained a similar result as FR II (resp. FR Ill(ii)). For example, Basic Inequality.

(Harris-Morrison [HM]): On M.g,i, ±g(g - \)K > 12Ai - A b d y .

Here K := relative canonical line bundle of Cg = Cgfi —* Mg. We point out that this Basic Inequality is equivalent to A ^ p < A®2 9~

+

' , i.e., our FR II with N = 1. (For details, please refer to

[Wel,2].) (3) Fundamental relations above in fact expose certain intrinsic relations between two different kinds of geometries for the moduli space: The discrete spectrum geometry represented by the Weil-Petersson line bundle and the continuous spectrum geometry represented via the Takhtajan-Zograf line bundle. (See below.)

4. Fundamental Relation on M9N- Arithmetic Story (4.1) Basic Relations For any [ ( M ; P i , . . . ,PN)] € M9,N, we may view the fiber of n at this point as a punctured Riemann surface

Deligne Pairings over Moduli Spaces

35

M° := A f \ { P i , . . . , PN}. Thus if 2g - 2 + AT > 1, by uniformization theorem, M° is covered by the upper half plane H. So M° ~ T\H with a certain T C PSL 2 (R). The Poincare metric on H given by ^ ^ is PSL 2 (R)-invariant, and hence can be descended to T\H ~ M°. Denote this metric by TtypOn the other hand, from M point of view, the metric Thyp may be better understood in terms of the logarithmic tangent bundle \,KM{P\ + • • • + PN)) • That is, we get then a natural singular metric on KM{PI + • • • + PN)- Gluing them together along M9tN, we obtain the nice hyperbolic metric on - ^ ( P i + • • • + PAT)- Denote this metrized line bundle by i ^ Q P i + ••• + ]? JV)By developing an arithmetic intersection for singular metrics (as a part of our ^/-admissible theory [Wei,2]), we obtain a natural smooth metric on A W p = (#,r(Pi + • • • +

PJV),

-MPi + • • • +

PAT))-

Denote the resulting metrized line bundle by AWP( F R IV) (Wolpert [Wo2], Weng [Wel,2]) On Mg%N, ci(Awpj = —2~-

Here c\ denotes the first Chern form, u^yp denotes the Weil-Petersson Kahler form. As for ATZ = (Kn,Pi + • • • + PJV), we may also get a very nice metric by some really very very hard work. (See 4.2 below.) Denote the resulting metrized line bundle by ATZ( F R V) (Weng [Wei,2]) On Mg
X

CI(ATZ) =

Here

CJWP

4

r^TZ-

denotes the Takhtajan-Zograf Kahler form.

36

K. Obitsu, W.-K. To k L. Weng

FRIV and FR V show that Weil-Petersson metrics and TakhtajanZograf metrics are all algebraic and also justify our terminology. (4.2) Backgrounds (i) According to Kodaira-Spencer's deformation theory, tangent vectors on the base may be viewed as elements in H1 (M, KM(—P\ — . . . — P/v)), o v e r which there is the Petersson metric. The Weil-Petersson metric on M9tN is defined by dx A dy ( M , ^ ) W Pv( [ M ; P I , . . . , P J V ] ) :=

'

/

Jv(T)

l&

Here V(F) denotes a fundamental domain of V in H. (So M ~ T\H.) Similarly, the Takhtajan-Zograf metric on M9IN is defined to be

Tz([M;P1,...,PJV]):= /

/^•f>(z;2)-^4^-

Here Ei(z, s) denotes the i-th Eisenstein series associated to T. (ii) In order to get a natural metric on ATZ = (Kir, Pi H r-Pjv), we need to construct nice metrics on KM as well as on the line bundle corresponding to P\-\ \-PN, starting from the (singular) hyperbolic metric on KM (Pi + h -P/v)Difficulty: P i , . . . , P/v are black holes for the hyperbolic metric Thyp: the metric becomes singular at these points; Solution: Use arithmetic construction; Keys: (a) KM has a natural metric, the so-called Arakelov metric ([Ar]), induced from M ^> J(M) = C s /A. This metric is Euclidean in nature. However we can use a fundamental result in cj-admissible theory, called the Mean Value Lemma, as a bridge to get a Thypadmissible metric, the so-called arithmetic hyperbolic metric on KM , which is then hyperbolic in nature. (b) Still, this new metric is not quite right: It is nice up to a certain constant factor, depending on punctured surfaces. To pin down this factor, we need to introduce a new invariant called Arakelov-Poincare

Deligne Pairings over Moduli Spaces

37

volumes for punctured Riemann surfaces. This new invariant is built from the so-called Faltings <5-function for M ([F]) and special values of Selberg zeta function for M°, motivated by a result of D'HokerPhong [DP] and Sarnak [Sa]. (iii) Till now, we have shown how arithmetic intersection can be used to study moduli spaces. To go further, we still need to develop an arithmetic cohomology for singular metrics. The details can be found in [Wei,2]. All in all, what we should say is that without using the analytic torsion approach ([Qu]), which is quite impossible for singular metrics, we successfully introduce a nice determinant metric and then prove the following Arithmetic Riemann-Roch Isometry for Singular Metrics (Weng [Wei,2]) Associated to a fixed normalized volume cu with at most quasi-hyperbolic singularities, there is an oo-admissible theory from which we have the natural isometry \{Lf12

~ (L,L-K)®6

® (K,K)

up to a constant factor depending only on the genus. (4.3) Application On Am = detT m , the determinant of the associated Petersson metric is denoted by p^i171)Moreover, following Takhtajan-Zograf ([TZ1,2]), set

v 7 ^ 1 )'

hQ{m) := pLi{m) •

JZMo{m)

if m = 1,

= , if m > 1.

Here oo

zM<s)--= n n^-^^'^)'

Res>i

7: primitive geodesic ra=0

denotes the standard Selberg zeta function, with I the length. (See e.g., [He].) Denote the metrized line bundle (Xm,hQ(m)) simply by

K. Obitsu, W.-K. To & L. Weng

38

( F R VI) (Deligne [De] N = 0; Weng [Wel,2] N > 1) On Mg,N, up to a universal constant factor, Xm®12 ~ A W P ® ( 6 m 2 - 6 m + 1 ) ® A T Z ® - 1 .

Hence, we recover the following fundamental result of TakhtajanZograf, which is the staring point for our study. Theorem. (Takhtajan-Zograf [TZ1,2]) ci[Xm,hQ(m)J

6m 2 — 6m + 1 wwp 12

7T2

—

.

LjTZ

9

Remark. Recall that as line bundles Am, A W P and A T Z make sense even over the compactified moduli M.g,N- Hence FR IV - VI indeed give a good way to understand the factorization of Selberg zeta functions, Eisenstein series, Weil-Petersson metrics and TakhtajanZograf metrics. (See also 6 below.)

5. Deligne Tuple in General For the universal curve 7TJV : Cg,iv —> -M9,N, define the i-th TakhtajanZograf line bundle by A* := Aj(JV) := (Kn,Fi)(Tr), i = 1,...,N. Then (i) Ai(N) = **,_! (A{(N - 1)) (Pi) for * = 1 , . . . , N - 1; (ii) A^v(iV) = K*^

(Pi + • • • + Piv-i) •

Now identify C9JN with Mg^+\ and set XN :Cg^+\ —• Cg,N be the map of 'forgetting the second to the last puncture'. Then (iii) X*N(KnN(PI

+ • • • + P J V ) ) = KVN+1 (p1

+

... + WNy

Let A = A (JV) be the Weil-Petersson line bundle on Mg^, then (iv) A(iV + 1) = ir*N+1 ( A ( A T ) ) + AN+1(N

+ 1).

More generally, for any m € Z>o, we have the natural map TTN+m,N '• MgyN+m —y M9tN of 'forgetting the last m punctures'. The relative dimension of TTN+m,N is m + 1, it makes sense to talk

Deligne Pairings over Moduli Spaces

39

about (Ai(N

+ m),...,A1(N

AN+m(N

+ m),A(N

+ m),..., + m),...,

AN+m(N

+ m),...,

A(N + m)^(7rJV+TOiJV),

where Ai(N + m) appears aj-times, i = 1 , . . . , N + m, and A appears 6-times. (Clearly aiH \-aN+m + b = m + 1.) For simplicity, denote this line bundle (on Mg^) by

(Ar>---.^+r.A-6)(^+m^). Line bundles ( A ] " 1 , . . . , A ^ ^ 1 , A' 6 j{^N+m,N) satisfy many nice properties ([WZ]). We will not try to list all of them here, but merely point out that (I) (String Equation) ( A * , . . . , A ^ m ) ( 7 r A r + m + 1 , J V ) N+m

= 53 (Ai*,...,A/'- 1 ,...,A^)(7r w + m,N); (II) (Dilaton Equation) ( A ^ 1 , . . . , A ^

m

, A J V + m + 1 )(7r^ + m + l i J V )

= ( 2 5 - 2 + iV + m ) ( A 1 d l , . . . , A ^ m ) ( 7 r A r + m , A r ) .

6. D e g e n e r a t i o n of T Z M e t r i c s : Analytic S t o r y (6.1) Pseudo-Metrics on Boundary of Tg^ MQ € Mg,N\Mg,N gives a stable Riemann surface with N punctures P i , . . . , P/v and m nodes Qi,..., Qm. Thus we may regard Mo as a point in ^i,...,ymTgtNWrite M0\{Qi,..., Qm) =: Ui< Q < r C a , Sllt...amTg
40

K. Obitsu, W.-K. To & L. Weng

cusps) of Sa by n'a (resp. n^), so that na = n'a-\- n'^. We index the punctures of Ca such that {Pa,i}i
V^ ? „ l
n(a,i) T

If none of the punctures of Ca are old cusps, then gTZ,a is simply defined to be zero identically. With this, the nodally depleted Takhtajan-Zograf pseudo-metric ^TZ>(Ti>-.7n) o n (S7li...)7mTfl)jv is defined to be the product pseudo-metric of the gTZ'a,s on the T Pai „ Q 's, i.e., (/)

T

r AT /7 T Z .(7l.-,7n)\ - T \ (T

hTZ'a\

i=i

(6.2) Degeneration Coordinates Each node Qj in MQ admits an open neighborhood Nj = {{ZJ,WJ) € C 2 : \ZJ\, \WJ\ < 1, Zj • Wj — 0} so that Nj = Nj U Nf. Here Nj and Nj are regarded as the coordinate discs in C 2 . For each a, we choose 3ga — 3 + na linearly independent Beltrami differentials v\a , 1 < i < 3ga — 3 + na, which are supported on Ca\ U!?=1 Nj, so that they form a basis of TcaTgoi!na. For simplicity, we rewrite {v^a'}i i<j<3 S _3 + ;v_ m . Then one has an associated local coordinate neighborhood V of Mo in b*n,...,'ymTg,N with coordinates r = (TI, . . . ,T3fl_3+/yf_m) such that Mo corresponds to 0. Shrinking V if necessary, we may assume V ^ A39~3+N~m. por a p0m^ T g ^ one has the associated Beltrami differential /J,(T) — J2i=^ + ~™ Ti vi and a quasi-conformal homeomorphism w^^ : Mo —» M T onto a Riemann surface M r satisfying 8 t gl T = /j,(z)dwgzT . The map u;^ T ) is conformal on each Nj, j — 1 , . . . , m, so that we may regard Nj C M r for each j . Then for each t = (ti,...,tm) with each |£j| < 1, we obtain a new Riemann surface Mt}T for M T by removing the disks {ZJ £ Nj : \ZJ\ < \tj\} and {WJ E Nj : \WJ\ < \tj\} and identifying Zj E Nj with WJ = tj/zj € Nh j = l,...,m. Then one ob-

Deligne Pairings over Moduli Spaces

41

tains a holomorphic family of Riemann surfaces {Mt>T} parametrized by the coordinates (t, r ) = ( i x , . . . , tm, n,..., r 3 3 _ 3 + A r _ m ) of A m x V ~ A m x A 3 s- 3 +^-" 1 . Moreover, the Riemann surfaces Mt,T with (t,r) € (A*) m x V are of type (g,N), where A* = A\{0}. The coordinates t = (£1,..., i m ) will be called pinching coordinates, and r = ( t i , . . . , *3g-3+jv-m) will be called boundary coordinates. For 1 < j < m i let ay denote the simple closed curve |ZJ| = \WJ\ = \tj\2 on XtiT. Shrinking A m and F if necessary, it is known that the universal cover of (A*) m x V is naturally a domain in Tg^ and the corresponding covering transformations are generated by Dehn twist about the ctj's. Since Dehn twists are elements of Mod9;jv, the Modg>;v-invariant metrics U for some A m x F a s described above, where % is a finite ramified cover. Obviously the metrics g W P and <7TZ on (A*) m x V C U may also be regarded as extensions of the pull-back of the corresponding metrics on the smooth points of U n A49tN via the map x(6.3) Degeneration Let MQ be a Riemann surface with N punctures P i , . . . , PN and m nodes Q\,..., Qm. A node Qi is said to be adjacent to punctures if the component of punctured Riemann surface Mo\{Q\,... ,Qi-i,Qi+i,... ,Qm} containing Qi also contains at least one of the Pj's. Otherwise, it is said to be non-adjacent to punctures. As an analog of Masur's well-known result [Ma] on degenerations of Weil-Petersson metrics, we have the following Theorem. (Obitsu-To-Weng [OTW]) For g > 0 and N > 0, let Mo € Mg,N\Mg,N be a stable Riemann surface with N punctures Pi,..., PN and m nodes Q\,..., Qm arranged in such a way that Qi, 1 < * < m ' a r e adjacent to punctures and Qi,m' + 1 < i < m are non-adjacent to punctures. Let U be an open neighborhood of Mo in Ai9tN; together with a local uniformizing chart tp : U ~ A m x V —> U, where V ~ ^g-s+N-m ^s a domain in an appropriate boundary Teichmuller space <^7i,...,7m?g,Ar corresponding to Mo, and

K. Obitsu, W.-K. To k L. Weng

42

let ( s i , . . . , s 3g _ 3+ jv) = (*i,..., <m, r i , . . . , T3g-3+N-m) = (t, T) be the pinching and boundary coordinates ofU~ A171 x V with each U and Si corresponding to Qi. Let the components of the Takhtajan-Zograf metric gTZ be given by

"J

*

ydsi'dsjj'

on (A*) m x V C U. Denote by ^TZ,(7i....,7m)

the

nodaUy

depleted

Takhtajan-Zograf pseudo-metric on the boundary J 7l ,...,7mTSiAT- Then (i) For each 1 < j ' < m and any e > 0, l™isup (tiT) _ ( o i o),(t, T )€(A*)'»xv(l*j| 2 (-log|*il) 4 ~ c ) -9jf(t,r)

= 0;

(ii) For each 1 < j < m' and any e > 0, liminf (tiT) ^ ( o,o),(t,T)e(A*)'"xy(l*j| 2 (-log|^|) 4+e J -gjf(t,T) (iii) For l<j,k<m

= +oo;

and j ^ k, as (t, r ) G (A*) m x V - • (0,0), 1

TZ 9*(t,T) = 0 Vl^ll4i(iog|t i)3(io |t |)3;' *jkK' ' j g fc

(iv) For each j , k > m + 1,

l i m i n f ^ ^ ^ ^ . ^ ^ e C A ^ - x y ^ l ^ l ^ - i o g l ^ l ) 4 ^ ) • gjffa'r) =

(v) For j <m andk>m

^Cn.-.^)(0>0);

+ l, as (t, r) G (A*) m x V -> (0,0),

*3?(*,r) = 0(-

1 l*jl(-log|*j|) 3

/"

Deligne Pairings over Moduli Spaces

43

R e m a r k s . (1) (i) (resp. (ii)) is equivalent to the following statement: For each 1 < j < m and any e > 0, there exists a constant Ci, e > 0 (resp. C2,e > 0) such that for all (t, r) € (A*) m x V, 9ii v(*>, wr ) < u 12/-

'" "

res

~" l ^ l ^ - l oi g' lu^ lM) 4- -,

V P"'- -^ |^| 2 (-log|^|) 4 + £

Accordingly, it is natural to ask whether the stronger estimate „TZ/, _\

(

^ '

J

2

^l,£

^fel (-log|^l)4

actually holds for 1 < j < m' and (i, r ) € (A*) m x V; (2) There are also discussions on degenerations of Green's functions (intersection), Selberg zeta functions, Faltings <5-invariants and Arakelov-Poincare volumes (cohomology). For more details, please refer to [Jo], [Wen], [TW1,2,3] and [Wei,2].

References [Ar]

S. Arakelov, Intersection theory of divisors on an arithmetic surface, Izv Akad. Nauk SSSR ser Mat., 38 (1974) no. 6, 1179-1192

[CH]

M. Cornalba, J. Harris, Divisor classes associated to families of stable varieties, Ann Sci. Ec. Norm. Sup 4 serie, 21 (1988), 455-475

[De]

P. Deligne, Le determinant de la cohomologie, Contemporary Math. Vol. 67, (1987), 93-178

[D'HP] E. D'Hoker and D.H. Phong, On determinants of Laplacians on Riemann surfaces, Coram. Math. Physics, 104, (1986) 537-545 [F]

G. Faltings, Calculus on arithmetic surfaces. Ann. of Math. (2) 119 (1984), no. 2, 387-424.

44

K. Obitsu, W.-K. To & L. Weng

[EM]

J. Harris, I. Morrison, Moduli of curves. GTM 187. SpringerVerlag, 1998.

[He]

D.A. Hejhal, The Selberg trace formula for PSL2(R), Springer LNM 1001, 1983

[Jo]

J. Jorgenson, Asymptotic behavior of Faltings's delta function. Duke Math. J. 61 (1990), no. 1, 221-254.

[Ma]

H. Masur, The extension of the Weil-Petersson metric to the boundary of Teichmuller space, Duke Math. J., 43 (1976) 623-635

[Mo]

A. Moriwaki, Relative Bogomolov's inequality and the cone of positive divisors on the moduli space of stable curves, JAMS 11, (1998) no. 3, 569-600

[Mu]

D. Mumford, Stability of projective varieties, L'Ens. Math., 24 (1977) 39-110

[Obi]

K. Obitsu, Non-completeness of Takhtajan-Zograf Kahler metric for Teichmuller space of punctured Riemann surfaces, Comm. Math. Phys., 205 (1999) no. 2, 415-420

[Ob2]

K. Obitsu, The asymptotic behavior of Eisenstein series and a comparison of Weil-Petersson and the Takhtajan-Zograf metrics, Publ. RIMS, Kyoto Univ., Vol. 37 (2001) 459-478

vol. 2

[OTW] K. Obitsu, W. To & L. Weng, The Asymptotic Behavior of the Takhtajan-Zograf Metrics, in preperation [Qu]

D. Quillen, Determinants of Cauchy-Riemann operators over a Riemann surface, Funk. Anal, i Prilozhen 19 (1985) 31-34

[Sa]

P. Sarnak, Determinants of Laplacians, Commun. Phys. 110, (1987) no. 1, 113-120

Math.

[TW1] W.-K. To & L. Weng: The asymptotic behavior of Green's functions for quasi-hyperbolic metrics on degenerating Riemann surfaces, Manuscripta Math. 93 (1997) 465-480

Deligne Pairings over Moduli Spaces

45

[TW2] W.-K. To &; L. Weng: Green's functions for quasi-hyperbolic metrics on degenerating Riemann surfaces with a separating node, Ann. of Global Analysis and Geometry (1999) no. 3, 239-265 [Tw3]

W.-K. To & L. Weng, Admissible Hermitian metrics on families of line bundles over degenerating Riemann surfaces, Pacific J. Math. 197 (2001), no. 2, 441-489

[TZ1]

L. Takhtajan &; P. Zograf, The Selberg zeta function and a new Kahler metric on the moduli space of punctured Riemann surfaces, J. Geo. Phys. 5, (1988) 551-570

[TZ2]

L. Takhtajan Sz P. Zograf, A local index theorem for families of 9-operators on punctured Riemann surfaces and a new Kahler metric on their moduli spaces, Commun. Math. Phys. 137, (1991) 399-426

[Wei]

L. Weng, fi-admissible theory II. Deligne pairings over moduli spaces of punctured Riemann surfaces. Math. Ann. 320 (2001), no. 2, 239-283.

[We2]

L. Weng, Hyperbolic Metrics, Selberg Zeta Functions and Ara- kelov Theory for Punctured Riemann Surfaces, Lecture Note Series in Mathematics, Vol. 6, Osaka, 1998

[WZ]

L. Weng & D. Zagier, Deligne Products of Line Bundles over Moduli Spaces of Marked Curves, in preparation

[Wen]

R. Wentworth, The asymptotics of the Arakelov-Green's function and Faltings' delta invariant. Comm. Math. Phys. 137 (1991), no. 3, 427-459

[Wol]

S. Wolpert, Noncompleteness of the Weil-Petersson metric for Teichmiiler space. Pacific J. Math. 61 (1975), no. 2, 573577

[Wo2]

S. Wolpert, Chern forms and the Riemann tensor for the moduli space of curves. Invent. Math. 85 (1986), no. 1, 119145

K. Obitsu, W.-K. To k L. Weng

46 [W63]

S. Wolpert, The hyperbolic metric and the geometry of universal curve. J. Differ. Gemo. 31 (1990) no. 2, 417-472

[Wo4]

S. Wolpert, Cusps and the family hyperbolic metric, preprint, 2005

[X]

G. Xiao, Fibered algebraic surfaces with low slope, Math. Ann. 276 (1987), 449-466

Kunio OBITSU Department of Mathematics and Computer Science, Kagoshima University Kagoshima 890-0065, Japan Email: [email protected] Wing-Keung TO Department of Mathematics National University of Singapore Singapore 119260 Email: [email protected] Lin W E N G Graduate School of Mathematics Kyushu University Fukuoka 812-8581, Japan Email: [email protected]

Vector Bundles on Curves over Cp Annette W E R N E R

1. Introduction This paper is a report on joint work with Christopher Deninger published in [De-Wel], [De-We2] and [De-We3]. We define a certain class of vector bundles on p-adic curves which can be endowed with parallel transport along etale paths. In particular, all those bundles induce representations of the algebraic fundamental group. This article is intended as a survey, explaining some results more leisurely than in the original papers. We focus mainly on the definitions and constructions in [De-We2], outlining the ideas rather than reproducing the formal proofs. The last section deals with Mumford curves. We explain results of Herz in [He] which relate our construction to the paper [Fal]. Note that in [Fa2], Faltings develops a p-adic non-abelian Hodge theory, which in some aspects is more general than the present theory.

2. Complex Vector Bundles Let X be a compact Riemann surface with base point x € X. Then every complex representation p : iri(X,x) —> GL r (C) of the fundamental group gives rise to a flat vector bundle £,, on I , i.e. a vector bundle with locally constant transition function. Namely, let IT : X —> X be the universal covering of X. Then Ep is defined as the quotient of the trivial bundle X x C r by the ni(X, x)-action given by combining the natural action of 7Ti (X, x) on the first factor with the action induced by p on the second factor. Conversely, if E is a flat vector bundle on X, its pullback ir*E is trivial as a flat bundle on X, i.e. TT*E ~ C r x X. Since iri(X,x) acts in a natural way on the pullback bundle TT*E, it also acts on the right hand side, which gives 47

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a representation p of 7Ti (X, x) on C r . This representation satisfies E ~ Ep. A similar construction gives for every flat bundle E and every continuous path 7 from x to x' in X an isomorphism of parallel transport along 7 from the fiber Ex of E in x to the fiber K,./. Namely, choose a point y in the universal covering X over x. Then 7 can be lifted to a continuous path in X starting in y. The endpoint of the lifted path is a point y' in X lying over x'. Since ir*E is trivial, there is a trivial parallel transport {it*E)y -^-> (n*E)yi. Since ir(y) = x and 7r(y') = x', there are natural isomorphisms (ir*E)y —> Ex and (ir*E)y> —> Exi. Putting all these isomorphisms together, parallel transport along 7 is given by Ex ^

(n*E)y ^

(n*E)y, - ^ Ex>.

Regarding Ep as a holomorphic bundle on X, a theorem of Weil [Weil] says that a holomorphic bundle £ on J is isomorphic to some Ep (i.e. E comes from a representation of iti(X, x)) if and only if E = 0 ^ Ei with indecomposable subbundles Ei of degree zero. A famous result by Narasimhan and Seshadri [Na-Se] says that a holomorphic vector bundle E of degree zero on X is stable if and only if E is isomorphic to Ep for some irreducible unitary representation p. Hence a holomorphic vector bundle comes from a unitary representation p if and only if it is of the form E — 0 ^ Ei for stable (and hence indecomposable) subbundles of degree zero.

3. Fundamental Groups of p-Adic Curves It is a natural question to look for a p-adic analogue of the results described in the previous section. We denote by C p the completion of an algebraic closure Q p of Q p . Besides, Z p and 0 denote the rings of integers in Qp and C p , respectively. By k = ¥p we denote the residue field of Z p and 0. We call any purely one-dimensional separated scheme of finite type over a field a curve. Let X be a smooth, projective and connected curve over Q p and Xcp its base change to C p . We are looking for a relation between vector bundles on Xcp and representations of the fundamental group of X.

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First we have to clarify what we mean by fundamental group. Of course, in the algebraic setting there is no topological fundamental groups defined with closed paths. However, there is an algebraic fundamental group TT\(X, X) for a base point x G X(CP). It is defined as the group of automorphisms of the fiber functor Fx. This fiber functor Fx maps a finite etale covering Y of X to the set of Cp-valued points of Y lying over x. Hence an automorphism of Fx induces in a functorial way for every finite etale covering Y —> X a permutation of the points in the fiber over x. Note that on a Riemann surface X, a closed topological path 7 on X induces such an automorphism 7 : Fx —> Fx of the fiber functor. Namely, for every finite etale cover Y —> X and every point y G Y over x we can lift 7 to a continuous path in Y starting in y. Its endpoint y' also lies over x, and one can define 7 by mapping y G FX(Y) to y' G FX(Y). Since the algebraic fundamental group involves only the finite etale coverings, it is in fact analogous to the profinite completion of the topological fundamental group on Riemann surfaces. There is also an analogue of non-closed paths. For two points x and x' in X(Cp) we call any isomorphism Fx —* Fx> of the corresponding fiber functors an etale path from x to x. Such an etale path associates for all finite, etale Y —> X to every point in the fiber of Y over x a point in the fiber of Y over x'. The etale fundamental groupoid of Hi(X) of X is defined as the category such that the points in X(CP) are the objects and such the set of morphisms from x G X(CP) to x' € X(CP) is the set of etale paths from x to x\ i.e. the set of isomorphisms of fiber functors Fx — > Fxi.

4. Finite Vector Bundles The algebraic fundamental group only involves finite etale coverings, hence in the algebraic setting there is no universal covering. Therefore we can imitate the constructions on Riemann surfaces in the p-adic situation only for finite vector bundles, i.e. for vector bundles E on Xcp such that there is a finite etale covering IT : Ycp —> Xcp for which TT*E is trivial.

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Namely, for every finite vector bundle E on Xcp we choose a finite, etale and Galois covering is : Y
-> 7Ti(XCp,x) -+ Gal(y C p /X C p ) -> 0.

Besides, since 7r*.E is trivial, the fiber map T(Yc ,7r*E) —> (ir*E)y is an isomorphism. As a pullback bundle, 7r*£7 and also its set of global sections F(Ycp,7r*E) carries a natural Gal(>c p /Xc p )-action. Via the fiber isomorphism, this action induces a Gal(l£ /Aic p )-action on (ir*E)y, which can be identified with the fiber Ex of E in x. Therefore we have defined a representation TTI(X,X)

= 7ri(X Cp ,x) -> Gal(y C p /^c p ) - • Aut(£ x ).

In fact, this construction works for curves over arbitrary fields. In [La-Stu], Lange and Stuhler investigate finite vector bundles on a smooth, projective curve C over a field of characteristic p. Let us denote by F the absolute Probenius on C, defined by the p-powermap on the structure sheaf. By [La-Stu], 1.4, a vector bundle E onC is finite if and only if for a suitable power Fn we have Fn*E -^-> E.

5. A Bigger Category of Vector Bundles Let us again consider a smooth, projective and connected curve X over Q p . We are interested in vector bundles on Xcp. Here we denote by Xcp the base change of X with C p . This kind of notation for base changes will be used throughout this paper. We have seen that finite vector bundles on Xcp give rise to representations of the fundamental group. However, in this way we only get representations factoring over a finite quotient of TT\ (X, x). The main idea for the construction of representations for a more general class of vector bundles is the following: We consider vector bundles with integral models (in a sense to be made precise below) which are "finite modulo pn" for all n. A similar construction as the one for finite bundles then gives for all n representations of 7Ti (X, x) modulo pn, i.e. over o/pno. In the limit we get a representation of 7Ti(X, x) over o which we can tensor with C p . In fact, we will more generally define parallel transport along etale paths.

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To be more precise, let us call any finitely presented, flat and proper Zp-scheme (respectively o-scheme) with generic fiber X (respectively XQP) a model of X (respectively Xcp)Note that any model of X descends to a finite extension of Z p and is irreducible and reduced by [Liu], 4.3.8. We assume that the curves and their models are finitely presented over Q p , repectively Z p , so that they can be descended to a finite extension of \ there is a finitely presented proper Xp~morphism ix : y —> X with the following two properties: i) The generic fiber n^ :Y = yQp^>Xis finite and etale. ii) The pullback bundle 7r*£ becomes trivial on yo after base change with o/pno. Hence 03xc can be viewed as the category of all vector bundles on XQP which are in some sense finite modulo all pn. Note however that the special fiber of the coverings TT : y —* X will in general be neither finite nor etale. Only the generic fiber has these properties. Note that every vector bundle E on Xcp can be extended to a bundle on a suitable model of Xcp, see e.g. [De-We2], theorem 5. Hence the important point in definition 1 is the existence of the coverings 7r. Although we omit it in our notation, -IT depends of course on n. This definition of the category 23xc i s useful for the construction of parallel transport, as we will see in the next section. However, it is difficult to check whether a bundle fulfils the conditions in definition 1. In sections 10 and 11 we give a more intrinsic characterizations of %Xc ( a n d also of a bigger category of bundles to be defined below).

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6. Parallel T r a n s p o r t on B u n d l e s in X descends to a morphism TTR : y^ —> A^ over a discrete valuation ring R finite over Z p . If R is chosen big enough, there is a semistable i?-curve y' over 3^ with geometrically connected generic fiber such that the induced map IT' : y' Z p —> X also has the properties i) and ii) in definition 1, see e.g. [De-We2], theorem 1. Hence we can assume that all the y in definition 1 are semistable with connected generic fibers. It follows that the structure maps A : y —> SpecZ p are cohomologically flat in dimension zero. This means that the formation of \*Oy commutes with arbitrary base changes. As a consequence, we have (A <8> R)*Oy®R = OR for every Z p -algebra R. Now let us fix two points x and x' in X(CP) and an etale path 7 from x to x', i.e. an isomorphism of fiber functors Fx -—> Fx>. Besides, we fix some n > 1. Then there is a morphism TT : y —> X as in definition 1 with a semistable 3^- We fix some y in y(Cp) with ir{y) = x. The fiber functor 7 maps y to a point y' 6 y(Cp) with ir(y') — x'. The points y and y' can be extended to o-rational points on the proper model y, and x and x' can be extended to o-rational point x0 and x'0 on X. By z n , x^, yn respectively y'n we denote the induced o n -rational points, where we put o n = o/p"o = Zp/pnZp. Besides, we write 7i"n : yn —* Xn for the base change of TT with on. In particular, Xn — X ®yj on and yn = y ^ o n . Let £n = £®0 on be the induced vector bundle on Xn. For the Z p -algebra R = on the equality above gives (A<8>on)*0yn = o„. This implies that the o„-rational point yn : Spec(o n ) —> yn induces by pullback an isomorphism y*n : T(yn,Oyn)

-^

r(Spec(on),y*nOyn)

= on.

Since 7r*£n is trivial on yn, pullback by yn also induces an isomorphism

y*n '• r(yn,K£n)

-—>r(Spec(o„) = r(Spec(o„),x*£ n ) =: £Xn.

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Similarly, pullback by y'n induces an isomorphism Vn '• r(yn,TTn£n)

^->r(Spec(o n)> Vn

Now we define pe^il) PE,n(i)

^n^n)

= r(Spec(o n ),z„*£n) =: £x>n. to be the composition

= Vn ° ( l / n ) m v : SXn

^ -

T(^n, < £ „ ) ^ +

£x>n.

Then the maps ps,n{l) form a projective system. We denote by p

lim p

the fiber of £ over x0 and by £x>o the fiber of £ over x'0. In the limit we get a map peirr) :€x. -+£x'0Tensoring with Cp we finally get a "parallel transport" map PE{I)

• Ex -> Ex>,

where Ex is the fiber of E in x. By construction, this isomorphism is continuous in the p-adic topology. In [De-We2], section 3, we show that the construction is independent of all choices.

7. Working Outside a Divisor on X 1 there is a finitely presented proper Zpmorphism 7r : y —> X with the following two properties: i) The generic fiber TT-Q : Y = y <8> Qp —• X is finite and its restriction to 7TQ inv(J7) is etale. ii) The pullback bundle TT*£ becomes trivial on ^ 0 after base change with o/pno.

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Thus 93 x c = 33xc ,0- Let 7 be an etale path from x G U(CP) to x' G U(CP) in U, i.e. an isomorphism of fiber functors defined on the open curve U. The same construction as above gives for every bundle E in 03x c ,D an isomorphism "of parallel transport" PE{I)

• Ex -> Exi.

8. Properties of Parallel Transport By [De-We2], theorem 22, the association p : 7 — t > PE(J) is functorial in 7, i.e. for E G 53xc ,D and etale paths 7 from x to x' and 7' from x' to x" we have />E(7' 07) = PEW) ° PE{I) as isomorphisms from Ex to £/x». Recall that for U = X\D the fundamental groupoid Hi(U) is the category with object set U(CP) and etale paths as morphisms. In fact, for x and x' in U(CP) the morphism set Mor(x, a/) = {etale paths from x to x'} = Iso(F x , i^') carries a natural topology, since it is profinite. A functor from Hi(U) to the category of finitedimensional Cp-vector spaces which is continuous on the morphism spaces is called a representation of Hi(U) on finite-dimensional Cpvector spaces. With this terminology, for every vector bundle E G 23xc ,D the functor PE : ni(C7) —> {finite dimensional Cp — vector spaces} given by x >—> Ex on objects and 7 1—> PE{I) on morphisms is a representation of IIi(£/). By [De-We2], theorem 28, we have Theorem 2. i) The association E *-* PE is functorial for morphisms in 23xc ,D> exact and commutes with tensor products, duals, internal horns and exterior powers. ii) If f : X —* X' is a morphism between smooth, projective, connected curves over Qp, and D' is a divisor on X', then pullback of vector bundles induces a functor f* : 93^' ,£>' —• 93x c ,f*D> which commutes with tensor products, duals, internal horns and exterior powers.

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Besides, for every bundle E in 05^' D' we have PJ*E — PE°f* where /* is the induced functor Ui(X\f*D') —> Ui(X'\D') on fundamental groupoids. Here we identify (f*E)x with •£/(£) • Hi) If X — XK ®K QP for some field K between Qp and Q p , then every element a in G&l(Qp/K) acts in a natural way on 23x c ,D- Besides, a acts on Ui(X\D) and on the category of finite-dimensional Cp-vector spaces and hence on the category of representations of IIi {X\D) • The functor p commutes with these actions.

9. Semistable Bundles Recall that for a vector bundle B o n a smooth, projective and connected curve over a field k the slope is defined by p(E) = deg(E)/rk(E) The bundle E is called semistable (respectively stable), if for all proper non-zero subbundles F of E the inequality p,{F) < p(E) (respectively fi(F) < p.{E)) holds. Let E be a bundle in 23x c ,D for some divisor D. By definition, there exists a model X of X, a vector bundle £ on X0 extending E and a finitely presented, flat morphism TT : y —> X which is generically finite and etale over X\D such that the special fiber of TT*S is trivial. Here we only use the condition for n = 1 in the definition of 23x c ,DUsing descent to a suitable discrete valuation ring and an argument due to Raynaud (see [De-We2], theorem 13) one can show that triviality of the special fiber of n*£ implies that the generic fiber 7r£ E is semistable of degree zero on y Cp. Since ircp is finite, E is also semistable of degree zero on Xcp •

10. A Simpler Description of 55xc ,D It turns out that the existence of some TT : y —> X as above such that 7r*£ has a trivial special fiber is also sufficient for a bundle to lie in 23xc ,£>• To be precise, by [De-We2], theorem 16 we have Theorem 3. A vector bundle E lies in 23x c ,D if and only if there is a model X of X, a vector bundle £ on X0 extending E and a finitely presented, proper morphism ir : y —• X which is generically finite and etale over X\D such that the special fiber of ir*£ is trivial.

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Let £ be a bundle in X be as in theorem 3. The image 7r(^) is closed in X and contains the generic fiber, since IT is generically finite. Since the model X is irreducible, IT must be surjective. Let ir^ : yk —> Xk be its special fiber (recall that k — ¥p is the residue field of Z p and o). Let C\,..., Cr be the irreducible components of X\. with their reduced structure. Since n is surjective, every Cv is finitely dominated by an irreducible component Dv of 34- Hence the restriction of £k to Cv is trivialized by a finite covering (which of course in general is not etale). Let Cv be the normalization of Cv. Then also the pullback of Sk to the smooth projective fc-curve Cv is trivialized by a finite covering, namely the normalization of Dv.

11. Strongly Semistable Reduction Let us denote by F the absolute Probenius in characteristic p. Definition 4. Let E be a vector bundle on a smooth, projective, connected curve C over a field of characteristic p. Then E is called strongly semistable if and only if for all n > 0 the pullback Fn*E is semistable on C. Let E be a vector bundle in the category 55xc ,D with an extension £ to a model X0 such that the special fiber of £ becomes trivial after pullback to some y —* X as above. We have seen in the preceding section that for all irreducible components Cv of the special fiber Xk the pullback of £k to the normalization Cv of Cv becomes trivial on a finite covering. As the trivial vector bundle is strongly semistable of degree zero, we deduce that the pullback of £k to any Cv is also strongly semistable of degree zero. One of the main results in [De-We2] shows that this property is also sufficient for a bundle to lie in *Bxc ,DTo be precise, we say that a vector bundle E on Xcp has strongly semistable reduction of degree zero, if there is a model X of X and a vector bundle £ on X0 extending E such that the pullback of the special fiber £k of £ to all normalized irreducible components of X\. is strongly semistable of degree zero. We denote the (full) category of all strongly semistable bundles on Xcp by 5 5 ^ •

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Theorem 5. For every vector bundle E on XQ with strongly semistable reduction of degree zero there is a divisor D such that E is contained in 23 x c ,D- In other words, we have *&sx = \JD 23x c ,DFor the proof see [De-We2], theorem 36. The idea is the following. We have just seen that all categories 03x c ,D are contained in ?8sxc • Hence it remains to show that for every bundle E on Xcp with strongly semistable reduction £\. of degree zero there is a divisor D on X such that E is contained in 23xc ,£>. There is a finite extension K of Qp such that the model X descends to a model X0K over the ring of integers in K and such that the special fiber £k of £ descends to a vector bundle £pq on the special fiber X¥q of X0K. By enlarging K if necessary, we can assume that the irreducible components Di,..., Dr of Xp are geometrically irreducible, hence they give the components C\,..., Cr of X^ by base change. Then the pullback of Sfq to all normalized components Du is strongly semistable of degree zero. Now on the smooth, projective curve Dv over the finite field ¥q there are only finitely many isomorphism classes of semistable bundles of degree zero. This implies that on Xp there are only finitely many isomorphism classes of bundles whose pullbacks to all normalized components Dv are semistable of degree zero (see the proof of [De-We2], theorem 18). For q = f we put Fq = Fr, i.e. Fq is the Frobenius fixing Fq. We regard Fq also as a Fg-linear automorphism of Xtfq and consider all pullbacks Fq*£^q. This gives an infinite collection of bundles whose pullback to all Du is semistable of degree zero. Since there are only finitely many isomorphism classes available, we find two bundles in this collection which are isomorphic. Hence there are natural numbers r > s satisfying Fq*£jq ~ Fq*£^q, i.e. for n = r - s we find Fg"*(Fgs*£irq) ~ F^*£¥q. By the theorem of Lange and Stuhler cited in section 4, this implies that Fq*£¥q is a finite bundle, i.e. it becomes trivial on a finite etale covering u) : YQ —> XY . Hence £p becomes trivial after pullback via u o Fg. Now we have to lift this covering of the special fiber Xpq to a covering of the whole model X0K. This is no problem for the etale part tu, which lifts to a finite, etale morphism UJ0K : y0K —>• X0K, but a non-trivial task for the Frobenius. As a first step we find a

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semistable, regular and projective o^-scheme y0K together with a morphism y0K —> y0K (we might have to enlarge K here). The map 0 • y 0 K —¥ y<>K —* XoK a l s 0 n a s * n e property that 0YQ O Fq trivializes the special fiber of £wq by pullback. We embed y0K in some projective space P ^ - Now we define a Frobenius lift fq* on P ^ . by [XQ,..., XN] I _ > [XQ ,... ,xqN] o n projective coordinates. The special fiber of fqs is the Frobenius Fq. Let y'0K be the base change of y0K by fqs, i.e. y 0 K s r t s m a cartesian diagram

I

IT

Of course, the generic fiber of this Frobenius lift fqs is etale only outside the union of the coordinate hyperplanes. But we can twist the projective embedding r of y0K into ¥^K by an automorphism in PGLN so that the morphism y'0 —* 3^,K —> ^ 0ft . is generically etale outside a divisor D on the generic fiber XK of X0K. Now we look at the special fiber of this morphism y'0K —> X, K -* Af0A:. There is a natural map i = (Tfq,Fq) : y^q —> y F • By definition, it gives the Frobenius F* after composition with the projection to y%q. Since y$ is semistable, it is reduced. Besides, i induces an isomorphism with the reduced induced structure of 3 ^ , see [De-We2], lemma 19. Now we dominate y'0K by a semistable OK~ scheme Z0K (possibly after enlarging K). Thus we get a chain of morphisms Z0K —> y'0K —> ^0i<. —> ,f0K. Let us look at the special fibers

Since the semistable curve i ^ is reduced, the first map factors through the reduced induced structure of y F , hence through i. But i composed with the projection }>F —> 3V9 is the Frobenius Fjj. Hence Zvq ~* Xvq factors through 6^q ° F* which is equal to Fq o 6¥q. Hence the pullback of £$q is trivial. Going up to o, we get a finitely presented, proper morphism 7r : Z —> X which generically is finite and etale over X\D such that the

Vector Bundles on Curves over C,

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special fiber of 7r*£ is trivial. Hence E lies in 2$xc ,D by theorem 3, which proves our claim. • In this argument, the freedom of changing the embedding of y0K into projective space by an automorphism in PGLN can be further exploited to find a second divisor D' on X which is disjoint from D such that E also lies in 2$xc ,£>' • Hence by the construction in section 2, we can define parallel transport on E along etale paths in X\D and along etale paths in X\D'. It can be shown that these two constructions fit together on the intersection of the open subcurves X\D and X\D' and hence give rise to parallel transport along etale paths in the whole of X (see [De-We2], proposition 34). Applying theorem 2 we then deduce the following corollary (see [De-We2], theorem 36). Corollary 6. There is a functor p from Q33cc ^° ^ e ca^egory °f representations ofHi(X) on finite-dimensional Cp-vector spaces, which is exact and commutes with tensor products, duals, internal horns and exterior powers. Besides, it behaves functorially with respect to pullbacks along morphisms of curves over Qp and is compatible with Galois- conjugation. Let 33^ be the (full) category of all vector bundles on Xcp for which there exists a finite etale covering a g : Y —> X over Qp such that a*E has strongly semistable reduction of degree zero on Ycp, where a : Ycp —> Xcp is the base change to Cp. HE lies in 93^ , we say that E has potentially strongly semistable reduction of degree zero. Let E be a bundle on Xcp with potentially strongly semistable reduction of degree zero, and let a^ : Y —» X be a finite etale covering as above. We can assume that Y —• X is a Galois covering. By corollary 6, a*E defines a representation of the fundamental groupoid IIi(Y). Let 7 be an etale path from x to x' on X. Once we choose a point y in Y lying over x, the path 7 can be lifted to an etale path 5 onY from y to some point y' over x'. Then we define PE{I) = Pa*E(S) : Ex = (a*E)y - • {a*E)y, = Ex>. Choosing another lift S amounts to choosing another point y over x as starting point. Hence there is a Galois automorphism a G Gal(Y/X)

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with a{y) = y. IfCT*: I l i ( y ) —> I l i ( y ) denotes the natural functor given by cr, then the path a*5 is the lift of 7 with starting point a (y) — V- Since a*a*E = a*E, it follows from corollary 6 that Pa*E(8) = Pa*E{o~*5). Hence PE is well-defined. Therefore the construction of representations of the fundamental groupoid extends to bundles in 23^ . All the properties stated in corollary 6 also hold for bundles in 23^ .

12. How Big are our Categories of Bundles? are It is easy to see that all the categories 23 x c ,D, ®3cc anc ^ ^ x closed under direct sums, tensor products, internal horns and exterior powers. The following theorem collects more information about these categories.

Theorem 7. i) The category 23^ zero.

contains all line bundles of degree

are ii) For every divisor D, the categories 23xc ,D, 23^ an^ ^x stable under extensions, i.e. if 0 —• E' —> E —* E" —» 0 is an exact sequence of vector bundles such that E' and E" are in the respective category, the same holds for E.

Hi) If E is contained in *&sx , respectively 03^ , then every subbundle of degree zero and every quotient bundle of degree zero is also contained in 23^ , respectively 23^ . A proof for i) can be found in [De-We2], theorem 12, ii) is proven in [De-We2], theorem 11, and iii) is shown in [De-We3], theorem 9. Theorem 7,iii) implies that 23^ together with its natural fiber functor E 1—> Ex for some fixed x G X(CP) is a neutral Tannaka category, see [De-We3], theorem 12. If X is an elliptic curve, then by Atiyah's classification [At] of vector bundles on Xcp the category 23^ contains all semistable bundles of degree zero, cf. [De-We2], corollary 15. Hence for elliptic curves 23^ coincide with the category of all semistable bundles of degree zero.

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For curves of higher genus it is an open question, if the category 55^ c coincides with the category of all semistable bundles of degree zero on XQ . In [De-We3], theorem 12, it is shown that this is the case if and only if the corresponding subcategories of polystable bundles of degree zero coincide. Here a vector bundle on XQP is called polystable of degree zero if it is isomorphic to a direct sum of stable bundles of degree zero. Denote by 1 ^ d (respectively 2 3 ^ ) t ^ ie c a tegory of all polystable bundles of degree zero on Xcp (repectively of all polystable bundles of degree zero contained in 2$^ )• I n [De-We3], theorem 16, it is shown that the Tannaka groups of T^ d and Q5^d have the same group of connected components.

13. Representations of the Fundamental Group If we fix a base point x € X(CP) and restrict the functor p to closed etale paths in x, then every vector bundle E in 23^ gives rise to a continuous representation 7Tl(.X»-»Aut(£x) of the fundamental group. In [De-Wel], we define an analogue of 23xc a n d P o n a n Abelian variety A over Qp with good reduction. We also show how our construction generalizes a map in [Ta] and how it is related to the Hodge-Tate decomposition of Hgt(A, Qp) ® C p .

14. Mumford Curves Finally, we consider the special case that X is a Mumford curve (see [Mu]), i.e. X descends to a curve XK over a finite extension K of Q p which has a rigid analytic uniformization as Q/F. Here T is a Schottky group in PGL(2, K) and 0. is the open subset of the rigid analytic FK where F acts discontinously. Let X0K be the minimal regular model of XK, see e.g. [Liu], section 9.3. Then all normalized irreducible components in the special fiber of X 0 K are P^s. In [Fa2], Faltings shows that there is an equivalence of categories between semistable vector bundles E of degree zero on XK and Klinear representations of the Schottky group F which satisfy a certain

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boundedness condition. In [Re-Pu], his results were generalized to non-discrete p-adic base fields, e.g. C p . In [He], Herz shows how Faltings' construction is related to the representations of the fundamental group given by bundles in 23x c • Namely, let E be a vector bundle on Xcp which can be extended to a vector bundle £ on the minimal model X0. By the generalization [Re-Pu] of Faltings' results, E gives rise to a representation TV of the Schottky group T. Herz shows that this representation comes in fact from a representation (also called n) of T on a free o-module. Moreover, he shows that E lies in 2$xc • Let i b e a base point in X(CP) and x0 in X0(o) its extension. By construction, the representation PE '• KI(X,X) —> Aut(.Ex) is induced from a representation PS • ni(X,x)

-» Aut(£ Xo )

on the o-lattice £Xo in Ex, where pe is the limit of representations pe,n '• ^i(X,x) ~^ Aut(£s n ). It is shown in [He] that p£tTl factors through a finite quotient of T for a suitably chosen model £, and that the induced representation on this quotient is isomorphic to the reduction of n modulo pn. Hence p£ factors through the profinite completion V of the Schottky group Y and is isomorphic to the profinite completion of Faltings' representation IT.

References [At]

M. Atiyah: Vector bundles over an elliptic curve. Proc. London Math. Soc. 7 (1957), 414-452

[De-Wel] C. Deninger, A. Werner: Line bundles and p-adic characters. In: G van der Geer, B. Moonen, R. Schoof (eds): Number Fields and Function Fields - Two Parallel Worlds. Birkhauser Progress in Mathematics, Vol 239, 2005, 101131. [De-We2] C. Deninger, A. Werner: Vector bundles on p-adic curves and parallel transport. Ann. Scient. Ec. Norm. Sup. 38 (2005) 553-597.

Vector Bundles on Curves over C

63

[De-We3] C. Deninger, A. Werner: On Tannaka duality for vector bundles on p-adic curves. Preprint unter http://arxiv.org/abs/math/0506263. [Fal]

G. Faltings: Semistable vector bundles on Mumford curves. Invent. Math. 74 (1983), 199-212

[Fa2]

G. Faltings: A p-adic Simpson correspondence. Adv. Math. 198 (2005) 847-862.

[He]

G. Herz: On representations attached to semistable vector bundles on Mumford curves. Thesis Miinster 2005.

[La-Stu]

H. Lange, U. Stuhler: Vektorbiindel auf Kurven und Darstellungen der algebraischen Fundamentalgruppe. Math. Z. 156 (1977), 73-83

[Liu]

Q. Liu: Algebraic Geometry and Arithmetic Curves. Oxford University Press 2002

[Mu]

D. Mumford: An analytic construction of degenerating curves over complete local rings. Compositio Math. 24 (1972) 129-174.

[Na-Se]

M.S. Narasimhan, C.S. Seshadri: Stable and unitary vector bundles on a compact Riemann surface. Ann. Math. 82 (1965), 540-567

[Re-Pu]

M. Reversat, M. van der Put: Fibres vectoriels semistables sur une courbe de Mumfort. Math. Ann. 273 (1986) 573-600.

[Ta]

J. Tate: p-divisible groups. In: Proceedings of a Conference on local fields. Driebergen 1966, 158-183

[Weil]

A. Weil: Generalisation des fonctions abeliennes. J. de Math. P. et App. (IX) 17 (1938) 47-87

64

A. Werner Annette W E R N E R Department of Mathematics Universitat Stuttgart D - 70569 Stuttgart, Germany Email: [email protected]

Absolute CM-Periods — Complex and p-Adic Hiroyuki YOSHIDA

1. I n t r o d u c t i o n This paper is written under the request of the organizers of the Karatsu conference who asked the author to give an exposition of absolute CM-periods in an easily accessible fashion. In fact, it reproduces the three given talks at the conference rather faithfully. The paper is divided into two parts. In part I, we will give an overview of the results in the author's book [Y4]. In part II, we will review a p-adic analogue of part I. The research was done in collaboration with Tomokazu Kashio. A paper which will give full perspectives of p-adic theory is in preparation [KY2], but the reader can see a preliminary version [KYI]. Now let us begin with the simplest case. Let K be an imaginary quadratic field. Let E be an elliptic curve defined over Q with complex multiplication by K and robea period of E. The ChowlaSelberg formula states d l

/„\wx(a)/4h

~

a=l

^ '

Here — d, h and w denote the discriminant, the class number and the number of roots of unity of K respectively; x denotes the Dirichlet character corresponding to K. For a, b € C, we write a ~ b if b / 0 and a/b E Q. The absolute CM-periods give a conjectural generalization of (1). Let F be a totally real algebraic number field and K be an abelian extension of F. We assume that K is a CM-field. For r e Gal(K/F), we define the absolute CM-period symbol #jr(id, T) using the multiple 65

H. Yoshida

66

gamma function. We conjecture that pniS^i r) ~ <7/s"(id,T) where px is Shimura's period symbol. The symbol gx is defined by (2) ^ ( i d > r ) = ^ W / = » e x p ( ] l | £

-&L

£

x(c)X(c)).

We refer the reader to §4 for the explanation of notation; we only note that X(c) is a certain ray class invariant attached to an ideal class c and has the following form. ^ log —)LJ—.—r—h correction terms. v Pr(j)\ j) jeJzeR(Cjtc) Here Fr^(z,Vj) is the r(j)-ple gamma function, J and R(Cj,c) are finite sets. A remarkable feature is that X(c) contains information on Stark-Shintani units. In this way we obtain a coherent picture unifying periods and units. We refer the reader to §6 for more details. Now let us explain the contents of Part II briefly. The p-adic counter part of the Chowla-Selberg formula is the Gross-Koblitz formula. Let p be a prime number which splits completely in an imaginary quadratic field K. Let (p) = ^J^3P be the prime ideal decomposition where p denotes the complex conjugation. We put y$h = (a). We choose an embedding K c Qp so that it induces ^3. Then d-i

(4)

logpfo"/a) = | E *(°) 1 O §P ( r * ( s ) ) '

Here logp is the p-adic logarithmic function and Tp is the p-adic gamma function. Fix an embedding F C Q p and let p be the prime ideal of F induced from this embedding. We define the p-adic (logarithmic) absolute CM-period symbol by (5)Z^(id>r) = - ^ l o g p + ] i

(6)

[

X ; ^ l

Xp(c) = 2__, 2^ LrPir(j)(z,Vj) jeJ zeR{Cj),c)

£

X(c)Xp(c),

+ correction terms.

Absolute CM-Periods — Complex and p-Adic

67

Here LTpr^{z,Vj) denotes the p-adic logarithmic r(j)-ple gamma function. We can make a very precise conjecture on lgv,K when p splits completely in K (see §8). The p-adic analogue of the StarkShintani conjecture is a conjecture of Gross. It predicts, if p decomposes completely in K, the value of the derivative of a p-adic L-function at s = 0, Lp(0,u;x)) where % £ G- a n d W is the Teichimuller character. We can show that our conjecture implies the Gross conjecture, but not conversely. In §10, we will discuss the meaning of the symbol lgPtK in the general case in terms of p-adic periods.

2. Notation For a, b G C, we write a ~ b if b / 0 and a/b is an algebraic number. We fix an algebraic closure Q of Q in C. By an algebraic number field, we understand an algebraic extension of Q of finite degree contained in Q. We denote by p the complex conjugation. Let F be an algebraic number field. The ring of integers and the group of units of F are denoted by OF and Ep respectively. We denote by Ep, the group of all totally positive units of F. The class number of F is denoted by hp. We denote by Jp the set of all isomorphisms of F into C. If F is a totally real algebraic number field of degree n, ooi, ..., oo„ denote all archimedean primes of F. For an integral ideal f of F, Cf denotes the ideal class group modulo fooi • • • oo n . By a CM-field, we understand a totally imaginary quadratic extension of a totally real algebraic number field. For a prime number p, Q p denotes an algebraic closure of Qp and Cp denotes the completion of

o;. Part I. Complex Theory §1. Gamma function For x > 0, we define the Hurwitz-Lerch zeta function by oo

(1.1)

C0», x) = Y, (* + m)~s, m=0

Re(s) > 1.

68

H. Yoshida

This function can be continued meromorphically to the whole splane. It is holomorphic except for a simple pole at s = 1. If x = 1, we have £(s, 1) = C(s)- It i s weU known that (1.2)

C(0,x) = 2 -x,

(1.3)

^ C ( S ' X ) s=0

= 10 g r(x)--log27T Z

In particular, we have C/(0) = - ^ l o g 2 7 T .

C(o) = - ^

Let x be a Dirichlet character to the modulus / . Then we have /—1 oo

n S

L(s, X) = Y1 X(n)n- = J2Y1

* ( a ) ( m / + a)~S-

a=l m=0

n=l

Hence (1.4)

£(*,*) = £ x ( a ) / - ' C ( « , y J .

Assume that the associated primitive character to x is nontrivial. By (1.2) and (1.3), we obtain 1/_1 L(0, X ) = - 7 X > ( a ) a ,

(1-5)

•^ a = l

/-I

(1.6)

,

v

,

L (0,x) = E x ( a ) l o g r ( 7 ) - l o g / . L ( 0 , x ) . o=l

^•''

Let if be an algebraic number field. Let (K(s) = Y/N(a)-s,

Re(S)>l

Absolute CM-Periods — Complex and p-Adic

69

be the Dedekind zeta function. The analytic class number formula of Dirichlet-Dedekind states that Res s= iCfc(s) =

|1/2

•

Here r\ and r2 are the numbers of real and imaginary infinite places of K respectively; hx, RK, DK and WK denote the class number, the regulator, the discriminant and the number of roots of unity of K respectively. Using the functional equation, we see that this formula is equivalent to (1.7)

CA-(S)

~ -^£

r i + r a s

-\

a -> 0.

Now let us consider the case where K is a quadratic extension of Q. Let d be the discriminant of K. Define a Dirichlet character \ by

x(») = (z using the Kronecker symbol. Then % is the primitive Dirichlet character to the modulus |d|. We have Ctf(s) = C(*)£(s,x)We distinguish two cases. (I) Assume d > 0. Then —s,

CK(S)

s -+ 0

by (1.7). We have L(0,X)=0,

L'(0,x) =

hKloge0

since £(0) = —1/2, RK — logeo- Here eo denotes the fundamental unit of K. By (1.6), we have d 1

/n\*(a)

"

a=l

^

'

70

H. Yoshida

We can transform the formula (1.8) to traditional Dirichlet's class number formula for a real quadratic field easily, using T(s)r(l — s) = (II) Assume d < 0. By (1.7), we have Cx(0) = - £W* . Hence K

2hx By (1.5), we obtain \d\-\

^ = - S E ^(fl)a-

(i-9)

' ' a=l

This is Dirichlet's class number formula for an imaginary quadratic field. Now it is natural to ask whether a similar expression of the left hand side of (1.8) has a meaning in the imaginary quadratic case. By (1.6), we have

(i.w)

exp (*m) _ i . n r ( ^ i(0,x)7

|d| • " \\d\J

Let E be an elliptic curve defined over Q with complex multiplication by K. Let w be a. period of E. Then the Chowla-Selberg formula states that |d|_1

(1.11)

/ n

r

II (^iJ

\^KX{o)/2hK

-TT-^-TTp^id)2.

Here a ~ 6 for a, 6 6 C means 6 / 0 , a/6 e Q; px is Shimura's period symbol which will be reviewed later. Now let us explain the multiple gamma function. Let r be a positive integer. Let u — (UJ\,<JJ2I • • • > wr)> ^i > 0 for 1 < i < r and x > 0. We define the r-ple zeta function by (1.12)

Cr(s,C0,X)=

(X + VTS.

Yl Cl=miuii +7712^2 H

\-mTu>T

Absolute CM-Periods — Complex and p-Adic

71

Here ( m i , m 2 , . . . , mr) extends over all r-tuples of nonnegative integers. It is known that (,r(s,uj,x) can be continued meromorphically to the whole s-plane and is holomorphic at s = 0. For £ r (0,w, x), we have an elementary formula: if Xfc € R and 5^fc=i ukXk > 0, then (1.13) = (-l)r

Cr( 0, (uUU2, • • • ,Vr),^2ukXk) ^ fc=l '

,ir-iBh(xi)

^,Ii-l,.,la-l 'u'

2_]

Bi2{x2)

1-

h,..Jr>0,h+-+lr=r

2

"

Blr{xr) r

'

Here Bm{x) denotes the Bernoulli polynomial defined by eXt _ v - Bm{x) t _ i ~ Z-,„ m ! e

x

m=0

We put -logpr(u)

= lim \ — (r(s,uj,x) x-*+0 (_ ds

S=Q

+logx>, J

9

r( \ — Cr(s,CJ,X) s =0

Mw)

The function r r (x,u/) is the r-ple gamma function introduced by Barnes ([Ba]). If r = 1, u = 1, we have

ri(x,i) = r(x),

Pl(i)

= V2^.

§2. CM-periods We begin with an example due to A. Weil [W]. Let I be an odd prime. Take an integer a such that 1 < a < (/ — l)/2 and consider a curve C:yl

= xa(l - x).

The genus g of C is g = {I - l)/2. Put C = e 27ri /'. The curve C has the automorphism a : (x,y) > (x,Cy)-

72

H. Yoshida

For x G R, let (x) denote the fractional part of x. satisfies 0 < (x) < 1, x - (x) e Z.

Namely (x)

For t € (Z/7Z) X , we consider a differential form x-1dx.

uH =

Then we can check that u>t is a holomorphic 1-form1 if and only if (at/I) + (t/l) < 1. Under the action of a, ut transforms as (2.1)

a*ut = C V

For the periods of ut, A. Weil showed that (2.2)

[ut~B((at/l),(t/l))

for every

ceHi(R,Z).

Here B denotes the beta function:

Let J be the Jacobian variety of C; J is defined over Q. From the existence of a € Aut(C), we see that End(J) D Z[(\, End(J) Q D Q(C). For t S ( Z / / Z ) x , define a(t) € Gal(Q(C)/Q) by C{t) - CWe put Ta = {t\l
The curve C has singularities except for the case 1 = 3, a = 1. We identify C with the corresponding compact Riemann surface R. By a holomorphic form, we mean a holomorphic form on R.

Absolute CM-Periods — Complex and p-Adic

73

Next let us explain Shimura's period symbol PK- Recall that an algebraic number field K is called a CM-field if it is a totally imaginary quadratic extension of a totally real algebraic number field F. Let K be a CM-field of degree 2n over Q. Let JK be the set of all isomorphisms of K into C and IK be the free abelian group generated by JK- A set $ consisting of elements of JK is called a CM-type if the restrictions of elements of $ to F give all the distinct isomorphisms of F into R. We have necessarily | $ | = n. We often identify $ with ^2ae$ cr € IK and with the representation of K into M(n, C), which it naturally defines. Let A be an abelian variety of dimension n defined over C. We say that A has complex multiplication by K if End(yl) Q D K. Then we obtain a representation $ of K on the space of holomorphic 1-forms. It can be shown that $ is equivalent to the representation which comes from a CM-type of K. In this case, we say that A is of type (K, 3>). A typical example is the complex torus CU/^{OK)If A and A' have complex multiplication by K, they are isogenous if and only if they belong to the same CM-type. Now let $ be a CM-type of K. Then there exists an abelian variety of type (K, $) defined over Q. For every o € $, we have a nonzero holomorphic 1-form rational over Q such that a*uc = aaua,

a € K.

Here we assume that a is in End(^4) and a* denotes its action. Then there exists a CM-period PK(C, $) € C X such that (2.3)

/ uja ~ irpK(v, $)

for

every

c € Hi(A, Z).

We can show that PK{&, $) mod Q does not depend on the choice of A and oja. Now the properties of the period symbol px can be summarized by the following theorem of Shimura. We denote by p the complex conjugation. For an extension L of K, Res L /x : II —> IK denotes the restriction map. IniL/K '• IK —* II denotes the inflation map: it is the homomorphism such that for every a G JK, InfL/K(P) is the sum of all elements in the inverse image of a under the restriction map.

74

H. Yoshida

Theorem 2.1 (Shimura [Sh3], Theorem 32.5). For every CM-field K, there exists a map px • IK X IK —• C x with the following properties. (1) PK(
a = 1 mod x q.

Here t^, cr e <& are nonnegative integers. In Weil's old terminology, A is a Grossencharacter of type AQ. Theorem 2.2 (Shimura [Sh3], Theorem 32.12). Assume that t„ mod 2 is independent of cr 6 $ . Let m be an integer such that m = ta

mod 2,

— ta < m
for every a 6 4>.

Then we have

L(m/2, A) ~ W V C ]£ ta • a, $), where e = m[K : Q]/2 + £ CTG<J) **•

Absolute CM-Periods — Complex and p-Adic

75

Now let us consider Weil's example again. Let J be the Jacobian variety of C : yl = xa(l - x). Put K = Q(C), C = e2ni/l- By (2.1), we see that J is of type (K, $ a ) . Prom the formula (2.2), we have (2.4) irpK(a(t), $ a ) ~ B((at/l), (t/l)) for 1 < a < (I - l)/2, a(t) G $ a . As an example, let / = 7, a = 2. Then we have T2 = {1,2,4}. Let KQ = Q(v / Z 7) C K. Taking o = 2, « = 1 in (2.4), we obtain Ptf(id > Inf* /J r 0 (id)) =PK(id,a(l)+a(2)+a(A))

~ ^iIWl^/I).

By Theorem 2.1, (5), we get

™ rw iH^ r(i/7)r(2/7) 7rp^ 0 (id,id)

——r

.

Using T(s)r(l -s) = g ^ j , we obtain r(l/7)r(2/7)r(3/7)-1r(4/7)r(5/7)-1r(6/7)"1 ~ 7rm(id,id)2, which is the Chowla-Selberg formula for the field KQ. In general, we can calculate PA;(id, a(t)), 1 < t < I — 1 in terms of the gamma function using (2.4). The proof is not very difficult. It is also possible to calculate the period symbol for K = Q(e 27ri /"). In this case, we use factors of the Jacobian variety of the Fermat curve xn + yn = 1. We refer the reader to [Y4], Chapter III, §2. The result is Theorem 2.3 (Anderson [A]). Let K be a CM-field abelian over Q. Put G = Gal(X/Q). Then we have

P ^>~^ n -(T§^).

re*

ueGHere G- denotes the set of all odd 1 -1 M(r) 0

characters of G and i f r = l, ifr = p, otherwise.

In view of this theorem, for non-abelian case, we naturally conjecture:

76

H. Yoshida

Conjecture A (Colmez [C], Yoshida [Y2]). Let K be a CM-field normal over Q. Put G = Gal(K/Q). Let c be a conjugacy class of G. Then we have

weG Here G- denotes the set of equivalence classes of all irreducible odd representations of G, Xw denotes the character of ui and { 1 -1 0

fj,(c)

if

c={l}, ifc={p}, otherwise.

R e m a r k . We remind the reader that p denotes the complex conjugation; p belongs to the center of G. A representation u of G is called odd if u(p) = —id. §3. Shintani's formula Let A = (dij) be an n x r-matrix whose entries a^ are positive real numbers. Let x = *(xi,X2,... ,xr) be a column vector such that x i > 0, 1 < i < r, x ^ 0. Shintani introduced the zeta function oo

(3.1)

C(s,A,x)=

n

Yl

,

r

ai

Il{5Z ^^

mi,...,mr=0 i = l

+m

) ~s

^} •

j'=l

The series converges when Re(s) > n/r and can be continued meromorphically to the whole complex plane. It is holomorphic at s = 0. Let A^> = (an, aj2,..., aj r ) denote the 2-th row of A. Then comparing with (1.12), we see C(S)^),x) = C r ( s , ^ ( i ) , ^ a ^ ) . V J j=1 Here, of course, the left hand side is defined by (3.1) regarding A^1' as an n x 1-matrix. Shintani proved that (3.2)

C(01A1x) = - V C ( 0 , A ( i ) ) x ) , n

•

1

Absolute CM-Periods — Complex and p-Adic

ds

a x

^r(l2j=l

= ^iog:

C(s,A,x) s=0

77

Kaili ai2i • • • > &ir))

ij ji

Pr(an,ai2,

i=\

...

(3-3) n

l

,dir)

IA

3=1

Here I = (h,h, • • • ,lr) extends over all r-tuples of nonnegative integers such that h + k + h lr = r and

Q(A)=

Yl ClJtk(A), l<j,k
ClJ,k(A)

= /

{ I I ( ° j m + akmu)lm-1 *• m—1

-

f j 4m"1} ^ m=l '

The formula (3.3) can be written as d £(s,A,x) ds d

s=0

=E^c(^«,x) + s=0 i=0

(-1)' n

The second term on the right hand side can be regarded as the correction term. The derivative J^C(s, >1, a:)

is decomposed as the s=0

sum of the contribution from every row. (The same observation is true for (3.2), except for the proportionality factor 1/n.) Prom our point of view, this is the most remarkable feature of Shintani's formula. The author generalized it to the second derivative of C(s, A, x) at s — 0 with the proportionality factor n ([Y4], Chapter I, §3). Now let F be a totally real algebraic number field of degree n. Let Jp = {01,02, • • • >0"n} denote the set of all isomorphisms of F into R. We denote by 001, 002, ..., oo n the corresponding archimedean primes of F. Let Ep be the group of all totally positive units of F. We embed F into R n by x - • (xcri,xff2,..., x°n). Then Ep~ acts on R " by componentwise multiplication, where R+ is the set of all

78

H. Yoshida

positive real numbers. For linearly independent vectors v-y, V2, • •., vr, we set C(vi,V2,...,Vr)

= I Y^^i

h,t2,...,tr

> 0>

^ i=l

>

and call it an r -dimensional open simplicial cone. Theorem 3.1 (Shintani [Shi]). open simplicial cones Cj = C(vjU Vj2, ..., fjr(j)),

There exist finitely many v

ji € Op

such that Here J is a finite set of indices. We call this decomposition a cone decomposition of R™. E x a m p l e . Let F be a real quadratic field. Then E^ = Z. The generator e > 1 of Ep~ is called the totally positive fundamental unit of F. A cone decomposition of R+ is given by taking Ci = C(l,e),

C 2 = C(1),

J ={1,2}.

Let f be an integral ideal of F. We denote by Cf the ideal class group modulo fooioo2 • • • oo„ of F. The conductor of this ideal class group is not necessarily equal to fooioo2 • • • oo n . The following formula for the order of Cf is well known in class field theory.

[EF

Here <£>(f) = iV(f) JTpif (1 — MSy)

• -E^fooi-ooJ ls

* n e Euler function and

£>,fooi-oon = {e € Ep~ | e = 1 mod f}. For c € Cf, let

Cf(S,c) = 5]iV(a)- s , age

Re( S )>l

Absolute CM-Periods — Complex and p-Adic

79

be the partial zeta function of the class c. Here a extends over integral ideals. This partial zeta function has meromorphic continuation to the whole complex plane, holomorphic except for a simple pole at s = 1. Let oi, 02, . . . , <Xh0 be integral ideals which represent the narrow ideal classes of F. For a fractional ideal a of F, we set r

r(j)

R(Cj, a) = < z € a n Cj z — ^

Xi(z)vji,

Xi(z) E Q, 0 < Xi(z) < 1, 1 < i < r(j) i . This is a finite set, being contained in the intersection of a compact subset of Cj with the discrete set o. For z € R(Cj,a), define the column vector x(z) by x(z) = t(xi(z),X2(z),..., xr^ Take a class c € Cf. We take aM so that c and oMf belong to the same narrow ideal class. Put R(Cjt c) = {ze R{Cj, (o^f)" 1 ) | (z)alij = c in Cf}. For Cj, we attach an n x r(j)-matrix Aj = (fjj1); the (I, z)-component of Aj is v°i\. Then we have (3.4)

(;F(s,c) = N(allf)-sY,

E

C(5,^-,x(z)).

Using (3.2) and (1.13), we have a closed formula of CF(0, C). Using (3.3), we have an expression of CF(0> C ) m terms of the multiple gamma functions. §4. Absolute CM-period symbol We understand that an algebraic number field is a subfield of C, which is of finite degree over Q. Let F be a totally real algebraic number field of degree n. We set Jp = {cri, oi, • • •, crn}. Proposition 4.1. Let K be a CM-Geld which is abelian over F. Put G = Gal(K/F). Let G denote the set of all characters and

80

H. Yoshida

G- the set of all odd characters of G. Assume Conjecture A. Then, for T € G, we have

(4.D

iW,,™,)-,-*>/> n ex P (#£M).

Here ^i(r) is denned by the same formula as in Theorem 2.3; we use the same letter U{ for its extension to an element of JK- By Theorem 2.1, (6), we see that Pxi^i^^i) mod Q does not depend on the choice of this extension. We also note that PK(VU

roi) ~ pK°i (id, o~xroi)

which follows from Theorem 2.1, (6). Now we are going to factorize the right hand side of (4.1). Let f be an arbitrary integral ideal of F. We choose G\ = id. For c € Cj, we put

(4.2)

G(c) = J2 E

lo T^A^)

j€JzeR(Cj,c)

§

j_

Pr(j)(Aj

)

Note that z G R(Cj,c) is a positive real number and that A(vji,Vj2, • • •, vjr(j)) is the first row of Aj.

V(c) = X)(-l) rW E ^ E ( C j , i l f c ( ^ ) n

(4.3)

=

+ Cj, M (^))

fc=2

~2

E

^Afc^j)

X

V

T7

«Gfl(Cj-,c)m=l

(4.4)

W(c) = --logiV(a M f)CF(0,c),

(4.5)

X(c) = G(c) + F(c) + W(c).

B

lm(xrn(z)) m

'

In (4.3), I = (li,h, • • •, lr(j)) extends over all r(j)-tuples of nonnegative integers such that l\ +12 + • • • + lr(j) — r(j). These quantities depend upon the choice of a cone decomposition {Cj}je j and the choice of the representatives {oM}. We write them as X(c;{Cj}j€j,afj), etc., when we have to be precise. Now we have

Absolute CM-Periods — Complex and p-Adic

81

Theorem 4.2.

C&JF

From our point of view, this theorem expresses the most remarkable feature of Shintani's formula reviewed in §3. Concerning the V-term, we have Theorem 4.3.

We can write V(c) = YALI

a

i l°g e i w^n

a

i

e

F>

- E1+ ei€E •i c ^p •

In [Y4], Chapter IV, §6, we proved the assertion with F in pi[ace of F, where F is the normal closure of F over Q. Theorem 4.3 was \v» left as Conjecture 6.5. The author proved it in collaboration with T. Kashio. Let K be as in Proposition 4.1 and let fooioo2 • • • oo„ be the conductor of K over F. By class field theory, we can regard x € G as a character of Cj. For x S G, let f(x) denote the finite part of the conductor of X- We have

xeG_

c€G

xeG_

Kx)

For r € G, we define (4.6) ^ ( i d , r ) = T - " W / a e x p ( 1 l [ £

^X-

xeG_

£ '

c

x(c)*(c)).

ec f ( x )

When we have to be precise, we write it as <7£-(id,r; {Cj}j
(4.7) UpK'i^a^rai) i=i

n

~ Hgie^,^1™*

{Cf } j e J , {<%}).

t=i

Now it is natural to conjecture a stronger version of this formula.

82

H. Yoshida

Conjecture B. Let K be an abelian extension of a totally real algebraic number Held F. Put G = Gal(K/F). We assume that K is a CM-Md. For r G G, define gK(id, r) by (4.6). Then PA"(id,r) ~

gK(id,r).

We stress the importance of the factorization process described above; not only that Conjecture B is stronger than Conjecture A, it suggests possibilities of further generalizations. We can show that the validity of Conjecture B does not depend on the choices of {CJ}J^J and {a M }. We call QR the absolute CMperiod symbol. Strictly speaking, gK (id, r) depends also on the choice of the field F and should be written as gK/Fi}^-,T! {Cj}jeJ, {a/j.}) m the full notation. For example, if K is abelian over F\ and F2 and r G Gal(.fiT/.Fi) n Gal(K/F2), it is a highly nontrivial problem to show that SK/i?!(id,r) ~ gK/F2(id,r). For partial results, see [Y4], Chapter III, §3.8. Conjecture B, if true, is powerful enough to give values of PR symbol of Shimura modulo algebraic numbers. In fact, let us consider r VK{PI T), CT,T G JR f° an arbitrary CM-field K. Let L be the normal closure of K over Q and choose a € JL SO that Res£/#-(c7) = a. By Theorem 2.1, (5), we have PR(O-,T) ~ p^(a,Inf^/x(r)). Put ^^L/K{T) = YliTi> Ti € JL- By Theorem 2.1, (6), we have pL(a,IniL/K(T))

~ J^JpL(5,Ti) ~ I^PL^d,? - 1 -/*). i

i

Let Fi be the fixed field by the group generated by T» and p. Then Fj is totally real and L is abelian over Fj. Therefore Conjecture B can predict the value of £>z,(id, CT-1TJ). Now we are going to state a refinement of Conjecture B which is amenable to numerical tests. In view of Theorem 2.2, we can find an integral ideal q and homomorphisms Ai, A2, ..., \ of Iq(K) into C x and integers ei, 62, ..., eq so that q

r ei

A

J J L ( m / 2 , Xi) ~ TT l[pK*i i=l

i=l

holds with en G Z, n G G, A G 2 _ 1 Z.

(id,

o^Tm)*

Absolute CM-Periods — Complex and p-Adic Conjecture C.

83

For a G Gal(Q/Q), we have nLi£(™/2,AQ«

A

« n i = i 9K*i (id, a," V i J {Cf }, {a**})* = <(*)•

T A m=i S*-i (id, a~ V W ! {Cf17}, { C } ) " '

where £(er) is a root of unity. It seems that we can control the denominator of the quotient in this Conjecture if e* > 0, 1 < i < r. See [Y4], p. 108, Remark 4.5. The determination of £ is a very interesting problem, the solution of which may be called a reciprocity law. For partial results, see [Y2]. §5. Numerical examples Let F = Q(\/d), 0 < d € Q be a real quadratic field. We take a totally positive element x + yVd e F , i , i/ 6 Q, |/ 5^ 0 and put £ = ^Jx + yVdi, £' = y/x - yVdi. Then K = Q(£) is a CM-field of degree 4. We define a G JK by g* — £'. Then $ = {id, a} is a CM-type of K. Let q be an integral ideal of K and we consider Grossencharacters of K such that

Ai>»=(^f(^y.

« = 1 mod*,.

Here a and 6 are positive integers such that a = b mod 2. We put Q_

L(l,A$)L(l,Ag) 7r 5 i f (id,id) 4 5 ^(id,id) 4 ' 8

L(l,Ag)L(l,A
L(l,A^)L(l,Ag) 10

7r ^(id,id)V"(id,id)8"

84

H. Yoshida

Assume that q = (1), hx = 1- Then, for a G Gal(Q/Q), Conjecture C states that Qa = CQ, Q? = CiQi, Q? = C1Q2, a

a

where £ = £( )> Ci = Ci( ) depend on a.

Q2=C2Q2,

i f a | F = id,

Q? = C2Q1,

if « | F ^ id,

an

d C2 = £2(0:) are roots of unity which

Now take F = Q(\/29). Then e0 = ^ ^ is the fundamental unit and e = e2, = 27 +5v 29 j s the totally positive fundamental unit. The narrow class number of F is 1. We take (1) as the representative for the narrow ideal class. We take C\ = C(l,e), C2 = C(l) for the cone decomposition. Let K = Ql y 9 +v 29 ^ J ^he conductor of if over F is fooioo2, f = ( £ ± ^ 1 ) . We have hK = 1, JV(f) = 13. By numerical computation, we find ^

292 '

2935 + 503\/29 _ _ 9 / 1 3 2935 - 503^29 2 Q2 C ~ ' 13-29 ' ' 13~292 ' Both sides coincide to the 50th decimal place. We have Cf = {c\, C2}, c\ = 1. For the V-term, we have 9/13

Ql €

V(ci)-y(c2) = -£p/291oge and the congruence

9 - ^ 2 9 = 0 modf,

f=(9

^ \

In this example, we have ((a) = 1, d (a) = (69/i3)
if a

j

F=

id)

ifa|F#id,

Absolute CM-Periods — Complex and p-Adic (2(0:) = Ci( a )

1

85

- We see that they are 13th roots of unity.

§6. Reflection on periods and units In this section, we will show that the invariant X(c) contains information of CM-periods and units of number fields. Let us first recall a conjecture of Stark-Shintani. Let F be a totally real algebraic number field of degree n > 2. We assume that 001 corresponds to the identical embedding of F into R. Let M be an abelian extension of F whose conductor is of the form foo2 • • • oo„. In other words, if <7j is the embedding F c R corresponding to oc>i and if a, is any extension of o-{ to an embedding M C C, we have <7i(M) C R and at(M) £ R for 2 < i < n. For a G Gal(M/F), let

((s,a)=

J2

Na

( )~"

be the partial zeta function attached to a. Here (—5—) denotes the Artin symbol. Conjecture of Stark-Shintani. exists a unit u € EM such that exp(-2C^(0, a)) = uc

In the above situation, there

for every a € Gal(M/F).

Now let F be a real quadratic field and f be an integral ideal. For i = l , 2 , we can choose v\ 6 OF SO that i/j = 1 mod f, vf' < 0 < v\ if j 7^ i. Let S{ € Cj be the class of the ideal (vi). A character x oi Cj ramifies at ooi if and only if x(si) = ~ 1Theorem 6.1. — 1, we have

Let eo be the fundamental unit ofF. IfN(eo)

=

X(c; {Cj}, dp) + X(cs2; {Cj}, o^) = aloge 0 with a e Q. If N(eo) = 1, write F = Q(Vd) with a square free integer d. Then we have X(c; {Cj}, a,») + X(cs2; {Cj}, (Vd)a^) = aloge 0 with a € Q.

H. Yoshida

86

Note that the narrow ideal class of S2 is trivial if and only if iV(eo) = — 1. We can give an explicit formula for the value of a. For the proof, see [Y4], Chapter III, Theorems 5.8 and 5.12. For c e Cf, we put X(c) = X(c]{Cj},aJ,

X(r, ^ = I

X(CS2; {C

if N

^ ^

= _1

^

'

*{CS2) \ X(cs2; {Cj}, (Vd)a„) if N(e0) = 1. Now let L be the maximal ray class field modulo fooioo2- Then we have Gal(L/F) = Cf. (We do not assume that the conductor of L over F is equal to fooioc>2.) We can show that the maximal totally real subfield FQ of L corresponds to the subgroup generated by si and S2- There exists the maximal CM-subfield K of L if and only if there exists a character of Cf which ramifies at ooi and 002. In this case, K corresponds to the subgroup generated by s\S2Let Mi be the fixed field of si for i = 1, 2. Then OOJ is unramified in Mi. Assume that the conductor of Mi over F is foo2- For c G Cf, the Stark-Shintani conjecture says (

(6.1)

exp(-2(C7F(0>c) + Ci-(0,cfli))) = «V

MT/F\

c

J

with a unit u of Mi. Let a be the nontrivial automorphism of F. Let E stand for the term of the form alogeo with a G Q. Put X(cff) = X(c C T ;{C J },ap,

ceCf.

Using Theorem 4.2, we can show that X ( C ) + X ( O = CF(0,C) + £7.

Now (6.1) can be written as X(c) + X{cs1) + X(ca) + X(c(7sl) = -^\ogUy

A

c c

/J+E.

Here the sum of the last two terms of the left hand side is of the form E by Theorem 6.1. Therefore, assuming (6.1), we obtain for c 6 Cf that (6.2)

X{c) + X(cSl) =

-^logu(^+E,

Absolute CM-Periods — Complex and p-Adic (6.3)

87

X(c) + X(cs2) = E.

On the other hand, it is not difficult to see that L contains the maximal CM-subfield if and only if si ^ 1. In this case, our Conjecture B says that the quantity X{c) + X(csiS2) is related to CMperiods. Thus our invariants X(c) contain information of CM-periods and the Stark-Shintani units in a clear-cut manner. On reflecting this situation, we conclude that the periods and units (algebraic numbers) are tied together very closely. This will become even more clear if we develop the p-adic theory. Remark. A careful reader may be curious about the asymmetry of oo i and 002. This is because we consider F as a subfield of R, hence 001 is the privileged infinite place.

Part II. p-Adic Theory §7. A p-adic analogue of Shintani's formula (Kashio's formula) Let p be a prime. For simplicity, we assume that p is odd. Let C p be the completion of an algebraic closure Q p of Q p . We normalize the absolute value | | p of Cp so that \p\p = p _ 1 . Put U = {u e Cp I |u| p = 1},

D = {u e Cp | \u - l\p < 1}.

Let V be the subgroup of U consisting of all roots of unity whose orders are prime to p. Then we have the direct product decomposition

U=

VxD.

For u E U, let 6p(u) be the projection of u to V. Let tp be a function on Zp taking values in Cp. We define the p-adic integration of ip by p'1-1

plr-l

ni=0

nr=0

whenever the limit exists. Let Op = {x 6 Cp \ \x\p < 1} be the ring of integers of Cp. Take u = (ui,..., u; r ), a;, G Op, ui / 0, x € Op. We consider a linear form r

f(X) = J2"iXi + x, t=l

X = (Xu...,Xr)

H. Yoshida and for s 6 Z p , define a Cp-valued function on Z„ by

,(ow ,N _ J F(xy{ep(f{x))-if(x))-

if /(x) e op,

J

if/(X)£0p.

l

, j

\0

We define the p-adic r-ple zeta function by Cp,r(s,U},x)

=

(-iyi(f(°)(x,s))

(r — s)(r — 1 — s) • • • (1 — s)u>i • • • uir

In analogy to the complex case, we define the p-adic logarithmic r-ple gamma function by LFPtr(x,Uj)

= —

Cp,r(s,UJ,x)

°S

s=0

We have Lr P ) 1 (x,l) = log p r p (:r). Where logp is the Iwasawa p-adic logarithmic function and Tp is the Morita p-adic gamma function. We can extend the domain of definition of £Ptr(s,u,x) and LTp^r(x,uj) to a; € C p , u 6 ( C * ) r in the following way. Suppose that x £ Ov or there exists Ui such that u>i $. Op. We take c 6 Cp so that ex € Op,

cu>i E Op,

1< i < r

and that |c| p takes the maximum value among the elements satisfying this condition. We write c = p°"u,

a E Q,

uEU

and put Cp,r(s,W,x) =

(8p(u)-1u)S(p:r(s,CU},Cx).

This is well defined and we obtain the extension of the domain of definition. A similar procedure can be applied to LTPir(x,uj). Now let F be a totally real algebraic number field. Let f be an integral ideal of F which is divisible by every prime factor of (p). For c E Cj, taking an ideal a E c, we put cu(c) = ep(N(a)).

Absolute CM-Periods — Complex and p-Adic

89

As shown by Barsky, Cassou-Nogues [Ca] and Deligne-Ribet [DR], there exists the p-adic partial zeta function (P,F(S, C). It is analytic except for a possible simple pole at s = 1 and characterized by the interpolation property CP,F(-m, c) = uj(c)-m(F(-m,

c),

0 < m E Z.

We fix an embedding F C Q C Cp. Theorem 7.1 (Kashio [K]). Let f be an integral ideal of F which is divisible by every prime factor of (p). Let F be the normal closure of F over Q and let ex, ..., em be generators of E%. Write t

CTGJF jeJ

zeR(CjlC)

rr(3)\3)

m i=l

with en e F. Here Vj — (VJX, • • • ,^>(j)) = ^ Then we have

^(0,c)=X;E C^JF

i&J

E

and v? = {A"- ')"'.

LTpAj)(z°,v°)

Z&R(CJ,C)

m

+ E fl*

lo

Sp €i ~ l0§P N(a^KP,F

(0, C).

§8. p-adic absolute CM-period symbol We fix an embedding F C Cp and let p be the prime ideal of F induced by this embedding. Let f be an integral ideal of F. For c E Cf, we define

(8.1)

Gp(c) = Y, j€J

E

LTpAj){z,Vj).

z€R(Cj,c)

Taking generators ei, ..., e„_i of Ep~, write V(c) in the form V(c) — Y^i=l ai ^°S ei w * t n ai e F- By a theorem of Baker, this expression

90

H. Yoshida

is unique. We define n-l

(8.2)

y

Vp(c) =

£2ailogpei.

i=l

For a fractional ideal a of F , we define k>gp(a) and log (a) in the following way. Let ho be the class number of F in narrow sense. Let q be a prime ideal. We write qh° = (/3) with a totally positive element /3. Then we define logp(q) = ^ - logp (3,

log(q) = — log (3.

For a E Jp, we formally define log p (q a ) = ^ logp /T,

log(qCT) = i - log /T.

(It may happen that q = qCT, but logq / logqCT. We regard logqCT as the symbol attached to q and a.) We extend this definition to fractional ideals so that \ogp(a
Wp(c) = - l 0 g p ( o M ) • (CF(0,C) - C F C O . P " ^ ) ) .

Here p is the class of p in Cj if p \ f; if p | f, the term (jr(0,p _1 c) should be dropped. For Wp(c), this definition leaves the ambiguity of the addition of elements of the form alog p e, a 6 Q, e e Ep. We put (8.4)

Xp(c) = Gp(c) + Vp(c) + Wp(c).

If f is divisible by every prime factor of (p), Theorem 7.1 can be written as

(8.5)

C;,F(O, C) =

Yl XP^)

- iogp N(f)CpAo,
Absolute CM-Periods — Complex and p-Adic

91

Now let K be a CM-field which is abelian over F. We fix an embedding K C C p and let ^ be the prime ideal induced by this embedding. We put

y>hK = (a). We define the (logarithmic) p-adic absolute CM-period symbol by /feJf(id,r) = - ^ l o g p ( p ) (8.6) xeG_

ceG

f(x)P

To state a precise conjecture on Z<7p7JFc(id, r ) , we define the logarithmic version of the absolute CM-period symbol: ^(id,r;{Ci},{aM}) = - ^ l o g 7 r

(8 ?)

'

4 E £ E

X(C)X(C;{C,})K}).

Here we modify the W-term in the definition of X(c) by W(c) = -log(a M f)CF(0,c). Prom [Y4], Chapter III, §3.6, §3.7, it follows that n-l

(8.8) lgK(id, r; {Cj}, {a^p}) - lgK(id, r; {C,-}, {aM}) = ^

Oi loge*

with generators e$ of £ ^ and a, 6 Q. Now we can state Conjecture P. Let e\, €2, •.., en-i be generators of E~p. Ifp is completely decomposed in K, then we have 1

lgp,K(id, T) = —— log (aT~lp/aT with a* € Q defined by (8.8).

n— 1

) + V a* log e*

92

H. Yoshida

R e m a r k . We can define another version of the p-adic absolute CM-period symbol, replacing J2cec by Y^cec • Both symbols IvX^P

1vX/

have almost same properties, but the version defined by (8.6) is somewhat simpler. Now let us recall a conjecture of Gross [G], which can be regarded as a p-adic analogue of the Stark-Shintani conjecture. Conjecture of Gross. Let F, K and the other notations be as above. Let pi, p2, ..., p r be all the prime ideals of F lying above p. We assume that p = pi splits completely in K. Then, tor x & G-, we have

^mw

=

n(1 - M

i

)

£ * ( r ) I°S P (^/Q> T 7« T ))-

We can show that Conjecture P implies the Gross conjecture. Moreover it produces interesting algebraic numbers (Weil numbers) even when L'p(0,ux) = 0. §9. Numerical examples (p-adic case) As in §5, we take F = Q(V29), K = Q[\f^^i)

and use the

same notation. Let x be the character of Cf which corresponds to the extension K/F. We write p = (ceo) with a totally positive aoSince L(0, x) = 2, (8.6) can be written as (9.1)

lgP,K(id, id) = - - logp ao + - J ^

x(c)Xp(c).

First take p = 79. Then p remains prime in F and decomposes completely in K as pOx — («)(o ;P ) with 1 + V^I

9 + V29.

« =^ — ( V ^ —^+

3 + \/29\

By (9.1) and ao = p, logpp = 0, we have teP,A-(id,id) = - Yl cef(P)

.

n

,—.

^ J -(5 + 2V^9).

x(c)Xp(c).

Absolute CM-Periods — Complex and p-Adic

93

We can check easily that the correction term (8.8) is 0. Hence Conjecture P states that

(9.2)

Z^*(id,id) = ilog p (a'ya).

Both sides are elements of OFP and we verified, by numerical computation, that they are equal modulo p 5 0 . Let a is the nontrivial automorphism of F. Since a gives a continuous automorphism of Fp, (9.2) yields £ X(c)Xp(c) cef(p)

= 21ogpK/a),

£ X(c)Xp(c°) cef(p)

=

2logp(a»°/a°).

By (8.5), we have L'p(0,Xu)=

£

x(c)(Xp(c)

+

Xp(c°)).

c€f(p)

Therefore Conjecture P gives L'p(0,Xu)

=

21ogp(NKv/Qp{af>/a)).

This is what the Gross conjecture states in this case. Conjecture P is finer since it gives a factorization of logp(Nx / Q (ap/a)). Next let p — 23; p decomposes in F as POF = pp' with

P=

/ l l + \/29\ ^ ,

,

/11-V29 P='

Then both of p and p' decompose completely in K. The Gross conjecture states LP(0,UX)

= LP(0,LOX)

= 0,

which is a special case of a theorem proved in [K]. We may take a =

\

/9 + V 2 9 . n ^ « + l,

ceo =

l l + \/29

•

94

H. Yoshida

We can show that the correction term (8.8) is equal to -\\ogeJ2x(c)

Yl

c€Cf

lWC2(0,(l,a0),*))

zefl(C(l,a 0 ),c)

109

,

Hence Conjecture P states that (9.3)

/ffP,x(id, id) = - \ogp{ap/a)

-

2

.13

logp e.

Both sides are elements of Z p ; by numerical computation, we verified that the equality holds modulo §10. Interpretation of lgPiK in terms of p-adic periods We wish to know the nature of the p-adic absolute period symbol ^<7p,tf(id>T)- Conjecture P clarifies it when p is completely decomposed in K. Now our problem is to drop this assumption. For the formulation of the conjecture in general case, it seems most advantageous to consider motives attached to Grossencharacters of type AQ. In fact, in the complex case, Conjecture C is formulated with respect to the critical values of the L-functions attached to such Grossencharacters, which eventually turns out to be amenable to numerical tests. Let K be a CM-field. Let x be a Grossencharacter of K of conductor f such that X((a))=

n

*(<*)'*«

a

=

l

modX

f-

a£jK

Here la are integers such that la + lap are independent of a. Put E = Q(x(<*) | a is a fractional ideal prime to f). Then E is an algebraic number field of finite degree. There exists a motive M — M(x) over K with coefficients in E; M is characterized by the property L(M,s) = (L( S ,<7(x))W £ ,

JE = Hom(£,C).

Absolute CM-Periods — Complex and p-Adic

95

We consider four realizations of M; de Rham, p-adic, Betti and crystalline. First M has the de Rham realization Hv&(M) which is a free E ® Q X-module of rank 1. For a finite place A of E, M has the A-adic realization H\(M) which is an S^-module of rank 1. We have an £Vlinear action of Ga\(K/K) on H\(M). It is related to the p-adic realization by HP(M) = © A|p

HX(M).

Let Q be the algebraic closure of Q in C. We fix an embedding of Q into C p and let *# be the prime ideal of K induced by this embedding. Below we take K<# as the basic local field. Using K C Q C C, we have the Betti realization HB(M) which is an ^-module of rank 1. We have the canonical isomorphism (10.1)

ip : HB(M) Q Qp 2* HP(M)

as E®Q Qp-modules of rank 1. The de Rham and p-adic realizations are related in the following way. Let 7?DR be the discrete valuation field introduced by Fontaine [Fo]. We have (10.2)

7 D R : HP(M) ® Q p BBR 9* HBR{M)

®K

BBR.

Both sides are isomorphic as free E ® Q SDR-modules of rank 1. The isomorphism is compatible with the action of Gal(Q p /Ky) and nitrations. Suppose that x i s unramified at ^3. Then M has the crystalline realization Hcris(M). Let k be the residue field of K<$ and -ftT
7 cris : HP(M) ® Q p B c r i s ^ Hct]a(M) ®w(k)

Bcris.

Both sides are isomorphic as free E <8>Q B cr i s -modules of rank 1. The isomorphism is compatible with the action of Gal(Q p /Ky) and

96

H. Yoshida

Frobenius. Here B cr ; s denotes the subring of BDR introduced by Fontaine. Let ip G Gal(Q p /Q p ) be a Frobenius and let

There is a i^tp^-structure on H D R ( M ) <S>K Ky$ which we denote by HBK(M/K^0)We have (10.5) (10.6)

70 : ^cris(M) ®w(k) KVfl ^

HDK(M/K^0),

7 D R = (io)DR ° (-fcris)DR-

Here (ICris)DR denotes the isomorphism obtained by taking ®Bcris-BDR in (10.3) and (JO)DR denotes the isomorphism by taking ®Km 0-^DR in (10.5). Hereafter we assume that ifQ Q p -module. Take a generator CDR € HBR(M) as an E Cg>Q /^-module. Then we define the period P(x) by (10.7)

/DR(* P (CB)

® 1) = P(x)(cDR

® 1).

We have P(x) £ (-E-^Q-BDR)*; it is determined up to the multiplication of elements of (E®Q K)x. Here we regard E®Q K as a subring of E ® Q -BDR- P(X) is the p-adic period in the standard sense. We can choose a generator ccris of Hcris(M) as an E ®z W(k)module so that (10.8)

7o(Ccris®l)=CDR®l-

We define P(X) G (E ® Q Bcris)x (10.9)

by

) ® 1) = P(x)(Ccris<8) ! ) •

Absolute CM-Periods — Complex and p-Adic

97

Since 5 c r i s c BDR, we can regard P(x) € (£<8>Q # D R ) X - Then, by (10.6) and (10.9), we have (10.10)

P(x) = P(x)-

Since F c r i s (M, W(k)) ®w(k) ^Q if^o-module of rank 1, we can write V i (c C ri S ®l) = g ( i ) ( c c r i s ® l ) ,

(10.11)

w i t h Q ^ € (E®QK
K i e Z

Applying the Frobenius i-times on (10.9),

* c r i s ( £ ( x ) V ( C c r i s ® 1) = F ( x ) ( c C r i s ® 1)-

Here $Cris denotes the action of Frobenius on -BCris- Therefore we obtain (10.12)


l
We put Q = Q(l>. In particular, we have (10.13)

$cris(P{x))Q

=

P{x)-

Applying <&cris on both sides noting that $Cris acts on Kqj,o by the absolute Frobenius
= $cris(P(x)) = H x ^

•

Hence we have Repeating this process, we get (10.14)

g « = ^-\Q)

• • •
1 < i € Z.

We note that (10.15)

QW = x(W ® 1 € {E ® Q

which follows from (10.14).

KVfi)x

98

H. Yoshida

Using QW, we can predict the nature of lgPtK in the general case. Suppose that K is abelian over a totally real number field F as in our construction of lgp,x- Take r G Gal(K/F). We write lgPtK/F(^T) for lgp,K (id, r ) . We have

Q^ e (E®Q K%0)x C (E®QQp~)x li J ]

Q^.

ITIECQ

Let Q^\cr) e Q p X denote the <j-component of QW. Let p be the prime ideal of F below ^3 and let /o be the degree of p over Q. We take a Hecke character x °f the form X((a)) = ( r ( a ) / p r ( a ) ) ' ,

a s l

mod x f

with 1 < I G Z. Let if be the normal closure of K over Q. We can take / and x so that E
We have

l9P,K/F(id, r _ 1 ) = - — logp Q ( / o ) (id) + J ^ a» logp e, + logp 6 i

with a, e F , £j € .E^, 6 G K. Since -P(x) is determined up to the multiplication of elements in x (E<S)Q K) , Q^°) is determined up to the multiplication of elements of the form
We can show, using (10.15), that Conjecture P implies that Conjecture Q is true with b = 1 if p is completely decomposed in K. The p-adic periods can be understood more explicitly in this case, but we will not go into details here. Remark. What is remarkable in the p-adic case is that the nature of the p-adic absolute period symbol lgPtK/F{^LiT) depends strongly on F. (This is not so when p is completely decomposed in K.) In the complex case, p^-(id, T)21 appears in place of Q^°)(id) - 1 . The author thanks Professor Don Blasius for discussions which are useful in writing §10.

Absolute CM-Periods — Complex and p-Adic

99

References [A] W. Anderson, Logarithmic derivatives of Dirichlet L-functions and the periods of abelian varieties, Comp. Math. 45(1982), 315-332. [Ba] E. W. Barnes, On the theory of the multiple gamma function, Trans. Cambridge Philos. Soc. 19 (1904), 374-425. [Bll] D. Blasius, On the critical values of Hecke L-series, Ann. of Math. 124(1986), 23-63. [B12] D. Blasius, A p-adic property of Hodge classes on abelian variety, Proc. Symposia Pure Math. 55(1994), Part 2, 293-308. [BO] P. Berthlot and A. Ogus, F-isocrystals and De Rham cohomology I, Inv. Math., 72(1983), 159-199. [Ca] P. Cassou-Nogues, Valeurs aux entiers negative des fonction zeta et fonction zeta p-adiques, Inv. Math. 51(1979), 29-59. [Co] P. Colmez, Periods des varietes abeliennes a multiplication complexe, Ann. of Math. 138(1993), 625-683. [DR] P. Deligne and K. A. Ribet, Values of abelian L-function at negative integers over totally real fields, Inv. Math. 59(1980), 227286. [DS] E. de Shalit, On monomial relations between p-adic periods, Journal fur reine und angew. Math., 374(1987), 193-207. [Fa] G. Faltings, Crystalline cohomology and p-adic Galois representations, Algebraic ananlysis, and Number theory (J. I. Igusa, ed.), Johns Hopkins Univ. Press, 1990, pp. 25-79. [Fo] J. -M. Fontaine, Sur certains types de representations p-adiques du grou-pe de Galois d'un corps local; construction d'un anneaux de Barsotti-Tate, Ann. of Math., 115(1982), 529-577. [Gi] R. Gillard, Relations monomiales entre periodes p-adiques, Invent. Ma-th. 93(1988), 355-381. [Gr] B. H. Gross, p-adic Z-series at s — 0, J. Fac. Sci. Univ. Tokyo 28(1981), 979-994.

100

H. Yoshida

[GK] B. H. Gross and N. Koblitz, Gauss sums and the p-adic T-function, Ann. Math., 109(1979), 569-581. [I] K. Iwasawa, Lectures on p-adic L-functions, Annals of Math. Studies, 74, Princeton University Press, 1972. [JKS] U. Jannsen, S. Kleiman and J-P. Serre (eds.), Motives, Proc. Symposia Pure Math. 55(1994), Part 1 and 2. [K] T. Kashio, On a p-adic analogue of Shintani's formula, J. Math. Kyoto Univ. 45(2005), 99-128. [Kat] N. Katz, p-adic L-functions for CM-fields, Invent. Math. 49(1978), 199-297. [KYI] T. Kashio and H. Yoshida, On the p-adic absolute CM-period symbol, Algebra and Number theory (ed. R. Tandon), Proceedings of the Silver Jubilee Conference, University of Hyderabad, 2005, 359-399, Hindustan Book Agency. [KY2] T. Kashio and H. Yoshida, On p-adic absolute CM-periods, in preparation. [O] A. Ogus, A p-adic analogue of the Chowla-Selberg formula, Lecture notes in Math., 1454, 319-341, Springer Verlag, 1990. [SI] G. Shimura, Automorphic forms and periods of abelian varieties, J. Math. Soc. Japan 31(1979), 561-592 (=Collected Papers III, [79a]). [S2] G. Shimura, The arithmetic of certain zeta functions and automorphic forms on orthogonal groups, Ann. of Math. 111(1980), 313-375 (=Collected Papers III, [80]). [S3] G. Shimura, Abelian varieties with complex multiplication and modular functions, Princeton Mathematical Series 46, Princeton University Press, 1998. [Shi] T. Shintani, On evaluation of zeta functions of totally real algebraic number fields at non-positive integers, J. Fac. Sci. Univ. Tokyo 23(1976), 393-417. [Sh2] T. Shintani, On values at s = 1 of certain L functions of totally real algebraic number fields, in Algebraic Number Theory, Proc. International Symp., Kyoto, 1976, 201-212, Kinokuniya, 1977.

Absolute CM-Periods — Complex and p-Adic

101

[Sh3] T. Shintani, On certain ray class invariants of real quadratic fields, J. Math. Soc. Japan 30(1978), 139-167. [St] H. M. Stark, L-functions at s — 1,1, II, III, IV, Advances in Math. 7(1971), 301-343; 17(1975), 60-92; 22(1976), 64-84; 35(1980), 197-235. [W] A. Weil, Sur les periodes des integrates abeliennes, Comm. on Pure and Appl. Math. XXIX (1976), 813-819 (^Collected Papers III, [1976b]). [Yl] H. Yoshida, On the zeta functions of Shimura varieties and periods of Hilbert modular forms, Duke Math. J. 75(1994), 121-191. [Y2] H. Yoshida, On absolute CM-periods, Proc. Math. 66, Part 1, 1999, 221-278.

Symposia Pure

[Y3] H. Yoshida, On absolute CM-periods II, Amer. J. Math. 120(1998), 1199-1236. [Y4] H. Yoshida, Absolute CM-periods, Mathematical Surveys and Monographs 106, American Mathematical Society, 2003.

Hiroyuki YOSHIDA Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan Email: [email protected]

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Special Zeta Values in Positive Characteristic Jing Y U

1

Introduction

This is the story of certain zeta values in positive characteristic. The arithmetic here starts with the ring A := ¥q[9], where F 9 is the finite field of q elements with q a power of prime number p, and 9 is a variable. This ring A plays the role of Z. It sits discretely inside F g ((^)) with respect to the ^-adic topology just as Z is discrete subset on the real line R. Moreover one has that F q ( ( | ) ) / A is compact as against the fact that Z is cocompact inside R. The number q — 1 is particularly relevant since there are q — 1 signs (i.e. units in A) in this positive characteristic world. In the simplest case q = 2, it is a world where sign does not matter. On the other hand for q > 2, one also needs the set of all monic polynomials in A to be denoted by A+. In 1935 L. Carlitz [4] initiated the study of the following zeta values: (1.1)

a(n):= £

^eF,(Q)Y

n = l,2,....

These special values are natural analogues of the classical Riemann zeta values Q{n) at integer n > 1. They contain very interesting information about deeper structures of the arithmetic of function fields, cf. [7] and [9]. Carlitz made the discovery that there is a constant 7TA algebraic over F g ((|)) such that C,\{n)/ir\ := bn falls in k := ¥q(6) if n is divisible by q — 1. This constant 7TA is later shown to be transcendental over k by Wade [10]. Moreover Carlitz has explicit formulas for these rationals bn in case n is divisible by q — 1. If q = 2 this explains perfectly the nature of CA(") f° r a u 103

104

J. Yu

n > 0. For q > 2 it also leads to the more interesting question of explaining the nature of CA(«) for all n which are odd, in the sense that n is not divisible by q — 1. A central feature of Carlitz theory is the introduction of the socalled Carlitz module. It plays the role of Gm for arithmetic of the rational field k. The crucial transcendental element ITA above occurs as a fundamental period of this Carlitz module. We shall survey Carlitz theory in Section 2, together with the Euler-Carlitz relations : C A ( ^ ) = &n""A f ° r a ^ n divisible by q — 1. As in the classical story of Riemann zeta values £(n) for integer n > 1, one is tempting to conjecture that all CA(«) are transcendental over k and these EulerCarlitz relations generates all the non-trivial relations among these zeta values. Because we are now in the characteristic p world, p-th power (Frobenius) relations are always there for all positive integers n and m : (1-2)

CA(pmn) = C A ( n r .

These relations certainly are not considered as non-trivial. Half century after Carlitz's discovery, a breakthrough towards understanding these zeta values takes place. First Anderson-Thakur [2] is able to relate the zeta values CA(^) to a special point on the n-th higher Carlitz module for all positive integer n. As a result we have proved [12] the transcendence of the zeta values CA(^) for all positive integer n, in particular for those odd n not divisible by q — 1. Later in [13], we are also able to prove that the Euler-Carlitz relations are the only linear relations (with algebraic coefficients) among these Carlitz zeta values and powers of TT\. In Section 3 we will sketch this theory of Anderson-Thakur and Yu, with n-th higher Carlitz module playing the central figure. These so-called ^-modules can be interpreted as the tensor powers of the old Carlitz module. The grand final question is to determine all the algebraic relations among the values C A ( ^ ) , n = 1,2,3,.... That is to prove that the Euler-Carlitz relations and the p-th power relations, account for all the existing algebraic relations. With the recent advances on transcendence theory of positive characteristic made by AndersonBrownawell-Papanikolas [3] and Papanikolas [8], this is completely settled by Chang-Yu 2005 [6]:

Special Zeta Values in Positive Characteristic

105

Theorem. The transcendence degree O/F 9 (0)(7TA, CA(1)) • • •, C A ( I ) ) over ¥q(6) is : n

- KPJ - [n/(q - 1)J + Ln/(p(g - 1))J + 1.

We will give a very brief account of this latest development. This round, the central figure on stage changes to t-motives (introduced by Anderson [1]) which is a structure dual to the previously predominant t-module structure. In Section 4, we outline the construction of t-motives whose periods are related to the values CA(W). The breakthrough in passing from linear independence to algebraic independence, is achieved by developing an analogue of classical differential Galois theory and relating it to the motivic Galois groups of the t-motives (cf. Papanikolas [8]). This last episode can be viewed philosophically as realizing a function field version of Grothendieck's conjecture on periods of abelian varieties. Short sketch of proof our main theorem will be given in the final Section 5.

2

Carlitz Theory

Recall the Riemann zeta function:

CW = E ^ .

Re( S )>l.

For even m > 1, Euler showed ^ 1 -(27ry^l)mBm = > —m = : , ' ^ n 2 m!

. t( C(m) sv

n=l

where Bernoulli numbers Bm are given by °°

~m

z

_ ST R

—

ez — 1

•*—' m=0

m!

Here ez is the exponential map for the algebraic group G m : C £ G m (C) = C x C

4


J. Yu

106 Fundamental period of G m is 27r\/—T which satisfies 0 -» Z27r v / Z l -» C ^ C x -> 1.

For arithmetic of the function field fc, Carlitz observed that instead of Gm one should let the key role be played by the additive group Ga together with the following i-action <j>c{t) : x i—• 9x + x9. This structure is now called the Carlitz module. One has the Carlitz exponential map to linearize the non-linear t-action: Coo

0(-)i

«J r a(,CooJ

^

=

^oo

lc(t)

ex —

Coo * G-a(Coo) = CooHere Coo is the completion of a fixed algebraic closure of koo := F g ((l/0)), on which the chosen absolute value will be denoted by I ' loo-

The commutative diagram above amounts to the following functional equation for the exponential: expc(6z)

+expc(z)q.

= 0expc(z)

Solving the recursive relation for the Taylor coefficients inductively gives the expansion 00

h

zq

f

exp c (z) = ^2 T p

o r all z € Coo,

where D0:=

1

A:= nt!,(^-^)Note that Di is in fact the product of all monic polynomials in A with degree i. One also has the exact sequence 0 -» ATTA - • Coo

e

^ - c Coo ->

0,

as a result of the following basic theorem of Carlitz:

Special Zeta Values in Positive Characteristic

107

Theorem 2.1. One has the product expansion (2.1)

expc(z)

= z TT ( l - — \

for all z e Coo,

a^OeA

where °°

(2.2)

irA =

_i

o(-9)T*H(i-o1-«i)

.

»=i i

We will fix from now on a choice of (—D)?-1 so that 7TA is a well-defined element in fcooProof. From the Carlitz t-action one sees that the zero set of expc(z) is a discrete A-submodule of rank one inside Coo with respect to scalar multiplication by A. Thus (2.1) is established by checking that all zeros are simple. We sketch here a proof of (2.2) from [2]. Consider the following generating function in variable t: i

s

v

y

OO

m=0

For each m, substitute the product expansion (2.1) of e x p c ( ^ ^ 1 ) into the above Taylor series defining E(t). We see that this generating function can be written as a function analytic at t — 8 plus the dominating term : TTA I £ \

En e \e) m=0

N

_ -7TA

~ t-e'

'

Thus E(t) has residue — n\ at t = 6. On the other hand we also contend that E(t) has product expansion i

(2.3)

°°

/

£?(*) = (-0)5=* J ! i=0

\

/

1-77 V

_ 1

• '

Formula (2.2) results from computing residue at 6 using this product expansion.

J. Yu

108

To prove the infinite product (2.3), let F(t) be the function on the right-hand side. "Twisting" the series for E(t) to q

*» (<>:-£ "ft, (|£r)<" m=0

m=0 ^

v

'

^

'

\

/ /

we find it satisfies the same equation as twisting F(i) to F^(t). follows that we must have E{t) = F(t).

It D

Carlitz also introduced "factorials" for the ring A. To define, say Tn for positive integer n, just write down the q-adic expansion oo

J2 niq1 of n — 1, and let i=0

oo

r„:=n^r1=0

With these Carlitz factorials, the Bernoulli-Carlitz "numbers" cn in A are then given by the following series expansion oo

- Y^ Cn ~n exp c (2) ^ rn+i The important point is that both the fundamental period 7TA and the Bernoulli-Carlitz numbers come from the Carlitz module structure. By taking logarithmic derivative of (2.1), Carlitz ([5]) arrives immediately at the Euler-Carlitz relations: Theorem 2.2. For all positive integer n divisible by q — 1, one has

(2-4)

CAH

= ^TTA.

It should be noted that as in the classical arithmetic, both the Carlitz factorials and the Bernoulli-Carlitz numbers enjoy good divisibility or congruence properties, cf. [7] and [9]. There are lots of arithmetic information behind the Euler-Carlitz relations.

Special Zeta Values in Positive Characteristic

109

The formal inverse to the Carlitz exponential is the Carlitz logarithm. It is the series

log

OO

nk

:=

^ £v fe=0

"

where Lo := 1, Li:=irj=l(0-0qj)(i

=

h2,...).

This is no longer an entire series. It only converges for all z G Coo _3_

satisfying |z|oo < |#|oox • It also satisfies the functional equation 01ogc(;z) = log c (03) + logcOz9) whenever these values in question are defined. As pointed out by Thakur, cf. [2] Sec. 3, Carlitz already noticed the following identity connecting logarithm with the zeta value at 1: Theorem 2.3. (2-5)

CA(1) = logc(l).

Proof Let y be a variable and consider the generating function: J2Bd(y)zqd,

exp c (2/log c (z)) = d=0

where

B

*w = E -£re k[y] U L

j=0

i d-i

satisfying, for all a € ¥q[t], that

J2Bd(*)xqd-

c(a)(x) = d

Here a \—> (f)c(a>) is the Fg[i]-action on G a determined by the Carlitz t-action.

J. Yu

110

Let Ad+ denote the set of monic polynomials in A having degree d. One checks that 1 — Bd(y) has elements in A^+ as simple zeros. Hence, by taking logarithmic derivatives, we have i

E

_

B;(Q)

Bd(y)'

y-a

aeA,j +

0, in fact one gets even:

The identity (2.5) follows by setting y 1

y

-=-

aeAd+

This identity brings us to the starting point toward understanding the Carlitz zeta value at integers not divisible by q — 1. Let k be the algebraic closure of k in Coo. We shall say that an element in Coo is transcendental if it is not in k. After giving ad hoc proof of the transcendence of 7TA, Wade went on in [11] to prove the following analogue of Hermite-Lindemann theorem for the Carlitz exponential: T h e o r e m 2.4. For every a / 0 £ fc, expc(a)

is transcendental.

This implies not only the transcendence of the fundamental period 7TA but also the transcendence of CA(1) because of (2.5).

3

Anderson-Thakur Theory

To understand those zeta values CA(^) for n > 1 not divisible by q — 1, Anderson introduced in [2] the concept of n-th higher Carlitz module. This is the n-dimensional vector group G^ together with i-action given by the polynomial map

/Xl\

(6

1

*n(<) =

0\ (x{\

•• 1 \Xn)

ej \XnJ

Vo

/0

+

V

/x\\

0\

w

Still one has exponential map exp„ to linearize the i-action exp. ^oo

dQn(t)

cn

| exp„

GS(Coo) = C ^

Special Zeta Values in Positive Characteristic

111

When n = 1 it is just the Carlitz module with exp 1 (z) = exp c (z). Moreover for any n, the entire analytic map e x p n ( z i , . . . , zn) satisfying the above commutative diagram can be easily solved as everywhere convergent power series with matrix coefficients: z\

" zzlqi

oo

'

=£ «

exp n

i=0

,

Pi€Mn(k).

z

n

The polynomial map $n{t) determines, via composition and addition, an action of ¥q[t] on G™ with constants in ¥q acting as scalars. Such a structure is called a t-module. The kernel of the exponential map is called the period lattice of the i-module. Regarding C ^ as tangent space at the origin of G™, Anderson-Thakur was able to prove for n-th higher Carlitz module the following fundamental Theorem 3.1. For all n one has exact sequence

d* n (F,[t])n n ~CSo^ n €£,-•<) where fin = (..., ir%) E C£,. Writing down the compositional inverse to exp n as formal series Zl lo

oo

' zqi

= £«*

Sn Zn

i=0

'

qi

z

n

with matrix coefficients from Qi £ Mn(k), one finds that this series converges for all (z\,.. .,zn) € C ^ satisfying

Moo < WC*^ • Moreover one has the following particular simple formula: 0 (3.1)

!og„

0 z

oo Ei=0

z^_ U>

112

J. Yu

We call the n-th Carlitz polylogarithm the series

po\ylogn(z)~^2

°°

fe=o

zqk

— k

nq

which converges for all z € Coo with \z\oo < l^loo"1 • Its value at particular z = a ^ 0 is called the n-th polylogarithm of a. As can be easily checked these polylogarithms always has no zero value. Also when n = l, one has polylog1(z) = logc(z). In transcendence theory we are interested in those polylogarithms of a e k, as analogues of classical logarithms of algebraic numbers. In [12], we have an analogue of Hermite-Lindemann Theorem for n-th higher Carlitz module: Theorem 3.2. If v = (h, . . . , / „ ) ^ 0 in C ^ , and exp n (v) € k , then ln must be transcendental. Given u £ Coo, if there exists elements Zi,...,Z n _i such that (li,. . . , / n _ i , u) = p is in k , then we also shows that the algebraic point p = p(u) on G" is determined completely by u. We call such u a last coordinate logarithm of algebraic point on the n-th higher Carlitz module. The key result of Anderson-Thakur in [2] is Theorem 3.3. For all n > 1, CA(TZ) is last coordinate logarithm of certain algebraic point on the n-th higher Carlitz module. Applying Theorem 3.2 to the algebraic points of Theorem 3.3, we obtain immediately the transcendence of all CA(TI)- The special algebraic point on the n-th higher Carlitz module corresponds to CA(^) can be written down explicitly. They are torsion points for fraction precisely when q is even, i.e. n = 0 (mod q— 1). Writing down these special points amounts to deriving nice "integral" formulas for C A ( " ) - To do this, let y n j be the column vector of length n with last coordinate 6l and all other coordinates zero. Define the algebraic point

Pn := 5Z $ «(V»)(yni), i

Special Zeta Values in Positive Characteristic

113

where the hn^ € A come from a generating function identity with variables x, y:

V

U

3

j=0

/

n=0

1

"+1

Hn-\(y) •= J^ ViJ/* G A bli

What Anderson-Thakur established is that TnC,x{n) is the last coordinate logarithm of this point p„. By checking that the coefficients hni vanish for i > - ^ j , AndersonThakur also obtained the following formula connecting CA(^) with polylogarithms of appropriate powers of 6. Theorem 3.4. Given positive integer n, one can find non-negative integer ln < ^ j such that hn^n / Q,and the following identity holds In

(3.2)

r„CA(n) = £

hn,i polylog n (^).

i=0

In other words, CA(«) can be written as linear combination of n-th polylogarithms of powers of 6 with rational coefficients from k. This fact eventually enables us to determine all the algebraic relations among the zeta values in question. Example. For 1 < n < q — 1, one has simply the formula C A W = polylog n (l).

4

^-Motives

Following Anderson [1] and Papanikolas [8] we now consider i-motives. Let k[t, a] be the twisted polynomial ring such that for all c e k: ct = tc, at = ta, ac = c 'qa. An Anderson i-motive is a left k[t,
J. Yu

114

Let m € Mrxi(M) be a fc[£]-basis of M. Then the multiplication by a on M is given as cr(m) = $ m , for some matrix $ G Mr(fc[fj), We also need the Tate algebra T in variable t. This is the subring in Coo[[i]] consisting of power series which converge on | t |oo< 1. The field of fractions of T is denoted by L. There is a twisting automorphism on T defined by:

/ ( - l) =E a r 1 *v f /=E^ iGT i

i

This twisting leaves invariant various natural subrings of L, e.g. k(t). It is convenient to introduce left fc(t)[cr, <j_1]-modules which are finite dimensional over k{t). These are called pre-i-motives. They form an abelian category. Each Anderson i-motive M corresponds to a pre-t-motive k(t) ®^m M, with a(f ® m) := / ( _ 1 ) am. A given pre-i-motive of dimension r over k(t) is said to be rigid analytically trivial if there exists ^ € GL r (L) such that (4.1)

a(^) = *(_1) = $ # ,

where $ represents multiplication by a with respect to a basis of this pre-t-motive. This \& is said to be a rigid analytic trivialization of the matrix
Special Zeta Values in Positive Characteristic

115

Let M be a t-motive, with $ € GLr(k(t)) nMr(k[t]) represents multiplication by o on M . It can be shown that entries of ^ are in fact everywhere convergent power series in t with coefficients lie in finite extensions of fcoo. One is allowed to substitute 0 for the variable t in \&. Call \ t ( 0 ) - 1 € GLr(fcoo) a period matrix of the motive M. The entries of \l/(#) generate a field over k whose transcendence degree over k is an analytic invariant of the motive M. Let E c T b e the set of entire power series with coefficients lie in finite extensions of k^. A fundamental Theorem of Papanikolas [8] relates the analytic invariant of the t-motive coming from \P with the algebraic Galois group FMTheorem 4.1. Let M and $ be given as above such that det
°U) = {t-0)f{-l\

fee.

Here $ = (t — 0) and r = 1. An analytic solution * of (4.1) is oo

*c(t) = ( - 0 ) - < ^ - D j ] ( i _ t/e?). Period of Carlitz motive is ~^c{0)1 = —TTA, where 7TA is fundamental period of the Carlitz module. One computes that Fc = G m thereby gets transcendence of 7TAExample 2. The s-th tensor power of the Carlitz motive C®s. Let s be a positive integer and

c®s =

c®m---®mc

116

J. Yu

with cr-action:

a(/) = (t-^r/ ( " 1 ) , /eC® 8 Here <& = (t — 9)s and r = 1. An analytic solution ^ of (4.1) is * ^ . Period of this motive is ^(Q)'1 = (—TTA)3- One also computes that Example 3. The t-motives for s-th Carlitz polylogarithms (case s = 1 is given by Papanikolas [8]). Given a\,...,am Ekx satisfying

Noo < I C 7 ' " 1 . Set

I {t-ey $s(ai,...,am)

=

o ••• o\

a[~ 1} (i-0) s 1 ... 0

y^ty-fl)* 0 ••• 1,

This defines a i-motive M$ s which is an extension of l m by the sth tensor power Carlitz motive C® s . Here 1 is the trivial rank one t-motive. Analytic solution to (4.1):

*s(ai,...,am) =

£«i,«(*c)a

1

•••

V^ m ,«(*c) s 0 ••• where

L

«*®-a*

+

'Lit

_#)....{t_tfy

One has

LizP = a™ + (t-ey Moreover for all i, one has Lai,s(e) = Polylog s (ai).

0

l)

Special Zeta Values in Positive Characteristic

117

To compute the Galois group T$ s for the ^-motive M$ s , consider

One checks that T$ C G, and T$ -» G m canonically. Let V be kernel of the later map. Then V can be identified as a linear subspace of G™ over ¥q(t). This observation leads to the following Theorem 4.2. Suppose polylog s (ai),..., polylog s (a m ) and TTAS are linearly independent over k. Then the Galois group F(f>s is an extension ofG™ by Gm and dimT$ 3 = m + 1. It follows that 7TAS and polylog s (ai),... ,polylog s (a m ) must be algebraically independent over k if they are linearly independent over k.

5

Algebraic Independence of the Special Zeta Values

Given n not divisible by q — 1, we let Nn := A;-span{polylog n (l),polylog n (0),... ,polylog n (0'")}. By Theorem 3.4 we have CA(^) S Nn. Since CA(") and ITA71 are linearly independent overfcoo,they are also linearly independent over k. For each such n we fix once for all a finite set of distinct exponents 0

so that both { p o l y l o g ^ M ) , . . . , polylogn(0im»)} and {C A (n), p o l y l o g j ^ 1 W ) , . . . , polylog n (^-("))} are bases of Nn over k.

118

J. Yu Let M$ n be the i-motive defined by the matrix

(5.1)

n =

n(eio{n\...,9im»W).

The Galois group of this ^-motive M$ n has dimension mn + 2 by Theorem 4.2 On the other hand, a rigid analytic trivialization of $ n is given by * c

0

0 0

*C^
*n:= ^cLe^nM,n

°

•••

!

Hence by Theorem 4.1, m n + 2 also equals to the transcendence degree over k of the field fc(7rAn, p o l y l o g j ^ " ) ) , . . . , P o l y l o g n ( ^ » W)) = HnAn, C A W , polylog„(0^ n )),..., polylog n (^"("))). In particular, the elements VTA", C A ( " ) ,

p o l y l o g ^ 1 (")),..., polylog n (^« W)

are algebraically independent over A;. Therefore, 7TAn and CA(^) are algebraically independent over k. This completes the proof of the following Theorem 5.1. For any positive integer n satisfying q — 1 \ n, 7TA and CA(^) are algebraically independent over k. Given positive integer s, we set U(s) : = { l < n < s | p | n , q— I f n,}. Define diagonal block matrices $ ( s ) := © n e t / ( s ) $ „ .

The matrix $(s) defines a i-motive M(s) := M$(s) which is the direct sum of the t-motives M$ n with n € U(s). The algebraic Galois group r<j,(s) can be computed, in particular it gives:

Special Zeta Values in Positive Characteristic

119

Theorem 5.2. Fix any s € N, we have

dimr^) = 1+ ^

K + 1 )'

neU(s)

Writing down rigid analytic trivilization \I>(s) of $(s) and applying Theorem 4.1 to r$( s ), we find that l + Snet/(s)( m n + 1 ) i s exactly the transcendence degree over k of the following field: *(*.

{Polylog n (^ o ( n ) ),...,polylog„(^«(n) ) }

(J n€U(s)

= k(it, ^

(J

{CA(n),polylog n (^( n )),...,polylog n (^"("))}

neU(s)

It follows that the set ( J { C A H , p o l y l o g J ^ W ) , . . . , polylog n (^"("))} neU(s)

is algebraically independent over k, hence also {^(n) \ n € U(s)} is algebraically independent over k. Counting cardinality of U(s) we obtain Theorem 5.3. The transcendence degree ofk^x, over k is:

CA(1)>

• • • •> C A ( " ) )

n - [n/p\ - [n/(q - 1)J + [n/{p(q - 1))J + 1. Finally it remains to compute the Galois group T^,,). Let l(s) := Xmec/(s)( mn + 2). Define G(s) to be the algebraic subgroup of GLj(s) over Fq(t) which consists of all block diagonal matrices of the form

' 1 0 " * 1

©•

•©

" 1 0 - • 0 * I • • 0 * 0

J. Yu

120

where the block matrix at the position corresponding t o n 6 U(s) has the size mn + 2. Thus G(s) is isomorphic to direct product of J2neu(s)(mn + 1) copies of G a canonically. Since the Carlitz motive C is contained in M$1, it is also contained in M(s) for any positive integers s. There is a surjection of algebraic groups over ¥q(t) pr : r*( s ) -» G m which coincides with the natural projection on the upper left corner position of the first 2 x 2 block matrix. Let V(s) be the kernel of pr so that one has an exact sequence of algebraic groups over ¥q(t) (5.2)

1 -» V(s) -» Tns)

-» G m -> 1.

From the equation *(s)( - 1 ) = ^(s)^(s) we find that V(s) C G(s). We claim that in fact V(s) = G(s). To prove this claim we intoduce a G m -action on G(s). On the block matrix at the position corresponding to n G U(s) it is defined by (5.3) 1 0 •• • 0 1 0 •• • 0 n * 1 • • 0 a* 1 • • 0 1—> , foraeGm(F,(t)). * 0 •

•

1 _

_ a"* 0 • •

1

Since ®neU(s)C®n is canonically contained in M(s), writing down coordinates for the Galois group implies that the restriction of the above G m -action to V(s) agrees with the conjugation action of G m on V(s) coming from (5.2). Taking into consideration of the G m -action, we are able to establish: T h e o r e m 5.4. Fix any s G N. Then we have exact sequence of algebraic groups over¥q(t): 1 - • V(s) -> T(s) -» G m -* 1 where V(s) is isomorphic to the vector group

c n+i

n"n€U{s)

Special Zeta Values in Positive Characteristic

121

Furthermore, the conjugation action by Gm on V(s) has multi-weight (n)neU(s)-

References Greg W. Anderson, t-mptives, Duke. Math. J 53 (1986), 457502. Greg W. Anderson and Dinesh S.Thakur, Tensor powers of the Carlitz module and zeta values. Ann. of Math. 132 (1990), 159191. Greg W. Anderson, W. Dale Brownnawell and Matthew A. Papanikolas, Determination of the algebraic relations among special Gamma-values in positive characteristic, Ann. of Math. 160 (2004), 237-313. L. Carlitz, On certain functions connected with polynomials in a Galois field, Duke. Math. J 1 (1935), 137-168. L. Carlitz, An analogue of the von Staudt-Clausen Duke. Math. J 3 (1937), 503-517.

theorem,

Chieh-Yu Chang and Jing Yu, Determination of algebraic relations among special zeta values in positive characteristic, preprint, 2006. David Goss, Basic structures of function field arithmetic , Springer-Verlag, Berlin, 1996. Matthew A. Papanikolas, Tannakian duality for AndersonDrinfeld motives and algebraic independence of Carlitz logarithms, http://arxiv.org/abs/math/0506078, 2005. Dinesh S. Thakur, Function field arithmetic, World Scientific Publishing, River Edge NJ, 2004. L. I. Wade, Certain quantities transcendental over GF(pn,x), Duke. Math. J 8 (1941), 701-720.

122

J. Yu

[11] L. I. Wade, Transcendence properties of the Carlitz ip-functions, Duke. Math. J 13 (1946), 71-78. [12] Jing Yu, Transcendence and special zeta values in characteristic p, Ann. of Math. (2) 134 (1991), 1-23. [13] Jing Yu, Analytic homomorphisms into Drinfeld modules, Ann. of Math. (2) 145 (1997), 215-233.

Jing Y U Department of Mathematics, National Tsing-Hua University Hsinchu, Taiwan 30043 Email: [email protected]

Automorphic Forms, Eisenstein Series and Spectral Decompositions Lin W E N G

In Memory of Serge And Serge did this sort of thing, through the decades, with many of the young: he would proffer to them gracious, yet demanding, invitations to engage as a genuine colleague not teacher to student - but mathematician to mathematician; he did all this naturally, and with extraordinary generosity and success. Prom For Serge, by B. Mazur

This note is prepared for the reader who wants to learn Langlands' fundamental results about Eisenstein series and spectral decompositions, say using Moeglin and Waldspurger's Cambridge tract. It results from the six lectures given at a special seminar on Automorphic Forms and Eisenstein Series at Department of Mathematics, University of Toronto in 2005. (Once a week, one and a half hours starting from October 25, after another set of six beautiful lectures on automorphic forms given by Henry Kim.) The author wishes to thank Arthur and Kim very much for their kind invitation and support which, together with a JSPS fund, make his long visit possible.

123

124

L. Weng

Contents D a y One: B a s i c s of A u t o m o r p h i c F o r m s

127

1 Basic Decompositions 1.1 Langlands Decomposition 1.2 Reduction Theory: Siegel Sets 1.3 Moderate Growth and Rapidly Decreasing 1.4 Automorphic Forms

127 127 134 136 139

2 Structural Results 2.1 Moderate Growth and Rapid Decreasing 2.2 Semi-Simpleness 2.3 3-Finiteness 2.4 Philosophy of Cusp Forms 2.5 L 2 -Automorphic Forms

141 141 142 142 145 146

D a y T w o : E i s e n s t e i n Series

147

3 Definition 3.1 Equivalence Classes of Automorphic Representations . . 3.2 Eisenstein Series and Intertwining Operators 3.3 Convergence

147 147 149 150

4 Constant Terms of Eisenstein Series

152

5 Fundamental Properties of Eisenstein Series

. . .

156

D a y T h r e e : P s e u d o - E i s e n s t e i n Series

158

6 Paley-Wiener Functions 6.1 Paley-Wiener Functions 6.2 Fourier Transforms

158 158 158

6.3 Paley-Wiener on
159

7 Pseudo-Eisenstein Series

161

8 First Decomposition of L2(G(F)\G(A)\ 8.1 Inner Product Formula for P-ESes 8.2 Decomposition of L2-Spaces According to Cuspidal Data 8.3 Constant Terms of P-SEes

163 163 165 166

Eisenstein Series and Spectral Decompositions

125

9 Decomposition of Automorphic Forms According to Cuspidal Data 168 9.1 Main Result 168 9.2 Langlands Operators 169 9.3 Key Bridge 171 D a y Four: S p e c t r u m D e c o m p o s i t i o n : Residual Process

173

10 W h y Residue? 10.1 Pseudo-Eisenstein Series and Residual Process 10.2 What do we have? 10.3 Difficulties

173 173 175 177

. . . .

11 Main Results 11.1 Functional Analysis 11.2 Main Theorem: Rough Version 11.3 Main Theorem: Refined Version 11.4 How to Prove?

177 177 178 179 181

D a y Five: E i s e n s t e i n S y s t e m s a n d Spectral D e c o m p o s i t i o n (II) 184 12 Eisenstein Systems 12.1 Relative Theory 12.2 Discrete Spectrum 12.3 Eisenstein Systems 13 Why or Better How to Get Theorem B?! 13.1 Bridge 13.2 Basic Facts

184 184 185 186 . . . .

186 186 187

14 Spectrum Decomposition: Levi Interpretation . . 14.1 Spaces A ^

190 190

15 Spectral Decomposition: Arthur's Version of Langlands 15.1 In Terms of Levi 15.2 Relation with Residual Approach: The Proof

191 191 193

126

L. Weng

D a y Six: Arthur's Truncation a n d M e r o m o r p h i c C o n tinuation 195 16 Arthur's Analytic Truncation 16.1 Positive Chamber and Positive Cone 16.2 Arthur's Analytic Truncation

195 195 197

17 Meromorphic Continuation of Eisenstein Series: Deduction 198 17.1 Preparation 199 17.2 Deduction to Relative Rank 1 200 18 The Situation of Relative Rank 1 18.1 Working Site 18.2 Constant Terms 18.3 Basic Properties 18.4 Functional Equation for Truncated Eisenstein Series . . 18.5 Application of Resolvent Theory 18.6 Injectivity 18.7 Meromorphic Continuation 18.8 Functional Equation

201 201 202 203 203 204 206 208 209

References

210

Eisenstein Series and Spectral

Decompositions

127

Day One: Basics of Automorphic Forms 1

Basic Decompositions

Of central importance here are the following two decompositions: (i) (Langlands Decomposition) G(A) = U(A) -^M(A) -M(A) 1 K; and (ii) (Reduction Theory) G(A) = G(F)-S(u;tQ) = P(F)-Sp(u;t0).

1.1 1.1.1

Langlands D e c o m p o s i t i o n Reductive Groups

Let G be a connected reductive group denned over a number field F. Denote by G(A) the associated adelic group. Recall that by definition, (i) an algebraic group is called reductive, if its unipotent radical is trivial; (ii) if Fv is the f-adic completion of F at its place v (with Ov the v-adic ring of integers when v is finite), then the associated adelic ring A of F is defined to be the restricted product A = JJ Fv of Fv with respect to (the maximal compact) subgroup Ov. In other words, m = (mv) E ]JV Fv belongs to A if and only if for almost all v, mv € Ov. Similarly, in order to introduce the adelic group G(A), we need to first introduce maximal compact subgroups Kv := G{Ov) for almost all places v, then define G(A) as the restricted product Y\'VG(FV) with respect to Kv. So the difficulty here is to introduce ring valued groups G{Ov) and can be, say, overcome as follows: First of all, there is no problem for this to be done for the general linear group GL. Then being an algebraic group, there exists a natural embedding i'G of G into the general linear group GLn for certain n. So by taking care of the inverse operation as well, the morphism %G • G —> GL<m defined by g H-> di&g(iG(g),iG(9)~t) will do the job.

128 1.1.2

L. Weng Parabolic Subgroups

Let P be a parabolic subgroup of G defined over F. (Recall that this means that as algebraic varieties, P is defined over F and G/P is complete, Grothendieck's notion in algebraic geometry generalizing the common concept of separated and compact.) We have the socalled Levi decomposition P = MU = UM for P, where U is the unipotent radical of P and M is what we call a Levi subgroup of P. For later use, fix a minimal parabolic subgroup Po of G and a minimal Levi subgroup Mo of P, both defined over F, and call a parabolic subgroup P and a Levi subgroup M standard if they are defined over F and contain Po and MQ respectively. Denote by UQ the unipotent radical of Po. It is known that (i) every parabolic subgroup P of G defined over F conjugates by an element of G(F) to a standard one; and (ii) every standard parabolic subgroup P contains a unique standard Levi subgroup of G. Ex. (a) With G = GLn, parabolic subgroups P correspond to partitions n = r\ + r-i + • • • + r\p\ of n. Normally, we choose the minimal parabolic subgroup Po as the Borel subgroup, i.e., the subgroup consisting of upper triangle matrices and the minimal Levi subgroup MQ as the subgroup consisting of diagonal matrices. As such, the associated unipotent radical UQ is the subgroup consisting of upper triangle matrices whose diagonal entries are all 1. In terms of partitions, Po corresponding to the partition IQ = ( 1 , 1 , . . . , 1) resulting from n = 1 + H hi; (b) More generally, let I = (ni, ri2,..., n;v) be the partition resulting from n = n\ + n^ + • • • + njv- Then the corresponding standard parabolic subgroup Pj is the subgroup of G consisting of the blocked upper triangle matrices, for which the diagonal blocks are square matrices of sizes n i , r i 2 , . . . , and nj^ respectively. Consequently, the corresponding standard parabolic subgroup Mi consists of blocked diagonal matrices in Pj, while the unipotent radical Ui consists of blocked upper triangle matrices whose diagonal matrices are identity matrices of sizes n\, ri2, • • •, n^ respectively.

Eisenstein Series and Spectral Decompositions 1.1.3

129

Maximal Compact Subgroups

Recall that even in the defintion of G(A), we need a maximal open compact subgroup K = YlvKv of G(A), for which Kv = G(0V) for almost all v. For our purpose, we also require that K satisfies the following conditions: (i) (Langlands Decomposition) G(A) = P{A)-K K; and (ii) (Admissible Condition) P(A)nK

= U(A)-M(A) •

= (u(A)nK)

• (M(A)!1K)

.

E x . With G = SLn, Ku = SO(n) and Kc = SU(n). 1.1.4

Decomposition M(A) = ^4M(A) • M(A) 1

This is supposed to be the first difficulty the reader meets. The basic idea is to separate infinite places from others so as to get a commutative real Lie group with which the unbounded quotient space we encounter later will admit a relative simpler description. Starting point is the decomposition of idele group I of F. (As a group, I can be understood as A* = G m (A) = GLi(A), where A* stands for the collection of invertible elements of the ring A, G m stands for the algebraic group representing the so-called multiplicative group.) We have the subgroup 1° of the so-called norm 1 ideles, in which multiplicative group F* consisting of non-zero elements of F is naturally embedded as discrete subgroup via the diagonal map / t-» i(f) : = ( . . . , / , . . . , / , . . . ) . (For / € F*, there are only finitely many v such that |/|,; ^ 1, so i(f) E I, while by the product formula, the norm oii(f) is simply one, hence, i(f) E I 0 .) As such essential points are that (i) F*\I° = GLi(F)\GLi(A) is compact; and (ii) I°\I ~ K.+ given by mapping g to the idele defined as follows: each component at a finite place is trivially 1, while for each infinite i

place the entry is simply ||(?||[F:ltM . Now a generalization to general reductive group is quite obvious: We need to

130

L. Weng

(i) introduce spaces corresponding to 1° and R>o, which are simply M(A) 1 and AM(Ay, and (ii) introcuce a generalization of the logarithmic map. Let Rat(M):={x:M^Gm} be the collection of morphisms between M and G m (as algebraic groups). This is a finitely generated Z-module. Set a*M := Rat(M) ® z C,

and

a ^ := Hom z (Rat (M), C J

which contain the real subspaces Rea*M := Rat(M) ® z R,

and

Rea*M := Hom z (Rat

(M),R\

respectively. For x € Rat(M), by definition, x is a character of M, i.e., a morphism from the algebraic group M to G m . Thus in particular, over the local field Fv, \ induces a morphism \v = X\G(FV) '• G(FV) -* Gm(Fv) = F*. Accordingly, set |x| : Af(A) -> R> 0 to be the continuous map defined by m'*' := (mvyx^ :— Y\v |ni v |*" where m = {mv) € M(A). With this, let M(A)1:=nx6Rat(M)Ker|x|. Next, we define AM^y (a) AMo(A) for the minimal Levi Mo. Here PQ = UQMQ is a minimal parabolic subgroup. Denote by ZM0 the center of MQ. Then ZM0 contains a maximal F-split torus To (of G), and there exists a natural number R, the semi-simple rank of G, such that To * G * .

As such, with the injection R>o *-+ I = G m (A) defined by t — i> ( 1 , . . . , 1; £,..., t), i.e., the idele whose finite components are all 1 while whose Archemidean components are all t, we obtain a natural inclusion R>o ^ G m (A) ~ r 0 (A)

Eisenstein Series and Spectral Decompositions

131

which is known to be split (as an entension of groups, since R>o is simply connected.) Therefore there exists a unique subgroup AMo^ of To(A) C ^Af0(A)» the image of the above morphism, such that it projects back onto M>0. (b) AM(ty for general (standard) Levi M . Being standard Levi, Mo (A) C M(A). Consequently, Z M ( A ) C ZMo(k) (bigger group admits smaller center). As such, set AM^ := AM 0 (A) n ZM(A)T h e o r e m . For any standard parabolic subgroup P = MU of G, (a) M(A) = M(A) 1 • AM(A) = AM(A) • M(A) 1 ; and (b) (Langlands Decomposition) (3(A) = U(A) • M(A) 1 • AM(Ay K. The reader should understand that while (b) is a structural result, (a) is merely a simple decomposition for our convenience. 1.1.5

Logarithmic M a p

Naturally associated to the decomposition M(A) = M(A) 1 • AM^ is the so-called logarithmic map log M : M(A) -»• ReaM m i-> log M (m) characterized by (x,logM m\x\

=

. e{x,\ogMm)^

m) := logm™ or the same, y

m

e

M(A)

&

% e

R a t

(My

Clearly (i) log M factors through M(A)*\M(A); and (ii) log M ( M ( A ) X \ M ( A ) ) = ReaM. For convenience, define also a map mP : G(A) - • M(A) : \M(A)

by

g i-» M(A) 1 • m(g)

where g = u • m(g) • k is a Langlands decomposition with u E U(A),m(g) E M(A) and ke K. (mP is well-defined since M(A)nK is contained in M(A) 1 and M(A) 1 is a normal subgroup of M(A).) Accordingly, we are able to talk about Hp(g) := \ogM mp(g) an element in ReaM associated to g € G(A), a notation used commonly in literature.

132 1.1.6

L. Weng Decomposition a*Mo = a M 0 (ajJJ0)

At the level of associated Lie algebras, there is a natural decomposition ,* M0

a

—

a

M

(naM V

[ M0)

(a) The space a*MQ. While defined as Rat (M) <%>% C, it may also be identified with the space generated by the root system R(To, G). (Recall that the adjoint representation gives a natural action of the group To on the Lie algebra g associated to the reductive group G. Being commutative, the non-trivial common eigenspaces of To in g give the elements, the so-called roots, in R(To,G).) As such, with respect to the parabolic subgroup PQ, we then further get the set of positive roots R+(TQ, G) in R(TQ, G) and also the set of simple roots Ao := A M 0 C R+(TO,G). AS a complex vector space, a*Mo admits a basis Ao; (b) The space a*M. Similarly, being commutative, maximal F-split torus TM in the centre ZM of M acts naturally on the Lie algebra 0 of G. Denote by R(TM, G) the set of 'roots' of G with respect to this action of T M , i.e., these with non-trivial eigenspaces. In general, R(TM,G) is not a root system. Set A M := the set consisting of elements of R(TM,G) obtained from the non-trivial restrictions to T M ( C TO) of elements in A 0 . (And similarly for R+(TM,G).) a*M admits a natural basis A M ; (c) The space (aMo J . The adjoint action for TM0 on the Lie algebra associated to M gives the set R(To, M), the root of M with respect to T 0 . Set A M o := A 0 D R(T0,M) be a subset of A 0 , consisting of only elements which come from the M-level, i.e., those operating only within M. The space ( aH ) admits a natural basis Aj$ • As all Ao, A M and A j ^ are defined over R, we have the decomposition Reo M o = R e a M © R e ( a ^ 0 ) . Similarly, we get decompositions of their duals aM0 - aM © aM0

and

Re aMo = Re a M © Re a%0.

Eisenstein Series and Spectral Decompositions

133

Set accordingly

Po = g

a

^

and

aeR+(T 0 ,G)

pp =

2

a>

5^

aeR+(TM,G)

Ex. In the case G = GLn and P = Pi. The subset Aj$Q of A 0 consists of simple roots which operate within the small blocks associated to M, while A ^ consists of restrictions of simple roots in Ao which operate exactly among different blocks associated to M. 1.1.7

Haar Measures

Key is that all have to be compactible with the Langlands decomposition G(A) - tfo(A) • AMo{A)

• Mo(A) 1 • K = U(A) • AM(A)

• M(A) 1 • K.

For compact groups K, there is no problem. Even for unipotent radicals U, since U(F) is discrete and U(F)\U(A) is compact, it is obvious. With this, for all compactly supported continuous functions / on G(A), f(uamk) dk (a~2poda)

/ H-> /

dmdu

Ju0(A)-AMo(AyM0{k)i-K

defines a Haar measure dg on G(A) where u € Uo(A), a G AM0(A)> m £ Mo(A) and k G K. Replacing G with M, we also obtain a Haar measure dm on M(A). Among all these Haar measures, we have the following Compatibility Condition. For any compactly supported continuous function f on G(A), / JG(A)

f{g)dg=

/

f (uamk) dk a" JU(h)-AM(AyM(A)l-K

where u € {7(A), a € AM^,m

G M(A) 1 and k e K.

Pp

da dmdu

134

1.2

L. Weng

Reduction Theory: Siegel Sets

Topologically, P(F) is a discrete subgroup of G(A). We want to understand the quotient space P(F)\G(A), which classically is realized via the so-called fundamental domain. A typical example is for the upper half plane H under the action of SL(2, Z), or equivalently, for the quotient space SL(2,Z)\SL(2,C)/SO{2), a model for SL(2, Q)\SL(2, A) modulo certain maximal compact subgroup K. According to Langlands decomposition, G(A) = U(A) • AM(A) • M(A) 1 • K and the Levi decomposition P = UM, we have P(F)\G(A)

= U(F)\U(A)

• AM{k)

• M ( F ) \ M ( A ) 1 • K.

Note that being unipotent radical, U(F)\U(A), being essentially copies of F\A, is compact. (In fact, here, a nitration and its associated graded quotients have to be used.) Consequently, the problem of reduction mainly reduces to the one for M(F)\M(A), or the same for AM(A) • M(F)\M(A)1, with K being compact. As such, then we need to know that M ( F ) \ M ( A ) 1 is of finite volume, while AM^ is a commutative real Lie group. With all this, then we can go as follows: (a) The case where P = PQ = UQMQ is minimal parabolic. Then MQ is a torus. So M 0 (F)\Mo(A) 1 is simple, being essentially copies of compact space F*\I° (at least when G is split). In any case, here, the reduction theory does not offer us any new thing as to a large extent both AMo^ and Mo(F)\Mo(A) are left untouched; (b) The case where P = G. There is nothing we can say about 1 AQ(A)- But for G(F)\G(A) , we use the maximal infinitesimal space a^ 0 , or the same AMo^. So let UJ be a compact subset of Po(A) and to € Mo (A). Then set ^M0(A)(*o) := {a e AMo(A)

| aa > tg Va € A 0 } ,

which is a unbounded subset of AMo(Ay ated Siegel set to be

Accordingly, set the associ-

S(uj;to) :~ S := Ipak \p € u,a E AMo(A)(t0),k

e

K\.

Eisenstein Series and Spectral Decompositions

135

R e d u c t i o n Theoryo. We may choose big enough to and small enough to such that G(A) = G(F) • S(u; to). (Here to small enough means that for all a € Ao, £Q a r e v e r y c l ° s e to zero.) As such, Siegel sets S(u;to), whose structure is rather simple, may be viewed as approximations of usually complicated fundamental domains. (For many purposes in the theory of automorphic forms, exact fundamental domains, which are very hard to obtain, only play a secondary role, while much simpler structured Siegel sets are already sufficient.) Ex. G = SL(2). We are lead to consider SL(2,Z)\H. Then the canonical fundamental domain is essentially given by D := < z = x + iy € H \ \z\ > 1 \x\ < - >, an intersection of the strip defined by a linear inequality |a;| < \ and a quadratic inequality \z\ > 1. On the other hand, a Siegel domain may be chosen to be a much simpler S := < z = x + iy E H | \x\ < 5, y > c \ for all 5 > \ and 0 < c < ^ defined by two linear inequalities |x| < 5 and y > c. (c) The case where P = UM is standard parabolic. As said, we only need to study the space J4M(A) X M ( F ) \ M ( A ) X . NOW the difficulty is that unlike the case F*\l°, M(F)\M(A)1 is no longer compact (otherwise the life would be very easy) - it is only of finite volume. Recall now the infinitesimal decomposition a*Mo — ( a ^ 0 J © a*MClearly, reduction theory is supposed to offer us a clear way to understand AM (A) x M(F)\M(A)1. Thus being very simple, we will leave the .,4-part AM(&) untouched. Or in terms of infinitesimal structure, leave a ^ wild. As for the part M ( F ) \ M ( A ) 1 , then we use infinitesimal space (aj$ ) which admits a basis Aj^ o . That is to say, set ^M0(A)(*o) := {a € AMo{A)

| aa > # Va € A $ 0 } .

136

L. Weng

This is a unbounded subset of AMo^ which contains the entire subspace corresponding to o^f or the same a*M. Accordingly, let the F-level Siegel set to be Sp(u;t0)

:= Sp := {pah \p € w,fl£ A^o{A)(t0),k

E #}.

R e d u c t i o n T h e o r y p . We may choose big enough u> and small enough to such that G(A) = P(F) • Sp(to;to). We will not say anything about proof. Instead we point out that theory of automorphic forms may be understood from a*Mo = (aMo) ® a*M o r t n e s a m e f r o m AM{k) x M(F)\M(A)1. Indeed, re1 duction theory uses only (cij^0) for M(F)\M(A) , while a*M gives new dimension of freedom. Say, automorphic representations are parametrized by o ^ ; Eisenstein series is with variable X £ a*M in which residue can be taken; pseudo-Eisenstein series, being integrations of ESes over I m a ^ , are square integrable, etc. Moreover, such freedom is dominated by a huge Hecke algebra and a nice algebra of Langlands operators (both of which are associated to a^). 1.3 1.3.1

Moderate Growth and Rapidly Decreasing Heights

Being unbounded, we need to measure the size of elements of G(A) or P(F)\G(A). Borrowing a concept from Diophantine geometry, we have the so-called (Weil) heights. There are many ways to define them, but it is a well-known theorem due to Weil that up to 0(1), all these heights are the same. Thus it is of less importance for many purpose to figure out the exact definition. However to start with, for any g € G(A), define its height by llsll := n s u p V 5 s t ^ : s,t = 1,2, . . . , 2 n | V

where (gst) = idg) is the element in GL-m corresponding to g under the map %G • G ^-> GL^n in 1.1.1. Lemma, (a) (Naturalness) There exist c, r > 0 such that, for any g, gi, gi € G(A),

Eisenstein Series and Spectral Decompositions

137

(0 \\g\\ > c; (ii) II5152II < c||5i|| • H52II; and (m)\\g-l\\ 0 and c € R such that, for any a

€ AM 0 (A)> C

I||

1O

SM 0

a

ll ^

lo

§ INI +

c

< c 2|| logWo a||;

(ii) For any A 6 Rea^ Q , there exists c, r > 0 such that mp0(g)X
VjGG(A);

(iii) there exist Ao € Re a*Mo and c > 0 such that \\g\\
V5eG(A)1n5;

(c) (Arith) There exists c > 0 such that

\\g\\ < c • hg\\

v 7 e G(F),

geS;

(d) (Geo) There exist c,r\,r2 > 0 such that c • IMP • I I S P < \\ag\\ 1.3.2

Va e A G ( A ) , g G G(A) 1 .

Moderate Growth

Denote by U the universal enveloping algebra of the complexified Lie algebra LieG!(A0O) <S>m C, where AQQ denotes the Archimedean components of the adeles. By definition, a function / : G(A) —> C is called smooth, if for any g e G(A) with the decomposition g = g/ • goo according to G(A) = G(Af) x G(Aoo), there exist a neighborhood V^ of g^ in G(Aoo), a neighborhood Vf of gf in G(A/) and a C°°-function /QQ : Voo —• C such that

fig'oog'f) = fooig'oo)

v<4 e t ^ , 5/ e vf.

For many purposes, a good control on growth of functions is crucial. We will need two types of them, that is, moderate growth and rapidly decreasing.

138

L. Weng

(a) G-Level. A function / : G(A) —> C is called of moderate growth if there exist c , r £ l such that \f(g)\
V5€G(A).

(Simply put, / is of moderate growth if f(g) is about the size of g.) Moreover a smooth function / : G(A) —> C is called of uniformly moderate growth if there exist r E l and for all X €U, there exists a constant ex > 0 such that \(6(X)f)(g)\
VjeG(A).

(b) P-Level (for P = UM a standard parabolic). A function / : U(A)M(F)\G(A) —> C is called of moderate growth if there exist c,r G R and A € Re(aj$Q)* such that \f(amk)\

< c - | | a | r - m P o ( m ) \ Va G A M ( A ) , m € i W X A ) 1 ^ , k e K.

Moreover a smooth function / : U(A)M(F)\G(A) -> C is called of uniformly moderate growth if there exist r £ l and for all X G U, there exists a constant ex > 0 and Ax G Re(a^ )* such that \(S(X)f)(amk)\

< cx • \\a\\r •

mPo(m)^,

for all a G A M ( A ), m G M(A) 1 n S p , k G K As it stands, moderate growth condition is the one on M(A) = ^M(A) • M(A) 1 -component, i.e., always in terms of ||a|| r for all a G ^M(A) with a fixed r, and m p 0 ( m ) ^ for m G M(A) 1 with a fixed A or Ax- (Simply put, moderate growth looks like polynomial growth.) 1.3.3

Rapidly Decreasing

(a) G-Level. A function / : G(F)\G(A) —• C is called rapidly decreasing if there exists r > 0 and for any A e R e a ^ there exists c > 0 such that \f(ag)\ < c • \\a\\r • mPo(g)\

Va G AG{A),

g G G(A) 1 n S.

Eisenstein Series and Spectral Decompositions

139

(b) P-Level. A function / : U(A)M(F)\G(A) -> C is called rapidly decreasing if there exists r > 0 and for all A G Re(aj|f )* there exists c > 0 such that \f(ag)\ < c • \\a\\r • mP(g)\

Va e AM{A),

g G M(A) 1 n Sp.

Here AM^-paxt plays only a secondary role. Simply put, the rapidly decreasing condition is the one on M(A) 1 -part, where functions decay like exponential functions, or better, decay faster than any rational functions. 1.3.4

Natural Pairing

Let (j),

C be two functions, one is of moderate growth and one is rapidly decreasing. Then we have the following L e m m a . The integral (<£>

JzMWU(A)M(F)\G(A)

converges absolutely. Sketch of a proof. After writing out using reduction theory, the integration looks like J*c°° Q(i)e _ i di where Q is a polynomial of t and c > S > 0, (since essentially moderate growth means polynomial growth while rapidly decreasing means exponential decay). Clearly, this latest integration converges. 1.4 1.4.1

A u t o m o r p h i c Forms Automorphic Forms

Let P = UM be a standard parabolic subgroup of G. For any k G K and a function / : U(A)M(F)\G(A) —> C, define a new function fk : M(F)\M(A) -> C by setting fk(m)

:= mrpp

• f(mk)

m G

M(F)\M(A).

With this, / is said to be K-finite, if the space of C-linear span of I fk I k G K \ is finite dimensional. Similarly, / is called ^-finite if the space of C-linear span of \s(X)f\X€3\

is finite dimensional.

L. Weng

140 Definition. A function : U(A)M(F)\G(A) automorphic form if (1) it is of moderate growth; (2) it is smooth; (3) it is if-finite; and (4) it is 3-finite.

-> C is called an

Denote by A(U(A)M(F)\G{A)) the collection of automorphic forms on the space U(A)M(F)\G(A), the so-called P-level automorphic forms. In particular, we have the G-level automorphic forms .AI G(A)\G(A) J hence also the M-level automorphic forms A(M(A)\M{A)\ Bridge Lemma, tf, e A(U(A)M(F)\G(A)) & is smooth, Infinite, and for all k € K, lM(A)-Part is a direct consequence of 3M-finiteness. 1.4.2

Constant Terms and Cusp Forms

For a function / : G(F)\G(A) defined by the integration

—> C, its constant term along P is

fp(g) •= /

f(u9) du.

JU(F)\U(A)

Let E A(U(A)M(F)\G(A)\

be a P-level automorphic form.

Then by definition

P, = 0. Denote by A0(u(A)M(F)\G(A)) the space

Eisenstein Series and Spectral Decompositions

141

of P-level cusp forms on U(A)M(F)\G(A). Similarly, we have the space A)(M(A)\M(A)) of M-level cusp forms.

2 2.1

Structural Results Moderate Growth and Rapid Decreasing

Here we want to construct rapidly decreasing functions out of (uniformly) moderate growth functions. As a by-product, we show that there is a natural pairing between automorphic forms and cusp forms. T h e o r e m A . Let f : G(F)\G(A) -> C be a smooth K-finite function. If f and its derivatives have uniformly moderate growth, then [/]:

S

-

9

~

C [/](9):=Ep(-irk("M/Zc)-/p(9)

is rapidly decreasing. Key to the proof is that unipotent groups have simple structures. Essentially U(F)\U(A) is a certain copies of F\A. As such, we only need to understand what happens for Z\R: it is then well known that over the compact space Z\R, non-constant terms of the Fourier expansion of a smooth function is rapidly decreasing. Ex.

Let

be a cusp form, then [] :

p

S —> C is simply <j> itself by definition, since if P = G, (f>p = while if P y£ G, by the cuspidality, 4>P = 0. As such, cusp forms are rapidly decreasing if we can somehow show that not only they are of moderate growth, their derivatives are also of moderate growth. (This is almost the case. See e.g., Lem 1.4.1 [MW].) Consequently, we have the following Natural Pairing. There is a natural pairing between automorphic and cusp forms: (•,•): A(U(A)M(F)\G(A)^J

x A0(u(A)M(F)\G(A)^j

-

C

L.

142

Weng

defined by () == /,

x


d

9-

J(ZMWU(A)M(F))\G(A)

2.2

Semi-Simpleness

This is a fundamental result due to Harish-Chandra. Note that A(M(F)\M(A) J is not invariant under the action of M(Aoo) even we have the 3M-finiteness. So it is not an M(A)-module. However, a weak version holds correctly: the space can be viewed as an M(Af) x (LieM(A 00 ) KC,M(A) H/H-module. This in literature is called the theory of (g — if )-modules. The center of this theory is in fact not about automorphic forms, rather it is about cusp forms. More precisely, we have the following T h e o r e m B . (Harish-Chandra) As an M(A/) x (LieM(Aoo) K C,M(A) n K \ -module, AQ is semi-simple. To understand this result, let us fix an irreducible representation a of 3 and an irreducible representation /i of K, and set A(a, /i) to be the isotypic submodule of Ao of type (a, /i). Then we have the following T h e o r e m B'. (a) Each space A(fi,a) is finite dimensional; and (b) A ) ( M ( F ) \ M ( A ) ) = ®aillA(
->

A0(U(A)M(F)\G(A))

such that for any cusp form o € Ao ([/(A)M(F)\G(A)), (
2.3

3-Finiteness

Key to the 3-finiteness in the definition of automorphic forms is that 3 is Noetherian for reductive groups. Say in the case of GLn, they are

Eisenstein Series and Spectral Decompositions

143

in fact polynomial rings. Clearly, it is then natural to understand this finiteness in more details, say to make the finite dimensional space even one dimensional. To give a good approximation to this, the first thing we want to do is to introduce a center character £M of Z M ( A )

and set A(M{F)\M{A))

:=

U e A ( M ( F ) \ M ( A ) ) | (zm) = z^M(j)(m) Vz € ZM{A),m

E

M(A)\

and accordingly A, (C/(A)M(F)\G(A)) \(zg) = z»p • z^

:= { 6 A0 (t/(A)M(F)\G(A)) • (g) V* € ZM(A),g

(In fact in Theorem B', we should work on

€ G(A)}.

A0(u(A)M(F)\G(A))

for a fixed £M-) The idea is that now we may consider a weaker condition: being •^M(A)~nnrfce> o r t n e same being AM^-Rnite. As such the problem then becomes simpler since the group involved is AM^ which is commutative. Consequently its infinitesimal version will be a polynomial ring over a vector space. So the corresponding finiteness condition means that for every automorphic form there exists a finitely codimensional ideal (of this polynomial ring) which annihilates (p. Consequently, using all the non-trivial polynomials in the resulting quotient space, which is of finite dimensional, we can write our automorphic form as a certain combination, (which is far from being canonical), in terms of these polynomials whose coefficients are again automorphic. These later automorphic forms are then expected to be much simpler in the sense that this time they behave according to certain fixed central characters. Put this in a more theoretic form, we have the follows: First, for any Q e C[Re o M ] and G A{0) (u(A)M(F)\G(A)) g>->Q(\ogMmp{g))

-4>{g)

,

144

L. Weng

is again a P-level automorphic form. Then, let A{0)(U(A)M(F)\G(A))

z

E A{0)(U(A)M(F)\G(A)) tMenom(zM(A),c*)

. iM

T h e o r e m C. (Global Version) The above natural map induces an isomorphism C[Rea M ] x A{0) (u(A)M(F)\G(A)"jz Set U.Q(M(A))^M

~ A ( 0 ) ([/(A)M(F)\G(A)).

:= the set of isomorphism classes of irreducible

representations of M(A/) x (LieM(AQQ) <8>RC, M(A) n i f J, or simply, of M(A), occuring as submodule of A0[M(F)\M(A))

, and

n 0 (M(A)) ? M := U CM6 Hom(z M(A) ,c*)n 0 (M(A)) ?M . Then for IT £ U0(M(A))^M,

(or better using (V,n) with V the real-

ization space of 7r), set AQ

:= the isotypic compo-

nent of type 7r in A0M(F)\M(A) Hom M ( A ) (V,A0(M(F)\M(A))CMJ. A0(U(A)M(F)\G(A))^

, that is, the image of V x Set also the P-level space by

:= { e A ( 0 ) (V(A)M(F)\G(A))

J

(M(F)\M(A));V/;eif}. T h e o r e m B". We have the natural decompositions Ao(M(F)\M(A)) - © weno(M (A)) A > ( M ( F ) \ M ( A ) ) ^ A)(V(A)M(F)\G(A)) = ewen0(M(A)) A 0 ( t / ( A ) M ( F ) \ G ( A ) ) / T h e o r e m C . (Micro Version) For <>/ € there exists a finite set D(M,4>) c C[Rea M ] x n 0 (M(A)) x

A0(u(A)M{F)\G{A)Y

A0(u(A)M(F)\G(A)^

Eisenstein Series and Spectral

Decompositions

such that (i) if(Q,ir,),

145

; and

l(

(ii) (g) = Y,{Q,*,v)eD{M,) Q( >gM™p(g)) • ¥>($)• Usually we call D(M, (j>) the cuspidal data for <j) and set UQ(M(A), (f>) be the projection of D(M, ) to IIo(M(A)), called the cuspidal support of . In particular, if now € A(G(F)\G(A) J is a G-level automorphic form, then its constant term (f>p along P is a P-level automorphic form. So it makes sense to talk about (f>pusp the corresponding P-level cusp form introduced in 4.2 and hence also IIo(M(A),<^J lsp ). Set and call it the cuspidal support of . 2.4

P h i l o s o p h y of C u s p Forms

This result will play a key role in the study follows: It exposes the reason why cusp forms are really the 'heart' of that for automorphic forms. Let C0(U(A)M(F)\G{A)\ := the space of functions on the quotient U(A)M(F)\G(A) which are linear combinations of functions of the form g t—> b(mp(g)) • <j)(g) where b are compactly supported <7°°-functions on M{A)l\M(A) and e A0(u(A)M(F)\G(A)) are P-level cusp forms. Now let / : G(F)\G(A) —* C be a function of moderate growth. By definiton, the cuspidal P-form of f is the linear form on the space C 0 (t/(A)M(F)\G(A)) given by /

IU{K)M{F)\G(k) JU(A)M(F)\G(A)

fp{g)-
Note that here P is allowed to be taken as the entire group G itself. T h e o r e m D . Let f : G(F)\G(A) -> C be of moderate growth. Then the following two conditions are equivalent: (i) For all standard parabolic subgroup P of G, the cuspidal P-form of along P are zero;

L. Weng

146

(ii) / — 0 almost everywhere. While important, the proof is a simple manipulation of integrations using a unfolding argument and a standard approximation argument. We leave it to the reader. Consequently, : G(F)\G(A) —» C is an automorphic form (in the sense of almost everywhere) if and only if all its cuspidal P-forms can be induced by automorphic forms. 2.5

L 2 - A u t o m o r p h i c Forms

Recall that by Theorems B and C, if cj> is an automorphic form, then l


.

^(mk)

m G M(A) 1 , k e K,

where Hp(a) := log M mp(a), and ipj is an automorphic form of level P on which the ^ ^ ( ^ - a c t i o n is simply given by is of moderate growth, so is tpj on M(F)\M{K)1. Thus by the finiteness of the volume M ( F ) \ M ( A ) \ ipj's are relatively easy. Consequently, the condition that is square integrable is really one for Q and the exponents Xj. Note that among two factors e^x+pp'Hp^ and Q(HP(a)), the formal one behaves like an exponential function while the second like a polynomial. Hence we are lead to / { a e A G ( A ) \ A M o ( A ) : a a > t ? V Q e A 0 } Q(HP(°)) " 2pp 2(Re\+pp)-2p ^ where -Ipp comes form Haar measure a~ da a P on the AM(K) part. So it is a question of exponential decay competing with polynomial growth. As such we are ready to understand the following T h e o r e m E. (Exponents of L2-Automorphic

Forms) Let £ be

a unitary central character of ^G(A)- Then (f> € A(G(F)\G(A) L

2

x

a

<=> W 6 UQ(M, 4>) with Re7r = X^QGA M a • >

we

J is

have xa < 0.

Eisenstein Series and Spectral Decompositions

147

Day Two: Eisenstein Series 3

Definition

3.1

Equivalence Classes of A u t o m o r p h i c R e p r e s e n t a tions

Let P = UM be a standard parabolic subgroup of G. Then the space A(M(F)\M(A)) is a M(Af) x ( L i e M ^ ) ®M C,M(A) n #)-module and

is a G(Af) x (LieG(Aoo) ® R

A(U(A)M(F)\G(A))

C, K J -module. For simplicity, we say that they are M(A) and G(A)modules respectively. Suppose that TTQ is an irreducible cuspidal representation, i.e., A(M(F)\M(A))

nA0(M(F)\M(A))^{0}.

Basic Hypothesis. If no is cuspidal, then A(M(F)\M(A))

C

A0(M(F)\M(A))

and A (u(A)M(F)\G(A)')

c A0 (t/(A)M(F)\G(A)).

Note that if <j) <E A[M(F)\M(A)) \

tion

and A € a*M, then the func/ 7T0

is again an M-level automorphic form. Denote by :={\\eA(M(F)\M(AJ)

A(M(F)\M(A)) \

/7T0®A

L

V

}, /7T0 i

which defines an automorphic representation of M(A) as well. Similarly, at the P-level, if 0 e A(U(A)M{F)\G(A)) , then the function \(f> : g H-> mp(g)x • (f)(g) is again a P-level automorphic form.

L. Weng

148 Denote by A(U(A)M(F)\G(A))

:= l\<j> U 6 A([/(A)M(F)\G(A))

1.

One checks that this space is the same as the P-level G(A)-module induced from the M-level automorphic representation -KQ®\. For this reason, we call two automorphic representations IT\ , 1T2 are equivalent if there exists A € a*M such that 7T2 = n\ ® A. Quite often, we use !p as a running symbol for equivalence classes of automorphic representations. Clearly, if 7r G ^3, then *# = TT® a*M. As such, for a fixed automorphic form 0, we get a family of automorphic forms by setting <j>\(g) = mp(g)x • <j>(g) parametrized by X € a*M. Roughly speaking, an Eisenstein series associated to with variable A is an average of such a function over the coset P(F)\G(F). Before going to that, let us give the following easy calculation on the change of central characters with respect to the shift from to \: (\)(zg) =mP(zg)x<j>(zg) = (zx • mP(g)x) . ztM+\. ^ } ) =zPP w e ZM(A):

• (Z"+€M
i.e., A0 G A(U(A)M(F)\G(A))^+X if 0 e A(U(A)M(F)\G(A))(M. On the other hand, for £ : ZG(A) ~> C*, and we set A(U(A)M(F)\G(A))t

:={(t> G A(U(A)M(F)\G(A))

\

4>{zg) = z* • {g), Vz G Z G ( A ) , g G G(A)}. Then by the fact that zx = 1 for all z G ZG(A) a n d A G a*M, we conclude that if 0 G A(U(A)M(F)\G(A))(. then so is A. So the reader should distinguish £ and £M as well as their associated spaces of automorphic spaces: In general, since ^G(A) C ^M(A) f ° r a fixed £ there are many extensions £M to ZM(A) a n d the space for £ is simply the summations of the corresponding spaces for £M for all possible (,M- As such clearly, when we talk about G(A)-modules, we should work over a fixed £ instead of a fixed £M- F° r o u r own convenience, an automorphic representation is called ^-admissible, if A{U{A)M{F)\G{A))1X c A(U{A)M(F)\G(A))V

Eisenstein Series and Spectral Decompositions 3.2

149

E i s e n s t e i n Series a n d I n t e r t w i n i n g O p e r a t o r s

Fix a central character £ and let no be a ^-admissible automorphic representation. Fix fa0 € A[U(A)M(F)\G(A)) and write fa := fao®\ : = ^fao a family of automorphic forms parametrized by IT = 7i"o ® A € ^3 (or equivalently by A € o ^ ) . View % (or A) as variable. Definition (1) The Eisenstein series associated to fa is defined by the following series E(fa,n)(g):=E(fa0;\)(g) <5eP(F)\G(F)

=

E

(*'&ro)(
V5€G(A);

<5eP(F)\G(F)

(2) For any it; G G(F) and ir e
y

:= ./(t/'^nwi/CFjtu-ijy/'tF)

fa{w~lug)du

where „• € ^ f [/(A)M(F)\G(A)) , C/' is the unipotent radical of the parabolic subgroup associated to the Levi subgroup M' := wMw~l. Quite often we also write M(w; A) = M(w, ITQ <S> A) when TTO is fixed to indicate that A is a variable. Naturally, for any automorphic representation, we have a corresponding automorphic representation WTT Of

wMw~l.

So far, we have not said anything about the convergence of the series and the integration above. This will be dealt with later. Assume this, easily we have: T h e o r e m B . (1) E(fa,n) e A(G(F)\G(A)} ; and (2) The intertwining operator gives a morphism M(W,TT):A(U(A)M(F)\G(A))

-> A(U'(A)M'(F)\G(A))

.

L. Weng

150 3.3

Convergence

Here we want to establish the convergence of E((f>nQ; A). T h e o r e m A . There is a cone C C Rea*M such that if Re A e C, then the series E^XXg)

:=

mP(Sg)x

]T

• wo(5g)

6£P(F)\G(F)

converges absolutely for any fixed g € G(A). Moreover if g is contained in a compact subset, the series converges uniformly. Sketch of a proof. There are two steps. In the first we show that when M ~ MQ is minimal, and is an Mo-level cusp form, the associated Eisenstein series converges as claimed. In the second, we show that the convergence problem of Eisenstein series associated to any automorphic form can be reduced to the case discussion in Step 1. Step 1. Special case for Eisenstein series of constant function over minimal PQ. Let 1 be constant function 1 over Mo (A). It is Z*2-automorphic. (Why?) Then for any A € a*M , we get the automorphic form Al over LT0(A)Mo(F)\M0(A), given by g H-> mP(g)x • 1 = mP(g)x. Form the so-called relative Eisenstein series EM/M°(l,\;g):=

£

mP(6g)x • 1

6e(M(F)nP0(F))\M(F)

Y^

rnP(6g)x.

<5e(M(F)nP 0 (F))\M(F)

Claim 1. //ReA G p0 + C$, then EMIM° is well-defined. To understand the proof, let us recall the basic convergence proof in calculus for the Riemann zeta £(s) := Y^n=\ ^- I* suffices to prove that f£° x~Resdx converges. But this late integration may be essentially written as x _ ( R e s _ 1 '|^°. Clearly, for convergence, it suffices that Res — 1 > 0. Similarly, for the associated Eisenstein series, we first use properties related with UQ to reduce the convergence problem to that for a

Eisenstein Series and Spectral

Decompositions

151

certain integration. With a suitable manipulation, the final problem becomes the one for the integration roo

,-(Ren+p0-2po,wa)

7,

Jc

where Rep+po comes from the exponent involved, while — 2po comes from the part a~2poda of the corresponding Haar measure. Therefore, for all a e Ao, it suffices to have 0 < {Rep + p0 -2/9 0 , a) = (Rep-

p0,a).

In other words, Re p € po + CQ . Step 2. Reduce to Claim 1. (a) Eisenstein series EM/M°(l,X;g) provides a standard model for all automorphic forms. Clearly, if is a P-level automorphic form, then so is \. Hence Xcj) is of moderate growth. Consequently there exist a constant c > 0 and a sufficiently positive p € Rea*M such that for all k € K and m e M(A) 1 n Sp, we have |A^(mife)| < c •

mPo(mky+Po

which is then < c • EM/M°(l,mk), since EMIM° is a summation of all positive terms mp0(5mk) a special case of which, i.e., 5 = 1, gives mp0{mk). Claim 2. \XM9)\ < c • mP(g)R™+ReX • EM/M°(l,g), Vp G G(A). From the above discussion, we have the following inequality works only for jfc € K, m € M(A) 1 n Sp, \\
(*).

Thus we have to see how it works when these special mk is changed to general g E G(A). By the Langlands decomposition G(A) = [/(A) • -AM(A) • M(A) 1 • K, and the fact that both sides of (*) is M(F)-invariant, it suffices to see how the part AM^x) P a ly s a r °l e But J4M(A) C ZM(A)- This then reduces the problem to the one for center characters involved. This is then done at the end of 5.1.

L. Weng

152

(b) As such, to justify the convergence of E((j>n, 7r), it suffices to study

mP(dgr+x-EM/M°(l,6g).

J2 6<=P(F)\G(F) M M By17 writing 6 out E f °

in terms of V* / „ . „ n /ri.\, „,.„., it is suf<^7e(M(F)nPo(F)J\.M(F)' ficient to justify the convergence of the series <5eP0(F)\G(F) which is done in Step 1, or better by Claim 1. This completes the proof. Remarks. (1) In Step I, if we replace the constant function 1 by a cusp form, a rapidly decreasing function, the same argument works. T h e o r e m B'. Let e A0(U0(A)M0(F)\G(A)) be a P0-level cusp form. Then its associated relative Eisenstein series EM^(
:=

mP(8g)x

£

• ftSg)

<5e(M(F)nP 0 (F))\M(F)

converges absolutely if ReA E po + CQ" . (2) There is a parallel theory of relative Eisenstein series EMl/M2 for M 2 c Mi. In particular, as indicated in Step 2(b), EM^Mi is M2 Ms M M naturally related with the composition of E / with E ^I ^ if Mi D M 2 D M 3 .

4

Constant Terms of Eisenstein Series

We will see the importance of constant terms of Eisenstein series later. Here let us give a nice formula for a special but most important case of them. That is, the one for Eisenstein series associated to cusp forms. T h e o r e m C. Let fa € A0(u(A)M(F)\G{A)) be a cusp form. Then for any standard parabolic subgroup P' — U'M', we have EP,(fa,n)

=

J2 weW(M,M')

EM'lM(M{w^)fa,w^))

Eisenstein Series and Spectral Decompositions

153

where W{M, M'):=iweW\

w~\a)

> OVa G R+(T0, M'), wMw~l

'—> M' standard levi \.

Remark. If W(M, M') = 0, we understand that the above summation is zero. Proof. By defmiton, £p'(07r,7r) = /

V

Ju'(F)\U'(A) =

4>7r(iug)du

76P(FAG(F)

4>TX{IU9) du

E / y<=P(F)\G{F)Ju'(F)\U'W

due to the absolute convergence property. Now using the Bruhat decomposition G(F) = UweWM,\w/WMP(F)w-1P'(F), 1 where WM and WM are the Weyl group of the Levi M and M' respectively, we can rewrite G(F) accordingly to get E

E

we (WM,\W/WM)

7e

/

4>irinug) du

[P{F)\P{F)w-l-P'(F))

Note then that WM,M' '•— {we

W\w(a)

> OVa G.R+(T 0 ,M), w'^a)

> OVa G

is a system of representatives for WM'\W/WM, get

E

E

,-w / M,M'1e{P(F)\P(F)w-^(M'{F)U'(F)))

- E

we can continue to

/

wSW

\

R+(T0,M')\

Alug)du

JU'(F)\U'(A)

E

™£ H 'M,M' m ' € ((M'(F)nwP(F)«)- 1 )\M'(F))

/ JU'{F)\U'(A)

yj .


,

m'u'ug)du

L. Weng

154

using the decompostion P' = M'U' and rewrite the summation over P' to the double sum of M' and U'. Thus note that m'v! ug — [m'u'u(m')~l\ • m'g with m'u'u{m')~l still unipotent. So by using an unfolding argument to combine the integration and summmation over the unipotents into a single integration, we arrive at

E

V>€WM,M'

E m'e ( M'{F)C\wP(F)w-1 \M'(F) J

k

(u'(F)r\wP(F)w-1j\U'(A)

'

=

Y,


EMl'M(i

™ZWM,M>

,

du

4>^w~lum'g)du^\g)

\J[u>{F)nwP(F)w-i)\V>(A)

J

which converges absolutely and uniformly if g remains in a compact subset. So far, the cuspidal condition has not been used. Yet still we obtain a general formula for constant terms in terms of relative Eisenstein series. Next, we use the cuspidality to simplify the above general formula. Note that if w~lU'wr\M ^ {1}, wMw~l is not contained in M'. So M contains certain unipotent elements, and we obtain a proper parabolic subgroup in P. Thus, by the cuspidality of „• we see that for such P"s, ,

Si

x

§K{W lum'g) du = 0.

I U'(F)nwP(F)w-1 J\U'(h) Consequently, weWMM,,wMw-^cM>

m

, e ( M '(F)n«,P(.F)™-i\M'(F))

(U'{F)r\wP(F)w- )\U'(A) J(u> l

^(w

1

um'g)du

Eisenstein Series and Spectral Decompositions

= E

weW(M,M')

155

E m'G ( M'(F)nwP(F)w-1

\M'(F) j

(f>n(w~1um'g) du, J\U' (F)nwP(F)w- J \U' (A) 1

since I we WM,M' I wMw~x C M'\ = W(M,

M').

Now for we W(M,M'), (i) M' n wPw"1 is a standard parabolic subgroup of M' whose Levi is wMw~l; and (ii) U'(F) n ^ P ( F ) w _ 1 = U'(F) n w t ^ F ) ™ - 1 since M' contains no unipotent elements. Therefore, Epi(<j)n,7r)(g)

= E w€W{M,M')

E

rn'e(M'(F)nwP(F)w-1\M'(F))

(f>7r(w-1um'g) du

1

V(c/'(F)n«)f/(F)ui- )\C/'(A)

=

E

E

(M(W,TT)^)K5)

w€W(M,M')

as required. As a direct consequence of the above calculation, we also established the convergence of the integral defining the intertwining operator. A special case of the above theorem is of particular importance. That is, if M' — wMw"1 is just conjugate to M, then EM>/wMw~l

b e c o m e s

trivial

a n d

W(M, M') = iwe

W(M) | wMw~l

= M' }

where W(M) := the set of elements w of the Weyl group W of G whose length is minimal in the left coset of WM and such that

L. Weng

156

wMw 1 is again a standard Levi. That is to say, we have obtained the following T h e o r e m C . Let fa e A0(u(A)M(F)\G(A)) be a cusp form. Then for any standard parabolic subgroup P' = U'M' whose standard Levi component M' is conjugate to M, Epl(fa,ir)

=

^2

M(w,ir)fa.

weW(M)

We end this section by point out that in the convergence statement, for any automorphic forms, we assume that Re7r is positive, and in particular if -K is cuspidal, Ren E pp +Cp. For certain purposes, the following stronger technical result proves to be useful. Proposition. For convergence of E(4>,ir) and M(w,n), it suffice to assume that (i) (Re7r,a v ) » 0 Va £ R(TM,G) such that a > 0 but wa < 0, when is a general automorphic form; (ii) (Reir - pP, a v ) > 0 V a G R(TM, G) such that a > 0 but wa < 0, when 4> is a cusp form.

5

Fundamental Properties of Eisenstein Series

All results are due to Langlands in this generality. T h e o r e m I. (Meromorphic Continuation) (a) Eisenstein series E(fa,ir) defined by

E(fa,Tr){g):=

Yl

Ml/9)

7eP(F)\G(F)

over a certain positive cone C in Re a ^ admits a unique meromorphic continuation to the whole complex space a*M; (b) For any w E W, intertwining operator M(W,TT) : A(U(A)M(F)\G(AJ) fa

-•

A(u'{A)-wMw-\F)\G(AJ)

—»

M(w,Tr)fa

Eisenstein Series and Spectral Decompositions

157

defined by 4>Tr(ug)du

IU'(F)nwU(F)w-1\U'(..,

over a certain positive cone C of Re a*M admits a unique meromorphic continuation to the whole complex space a*M. T h e o r e m II. (Functional Equations) Meromorphically on*# = (a) E(M{W^)(().1X,W'K\ (b) M(W',WK) W(wMw-x).

O

= E((j)^,ir) for allw 6 W(M); and

M(w,n)

= M(w'w,ir)

for all w € W(M),w'

G

T h e o r e m III. (Holomorphicity and Singularities) Assume that 7To is cuspidal. (a) (i) / / R e A belongs to the pp-shifted positive Weyl chamber pp + C+ E\TTo®\i no ® -M

an(

i

M(w, TT0 ® A)

are holomorphic in A; (ii) / / A belongs to I m a ^ , E ( A J and M(W,TTO A] and M(W,TTO A) are without multiplicities at (ReA, a v ) > 0 Va € ^M, i-e-, the generic multiplicities are at most one; (iii) There are only finitely many singular hyperplanes of E

(-K0®\i ^0 ® AJ

and

M(w, 7r0 ® A)

which intersect < TTQ ® A | (ReA, av) > 0 Va € A A/ >.

158

L. Weng

Day Three: Pseudo-Eisenstein Series 6 6.1

Paley-Wiener Functions Paley-Wiener Functions

Over the complex space X^ := (o^-)*, denote by P(X^):= the space of C-valued functions on X^ which are Paley-Wiener, i.e., the space consisting of C-valued holomorphic functions on X^ such that s u p A 6 ^ { l / ( A ) | - e r | | R e A | l ( l + l|A|| n )} 0, set DR := { A € Xg | ||ReA|| < R } and PR(X^):= the space consisting of C-valued holomorphic functions on DR such that su

6.2

P A e ^ { l / ( A ) | • (1 + HAH")} < oo.

Fourier Transforms

For any / G P^R>(X^),

its Fourier transform is defined by /(A) • mx d\

f(m) := /

Vm € Z G ( A ) M(A) 1 \M(A).

J\€X°l,Ke\=X0

It is well known that (i) / becomes a C°°-function with compact support; and (ii) (Fourier Inversion) there exists a constant depending only on the Haar measures involved such that for all Paley- Wiener f, /(A) = c- f

f(m) • wTxdm.

JzG{A)M(A)i\M(k)

Denote by P^R\X^) := the space consisting of all Fourier transforms of Paley-Wiener functions. We then also easily have

Eisenstein Series and Spectral Decompositions

159

(iii) (Estimations) PR(X^) C PR(X^):= the space consisting of 1 C°°-functions f on ZG^M(A) \M(A) such that there exists c > 0 satisfying | / ( m ) | < c • e -RP°SMHI

6.3

Vm

6 M(A).

Paley-Wiener on
Let ^P = 7To (8> XM be an equivalence class of automorphic representations of G(A). We are interested in the functions on *P satisfying the following conditions: (i) for any TT E ^3, <&(7r) E

; and

(ii) $ is if-finite. Such functions are in one-to-one correspondence to functions $ over Xfy satisfying (i;) for all A e X%,

$(A)

6 A(U(A)M(F)\G(A))

; and / 7T0

V

(ii') the vector subspace of A[ U(A)M(F)\G(A))

generated by the

functions 3>(A) where A runs over all Xfy is finite dimensional. More precisely, if IT = TTQ ^ then $(TT) := (M**)(^)

-

$(M7r):=(A- 1 $)(7r)

where as usual (X(f>)(g) := mp(g)^ • <j)(g) for a P-level automorphic form 0. With this, define the space of Paley- Wiener functions on ^3 to be (a) -P(M,«p) : = the space of functions $ over the total space ^J such that $ € i4ft/(A)M(F)\G(A)) ( b ) P(M,
of

® P(X&); and

functions $ over <£* := { 7r € ^P|||Re7r|| <

such that $ 6 A(U(A)M(F)\G(A)\

P*(.Y&).

As such, for any function $ € P(V
160

L. Weng given by

A(U(A)M(F)\G(A)^

Fn($)(m')

: g i-> /

( m ' ) " ( A + p p ) • ($(TT ® A))(m'mA;) dA

for all g = umk G G^A) with u G J7(A),m G M(A),k G X. Clearly, (i) F,r($)(•)(#) G A(M(F)\M(A)) n - := the space obtained as the left-translation under the adelic group M(A) of A(M(F)\M(A)) (Recall that the difficulty as always is that A(M(F)\M(A))

is not

closed under the action of adelic group as said earlier); (ii) F^x($)(»)(g) = \Fn($)(*)(g) VA G X°; and (iii) F , ( $ ) W ( u m j ) - 5{m) • Fw(fc)(«)(s) Vm G Z G ( A ) M(A) 1 and ue U(A). In particular, we conclude that (a) At the unit element e of M(A), (Fw($)(e))(g

= umk) = / = /

e-^+^Qfr

®

X)(emk)d\

$(7rA)(mfc)dA

JxelmXfa

= /

$(7r®

X)(mk)d\

JXeX%I,ReX=ReXo

for any Ao G X^ since $ is holomorphic. It is independent of n; (b) We have a commutative diagram i([/(A)M(F)\G(A))^c?W(4)

*•

Id®Fj° A([/-(A)M(F)\G(A))

Jfj |F

1

® C C C °°(Z G ( A ) M(A) \M(A))

A

P ^

where the morphism /z is defined by 1/ ( Jli Vi ® /i 1 ( m 0 '• 9 ^ £ / i ( ™ p ( f f ) ) • K ) " P P V t ( m ' g ) Vm' 6 M ( A ) , 5 e G(A);

w

,

T }

Eisenstein Series and Spectral Decompositions

161

In particular, we have the following result which controls the growth of the Fourier transform of elements in PJ^ ~.\ (c) (Estimations) P{RM^ c Im(A(tf(A)M(F)\G(A)) w o ® c P*(X&)).

7

Pseudo-Eisenstein Series

In this section we assume that ^3 is an equivalence class of cuspidal representations. Definition. For $ 6 Pn^w,-, its associated pseudo-Eisenstein series is a function on G(F)\G(A) denned by

Ms)==

E

(Fv(*)(ej)(6g)

6eP(F)\G(F)

=

E

/

=

E 6€P(F)\G(F)

*(n®\)(6g)d\ x X

^ M^e>['-KeXo

SeP(F)\G(F)

/

> 7r

*(ir)(6g)dn.

' eq3,Re7r=Re7ro

Clearly, in the convergence region Re7r € C of E( $(7r),7r )(<7) we have 0*(g)=

f

EU(7r),ir)(g)d7r.

This explain why #$ is called a pseudo-Eisenstein series. T h e o r e m A . (a) 0$ is well-defined; (b) (i) 0$ is square-integrable over ZQ^G(F)\G(A) (ii) 6<& is rapidly decreasing over ZG^G(F)\G(A) and (c) <0* | S e P(M
if 3> € P{MW)> if $ G -P(Af,qj)/

The moral of this result is that it provides a natural bridge between general square-integrable functions and Eisenstein series: the decomposition of L2 functions may be transformed to the one for pseudo-Eisenstein series.

L. Weng

162

Proof. Let us start with (c). It is a direct consequence of the philosophy of cusp forms. Indeed, for any / € L2(G(F)\G{A)),

(M,<£)

and $ G P(M,<8),

/

J(9)-0*{9)dg

JZG(A)G(F)\G(A)

= /

A5)-

•IZG(A)G(F)\G(A)

= /

E

/

5eP(F)\G{F)

f(g) • /

= /

**{5g)d*dg J

^V,^^=^>^o

®n(g)dirdgdg

fp(g) • I

J ZG(k)U(A)M(F)\G(A)

<M#) dn dg dg

,/7r€
(by an unfolding argument and the definition of constant terms fp of / along P ) . Clearly, by the convergence property, this latest expression can be rewritten as /

/

Aeq3,Re7r=ReA0

L

fp(g) • $w{g)dg dg. J

JZG{A)U(A)M(F)\G(A)

Thus if / is orthogonal to all pseudo-Eisenstein series, then /

L

fp{g) ' *(9)dg

0

JZ GWU(A)M(F)\G(A) JZr? r^U(A)M(F)\G(A)

for any • e Co[U(A)M(F)\G(A)j since $ can be taken to be any Paley-Wener functions. In other words, all the cuspidal P-forms of / are identically zero. Consequently, by the philosophy of cusp forms, / = 0 almost everywhere. (a, b) By the estimations on Fourier transforms of elements in -P/M tp)) we know that if / € PR(X^j), there exists fi very near pp such that | / ( m ) | < c • mtiM Vm e M(A). Consequently |F w (*)(.)(ff)| < c • m%+»+^

vgESp.

Thus using the convergence proof for Eisenstein series, we conclude that 6$ is well-defined. This gives (a).

Eisenstein Series and Spectral Decompositions

163

Moreover, we have \e*(g)\
J

\0*(S)\-EG'M°(l,g)dg
lA G(F)\G(k) J A,G(A)

By an unfolding argument, we need to show that

/

/

/

\(Fn(^)(e))(amk)\

• EP(amk) dk

(am)-2ppdmda

converges. Recall that we have a natural upper bound for F 7r ($)(e). Using it, we are lead to show the convergence of the integration

f

a' i 'e- R H ,og " a|| da

JAG(A)\AMW

for fixed fi'. Clearly, it is a comparison of two exponential functions: with R sufficiently large, the second term dominates the first. This gives b.i) With the proof above, for (b.ii), what is left here is to apply the fact that cusp forms are rapidly decreasing.

8 8.1

First Decomposition of

L2(G(F)\G(A)^

Inner Product Formula for P-ESes

The following inner product formula for pseudo-Eisenstein series plays a key role in the first level decomposition for the space of L 2 functions on ZQ(A)G(F)\G(A). It also gives us a first example to realize why the constant formula for Eisenstein series associated to cusp forms is fundamental to all the following works.

L. Weng

164 M a i n T h e o r e m A . For $ e

-P(M,

,

e

-P(M', \ ^ * ' i ^ * )

=

/M(w_\-w#)$'(-ii;7f),$(7r)\

^ / 7re
$'

N

x

'

cfar M

wewi (M,q3),(M',€ W | u ; € W\w(M,y)

=

(M',¥')\

and

' W

7X

JzMWM(F)\M(k)

Remarks, (i) As usual, if w((Af,«p), (M',«P')) = 0. i-e-> if ( M > ^ ) and (M', ^3') are not equivalent to each other, then we understand the above summation as empty which gives the value 0. (ii) We also would like to reminder the reader that by using PaleyWiener functions, automorphic forms involved are all cuspidal. Accordingly, Ao € pp + CM can be taken to be any vector in the shifted positive Weyl chamber in Re a*M. Proof. The proof of this formula is surprisingly easy. In fact, by a standard unfolding argument, we have 0$/,0$) := / '

0&{g) • 0*(s) • dg

J {ZGWG(F)\G(A)

= I J (ZGWU(A)M(F)\G(A)

e^rpJg^-(F^)(e))(g)-dg. V

'

This leads to a calculation on the constant term of pseudo-Eisenstein series. Now Ao is in the absolutely and uniformly convergent region, so the problem may be transformed to that for constant terms of Eisenstein series. Also it is cusp form that shows up in Paley-Wiener functions. Hence, we only need to use the formula of constant forms for Eisenstein series associated to cusp forms. Recall then that by

Eisenstein Series and Spectral Decompositions

165

the formula of Day 2, for such a constant form to be non-zero, standard parabolic subgroups involved should satisfy the condition that one is conjugate to a standard sub-Levi of the other. Consequently when apply to our situation here, to get a non-zero inner product, by switching $ ' and $ if necessary, we conclude that M and M' must be in the same association class, i.e., there should be a w € W such that wMw~l — M'. But, constant terms for the associated Eisenstein series become extremely simple - it can be written in terms of the associated intertwining operator directly even without using relative Eisenstein series. All this, plus a simple application of Fourier inversion formula then give the result above, except the following easy point: if ^3 is not transformed to ^3', then the corresponding cusp forms (appeared in $ ' and $) are orthogonal.

8.2

Decomposition of L2-Spaces According to Cuspidal Data

A direct application of the above inner product formula (and the fact that P-ESes are dense) is the following first decomposition of the L 2 -space. As above, cuspidal data (M, ^J), (M',^J') are called equivalent if there exists w such that w(M, *P) = (M', ^3') or the same, wMw~l — M' and wty = <£'. Denote by X:=

(M,«P)]=:{(M,«P)}/

be an equivalence class of cuspidal data. Denote by £ the unique central character associated to X. (How?) We call then X as £admissible. Denote by © the collection of all ^-admissible X's and set L2(G(F)\G(A))X

:=

(e* I $ 6 p{MtV),

(M,«p) €

x)c.

M a i n T h e o r e m A'. There is a natural orthogonal decomposition L 2 (G(F)\G(A))

=

®Xe&L2(G(F)\G(A))x.

166 8.3

L. Weng Constant Terms of P-SEes

The description of a general formula for constant terms of pseudoEisenstein series in literature appears to be unnecessarily complicated: As stated above, it is a simple translation of that for Eisenstein series associated with cusp forms. To help the reader, and also for the purpose to introduce the so-called relative pseudo-Eisenstein series, we add this subsection. Recall that

MS) := /

E(
V

y

Thus to write down the constant terms of 9$ along say P', the key is to apply the following result for Eisenstein series associated to cusp forms: JZ EM'/M(M(W,TV),wirY weW(M,M') As such, relative pseudo-Eisenstein series should be introduced. We will not give the most original one, but their cousins. (Pay attention to the notation we use.) Fix an extension £^v of £ to ZMir&\. View M as a subgroup of M' (instead of G), set Theorem. Ep,(&(n),ir\

=

^PM' :— 17T € ^P | 7r is £MI — admissible >, which is an equivalence class of automorphic representations of M(A). If *$M' 7^ 0) we can have the associated Paley-Wiener functions P (M

X^Mw_x =: [K'W}M'- F° r a fixed R' > 0, and a G-level Paley-Wiener function $ e P?^ ~<, we have then a subset {V €

[TT'W]M,

I ||Re7r' -

A*O||

< R'} =•

and hence a Paley-Wiener function in Pfi,Mw-i WW]M';R'

B TT' !-*• M(w,w

l

ir')$(w

WW}M';R'

rni i \ given by 1

TT').

Eisenstein Series and Spectral Decompositions

167

(Here, to make sure that everything is well-defined, fio € ReX^j, a fixed vector satisfying \\to\\+tf
and

(-//0,av)»0

is

R+(TM,,G).)

Va €

Set also Pw be the standard parabolic associated to wMw"1 with unipotent radical Uw and UQ € R s l ^ . ! satisfying ||z/0|| < #

and

(uo-ppw,^)

> 0 Va € / ^ ( T ^ ^ - i . M ' ) .

Then define a relative pseudo-Eisenstein series by nM'/wMw"1 , l \( \ _ M(u,,w-UMw-^nw){9)--

V

f JTT'€:[lT'w\Ml,ReTT'=llo

+ Uo

(M(«;,u;-V)$(i«-V))(^)d7r/.

J2 (5G(M'(F)nP™(F))\JW'(F)]

Lemma. 7/TT^ e w1^+ImX^Iw-1, (

then

\M(u;,'iu- 1 .)$(u;- 1 »)V 7 I W ~~ ^ ( w . i u - 1 * ) * ^ - 1 . ) V^wJ-

Consequently, fl^^l?.)^-!.)^) the quotient space

ma

y

be viewed

as a function on

D

w,»0 : = { < ; <= urfP | Re<, = ^ o j / I m X ^ - L

With this, we are now ready to state the following easy consequence of the formula on constant terms of Eisenstein series: Proposition, D 'w,w and

(i)

0M(^?.)*(U,-I.)(O

(ii) 0<j>,p' = Y,wew{M,M') JD'

is

rapidly decreasing on

^M(u),tu-1.)$(w-i.)(7r^)^7r^-

L. Weng

168

9 9.1

Decomposition of Automorphic Forms According to Cuspidal Data Main Result

Prom the discussion above, the key to the success of our first decomposition of L2-space is that there is a natural pairing (, ) on L2-functions. On the other hand, general automorphic forms are only of moderate growth, so they are in general not integrable. This posts a huge difficulty for the study of (space of) automorphic forms. Luckily, we have a way to overcome this, still using a certain pairing. The essential points then may be summarized as follows: (i) Pseudo-Eisenstein series associated to Paley-Wiener functions (or better, to cusp forms) are rapildy decreasing; (ii) There is a natural pairing between rapidly decreasing functions and moderate growth functions; and (iii) There is a huge algebra of operators there ready to be used. Simply put, by (i) and (ii) we can talk about the pairing between pseudo-Eisenstein series and general automorphic forms, which gives us a way to understand automorphic forms in terms of cusp forms; and by (iii), with the common eigen-space decomposition for ceratin operators we obtain a decomposition for the space of automorphic forms as well, since operators acting on P-ESes can be 'transformed' to that for automorphic forms via the 'adjoint' operation w.r.t. the pair in (ii). Fix then a central character £. Two ^-admissible cuspidal representations (M, TT) and (M1, TT') are said to be equivalent if there exists a w € W such that w(M, TT) = (M',7r'). Let 77 := [(M, TT)} be the equivalence class associated to (M, TT) and S be the collection of all ^-admissible equivalence classes. Recall that from Day 1, for any <j> £ A(G(F)\G(A)

J , and any

standard parabolic P = UM, there exists a finite subset D{M,) c C[Rea&] x n 0 (M(A)) x such that

A0(u(A)M(F)\G(A))

Eisenstein Series and Spectral Decompositions

169

(i) For all (Q,ir,), ip G j4o(t/(A)Af(F)\G(A)) ; and (ii) cf>cpsp(g) = E(Qw)£D(M,4>)Q(lo&Mmp(9)) • G A(G(F)\G(A))

I (M )7 r) e r, V(Q,7r,) € D(M, )}.

M a i n T h e o r e m B . There is a direct sum decomposition A(G(F)\G(A))€

9.2

=

(Br,eEA(G(F)\G(A))^

Langlands Operators

Fix R > 0 and a ^-admissible equivalence classes 36 of cuspidal data say associated to (M, ^3). Set X :={(M, TT) 17T G

^AM;W(X)

: = z A o A(M, TT A) o A

G End M ( A ) where (A)(m) := mx • 4>{m). As such, (i) means that operators AM,IX{^) depend holomorphically on A;

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(ii) means that there exist positive constants c and N such that (for one and hence all operator norms) |iWA)||
: A0(M(F)\M(A))

1

,7r/ :— WTT. Then we have a

-> A0(M'(F)\M'(A))

#.

The admissibility says that the following diagram is commutative: 4,(M(F)\Af(A)) A(M,TT)

i

^

^lo(M'(F)\M'(A))

^

i

A)(M(F)\M(A))

" ^

;

A{M',TT')

^0(M'(F)\M'(A))

#.

Denote by TlR the collection of all Langlands operators. Clearly, under composition, H% becomes a C-algebra, and if A € H%, A is determined by its value at a single (M, TT) £ XR. Ex. A special type of Langlands operators is of particular interests. They are Langlands operators such that for each fixed (M, 7r) € 3tR, the endmorphism A(M, TT) on the space Ao ( M ( F ) \ M ( A ) ) is induced by a scalar, say / ( M , 7r), multiplication. We will denote the collection of all such Langlands operators by Wf. c , which is clearly a subalgebra of H^. For example, fixed a TTQ, and write TT = TTQ <8> A^. Set, in particular, the scalar f(M, TT) above to be (A^A,,-). We will denote the associated Langlands operator by AQ. Langlands operators A naturally act on Paley-Wiener functions $

e

P

(M,
Via

$ H-> A$

(ir H-> A(M,

TT)$(TT) J .

This then also induces a natural action of Langlands operator on pseudc-Eisenstein series via A{A)

: 6$ »

( A ( A ) ) ( 0 * ) :=

0A$-

Eisenstein Series and Spectral Decompositions

171

For later use set A := A(AQ). A, which is essentially the standard Laplace operator in question, can be extended to the entire L2-space over which it is self-adjoint. As such, the so-called spectral decomposition of the L 2 -space is indeed a decomposition with respect to this A. 9.3

Key Bridge

If A is a Langlands operator, set A* to be the Langlands operator defined by A*(M,-7r):=A(M,7r).

Set also nf = ®xn%. The reason why Langlands operator plays a central role in the decomposition of space of automorphic forms is the following Bridge L e m m a . Let <\> e A(G{F)\G{A)) and R »<^ 0. Then for any A € Ti?, there exists a unique element A(A)(f> € A( G(F)\G(A) J such that for any pseudo-Eisenstein series 9<&, we have A(A*)6*t) = (0*,A(A)). (The notation A(A)(f> is copied from [MW] and reminds us of Chinese characters.) Proof. Two main points: S u b l e m m a 1. For a fixed g £ G(A), and write mp(g)° for the function on X^ whose value at A is given by mp(g)x. Then for any polynomial Q in R e a ^ , (d Q m P ( 5 )*)(A) = Q(log&(mp(s)) • m P ( 5 ) A . S u b l e m m a 2. For $ € P^^

and ip 6 A0(u(A)M{F)\G(A)^

set

lU,
J

JZG{A)U(A)M(F)\G(A)

v

(F7r^)(e))(g)-mP(g)xcp(9)dg. '

,

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L. Weng

Then l(cf>,
i/ -7r ® A g (-7r(g) A, \ocp),

if —K A € ^3.

where C := Vol(z G ( A ) (ZM(F) D Z M ( A ) ) \ Z G ( A ) (Z M ( A ) n M ( A ) 1 ) ) . The reader should realize that H.% is a very powerful tool in studying automorphic forms. In Day 4, we will use it to get 'extra' flexibility which then helps us to obtain a spectral decomposition for -spaces. We too expect that an alternative proof of meromorphic continuation and functional equations for general Eisenstein series can be established by pushing the study here on the pairing between pseudo-Eisenstein series and automorphic forms.

Eisenstein Series and Spectral Decompositions

173

Day Four: Spectrum Decomposition: Residual Process 10

Why Residue?

10.1

Pseudo-Eisenstein Series and Residual Process

For $> 6 -P(j^(T})> ^s associated pseudo-Eisenstein series 0$(g) is defined by SeP(F)\G(F)

•/^e
for any A € pp + C^- As such, the series converges absolutely and even uniformly for g in a compact subset of G(A). Consequently,

77req3,Re7r=A J G P ( F ) \ G ( F )

V7re»,Re7r=A

which belongs to L2

v

'

(G(F)\G(A)Y

Moreover, if $ G P?MB\,

then E($n,Tr;g)

is holomorphic on

< 7r € ^3 : Re7r = 0 >•. So it too makes sense to talk about the integration fnemn^T=oE(^1T,Tr;g}

dir, which can be easily seen to

2

be L as well. As they stand, the difference we should notice here is that this latest integration is canonical since now for the integral domain, we replace an arbitrary XQ € pp + C^ above by 0, the zero element of (a^j . It is quite natural then to use these latest integrations to generate the L 2 -space. On the other hand, by shifting the contour from Xre
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L. Weng

Clearly, a far-reaching generalization of the classical residue theorem is needed in order to cover the generality encountered here. As one may imagine, while complicated in stating it, the main idea is rather simple. In literature, this is achieved via the so-called residue theorem. We are not going to give all the details here, instead let us concentrate our attention to the outcome. Assume then Sj is a singular root hyperplane of J5($(7r), n) which sits between |TT €

Resj£7($,r,7r, 0)d7r.

JireS),Ren=z(Sj)

Similarly, as above, this time at jj-level (instead of the <}3-level at the very beginning,) the integration is again not canonical. To make it canonical, we need to shift the contour from J . R _ ,-•> t o Inef) Ren=o(f))'

wnere

°C#) represents the so-called 'origin' of ft, pro-

vided that Res]*E(&n,Tr,g) is holomorphic at every point of I -K € ft | Re7r = o(S)) >. (See the precise definition below.) Accordingly, the residue of Res^E($n,ir,g) along singular hyperplanes, which is of codimension 2 in ^3, of S) have to be added. With this informal discussion, we are now ready to follow Arthur [Ar4] to give the general picture as follows (but we reminder the reader here that not all singularities are of interests to us - we only care about these 'sitting' between A and 0): (a) There are a pair of standard parabolic subgroups P' C P and a filtration of affine spaces to C ti C • • • C ir = Re dp, with U = Ai+Xi

Eisenstein Series and Spectral Decompositions

175

where tj is a linear subspace of Re ap, and Aj is a vector in Re ap, which is orthogonal to tj. Assume that to = a*P and that for any i, U-i = {A € U : A(/3V) = 0} for some root j3 in R(Tp>,P') such that /3V does not vanish on t{. Suppose that for each i we have also chosen a unit vector vi in ti which is orthogonal to £j_i. It is uniquely determined up to a sign. We will denote this sequence of affine subspaces together with the choices of unit normals by the letter S and denote the smallest space to by ts •=•• As + Re dp. (b) Suppose now that ip is a meromorphic function on ap, whose singularities lie along hyperplanes. Then we can obtain a meromorphic function on tsyc = A5 + ap by taking successive residues with respect to S. More precisely, let Ao be a point in t^c with the properties that any singular hyperplane of

1- zTvr

r

for z = (z\,..., zr) G C . Let Ti,..., Tr be small positively oriented circles about the origin in the complex plane C such that for each i, the radius of Tj + i is much smaller than that of IV Then

is a meromorphic function of Ao- We will denote it by Res^A-^Ao^KA) P/B or simply R e s e if © is the subspace in ^3 corresponding to t^. In particular, if P' = PQ and P = G, we write it as Res@ or even simply Res©. 10.2

W h a t do w e have?

(a) Singularities The above discussion on residue is clearly based on the fact that meromorphically extended Eisenstein series (associated to cusp forms) admit singularities only along root hyperplanes. Accordingly, let S(M,ys) : = be the collection of all hyperplanes which are

176

L. Weng

singular for at least one such Eisenstein series (associated to M-level cusp forms of type ^J), their intersections and the Weyl-conjugations of these intersections. <S'(M,q3) i s n ° t o, priori locally finite, so we need to introduce S?M~s to get the locally finiteness by assuming that cusp forms are of fixed finite K-type J . (Here, locally finiteness means that for any compact subset C, there are only finitely many hyperplanes which intersect C.) For later use, for & E S/M<^\, set Norm© := {w G Stab(M,'s can be studied using approximation. The choice of these special Paley-Wiener is surprisingly easy: We will assume that $ are such that they vanish very high order along singular hyperplanes of Eisenstein series. To end this discussion, let us recall the following starting point: Main Theorem A of Day 3. We have 0$,,0$\ = / \

where

I

4 ( * ' , $)(*•) cfrr

./7re
A($',$)(TT) Y^

M(W~1,-WTT)^\—WTt),$(ir)

weW(M)

••= Yl

Yl

(M',q}')e£uj€W'((M,
(^"(^_1>-^)*(M',«p')(-u;^)'$(7r))

Eisenstein Series and Spectral Decompositions

10.3

177

Difficulties

(a) Compactness. Note that integration domains {ir G 0, set e
^&esfM^,&C66'-

That is to say, by first removing any possible singularities which are 'trouble makers', while wishing that somehow at a certain later stage, approximation argument with allow us to deal general cases as well.

11 11.1

Main Results Functional A n a l y s i s

For £ a unitary central character and consider the Hilbert space H := L2(G(F)\G(A)){ Identify ^3 with X*^ and define an operator AQ on Paley-Wiener functions by $ H-> A0$

: TT i-> (Aw,

K)^-K-

AQ then induces a natural operator A on pseudo-Eisenstein series A(0$)

= 0A O (*)-

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L. Weng

A is self-adjoint and can be extended to L2( G(F)\G(A) J . (Essentially, it is the standard Laplace operator.) As such, if Q denotes the set of z G C such that A — z • Id is bijective and admits a bounded inverse, then Q is open and contains C — R. The set C — Q is called the spectrum of A and ( A — z Td J for z G Q is called the resolvent of A. It is well known that for every x G R, there exist a closed subspace Hx C H together with an orthogonal projection px : H —> Hx such that (i) Hx C Hx> if x < x'; (ii) Hx = Dxi:x'>xHxr, (hi) C\XHX = {0}, \JXHX is dense in H; and (iv) There exists a real number R > 0 such that the spectrum of A is contained in (—oo, R]; Set also Hx- := Ux>:x><xHx> and px- : H —> i7 x - the associated orthogonal projection. Then x G R is called a point of continuity of (Hx) or simply of A if Hx = i ? x - . (v) For T ^> 0 toift —T a point of continuity of (Hx), (h', qTh) = Jim - — ^ = = / (/»', (A - zld)-lh) £^0 27TV-1 Jr(~T,R';s)

dz,

where qx •— Id — p-r and F(—T,R';e) is the contour consisting of sides {(x, y) G C : x = - T , e < \y\ < c] U {(x, y) G C : x = R', e < \y\ < c} U {(x, y) G C : — T < a: < R, \y\ ± c} mi/j counter-clockwise orientation. (Do draw a picture for T(—T,R';e).)

11.2

Main Theorem: Rough Version

Two elements 6 and & are called equivalent if there exists w € W such that wB = & and w(M,<#) = ( M ' , ^ ' ) where (M,«p) and (M', ^3') are associated cuspidal data for S and &f respectively. Denote by Sf = U(M,q^:(Af,!p)63E'S,(Af,qj) and let [S~\ := 5 2 / ~ be the collection of equivalence classes.

Eisenstein Series and Spectral

Decompositions

179

Fix T, R > 0 and T" » T | R 0 be such that - T is a point of continuity of A. M a i n T h e o r e m . For any <£ e [S%] there exists an orthogonal projection P r o j | T onto a closed subspace of L2(G(F)\G(A))

:=<*•!* e i * >

satisfying the follows: (0) Projf

T

= 0 t//or a» 6 € C, {TT € 6 < T | ReTr = o(&)\ = 0;

(1) P r o j | T J. Projf, r if£^<£'

€ [Sf];

(2) tfr0* = E t e [ S | ] P r o j | T 5 * for all $ € Pf. Before we give a refined version of this theorem in which we describe the projection Projf r in details, let us give an application of this result. For this, set then 4?:=Proj|rL2(G(F)\G(A))^. Lemma. L2^ c L2^ Consequently, let

ifTx < T2.

L2 (G(F)\G(A))^

:= U r : T G R > 0 L g .

S p e c t r u m D e c o m p o s i t i o n . (Rough Version) There exists an orthogonal decomposition L2(G(F)\G(A))^

11.3

=

®€€m]L2(G(F)\G(A))^.

M a i n T h e o r e m : Refined Version

The projection P r o j £ r and hence the spaces L^ in the decomposition theorem may be understood in great detail. There are two stages of this. Here is the first one using residues. (The second, using Levi subgroups, will be one of the main topics of Day 5.)

L. Weng

180 Recall that here :=( =

A($',$)(7r)

^^ M(w~1,-v)Tt)$'(—WTt),$(ir) ^wew(M) J2

S

\ : '

(M(u>-\-w)$(M/iV)(-U/7r),*(?r)

(Af',«p/)eXiuew((M,
1

M

toeiv(M)

is holomorphic at every point of < TT E &^,; (ii)

(e^,Pvo)lTe^ = m c ( $ ' , $) := /

r c ( $ ' , $) (TT) d S e TT-

•'7r66£ i
Back to pseudo-Eisenstein series and hence Eisenstein series, set

It is a meromorphic function with polynomial singularities over S%. T h e o r e m B . For £ e [Sf], $ € P * ' 5 , and 6«r e €, (i) TTje function e
e
ee;($,7r)d ee 7r.

J7re6£,
With this, to see the structure Lg:=Proj|rL2(G(F)\G(A))

Eisenstein Series and Spectral Decompositions

181

more closely, for every ir £ I TT e &c,, ir) above.

G(F)\G(A)

T h e o r e m C. (0) A^n ~ ^ | W 7 r for all w e NormS^; (i) For all IT G ©C,
c
$)(TT);

(ii) There is a natural isometry

L J ! ~ Hilbf T where H i l b | T := the Hilbert space consisting of all measurable functions F on < 7r € 6c,;) > suc/i that • F(ir) € -A^,,. almost everywhere; •

F(WTT)

= F(n) for all w € Norm 6 c ; and 2 F'7r)ll ds,7r
/ Jire&
In particular, let T —• oo, we get the spectrum decomposition for Lg. stated above.

11.4

How to Prove?

It is based on an induction on the semi-simple rank of G. As stated above, we first deal with a narrow class of Paley-Wiener functions $, i.e., those $ which vanish to sufficiently high order along singular hyperplanes; then work over general $ via an approximation argument. Indeed, if we let ( H i l b ^ r ) := be similarly defined as that for H i l b | r above but with the additional condition that F vanishes to a sufficiently high order along singular hyperplanes, then we have the following

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L. Weng

Key Approximation Lemma. ( H i l b | T j

= H i l b | T . That is to

say, (HilbJr T ) is dense in Hilb^ T . The reader can see at least that with this vanishing condition to high order along singular hyperplanes, the statement in (i) of Theorems A and B on the holomorphicity then becomes trivial. As for the rest of statements, we need the following M a i n Theorem'. There exists z(&) e Re(6) and residue datum Resg such that for all $ ' € Pf, $ € PfM^y

(0Vt9rO*)= £

j^JL-

x f

(

S

Res§^l($',$)(u;7r)] den.

This theorem is indeed the heart of all theorems stated above. The start point is of course the following inner product formula for pseudo-Eisenstein series 0*',0*) = /

A($',$)(ir)dir

where Ao € pp + C^. Prom here, to shift the integral domain to that with Re7r = 0(6), we need to apply the residue theorem. With all this done, then we use functional analysis to have the formula 0&, 9r0* > = lim - -—7== / {0&, 6$z) dz I e->0 27TV-1 Jr(-T,R';e) where

Since

( z l d - A W = 0$, we are then in a position to apply the following

Eisenstein Series and Spectral

Decompositions

183

Key Technical Lemma. Let V » T ) f l 0 with T » 0 such that -T is a point of continuity of A. Suppose that for all © € SfMm), (1)

)»

(Res^@A($',$) )(wir) is holomorphic at every point

tuSStab(M,
of the imaginary space < IT € 6 < r | Rew = o(©) V; and (2) For allF € tf|c, ResgA($', F $ ) = F | s • ResgA(*', $ ) .

<3e S

' (M,!p)

x /

(

J2

Resg^($',$)(u;7r))d e 7r

Ae6
/

where the origin o(@) is defined by o(&) = Re© D (Re© 0 ) -1 with 6°:={\eX%I\&®\ = &}. Clearly, to verify the theorem, the point is to verify the two conditions listed in this lemma. The first is about holomorphicity while the other is about why taking residue process is compactible with the action of 'scalar' Hecke operators. As such, to say the least, by direct applying the residue theorem, which is long and hence tedious, we have

Norm© x / iireS< T ,R«r=j(6)

(

X)

Res§,4($',$)(w7r))de7!"

weNorm©

where =T means that up to certain terms outside the compact subset 6 < T , we have the equality. But then with the use of &z, we can get a holomorphicity out from this = r relation, which by the functional analysis relation togeher with condition (2) then gives a real equality for (#$',<7T#), since y.1 , t, may be taken out from Res^4($',$ 2 ) to get Resj4($', $) which is what we want in the lemma and the socalled horizontal lines are well-behaved. In this sense the key is to establish the relation Res%A($',F$) = F\6 • ResgA($',$) for the narrow class of Paley-Wiener functions used.

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L. Weng

Day Five: Eisenstein Systems and Spectral Decomposition (II) 12 12.1

Eisenstein Systems Relative Theory

Let PL = L • UL be a standard parabolic subgroup of G. For any unitary character £x, of ^L(A) > s e t L2(UL(A)L(F)\G(A))

:= {/ G L 2 (c/ L (A)L(F)\G(A))

| / ( ^ ) = z"" • z^ • f(g),Vz G ZL{A),g

G G(A)}. L M

A parallel discussion with relative pseudo-Eisenstein series 6 J for Paley-Wiener functions <£ € P(M«\ \ gives a spectral decomposition for L2 (uL(A)L(F)\G(A))

in terms of residues

9LJM = lim £

Proj^*

<=i€[S| 1

^ e J t s * , Ae6Ci.;
J]

Res4s£^L/M(*,^)^,7r

with the parallel notation index by L (instead of often omitted G), Ex. For every (M, ^3) G X with central character £ : ZG(A) -* C*, let £L be an extension of £ to ^L(A) • Set

^L:={7rG^|7r|ZL(A)=£L}. Assume that *P^ 7^ 0. Then we have (i) (Compatibility) For w G W(M) such that wLw-1 Levi, W<#L = W ) » L » - I ; and

is a standard

Eisenstein Series and Spectral

Decompositions

185

(ii) (Admissibility)

12.2

Discrete Spectrum

Note that if dim©£ 7^ 0, then

/

J2

Resi&CLEL/M($,wn)d6€LTr

w€WL(M)

is not an automorphic form (since the ji-finiteness is not satisfied). We conclude that dim&L has to be zero if the resulting residue is indeed an automorphic form. Viewing cusp forms as trivial Eisenstein series for which we take the residue process to be the identity map, then we have the following: Theorem A. Automorphic forms in L2 (UL(A)L(F)\G(A)

)

are

the residues of relative Eisenstein series associated to cusp forms. Equivalents,

A2(uL(A)L(F)\G(A))

:= L2(UL(A)L(F)\G(A))

n

A(UL(A)L(F)\G(A))

is spanned by cusp forms and Res^ 7 r i £ L / ' M ($, WIT) for all M C L and all Paley-Wiener functions $ which we remind the reader are induced from cusp forms. This explains why in literature A 2 = 4 0 ©Res(ES(>4o)) whose corresponding part in L2-space is often called discrete.

L. Weng

186 12.3

Eisenstein Systems

Usually we call Res^@

EL'M($,

wn) an Eisenstein system or better

LM

but more narrowly Res%igLE ' ($, urn^) an Eisenstein system where fl> : = nL ® M f ° r M e - ^ L Eisenstein system is the essential part in understanding the residue spectrum. Osborne and Warner devote a whole volume [OW] on the beautiful theory involved. Key properties of Eisenstein systems may be summarized as the following Theorem B. (i) (Compatibility) For •n^ :— TTL® [i, \i G Xj? and

6:=TrL®XZ,w€WL(M), E^^e^E^^WK^u)

=

Res^eEG^^,w^);

(ii) (Functional Equations) For t € W(L),w € WL(M), Px,

and $ €

M(t,n)(Re£ EL/M(*y)) , = (Res^;>tLt-1/^-1(M(i,/i)$,7r')) ,

.

Consequently, we have the following Theorem C. Eisenstein series associated to 1?-automorphic forms admit unique meromorphic continuation and satisfy the functional equations (as stated in Theorems I and II of Day 2). Proof. As said, every L2 automorphic form can be written as a summation of a cusp form and a residue form obtained from an Eisenstein series associated to a cusp form of lower level. For the cusp form part, it will be proved in Day 6, while for the residue part, it is a direct consequence of Theorem B based on that for cusp forms just mentioned.

13 13.1

Why or Better How to Get Theorem B?! Bridge

Let L C M. For a center character £ and a ^-admissible cuspidal datum (M, ^3) € X, fix a unitary character £x, of ^L(A) extending £

Eisenstein Series and Spectral

Decompositions

187

and trivial on J4W A >. Fix a nontrivial ^L- Also for any &L G 5(M, set 6 := &L + ReXjpf € 5(M
A($',$):=

Y^

(M(T-\-T7f)$'(-r7f),$(7r)y

reW(M)

Key Bridge. For all &L & S% , we have

ResgA($', $) = Re4L 4(*', $). That is to say, for functions A($', $), the residue taken with respect to (^3, 6 ) at the G-level is the same as that with respect to (<$£, &L) at the L-level. Clearly, this then essentially gives Theorem B(i).

13.2

Basic Facts

Let Sing T

:— the set of elements 6 of S% such that

(i) {TT G e
and

Sing G := l%Sing G ' 5 .

Now for any &, denote by (M, ^3) its associated cuspidal datum, set Me •= the smallest standard Levi D M such that 6 is ReXM

—stable.

L. Weng

188 Also as usual, set

©°:= {Aex£|6 is a linear subspace of ReXfy,

and

R e 6 := I Ren \ IT 6 6 [ is an affine subspace of ReX^- In particular, o(&), the so-called origin of ©, is simply an element of Re© defined by o(&) := Re© n (Re© 0 ) . (We here suggest the reader to draw a picture in R2 for the subspace Re© 0 = {x + y = 0} and the affine line Re© = {x + y = 1} to find what is o(©).) Clearly, for any L D M e , we have Re© 0 = (Re© 0 n ReX^)

+ ReXf

= (Re© 0 n R e X ^ e ) + ReX&e in particular. We will call ReXfy following

the fixed part of ©, due to the

K e y L e m m a . Let X C 6 be a singular hyperplane of a certain Resg£($,7r),$ €.P(M,q3) such that %n{7r € ©
(ii) M T = M s ; and (iii) There exists r G WM@ such that (a) T7r = —ff for any n € © mi/i Re7r = o(©); and (b) 1 is a singular hyperplane for at least one of the functions 7T € 6 H+ Resg/M(T,-T7f)$'(-r7r),$(7r)y Indeed, for the singular plane X, we have Re X =: Re©? = (Re©? D R e X ^ s ) + ReX^e .

Eisenstein Series and Spectral Decompositions

189

That is to say, the part R e X ^ kept unchanged from the level (3 to the level Z, while ( R e 6 ° n R e X ^ s ) is changed to ( R e 6 ? n R e X ^ s V which is one dimension lower. Hence ideally, we are finally supposed to get, after say N steps, R e 6 ^ = ( ( R e 6 ^ n R e X ^ e ) + ReX°@ ) = {0} +

ReXge.

Thus 0 corresponding to o(&) which gives a discrete spectrum, while ReXfyf kept unchanged even after N steps. All in all, the upshot here is the following fundamental T h e o r e m D . If 6 € Sing G . Then (i) & 3 7ro <8> o(6) where TTQ is the unique element of ^3 which is trivial on A*^,^; and (ii) o(<5) is in the interior of the cone generated by RT(TM,MQ). Reasoning. We start with (ii). There are two main reasons. (ii.a) We have the following Geometric Consequences of the Residual Argument. For any & <E £ (~1 Sing G , we have (a) &* : = - 6 € £ n Sing G ; {(3) There exists WQ £ Stab(M, ^}) such that

we o{e) = -o(e), o(e) \im& = id, 0(G) |ReSo = id; (7) Let $ 6 .P^'5,7Ti e & such that Resg£'($, TT) ^ 0 and is holomorphic at TT\ . Then the set of cuspidal supports o/Resg.E(,7r) is contained in {wit! I w <E W(M), -w& € Sing G ' 5 }. (ii.b) With (a), being L 2 , TT has to be 'real' and the I? criterion of automorphic forms, i.e., the negativity of the cuspidal supports transforms to a positivity statement here. As for (i), we have the following Theorem D' L\ C Discrete Spectrum <^> M e = G V6 e £ » 6 = {TTL := vro ® 0(6)} V6 6 £.

190

L. Weng

14

Spectrum Decomposition: Levi Interpretation

14.1

Spaces A^ TXfj.

Let (£ £ [Sx]- We want to attach it an association class of Levi subgroups. (Recall that two standard Levis are called associated if they are ^-conjugated.) This is given by Existence Lemma. If o(@) which we now know is always in 6 by Theorem D(i). In particular, {TTL} £ Sing1'. Consequently, let \i £ 6 , and n is called generic if (i) none of conjugates of 7^ is in the cuspidal supports of any L2automorphic forms; (ii) iTfj, does not belong to any hyperplane of & contained in S^. Set A(L(F)\L(A))^

:= A(L(F)\L(A))

nL2(i(F)\L(A))^.

Then as usual, we get the spaces A(UL(A)L(F)\G{A))

and it

cousin A(UL(A)L(F)\G(A))

^

:=(/*omPl).>lfc(A)L(F)\G(A))

For $ € Px, ^ € l m i f generic, set |Norm^(M){7TL}[^M) Recall that A ^

:= (Res^E^iUm^w

£ W(M),&

£ £ , $ e Pxj

.

.

Eisenstein Series and Spectral Decompositions

191

Proposition, (i) e L ($,7r„) G A(UL(A)L(F)\G(A)) (ii) A(UL(A)L(F)\G(A))

^

= ( e ^ , ^

; and €

Px)£.

With this, we are ready to state the following M a i n T h e o r e m , (i) For® G Px, S G / i (e L ($,7r M );/i) G A ^ ; and (ii) Up to a constant multiple, we have an isometry EG'L{.-^)

: A(UL(A)L(F)\G(A))

~ A^.

Proof, (i) Direct consequence of Thm B, or better of the Key Bridge; and (ii) The injectivity comes from the fact that EGIL{»; //) preserves the scalar product; while the surjectivity is a bit difficult, which may be established using an induction on the semi-simple rank. It breaks into the following Sublemma 1. Let w G W(M), $ G Px. Then Res^ e £'($, WTTM) G (Res£ e £($', a7rM) | a G W{M) s.t. Ma& # G, $ ' G PX)CSublemma 2.(Major One) Let a € W{M) be such that Ma& ^ G and $ ' G Px. Then Res%6E(&, air^) G (Res£ 6 £($", TTTM) | r G W(M) s.t. M,® G L(<£), $ " G P X ) C Sublemma 3. Let r G VT(M) be such that MT& € L(£) and $ " G P x - Then there exists $"' G P x , r L G Wj,(M) sucft tfiai Res£ e £(S>", TTT^) = Res^ e E($'" ) T i 7r / i ).

15 15.1

Spectral Decomposition: Arthur's Version of Langlands In T e r m s of Levi

Let L be a standard Levi of G. Denote by L2[L(F)\L(A))

:=

the part of the discrete spectrum of L (modulo Z ^ ) ) whose central

192

L. Weng

character is trivial on A^,^ Denote by

and whose restriction to ^G(A) i s £•

A(L(F)\L(A))^:=L2(L(F)\L(A))^n^l(L(F)\L(A)). As before, set the associated P-level space by A(UL(A)L(F)\G(A))

:= [ e ^ ( j / i ( A ) L ( F ) \ G ( A ) ) | 0fc € A(L(F)\L(A)),Vk

6 ATJ.

Denote by P ^ d i s := the space of holomorphic functions on ifj, G Xl|||Re/i||

< fl} with values in A(UL(A)L(F)\G(A))

satisfy-

ing the growth condition (required in the definition of Paley-Wiener functions). Then we have Lemma. For all $ e jP£ dis , lim /

E($, fj.) du

T

^°°JlielmXG,\\n\\
defines an element of L2(G(F)\G(A) Let L2 (G{F)\G{A)\

J .

be the closed subspace of L2

(G{F)\G(A))

generated by the elements of the lemma above. Lemma and Definition. Suppose that L ~ L' are associated, then L2(G(F)\G(A))

which we denoted by

L2(G(F)\G(A))

=

L2(G(F)\G(A)j

Let also WL :— the set of elements of W of minimal length modulo WL and introduce its subset S t a b ^ i L := {w € WL\wLw~1 = L}. With this, we are ready to state the following Main Theorem',

(i) L2 (G(F)\G(A))

is isomorphic to the

Hilbert space Hilb^ ? of measurable functions F on ImX^ with val2 ues in the space L (UL(A)L(F)\G(A)

conditions

)

satisfying the following

Eisenstein Series and Spectral Decompositions • M(t

1

,tfi)F(tfi)

193

= F(fi) almost everywhere Vt € St&bwLL; \stJwLL\ WF(»)\\2 dfi < +00;

• limr^oo J^imXG,y\\
(ii) There exists an orthogonal decomposition L 2 (G(F)\G(A))^ = © [ L ] L 2 ( G ( F ) \ G ( A ) ) ^ . 15.2

R e l a t i o n w i t h Residual A p p r o a c h : T h e P r o o f

Let X be an equivalence class of cuspidal data and <£ be an equivalence class in S% such that C n Sing G / 0. Then there exists an associated subspace L2 (G(F)\G(A)) of L 2 ( G ( F ) \ G ( A ) ) by the residual approach of the spectral decomposition. Now let (L, 6) be a discrete parameter for <£, that is to say, L £ L(<£) with L = M e for some 6 e C and L2(L(F)\L(A)){

6 =

7r0®o(e)}=:<5

so that R e 6 = (7r0 o(&)) ® ReX^. (This notation is well-defined as S is uniquely determined by no <S> 0(6).) Denote also by £ L the singular class attached to 6, i.e., that attached to {7ro®o(6) =: TXL). Define StabwL(L,

6) := {w 6 W(L) \ wLw~l = L,w£ = €}

which is known to be the same as < w e StabWLL I 3WL € WL(M), Proposition',

(i) Let $ be a function on Xj* with values in the

space A(UL(A)L(F)\G(A) points oflvaX^

tiXL = WLITL [•

J holomorphic in a neighborhood of every

and is rapidly decreasing on this set. Then e($) := lim / T->co

E($,n)dn Jneimx?,\\n\\
194

L. Weng

lies in

L2(G(F)\G(AJ)

(ii) L 2 ( G(F)\G(A) these e($) 's. M a i n Theorem".

; and J

is the closure of the subspace generated by

L2(G(F)\G(A))

is isomorphic to the Hilbert

space H<£ of measurable functions F on lvaX^ with values in the space L2(UL(A)L(F)\G(A))S satisfying the following conditions • M(t~1,t/j,)F(tfi) #

lim

= F(n) almost everywhereVt e Stab w z,(L, 5);

T - o o J ^ I m X f . l M K r | S t a b ^ L | H ^ ( M ) I | 2 <*M < +OO.

Proof. The point here is not really about the proof, rather is about how to rephrase the statement as essentials have already been given in Day 4 and the Main Theorem.

Eisenstein Series and Spectral Decompositions

195

Day Six: Arthur's Truncation and Meromorphic Continuation 16 16.1 16.1.1

Arthur's Analytic Truncation Positive Chamber and Positive Cone Dual Basis

Let P\ = U\M\ D PQ = UQMQ be a standard parabolic subgroup. For simplicity, set Oj := a^. for i = 0,1 and <XQ := aMl. (a) Level (ao) . Ao gives a natural basis for (ao)*. For any a G A, denote by a v 6 ao the corresponding coroot. That is to say, ay is the element of ao such that as a linear form on ( ao J , it is characterized by av(a) = 2 and Ker(a v ) = Sa, the hyperplane fixed by wa € W, the element of the Weyl group corresponding to a. (Recall that for simple root a, wa is the unique reflection of ( ao) determined by the property that it maps a to —a while keeps all points on a unique hyperplane Sa := {a}1- unchanged.) Denote the collection of all coroots by

A^{av|aeA0}, which gives a basis for ao- Set A 0 := \wa | a G A 0 | the basis of a0

dual to AQ

(b) Level ( a i ) . Then we have the decomposition

(a0)

= (a£) © (oi)

u A0

D

u

u

Aj

Ai.

196

L. Weng

Here A's below are bases of the corresponding spaces above. Recall that Ai may be obtained from non-trivial restrictions of elements of Ao to oi- Consequently, there is a natural bijection A 0 \ A j <-• Ax. Set then

A^ := { a v | a e A 0 \Aj} be the collection of coroots corresponding to elements in A O \ A Q . View A^ as a basis of Oi, then [wa\ae

A0\Aj},

i.e., the collection of tJ7a's for a € Ao\Aj gives a basis of ( Oi) dual to A)(. (Surely, then elements zva's are viewed as linear forms on ai via the restriction to oi.) Equivalently, compactible with the correspondence A O \ A Q <-> Ai, Ai may also be constructed as follows. For any (3 £ Ai, let a € Ao be the unique element such that a\ = (3. This /3 then gives the coroot a v € ao and hence also element wa in (ao) by (a). Restricting it to di, we get then an element mp := wa\ correspondence

. Clearly, we have a natural

{ro a | a e A 0 \A£} <-• {ro/31 p e A i } which is compactible with the correspondence A O \ A Q <-> Ai. For this reason, we also denote the set < wp | (3 e Ai > by Ai. Moreover, for late use, write /3V for the projection of a v to the space ai. It is clear that

Ay^l^l/JeAi} is the basis of ai dual to the basis Ai (of ( ai J ). One of the basic fact here is the following Lemma. Ai is in the positive Q-span of Ai.

Eisenstein Series and Spectral Decompositions 16.1.2

197

Positive Chamber and Positive Cone

In the space Re a\, define the so-called positive chamber C\ and positive cone 0, VaG A i j

€i := lteReai\m(t)

> 0,

VwEA'A.

Denote by T\ and T\ the characteristic functions for the subset C\ and for
A r t h u r ' s A n a l y t i c Truncation

As seen earlier, automrphic forms are only of moderate growth, hence not integrable over ZG^G(F)\G(A). Arthur's analytic truncation is a universal construction which help us to obtain integrable functions out of these only with moderate growth. An element T 6 Oo is said to be sufficiently positive and denoted by T > 0 if for all a e A 0 , ( a , r ) » 0 are large enough. Fix such a T. Let / : G(F)\G(A) —> C be a smooth function. We define Arthur's analytic truncation A r / (for / with respect to the parameter T) to be the function on G(F)\G(A)

:=

J2 P: standard

(-l)dim(^G)

given by ( A r / V 5 )

J2

fp(S9)-tp(Hp(6g)-T),

5eP(F)\G(F)

where fp denotes the constant term of / along with the standard parabolic subgroup P and Hp(g) := log M mp(g) is an element in dp. (As noticed above, since fp(A) depends only on the projection Xp of A in ap, it does not matter whether we view Hp as a map onto Op or a map to ao.)

198

L. Weng

Remarks. (1) If P = G, the corresponding term on the RHS is simply (g). So

AT/(S) = fig) ~

E

fp(9) • rp (H(g) - T) + • • • .

P: maximal

That is to say, ATf is obtained from / by removing its constant terms along proper parabolic subgroups; (2) Since every parabolic subgroup is conjugated with a standard one, the second summation J2seP(F)\G(F) si m Pty s a Y s that in the definition, constant terms along all parabolic, not just standard parabolic, have to be properly removed. In other words, we have

(Ar/)(s) :=

J2(-l)dim^ZG)fP(g)-rP(HP(g)-T). P

Fundamental properties of Arthur's truncation may be summarized as Theorem. (Arthur) For T > 0 , i.e., T sufficiently positive, in do, and f an automorphic form on G(F)\G(A), (i) A T / is rapidly decreasing; (ii) A T o A r = A r ; a n d (hi) A T is self-adjoint. The proof of these properties is by no means simple: a refined reduction theory and certain fine estimations, of which Theorem A of Day 1 is a starting point, must be worked out. We will omit it entirely due to two reasons: Arthur explains basic ideals behind the truncation in [Ar5] beautifully; and at least for this brief note we only need such a truncation for relative rank one which is indeed a toy model (so very simple).

17

Meromorphic Continuation of Eisenstein Series: Deduction

This part is beautifully presented in [MW], following Jacquet.

Eisenstein Series and Spectral Decompositions 17.1

199

Preparation

Let £ be a central character of G(A), and (M, ^3) a ^-admissible cuspidal datum. Set

Af := A(U(A)M(F)\G(A)Y,

L™ := L2(tf(A)Af(F)\G(A))?,

2 ^

and ^'io C := the space of C-valued functions on G(F)\G(A) of central character £ which are locally square integrable modulo ZG(A) • Set v

0 Va € A P }

C

be the shifted positive chamber. Then for 0 e A( U (A) M(F)\G(

A)j

with 7r € ^J, the Eisenstein series E((f),ir) and the intertwining operator M(u;,7r) are holomorphic at IT whenever Re7r G c, n) and M(W,TT) to the whole C-space ( a ^ J

~: X^-

Denote by jfi := the space of smooth functions h : G(A) —> C with compact support such that the subspace generated by the translations of h under K x K acting on both right and left decomposes into a sum of irreducible subspaces under the action of K x K each of whose isomorphism classes belongs to $ x g". It is an algebra which acts on Af, L? , Lsfoc and so on via

S(h)(p : x i-> /

h(y)ip(xy) dy.

JG(A)

Accordingly, for n 6 ^J, denote by i(7r, •) the associated representation of W? on A(M,

TT)5 := A(U(A)M(F)\G(A)J

.

Recall that W(M) := the collection of w € W such that (i) wMw~l is a standard Levi of G; and (ii) w is of minimal length in its right

200

L. Weng

coset modulo WM- For any

TT)*

Lgc).

Similarly, for w G W(M), we have the operator M(w, •) : 7r (-»• M (to, 7I-) G Hom c M ( M , 7r)5, A(wMw~l,

wir)^).

By definition, easily, we obtain the following Admissible Lemma. For any ir G c«p, hen* and we W{M), (i) £7(TT) O i(7r, /i) = 5(h) o S(7r); and (ii) M(w, 7r) O i(7r, h) = i(wir, h) o M(io, 7r). Moreover, by the convergence proof, we have the following Proposition. The function E and the operator M are holomorphic over c
Deduction to Relative Rank 1

Basic idea for proving Theorems A and B is to reduce them to the case when P is maximal, i.e, the case of relative rank one. This is based on the following elementary consideration. For w G W(M), let I = l(w) be the length of w. Then we can find a decomposition w = wai---

wai

of w into product of reflections wai such that (0) I is the minimal length among all such decompositions; (i) If we set Wi — wai • • • wai, Mi = WiMw^1, then CCJ+I is a simple root of R(TMt,G) satisfying w~1(ai+i) > 0; and

Eisenstein Series and Spectral Decompositions

201

(ii) The set of positive multiple roots of an element of lw~1(ai+i)

|i =

0,l,...,l — l\ coincides with l a € R(TM, G) | a > 0, wa < 0 >. Denote by Ui the unipotent radical of the parabolic subgroup Pi of Levi Mi. Note that in particular, Pw — wMw~x • Uw coincides with P[ = MiUi. Accordingly, we can decompose integration over (Ui(A) DwU{A)w'1)\Ui(A) into I integrals over (Ui(A) n «;u;r. 1 1 t7 i _ 1 (A)u; i _it«- 1 ) \ (ut(A) n

ww^U^WiW'1^

which by u i-> Wiwuww~l is isomorphic to (Ui(A) n

waiUi-1(A)w-^\Ui(A).

This decomposition and some variable changes then lead to the following Bridge Decomposition. With the same notation as above, M(w, IT) = M(wai, wi-in) • M(wai_l, wi-2ir) • • • M(wai, TT) . Consequently, in order to meromorphically continued M(w, IT), it suffices to do it for the operators M( wai, Wi-in ), which comes from the case of relative rank 1 corresponding to {aii}'s.

18

T h e Situation of Relative R a n k 1

18.1 Working Site The advantage in this case is that now everything can be written down quite precisely. Indeed, being relative rank one, dimX^j — 1, and W(M) only contains of two elements: 1 for which M(1,A) = Id for every A; and a non-trivial element which we denote here by w. Accordingly, denote by WP —w M • WU the standard parabolic subgroup associated to WM := wMw~1. (It can happen that WP = P. e.g., G — SL(2) and P = PQ = B.) Denote by a the unique element of AM- It is known that — wa is the unique element of

202 18.2

L. Weng C o n s t a n t Terms

For A € c := Cm,

A) is well-defined and gives an automorphic form in A^ C Lc j o c . In particular, for every standard parabolic subgroup P' of G, its associated constant term is precisely given by Lemma, (i) E(ip,X)P, = 0ifP'^G,P,wP

(ii.a) J / P ^ P , E(ip,\)P

=Ay>

E(tp, X)wP =(w\) (M(W,

\)tp);

(ii.b) IfP = WP, E(C T € OM0 sufficiently positive, Arthur's analytic truncation ATE(ip, A) becomes A T o E(X) = E(X) - ET(X) - El(wX) o M(w, A), or equivalently AT o E(X) - E(X) = -ET(X)

- El(wX) o M{w, A),

where ET : X%, -

Hom c (A(M,7r)S,4f o c )

is given by ET{^

A) :=

£

(Ap)( 75 ) • f{HP{19)

- T)

jeP(F)\G(F)

while its counterpart

El •• ^ 1 - -> Homc^^M^;- 1 ,^)^ Lg c ) is given by

f^M)0, ™A) :=

£

7€P(F)\G(F)

((«A)^) (75) • tfapCrg) - T) .

Eisenstein Series and Spectral Decompositions 18.3

203

Basic Properties

Lemma, (i) The map ET (on Xfy not just on c) is holomorphic; (ii) For every A G Xfa, AT o ET(X) = 0; and (iii) There exists c > 0 such that if\€ X^ satisfying (Re A, a v ) < c, then ET{, A). 18.4

Functional E q u a t i o n for T r u n c a t e d E i s e n s t e i n Series

Recall that for TT G SJ3, we have the natural action i(7r, •) of the algebra r& on A(M, n)^ (to make an artificial distinguish of the general action 5 of Tfi). Set FT{4>,X):=kT(6{h)ET{
Lemma. For A G c, h G H^, (A T

O

8(h) o A r ) (A T

=

O

£(A)) - (A T

O

£(A))

O

i(\, h)

-FT{\)-FZ{W\)oM{w,\).

Proof. A simple manipulation based on the fact that AT is idempotent and Admissible Lemma of 17.1.

204

18.5

L. Weng A p p l i c a t i o n of R e s o l v e n t T h e o r y

The advantage of using the operator AT o 5(h) o AT is that it is a compact operator. In particular, if h = h* : g H-> h(g~l), AT o 5(h) o A T is self-adjoint. So the resolvent theory can be applied. Indeed, with a fixed K-type, the finite dimensional representation I(TT, •) of l~fi on A(M, ir)^, it makes sense to introduce the determinant detz(7r, h) for h 6 Tfi. Set

Vh:=^\eX^:deti(7T,h)=0y Note that now X^ is just a complex plane, so V^, consisting of only zeros of detz(-7r, h), is a discrete subset of X^K e y L e m m a . T/iere exists a unique meromorphic function eT : X% - Vh - Home ( ^ ( M , TT)5, L | ' * ) i/ia£ /or every A € X^ — Vh, (1) A T o e r ( A ) = e r (A);and (2) (A T O 5(h) o A T )e r (A) - e r (A) o i(\, h) = - F T ( A ) . SUC/I

Proof. (1) is a direct consequence of the uniqueness in (2) since AT o AT = AT. As for (2), first, using the self-adjoint compact operator A T o5(h) oAT to decompose the space L, into its eigenspaces Lie'

=

®
Hence Home (i4(M, TT)?, L?,ff) = e 7 e r H o m c U(M, irf, H^ Accordingly decompose x € Home (A(M, 7r)5, L?'* j as z = X ^ e r x 7 ' and set K := {A 6 X ^ : det (7 - i(\, h)) = 0 3 7 e T U {0}}. Then X ^ — Vj( is open and dense in

X^.

Eisenstein Series and Spectral Decompositions

205

Now let U be a relatively compact open subset of Xfy whose closure U does not intersect V^. Then for any A 6 U and 7 6 T, the equation (A T O 8(h) o A T )e T (A) 7 - e T (A) 7 o i(X, /1) = - F T ( A ) 7 has a unique solution in Home T e

namely

(A)7:=-Fr(A)7(7-z(A,h))-1

using the resolvent (7 — i(X, /J)) . With this, it is now standard to show that the series eT(X) := X] 7 er eTWi converges and gives the solution of the lemma. Now define E : X& - Vh - Home (A(M, nf,

L|;£ c )

by setting £ ( A ) : = £ T ( A ) + e r (A). Similarly, we obtain a meromorphic function e j using the set Vw,h := {A € X%Mw-! : detiw(X, h) = o } , and hence the w-variation

Ew.

Corollary, (i) Functions E on X*^ - Vh and Ew on X^Mw_x - VWth are meromorphic; (ii) For every E(X) = E(X) + Ew(wX) o M(w, A). As such, using the relation E{ir)oi(ir,h)

=

6(h)oE(ir),

we find that for every A e c - f c f l (Vh U u>~1VW}h) J, 8(h)(E(X)+Ew(w\)oM(w,X)^

= (E(X)+EW(WX)OM(W,

A))oi(A,/i)

206

L. Weng

So with the relation M(w, 7r) O i(n, h) = i(wK, h) o M(w, TV), we finally arrive at the following crucial expression involving the intertwining operator M(w, A). Proposition. For every A G c — ^c PI (Vh U w~ («(/i)oE(A)-E(A)o»(A,/»)) = - (<5(/i) o Ew(wX) - Ew(wX) o ^ ( I O A , h)j o M(i<;, A). 18.6

Injectivity

The point of the above proposition is that first of all besides M(ir, A), all other terms, in particular, (d(h)oE(X)-E(X)oi(X,h)] and (5(h) o Ew(w\)

— Ew(w\)

o iw(wX, h) J

are defined over Xfy — (Vh U vj~1Vw,fl); and secondly, we have the following Injectivity Lemma. There exists \i e X^ — (Vh U w~1VWih) such that the operator (6(h) o Ew(w\)

- Ew(wX)

oiw(wX,h)j

in Home (A(M, 7r)^, Lc'foc) is injective. Assume this, being injective, a left inverse exists. That is to say, we can choose a continuous map p € Homefebc

A(wMw~1,ww)A

such that p o (d(h) oEw(wX) — Ew(wX)

oiw(wX,h)j

Eisenstein Series and Spectral Decompositions

207

is an isomorphism. Define then a meromorphic map Hom c (,4(M, TT)5, A(WMW~\

R:X&-Vh^>

WIT)*)

by R(X) : = p o (5(h) o E(\) - 25(A) o i(\, hj\ and similarly Rw

:X

wMw~\ ~ Vw,h - • E n d c ^ w M u T 1 , ^ ) ^ .

Then we have for A G c - (c D (V^ U w~1Vw>h)\ -i2(A) = ii u ,(ii;A)oM(ii; > A). Now since Rw(wfj,) is invertible, the function det Rw(») is not identically zero, consequently the operator 4 r > A

Endc^^Mii;-1,^)^ H-> i?w(A)_1

is still meromorphic. Thus, we can continue the function M(w, •) meromorphically to X^ — (Vh U w~lVw^) by setting M(w, A) = - ^ ( w A ) - 1 o R(X), and hence continue E to the same domain by letting E(X) := 25(A) + Ew(w\)

o M(w, A).

Thus to complete the proof, it suffices to show that by choosing different h, Xfy — (Vh U w~lVWth) cover the whole Xjfr, since the continuations obtained by changing /i's can be glued together by the uniqueness of the meromorphic extension.

208

18.7

L. Weng

Meromorphic Continuation

So the only essential problem left is to establish the injectivity for the operator 5(h) o Ew(wX) - Ew(wX) o iw(wX, h) G Home (A(M, TT)5, l | £ c ) . This clearly depends on the choice of h in the Hecke algebra. For this, we do it in general setting (i.e., without assuming that Xfy is of dimension one). Lemma. Let A G X^- There exists h G 1-fi such that (i) h = h*, where h*(g) :— h(# - 1 ); (ii) i(X, h) = \ for all w G W(M); (iii) There exists [i, G X^j such that (a) (Re/u, a) 3> 0; and (b) iw(w\, h) = y/=\ for all w G W(M). Proof. T-fi is large enough to allow such a choice [MW], Injectivity Lemma'. There exists ^ G Xfy — (Vh U w~1VWjh) such that the operator (5(h) o Ew(wX) — Ew(w\)

oiw(w\,h)\

in Home (A(M,ir)3,L/foc) is injective. Proof. Let /x be as in the previous lemma, then by definition, JJL GVhUw~1VWih- Moreover, by the choice of h, iw(w/j,,h) = \ / ^ T . Hence the operator 5(h) o Ew(w\)

— Ew(w\)

oiw(w\,h)

Sublemma. (i) For every A € X^, tive; (ii) (l-AT)oEw(wti) = El(wti).

= (5(h) — %/—I-Id J OEW(WJJL), ET(X) and E^(wX) are injec-

Proof, (ii) is simple and comes from the proof of A T o A T = A T . As for (i), even injectivity of ET(X) is crucial to the proof, it is still a rather 'weak' statement: After all, it suffices to check that if (f> is not identically zero, then the function ET((f)) is not identically zero. See pp.150-151 of [MW].

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Clearly, then the injectivity of E^(wn) implies the injectivity for EW(W^J,), since the formal is the composition of the latter from the left by 1 — AT. (A simple fact about maps.) Moreover, by definition, Ew(wfi) = e^(w^t) + E^(wfi). Thus from the fact that e^wji) is in Ls , by 18.3, which applies here since we assume that (Reto/x, — wot) -C 0, Ew(wu) € Ls as well. As such note that 5(h) is self-adjoint, so the restriction to the space L, of 5(h) — ^/—T • Id is injective. Consequently, the composition (5(h) — y/^l-ld) oEw(w[i) is injective. This establishes the meromorphic continuation. 18.8

Functional E q u a t i o n

Consider the meromorphic function e:

X° A

->

Homc(A(M,ir)Z,Llfoc)

H->

E(X)-Ew(wX)oM(w,X).

Let ip € A(M, 7r)^ such that e(
sp

= 0

if

P'±P

u sp

e( - Mw(w-\

wX)M(w,

\)ip).

Thus 7r (8> A € U0(M,

e(
That is, 7r ® A is in the cuspidal support of e(
Ew(wX)oM(w,X).

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Consequently, e(
= 1.

This gives (all) the functional equation (s) in relative rank one since W(M) = {1, w} only consists of two elements.

References [Arl]

Arthur, J. Eisenstein series and the trace formula, Proc. Sympos. Pure Math., XXXIII, 253-274, AMS, Providence, R.I., 1979.

[Ar2]

Arthur, J. A trace formula for reductive groups. I. Terms associated to classes in G(Q). Duke Math. J. 45 (1978), no. 4, 911-952

[Ar3]

Arthur, J. A trace formula for reductive groups. II. Applications of a truncation operator. Compositio Math. 40 (1980), no. 1, 87-121.

[Ar4]

Arthur, J. On the inner product of truncated Eisenstein series, Duke Math. J. 49 (1982), 35-70

[Ar5]

Arthur, J. An introduction to the trace formula, to appear

[La]

Langlands, R. On the functional equations satisfied by Eisenstein series, Springer LNM 544, 1976

[MW]

Moeglin, C. &; Waldspurger, J.-L. Spectral decomposition and Eisenstein series. Cambridge Tracts in Math, 113. Cambridge University Press, Cambridge, 1995.

[OW]

Osborne, M.S. & Warner, G. Theory of Eisenstein Academic Press 1981

Lin W E N G Graduate School of Mathematics Kyushu University Fukuoka 812-8581, Japan Email: [email protected]

Systems,

Geometric Arithmetic: A Program Lin W E N G

In this article, we originate a program for what we call Geometric Arithmetic. Such a program would consist of four parts, if we were able to properly understand the essentials now. Namely, (1) NonAbelian Class Field Theory; (2) Geo-Ari Cohomology Theory; (3) Non-Abelian Zeta and L Functions; and (4) Riemann Hypothesis. i+i(=-+-)+i

However, here we could only provide the reader with — \ 2 — of them. To be more precise, discussed in this article are the following particulars; (A) Representation of Galois Group, Stability and Tannakian Category; (B) Cohomology, Truncations and Non-Abelian Zeta and L-Functions; and (C) Explicit Formula, Functional Equation and Geo-Ari Intersection. So what are these ABC of the Geometric Arithmetic?! As stated above, (A) is aimed at establishing a Non-Abelian Class Field Theory. The starting point here is the following classical result: Over a compact Riemann surface, a line bundle is of degree zero if and only if it is flat, i.e., induced from a representation of fundamental group of the Riemann surface. Clearly, being a bridge connecting divisor classes and fundamental groups, this result may be viewed as and is indeed a central piece of the classical (abelian) class field theory. (See e.g., [H] and [Wl].) Thus it is then only natural to give a non-abelian generalization of it in order to offer a non-abelian class field theory. This was first done by Weil. In his fundamental paper on generalization of abelian functions [Wl], Weil showed that over a compact Riemann surface, a vector bundle is of degree zero if and 211

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only if it is induced from a representation of fundamental group of the surface. Thus far, two new aspects naturally emerge. That is, unitary representations and non-compact Riemann surfaces, reflecting finite quotients of Galois groups and ramifications in Class Fields Theory, CFT for short, respectively: In a (complex) representation class of a finite group, there always exists a unitary one, while a discussion for compact Riemann surfaces results only unramified CFT. Thus mathematics demands new results to couple with them. It is well known that to this end we then have (i) Mumford's stability of vector bundles in terms of intersection; (ii) Narasimhan-Seshadri's correspondence; and (iii) Seshadri's parabolic analog of (i) and (ii). That is to say, now the above result of Weil is further refined to the follows: Over (punctured) Riemann surfaces, (Seshadri) equivalence classes of semi-stable parabolic bundles of parabolic degree zero correspond naturally in one-to-one to isomorphism classes of unitary representations of fundamental groups. On the other hand, the results above, while central, do only parts of the CFT - at its best, the Weil-Narasimhan-Seshadri correspondence reflects a micro reciprocity law. What CFT really stands should not be a relation between a single representation and an isolated bundle, instead, CFT should expose Galois groups intrinsically in terms of bundles globally. Thus an integration process aiming at constructing a global theory becomes a great necessity. It is at this point where the theory of Tannakian category enters into the picture. Recall that a typical theory of Tannakian category takes the following forms: (i) groups may be reconstructed from their associated categories of representations; (ii) fiber functors equipped Tannakian categories are clone categories of (i), i.e., are equivalent to the categories of representations; and (iii) original groups may be recovered from the automorphism groups of fiber functors. At it turns out, with this strongest form of the standard theory of Tannakian category, we have little hope to match it perfectly with the CFT we are looking for. Fortunately, there are still room to manoeuvre, since in CFT we only care about finite quotients of the

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Arithmetic:

A

Program

213

associated groups, and in terms of representations finite quotients correspond to what we call finitely completed Tannakian subcategories according to Tannaka duality and van Kampen completeness theorem. In this way, we finally establish a non-abelian C F T for Riemann surfaces, or better, for function fields over complex numbers successfully. Main results include the Existence Theorem, the Conductor Theorem and the Reciprocity Law. See e.g. the Main Theorem in A.2. By establishing a C F T for Riemann surfaces as above, possibly, we give the reader an impression t h a t everything works smoothly. No, practically, it is not the case. For example, we do not need all unitary representations. Or put this in another way, all semi-stable parabolic bundles of parabolic degree zero lead us to nowhere. Consequently, we must carefully select among these semi-stable objects a handful portion so t h a t (i) the theory of Tannakian category can be applied; and (ii) there are still enough ingredients to apply Tannaka duality and van Kampen completeness theorem. This then leads to what we call geo-ari representations and geo-ari bundles, for which the Narasimhan-Seshadri correspondence still holds. (For curves over finite fields, besides deformation theory, we need to introduce a new principle, called the Harder-Narasimhan correspondence.) T h e experienced reader here naturally would ask how we overcome the difficulty about tensor products of geo-ari bundles, since, generally speaking, to show the closeness under tensor operation is the only point to check when applying the theory of Tannakian category. Here for Riemann surfaces, two approaches are available. For one, we use the Narasimhan-Seshadri correspondence, as easily tensors of unitary representations are again unitary. But this analytic approach is not a genuine one, since a micro reciprocity law, i.e., the Weil-Narasimhan-Seshadri correspondence, is used. Thus a purely algebraic proof should be pursued. This then leads to the works of Kempf and R a m a n a n - R a m a n t h a n on instability flags, which we call the i^i? 2 -trick. Moreover, as the original KR2-trick only works for bundles without parabolic structures, so to stylize the non-abelian C F T (for Riemann surfaces), we ask for a parabolic version of the

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RK2-trick. Motivated by such a success in non-abelian CFT for function fields over C, we anticipate that in principle, the non-abelian CFT for local and global fields works similarly. So the building blocks of our program for a non-abelian CFT then are the follows: (1) There should be a suitable type of representations of Galois groups, which we call geometric representations and a suitable type of intersection stability for bundles which we call geometric parabolic bundles such that an analog of the Weil-Narasimhan-Seshadri Correspondence holds; (2) There should exist subclasses of geometric representations and geometric parabolic bundles, which we call geo-ari representations and geo-ari bundles, respectively, such that (i) these classes form naturally two abelian categories, (ii) an analog of Hader-Narasimhan Correspondence holds; and (iii) an analog of KR2-trick works. Thus in particular, by (1) and (2), we obtain two equivalent (generalized) Tannakian categories together with natural fiber functors; (3) The (generalized) Tannakian categories contain systems of the socalled finitely completed Tannakian subcategories, so that via an analog of Tannaka Duality and van Kampen completeness theorem, we obtain the so-called fundamental theorem of non-abelian CFT consisting of existence theorem, conductor theorem and the reciprocity law. To end this brief discussion on Part (A), we would like to point out that to realize the above mentioned 123 for our non-abelian CFT, standard theories on GIT, Tannakian category and representations of Galois groups are far from being enough. For examples, to achieve (1), we require (i) a Geometric Invariant Theory over integral bases in the spirit of Arakelov; (ii) a deformation theory for geometric representations of Galois groups; and (iii) a suitable completeness for representation and stability along the line of Fountaine and Langton, respectively; and to achieve (3), we require a theory of Tannakian category over integral bases. Our next main scheme is devoted to non-abelian zeta functions.

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As stated at the very beginning, Part (B) is a combination of our partial understanding of our new non-abelian zeta functions and what we call geo-ari cohomologies. This part to a large extent is practical rather than theoretical, due to the fact that not only all studies here are based on practical constructions, but we have not yet understood the mathematics involved theoretically. Unlike for classical Weil zeta functions, instead of working on general algebraic varieties and counting their rational points (over finite fields) in a very primitive way, for our non-abelian zeta functions, we concentrate our attentions to moduli spaces of semi-stable bundles and count their rational points from moduli point of view, in a similar way as what Shimura does for Shimura varieties. To be more precise, consider function fields over finite fields first. Then, for each fixed natural number r, we, by using a work of Mumford-Narasimhan-Seshadri, obtain the associated moduli spaces of rank r semi-stable bundles. In particular, with the so-called Harder-Narasimhan correspondence, which claims that the rationalities of bundles and moduli points coincide, we could then introduce a new type of zeta functions by considering rational points of the moduli space as moduli points associated with rational semi-stable bundles. This approach, while different from that of Weil, is indeed a natural generalization of that of Artin: When r = 1, our construction recovers classical Artin zeta functions. Moreover, just like classical abelian zeta functions, our non-abelian version satisfies rationality and function equation as well. Since we even can give uniform bounds for the coefficients of these (local) zeta functions, so via an Euler product, we further introduce a more global non-abelian zeta functions for curves defined over number fields. Needless to say, when r — 1, these global zeta functions are nothing but the classical Hasse-Weil zeta functions. So non-abelian arithmetic aspect of curves is supposed to be reflected by these new zeta functions. Well, while this latest general statement should finally lead us to a mathematics wonderland, we have not yet found our theoretical feet: the non-abelian reciprocity law. For this purpose, we then

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turn our attention to some concrete examples. This directs us to the study of what we call refined Brill-Noether loci and their intersections: Beyond traditional considerations, refined Brill-Noether locus measures how automorphisms of the associated bundles change too. Consequently, we obtain a concrete reciprocity law for elliptic curves in ranks 2 and 3. All this is done in B.l. It now becomes quite apparent that to introduce non-abelian zeta functions for number fields, we need to introduce (i) stablity for lattices with which the associated moduli is compact; and (ii) a canonical cohomology theory which satisfies the Serre Duality and Riemann-Roch. All this is done in B.2. Indeed, with Arakelov's approach towards arithmetic, intersection stability for lattices becomes more or less trivial. (This explains why many mathematicians, including Stuhler and the author, introduce stability for lattices totally independently.) As for a cohomology theory associated to number fields, based on Tate's Thesis [T], we are able to offer a practical definition in terms of Chevalley adelic language. More precisely, our cohomology groups are locally compact groups with H° discrete and Hl compact and their counts are given with the help of Fourier analysis. In particular, the Serre duality comes from Pontragin duality and the RiemannRoch comes from the Poisson summation formula and Serre duality. (As such, it would be extremely interesting to compare our approach with Deninger's approach using infinite dimensional spaces.) Consequently, non-abelian zeta functions for number fields can be introduced. Being a natural generalization of classical Dedekind zeta functions, these new zetas are canonical as well. That is, they admit natural meromorphic continuations, satisfy 'the' functional equation, and the residues of them at simple poles are nothing but the Tamagawa type volumes of the associated moduli spaces of semi-stable bundles (over number fields). In particular, when rank is one, our work essentially recovers Iwasawa's ICM talk at MIT about Dedekind zeta functions. For details, please refer to B.2. By saying this, we have no intention to claim that we are satisfied with what we have achieved. Far from being it - we have little

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217

understanding of these new zeta functions. For examples, (1) we have no idea now on how the non-abelian reciprocity law, which, by (A), are supposed to hold naturally, can be read from our non-abelian zeta functions; (2) generally speaking, we are less sure about the meaning of special values of our non-abelian zeta functions, except their residues at 8 = 1. On the other hand, as to (i) and (ii) above for number fields, what is accomplished here is the introduction of intersection stability and hence moduli spaces of semi-stable lattices, and a practical formation of one-dimensional geo-ari cohomology for which the duality and Riemann-Roch hold. Not only our non-abelian zetas satisfy stereotype properties, they can also be written as integrations of Epstein type of zeta functions. This then leads us to introduce more general non-abelian L-functions using Langlands' fundamental theory of Eisenstein series. Roughly speaking, non-abelian L-functions are integrations of Eisenstein series associated to L 2 -automorphic forms over certain geometrically truncated domains. For more details, please go to B.3. On the other hand, in the theory of automorphic forms, or better, in Arthur's study of trace formula, there is another way to get integrations from Eisenstein series. Recall that since constant terms are usually only of moderate growth, Eisenstein series are not integrable over entire fundamental domains. To overcome this, by suitably truncating constant terms, Arthur then is able to create new yet rapidly decreasing functions, the so-called Arthur's truncation (at least for sufficiently regular parameters). A basic observation here is that integrations over fundamental domains of Arthur's truncated Eisenstein series can also be expressed as integrations of the original Eisenstein series over certain truncated domains. Moreover, following Lafforgue [Laf], we are able to build up a global bridge between algebraic truncation obtained using stability and analytic truncation of Arthur associated to the constant function 1. All this then transforms our non-abelian L-functions to a special kind of Arthur's periods, or better Eisenstein periods (at least in the case where parameters are sufficiently regular).

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As such, we are in a position to apply an advanced version of Rankin-Selberg method ([JLR]). This then helps us to see a much clearer structure of (a special kind of) these non-abelian L-functions. For more details, please go to B.4 and B.5. As a by-product, we are able to expose a fundamental relation among special values of abelian and non-abelian zetas (see e.g., B.5.6). As said, our practical cohomology works only in dimension one. However, based on linear compacity of Chevalley, as given in Iwasawa's Princeton lectures notes, and Parshin's approach to duality and residue in dimension two, we in Part (C) offer a program for what we call a half-theoretical geo-ari cohomology in lower dimensions. Part (C) of the program is designed to give a geometric justification of the formal summation X)p:£(p)=o ^p f° r t € ~R. For this purpose, we propose a model for two-dimensional geo-ari intersection theory. This geo-ari intersection turns out to be extremely interesting, since the Riemann Hypothesis may be naturally studied within its framework. Since the Cramer formula is behind the above summation for zeros of Riemann zeta and since the Explicit Formula of Weil is behind the above proposed geo-ari intersection, our approach to the Riemann Hypothesis is in appearance different from but in essence related to that of Deninger and Connes-Haran. More precisely, this model of two-dimensional geo-ari intersection is based on six elegant axioms for intersections of what we call micro divisors, which themselves are motivated by standard properties of intersections in dimension two, the Riemann-Roch in dimension one, adjunction formula for curves in surfaces, global functional equation, and Weil's explicit formula. They are parametrized by R and can be used to build up global divisors using an integration process. (This is a huge step: in algebraic geometry, divisors are defined as finite Q-combination of prime divisors, and in Arakelov theory, only finite R-combination of certain prime divisors are allowed.) The advantage of having this mathematics model on geo-ari intersection is that, as in geometry, then the Riemann Hypothesis may be deduced from an analog of the Hodge index theorem. Accordingly, with our experiences in algebraic geometry and in Arakelov theory, we are led to search for the corresponding geo-ari cohomology in di-

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219

mension two for which the Riemann-Roch holds, a challenging yet reasonable task. (However, it is left as everyone's guess that whether geo-ari ampleness and hence a natural positivity can be obtained.) While the Riemann Hypothesis for abelian zeta functions, the main driving force for the above consideration, is still far from being understood, surprisingly its analogue for rank two zetas can be established. This is treated in C.4, based on certain algebraic, analytic and geometric consideration. This article is a revised version of 'A Program for Geometric Arithmetic' written around 1999-2001: We rewrote sections B.2.2 and B.2.4 and added chapters B.3-5 and C.4, based on important discoveries described in (1) Non-Abelian L-Functions for Number Fields; (2) Rank Two Non-Abelian Zeta and Its Zeros, and (3) Arthur's Periods, Regularized Integrals and Refined Structures of Non-Abelian L-Functions1. So as Weil once said: Number Theory Is Not Standing Still.

1

Since essentials are copied in Chapters B.4-5, this paper will not be published elsewhere.

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220

Contents A . R e p r e s e n t a t i o n of Galois G r o u p , Stability a n d Tannakian C a t e g o r y 221 A.l. Summary A.2. Non-Abelian CFT for Function Fields over C . . . A.3. Towards Non-Abelian CFT for Global Fields . . . B . N o n - A b e l i a n L-Functions

221 221 243 251

B.l. Non-Abelian Zeta Functions for Curves Appendix to B.l: Weierstrass Groups B.2. New Non-Abelian Zeta Functions for Number Fields B.3. Non-Abelian L-Functions for Number Fields . . . B.4. Geometric and Analytic Truncations: A Bridge . . B.5. Rankin-Selberg Method

251 262 269 292 302 319

C. Explicit Formula, Functional E q u a t i o n a n d Geo-Ari Intersection 333 C.l. C.2. C.3. C.4.

The Riemann Hypothesis for Curves Geo-Ari Intersection: A Mathematics Model . . . Towards a Geo-Ari Cohomology in Lower Dimensions Riemann Hypothesis in Rank Two

References

333 336 350 359 382

A p p e n d i x : N o n - A b e l i a n C F T for F u n c t i o n Fields over C 392 1. Refined Structures for Tannakian Categories 2. An Example 3. Reciprocity Map 4. Main Theorem 5. Proof 6. An Application to Inverse Galois Problem References

. . . .

392 392 393 394 394 396 399

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A. Representations of Galois Groups, Stability and Tannakian Categories A . l . Summary Non-Abelian Class Field Theory: A Program Galois Aspect Principle Bundle A s p e c t Narasimhan- Seshadri Geometric Reps S.S. Para. Bdles 4 Rationality Vanishing of Brauer -IJ. Rationality-!! Geo-Ari Reps Harder-Narasimhan Geo-Ari Bundles KB? Trick -D-® ®\\Tannakian Cat Clone Category Tannaka Duality Galois Group Reciprocity Map Auf» Finite Quotient 3 Thm, Reciprocity Law F. Completed Mod

A. 2. Non-Abelian C F T for Function Fields over C A.2.1. Weil-Narasimhan-Seshadri Correspondence A.2.1.1. Unitary Representations of Fundamental Groups Let M° be a punctured Riemann surface of signature (g, N) with M the smooth compactification. Then, M° = M\{Pi,.. .,PN}, M is of genus g with P\,...,PN pairwise distinct points on M. Suppose that 2g — 2 + N > 0. From the uniformization theorem, M° can be represented as a quotient F\H of the upper half plane Ji = {z G C : I m z > 0} modulo an action of a torsion-free Puchsian group T G PSL2(R), generated by 1g hyperbolic transformations A\,Bi,..., Ag, Bg and N parabolic transformations Si,..., SN satisfying a single relation AiBiA^B^1

• • • AgBgA-lB-lSi

•••SN = 1.

Denote the fixed points, the so-called cusps, of the parabolic elements Si, •.., SN by ZI,...,ZN respectively. Then, images of the cusps

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G R U {00} under the projection p : 7i* := H URU {00} —> T \ ? f = M result the punctures Pi,...,PN e M. For each « = 1,...,N, denote by Tj = TZi the stablizer of Z{ in T. Then Tj is a cyclic subgroup in T generated by 5,. Moreover, for an element <7j G PSL2(R) such that aioo = Z{, we have a^SiUi — 1 ± 1 0 1 and hence (a~lSiai) — Too- (For simplicity, from now on, when only a local discussion is involved, we always assume that zi = 00.) For a representation p : 7ri(M°) ~ T —> GL(n,C) of T into a complex vector space V, the vector bundle V := W x V on H admits a natural T-vector bundle structure via ~f(z,v) = (7(2), /o(7)u) for 7 £ r , 2 S H and » 6 K The quotient o£ H x V modulo the action of T is then a vector bundle of rank n over M°. Moreover, since the same representation p defines also a T-vector bundle structure on H* x V, we obtain a vector bundle Vp on M — T\H* as well. Next assume that p is unitary. Then with respect to a suitable basis of V, ZI,...,ZN

p(Si) = diagf exp(27naji),..., exp^Tria^)) where atj 6 [0,1) for all i = 1 , . . . , N, j = 1 , . . . , n. Hence, Vp, or better, the associated sheaf p£(V) of sections may be interpreted as follows: On M°, it corresponds to the T-invariant sections of V, while near parabolic punctures P = Pi € M, over a neighborhood U of P of the form Hs/Yoo where Hs := {z = x + iy : y > S > 0}, the sections are all bounded Too-invariant sections of V on Tis- Thus, as an O^^-module, a basis of p£(V) at P is given by the Y^ sections 0j : z >-> exp(2-iriajz) • tj where { e i , . . . , e n } is a basis of V such that Si(ej) = exp(2iriaij) • ej. As a direct consequence, in addition to the associated bundles Vp on M, there exist as well the following structures on the fibers of Vp at punctures P i , . . . , PN- Over P = Pi, we obtain real numbers an = ai2 = • • • = aikl

< aitkl+i

— a i ) f c l + 2 = • • • = aik2 < • • • =

and a decreasing flag of Vp\p defined by

(OJMW-Vplp;

aitkr.

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(ii) F2(Vp\p) the subspace spanned by 6^+1, • • •, On\ (iii) Fz{Vp\p) the subspace spanned by #fc 2 +i,... ,On, etc. Clearly k\ = dimFi(V r p |p) - d i m i ^ V ^ p ) , . . . ,kr = dimFr(Vp\p), and these additional structures are indeed determined by a^- := aikpj = 1, . . . , r j , and fc/s. Proposition. (Seshadri) With the same notation as above, deg(Vp) = A.2.1.2. Semi-Stable Parabolic Bundles Following Seshadri, by definition, a parabolic structure on a vector bundle E over a compact Riemann surface is given by the following data: (1) a finite collection of points P i , . . . , P/v € M; and for each P = Pi, (2) a flag EP = FiEP D F2EP • • • D FrEP; and (3) a collection of parabolic weights a\,... ,ar attached to the filtration F\Ep,..., FrEp such that 0 < ot\ < a2 < • • • < ar < 1. Often k\ = dim F\Ep — dim F2EP,..., kr = dim FrEp are called the multiplicities of a\,..., ar; and a bundle E together with a parabolic structure (P = P^ EP = FiEp D F2EP • • • D FrEP; «i = an, ...,ar

= airi)f=1

is called a parabolic bundle and is written as £ ( £ ) := E := (E; (P = Pf, EP = F i £ P D F2EP • • • D F r £ p ; a i = a i l , . . . , a r = a ir-i) i=1 J • Trivially, if W is a subbundle of E, then E induces a natural parabolic structure E(W) on W. For parabolic bundles, its associated parabolic degree is defined to be paradeg(E) := deg£ + ] P f ^ fc^-ay J . So, in particular, if E is induced from a unitary representation of fundamental group of a Riemann surface, its associated para degree

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is zero. By definition, a parabolic bundle S is called (MumfordSeshadri) semi-stable (resp. stable) if for any subbundle W of E, paradeg(£(WQ) rank(W)

S

lr6SP

'

<j

paradeg(£(E)) rank(£) '

Proposition. (Seshadri) With the same notation as above, if £ is induced from a unitary representation of the corresponding fundamental group, then £ is semi-stable of degree zero. Moreover, if the representation is irreducible, then S is stable. A.2.1.3. Weil-Narasimhan-Seshadri Correspondence: A Micro Reciprocity Law The real surprising result is the inverse of Proposition 2.1.2. The starting point for all this is the following classical result on line bundles: Over a compact Riemann surface, a line bundle is of degree zero if and only if it is flat, i.e., it is induced from a representation of the fundamental group. It is Weil who first generalized this to vector bundles in his fundamental paper on 'Generalisation des fonctions abeliennes' dated in 1938: Over a compact Riemann surface, a vector bundle is of degree zero if and only if it is induced from a representation of the fundamental group. (The reader may find a modern proof in Gunning's Princeton lecture notes in which all fundamental results on Riemann surfaces are used.) It is said that Weil's primitive motivation is to develop a nonabelian CFT for Riemann surfaces. Granting this, clearly, the next step is to study what happens for general Riemann surfaces, which need not to be compact. This then leads to Weil and Toyama's theory on matrix divisors. While all this seems to be essentially in the right direction, still many crucial points are missing in these pioneer studies. Recall that the reciprocity law in CFT is essentially the one for finite field extensions and that for a finite group, in any equivalence class of (finite dimensional complex) representations there always exists a unitary one. Thus naturally we should strengthen Weil's theorem from any representation to that of unitary representation. For doing so, we meet a huge difficulty on algebraic side. That is, how to

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give a corresponding algebraic condition?! This is solved by Mumford with his famous intersection stability. In fact, not only Mumford introduced the intersection stability, he also studied the associated deformation theory via his fundamental work on GIT stability. On the other hand, for Riemann surfaces, fundamental groups can be described very precisely, and hence deformations of the associated unitary representations can be quantitatively studied. All this, together with certain completeness for both unitary representations and Mumford's intersection stability, the so-called Langton's Principle, then leads Narasimhan and Seshadri to prove that over a compact Riemann surface, Mumford's semi-stable vector bundles of degree zero are naturally associated with unitary representations of the fundamental group of the surface. Later on, Seshadri first generalizes this result to 7r-bundles, and then to parabolic bundles. Theorem. (Narasimhan-Seshadri, Seshadri) There is a natural oneto-one correspondence between isomorphism classes of unitary representations of fundamental groups of punctured Riemann surfaces and Seshadri classes of semi-stable parabolic bundles of parabolic degree zero. We would like call this result the Weil-Narasimhan-Seshadri correspondence. Due to the fact that it reveals an intrinsic relation between fundamental groups and vector bundles, often we call it a micro reciprocity law as well. (Over higher dimensional compact Kahler manifold, a related correspondence is named the KobayashiHitchin correspondence, which is established by Ulenberk-Yau. See also Donaldson for projective manifolds.) Note also that, as stated above, the existing proof is based on (a) Geometric invariant theory; (b) Deformation of representations; and (c) Completeness of semi-stable parabolic bundles - the Langton principle. Consequently, all this is supposed to play a crucial role in our program for non-abelian CFT, the details of which will be discussed

later.

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While the Narasimhan-Seshadri correspondence is a kind of Reciprocity Law, it is only a micro one. Thus to find the genuine one, we need to do it globally. Thus, the following result for finite groups naturally enters into the picture: Any finite group is determined by its characters and vice versa. Hence, if we have a similar result for more general groups, we can establish our non-abelian C F T for Riem a n n surfaces. As usual, we are led to the theory of Tannakian categories. But before t h a t we still need to make sure t h a t coverings and parabolic bundles are under control. It is for this purpose, we introduce the Rationality discussion in our program. A.2.2. Rationality: Geo-Ari Representations and Geo-Ari Bundles A . 2 . 2 . 1 . B r a n c h e d C o v e r i n g s of R i e m a n n Surfaces T h e advantage in developing a non-abelian C F T for Riemann surfaces is t h a t all the time we have concrete geometric models ready to use. As there is no additional cost and also for a possible development on non-abelian C F T for higher dimensional function fields over complex numbers, we next recall some basics for branched coverings of complex manifolds. Let M be an n-dimensional connected complex manifold. A branched covering of M is by definition an n-dimensional irreducible normal complex space X together with a surjective holomorphic mapping IT : X —> M such t h a t (1) every fiber of IT is discrete in X; (2) i?7r := {q E X : IT* : On^tM —> Oqtx is not isomorphic} and B-K '•= ir(RK) are hypersurfaces of X and M respectively. As usual, we call R^ and B^ ramification locus and branch locus respectively; (3)

IT

: X\TT^1(BTT)

—> M\Bn

is an unramified covering; and

(4) for any p € M , there is a connected open neighborhood W of p in M such that for every connected component U of TT~1(W), -1 (i) 7r (p) fl U — {q} is one point and (ii) TT\J := TT\\J : U —> W is surjective and proper. Thus in particular, the induced m a p IT : X \ 7 r _ 1 ( S 7 r ) —• M\BW is a topological covering and TT\J is finite. For example, if IT : X —> M is a surjective proper finite holomor-

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phic map, ir is a finite branched covering of M. Two branch coverings TT : X —> M and rr' : X' —> M are said to be equivalent if there is a biholomorphic map <j) : X —> X' such that n = n' o (j>. In this case we write w > TT' or n' < IT. The set of all automorphisms of 7r forms a group G^ naturally. One checks easily that if we denote by ir\ the restriction of IT to X\TT~1(BW), then Gw is canonically isomorphic to GWl. By definition, -zr is called a Galois covering if G^ acts transitively on every fiber of IT; and n is called abelian if n is Galois and G^ is abelian. Theorem. If IT : X —• M is a Galois covering, then (1) for every subgroup H of G^, there is a branched covering -KH '• X/H —> M such that -KH < TT; (02) i/ie correspondence H —> 7r' = 717/ gives a bisection between subgroups H and equivalence classes of branched coverings IT' of M such that n' < 7r; and (3) H is normal if and only if TTH is a Galois covering, for which Gnu is isomorphic to G^/H. Note that for any branched covering ir : X —> M, if p, q are points of Bw and ir~1{B1T) respectively such that (i) Bn is normally crossing at p, q £ ^~1(p)', (ii) X is smooth at q; and (hi) 7r_1(B7r) is normally crossing at q, then, for a sufficiently small connected open neighborhood of p with a coordinate system (wi,..., wn) such that p = 0 and B^ CiW = {(wi,...,wn) : Wk---wn = 0} for some k, there is a coordinate system ( z i , . . . , zn) in the connected component U of 7r -1 (W) with q EU such that 9 = 0, 7r _1 (S 7r ) (~l U = {(z\,..., z„) € U : Zfc • • • zn = 0} and iru(zi, ...,zn) = {z\,..., ^ _ i , zekk,..., z%>). Often we call e^, j = k,... ,n the ramification index of the irreducible Gj of 7r~l(Bn) such that Cj DU = {(z\,..., zn) : Zj = 0}. Suppose now that B^ = D\U • • -U DN is the irreducible decomposition of B-n- Let D — 5Z i = 1 ejZ)j be an effective divisor on M. By definition, the branched covering IT : X —• M is called branched at .D (resp. at most at D) if for every irreducible component C of 7r _1 (S)

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with 7r(C) = Dj, t h e ramification index of TT at C is ej (resp. divides ej). In particular, a branched at D Galois covering ir : X —> M is called maximal if for any branched covering IT' : X' —> M which branches at most at D, TT > n'. While, in general, it is complicated to describe maximal branched covering for higher dimensional complex manifolds (see however [Na]), for Riemann surfaces, it may be simply stated as follows. A result of Bundgaard-Nielsen-Fox says t h a t there is no finite Galois covering IT : X —> M branched at D = 5 3 J = 1 ejPj if and only if either (i) g = 0 and N = 1 or (ii) g = 0 N = 2 and e\ ^ e 2 . Here we write Dj as Pj, j = 1 , . . . , N. T h u s from now on, we always assume t h a t we are not in these exceptions. Also we assume t h a t ej > 2. (Otherwise, we may omit it from the beginning as there is no ramification for the corresponding points Pj then.) Let J(D) be the smallest normal subgroup of 7Ti(M°) containing S^1,..., Se^. Then (as used in the proof of the above result of Bundgaard-Nielsen-Fox,) the normal subgroup J(D) satisfies the following: C o n d i t i o n (*):

If Sf € J(D),

then d\ej for j = 1 , . . . , N.

As a direct consequence, we have the following T h e o r e m . (Bundgaard-Nielsen-Fox) (1) there is a maximal covering fr : M(D) —* M which branches at D. In particular, M(D) is simply connected; (2) there is a canonical one-to-one correspondence between subgroups (resp. normal subgroups) H of the quotient group TTI(M0)/J(D) and equivalence classes of branched coverings (resp. Galois coverings) 7r : X —> M which branch at most at D; (3) 7r : X —> M branches at D if and only if the condition (*) holds for K (here H=K/J(D)). That is, if Sf € K, then d\ej}i = 1,...,N. A.2.2.2. Geo-Ari Representations and Geo-Ari Bundles Motivated by the above discussion, we now introduce what we call geo-ari representations of fundamental groups. As before, let M be a compact Riemann surface of genus g with marked points P i , . . . , PN- Set M ° = M\{PU . . . , PN}. Then,

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the fundamental group 7Ti(M°) is generated by 2g hyperbolic elements A\, B\,..., Ag, Bg and n parabolic elements S i , . . . , SAT such that [A\, Bi] • • • [Ag, Bg]Si • • • SJV = 1. Fix an effective divisor D = ^2i=i ejPj on M. By definition, a geometric-arithmetical representation, a geo-ari representation for short, of 7ri(Af°) along with D is a unitary representation p : -K\{M°) —• U(l) such that (i) p(Si) = diagf exp(27ri Ai), • • •, exp(27ri /3u)j for alH = 1 , . . . , N; and (ii) there exist integers jij > 0 and 6ij > 0 such that (a) jij\ej] (b) (jijjij) = 1; and (c) Pa = | f for alH = 1 , . . . , iV and j = 1 , . . . , L In parallel, we define a geo-ari bundle on M along D to be a parabolic degree zero semi-stable parabolic bundles E =[E- (P = P^EP

= FiEp D F2EP • • • D FrEP;

oil = an, ...,ar

=

ain)i=l\

such that the following conditions are satisfied: there exist integers •jij > 0 and 5ij > 0 such that (a)

,

jij\ej;

(b) (jij,6ij) = 1; and (c) oiij = -^- for all i = 1 , . . . , N and j = 1 , . . . , r,. Obviously, by Theorem 2.1.3, we obtain the following Theorem'. There exists a natural one-to-one correspondence between isomorphic classes of geo-ari representations of 7Ti(M°) along with D and (Seshadri) equivalence classes of geo-ari bundles along D over M. Remark. We call this result the Harder-Narashimhan Correspondence, despite the fact that in the situation now, i.e., over complex numbers, the Harder-Narasimhan correspondence is simply a direct consequence of Narasimhan-Seshadri correspondence. Later we will see that when the constant field is finite, such a correspondence,

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first established by Harder-Narasimhan, is based on the vanishing of related Brauer groups. Clearly, if D' — X^=i ejPj ^s a n effective divisor such that ej\e'j for all j = 1,...,N, then geo-ari representations of 7Ti(M°) (resp. geo-ari bundles) along with D are also geo-ari representations of 7i"i (M°) (resp. geo-ari bundles) along with D'. (Usually, we write D\D'.) Thus if we denote U(M;D) the category of equivalences classes of geo-ari representations of 7Ti(M°) along with D, (see, e.g., 2.3.1 below for a brief discussion on categories,) and M{M;D) the category of geo-ari bundles along D over M, then by using a result of Mehta-Seshadri, see e.g., Prop. 1.15 of [MS], we have the following Proposition. U(M;D) and M{M;D) are equivalent abelian categories. Moreover if D\D', then U(M;D) and M(M;D) are abelian subcategories ofU(M;D') and M(M;D'), respectively. A.2.3. KR2-Trick

and Completed Tannakian Categories

A.2.3.1. Completed Tannakian Category and van Kampen Completeness Recall that a category A consists of two sets Obj and Arr of objects and morphisms with two associates dom and cod from morphisms to objects. Often an arrow / with dom(/) = x and cod(/) = y is written as / : x —> y. There is also a map called composition Arr x Arr —> Arr defined for (/, g) when dom(g) = cod(/) such that dom(3o/) = dom(/) a n d c o d ^ o / ) = cod(g), fo(goh) = (fog)oh, and for every object x there is a morphism lx : x —> x such that folx = lxof = f. Thus we may form the set Hom(:r, y) by taking the collection of all / : x —* y. By definition, if moreover the hom sets are all abelian groups such that compositions are bilinear, we call it a preadditive category. Among two categories, a functor T : A —> B is defined to be a pair of maps Obj (A) —> Obj(S) and Arr (.A) —>• Arr(B) such that if / : c —> c' is an morphism in A, then T(f) is a morphism T(f) : T(c) -> T(c') in B, and that T(l c ) = l r ( c ) , T(g o / ) = T(g) ° T ( / ) ; and a natural transformation r between two functors S,T : A —> B is denned to be a collection of morphisms r c : T(c) —> S(c) in B such

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that for any / : c —> d in A, rcT(f) = S ' ( / ) v ; if moreover all r c have inverses, then the natural transformation is called a functorial isomorphism. Among categories, abelian categories are of special importance. By definition, an abelian category is a preadditive category such that (1) there is a unique object called zero object such that it is the initial as well as the final object of the category; (2) direct products exist; (3) associated to any morphism are kernel and cokernel which are objects of the category as well; moreover, every monomorphism is the kernel of its cokernel, every epimorphism is the cokernel of its kernel; and (4) every morphism can be factored into an epimorphism followed by a monomorphism. To facilitate ensuing discussion, we introduce some new notion in category theory. By definition, an object x in an abelian category is called decomposable if there exist objects, y, z different from zero, such that x = y © z; and x is called irreducible if it is not decomposable. Moreover, an abelian subcategory of an abelian category if called completed if (i) for any object x, there is a unique finite decomposition x = ©Xj with Xi irreducible, the irreducible components of x; (ii) the subcategory contains all of its irreducible components of its objects. Proposition 1. The categories U(M;D) pleted abelian categories.

and M(M;D)

are com-

The reader may prove this proposition from the following facts: (a) Among two stable parabolic bundles of the same parabolic degree, homomorphisms are either zero or an isomorphism; (b) There exist Jordan-Holder nitrations for parabolic semi-stable bundles; and (c) The associated Jordan-Holder graded parabolic bundles in (b) is unique. R e m a r k . Motivated by this proposition, the reader may give a more

232

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abstract criterion to check when an abelian subcategory is completed. Next, let us recall what a tensor category should be. By definition, a category is called a tensor category if there is an operation, called tensor product A®B for any two objects A, B of the category, such that the following conditions are satisfied: (1) There are natural isomorphisms S : A® B —> B ® A and T :

(A® B) ® C -^ A® (B ® C); (2) S and T satisfy the so-called pentagon and hexagon axioms; and (3) There is a unique identity object 1 such that A~ A®1~1® A for all object A. Clearly, for tensor categories, we can introduce tensor operation for finitely many objects. Usually, there are many ways to do so, but the above conditions for tensor category implies that resulting objects from all these different ways are the same. Moreover, if for any objects x, y of A, Hom(x, y) is again an object in A satisfies the following conditions, we call A a rigid tensor category: (4) there exists a morphism, the evaluation map, evx>y : Hom(x, y) ® x —> y such that for any object t and any morphism g : t® x —> y, there exists a unique morphism / : t —> Hom(x, y) such that the following commutative diagram commutes t ®x 9i y

—>x Hom(x, y) x; / ev x , y

(5) There exists natural isomorphism Hom(xi, yi) Hom(x2,2/2) — Hom(xi x 2 , yi ® y2) which is compactible with (4); (6) For any object x, by (5), if we set x v := Hom(x, 1), then there exists a natural isomorphism x ~ ( x v ) v . Now we are ready to introduce Tannakian categories. By definition, a Tannakian category is a category which is both an abelian category and a rigid tensor category such that the tensor operation is bilinear. Clearly then in such categories, Hom(x, y) ~ x ® y v .

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For example, the category Vecfc of finite dimensional vector spaces over a field A; is a Tannakian category. By definition, a functor u> : A —> VeCfc from a Tannakian category A to the category Vecfc is called a fiber functor, if u> is an exact faithful tensor functor. (Recall that faithful means there is a natural injection w(Hom^(3;, y)) <—> HomyeCk(u)(x),u(y)) which is induced from to.) Associated to a fiber functor u> is naturally its automorphic group Aut®w. Essentials of Tannakian categories may be summarized as the following: Theorem 2. (Tannaka, Grothendieck, Saavedra Rivano) Assume that (A,LO : A —> Verfc) consists of a Tannakian category and a fiber functor such that the field k is canonically isomorphic to 1^, then A is equivalent to the category of representations of the group Aut®o>. A proof of this theorem may be deduced from the following facts: (1) Category of all representations forms naturally a Tannakian category together with a fiber functor; (2) Knowledge of the representations of a group is equivalent to the knowledge of the group; and (3) Any Tannakian category is in fact a clone of (1), via the so-called Tannaka Duality principle. Now we are ready to introduce our own Tannakian categories which are completed and equipped with natural fiber functors. There are two of them, i.e., the one for geo-ari representations and the one for geo-ari bundles. By Proposition 2.2.2, it is enough to introduce the fiber functors and show that geo-ari representations and geo-ari bundles are closed under the tensor operation. But all this is quite straightforward: By definition, tensors of unitary representations are again unitary, while for geo-ari bundles, the fiber functor may be defined by taking special fibers for the bundles at any point which is not a marked one. Note that morphisms between geo-ari bundles are either zero or isomorphisms, so the latest defined functor is faithful. Therefore, we obtain the following Theorem 3. U(M;D)

and hence M.(M;D)

are completed Tan-

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nakian categories equipped with natural fiber functors to Verc • We end this brief discussion on Tannakian category by the socalled van Kampen completeness theorem, which plays a key role in the proof of the fundamental theorem of CFT for Riemann surfaces later. For any group G, denote its associated Tannakian category of equivalence classes [p] of unitary representations p : G —> Aut(Vp) by U{G). Fix for all classes [p] a representative p once and for all. A subset Z of U(G) is said to contain sufficiently many representations if for any two distinct elements 51, gi of G, there exists [p] in Z such that p{gi) # p(g2). van Kampen Completeness Theorem. / / G is compact, then, as a completed Tannakian category, U(G) may be generated by any collection of objects which contains sufficiently many representations. A.2.3.2. KR2-Trick While it is quite nice to have a proof of Theorem 3 in 2.3.1, we are by no means satisfied with it. Recall that Theorem 3 consists of two aspects: the algebraic one for bundles and the analytic one for unitary representations. So, what we need to do should be the follows; (1) From analytic point of view, to prove that (a) the tensor operation is closed; and (b) the so-called forgetful functor is faithful; and (2) From algebraic point of view, to show that (a) the tensor operation is closed; and (b) the functor introduced in 2.3.1 is faithful. However in our approach outlined in 2.3.1, only (l.a) and (2.b) are shown. That is to say, with the help of the so-called NarasimhanSeshadri correspondence, a micro reciprocity law, we make no distinction between algebraic and analytic aspects. Thus, a purely analytic proof for (l.a) and a purely algebraic proof for (2.a) should be pursued. The proof of (l.a) is simple since unitary representations for fundamental groups, or better for quotients of fundamental groups by

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(normal) subgroups generated by weighted parabolic generators have semi-simplifications. We leave the details to the reader. Thus the real challenge here is an algebraic proof of (2.a), i.e., a proof for that tensor products of geo-ari bundles are again geo-ari bundles. It is here we should use another central concept in GIT, the so-called instability flag of Mumford-Kempf (with the refined version given by Ramanan and Ramanathan). This goes as follows. In his study of GIT stabilities, Mumford conjectures that if a point is not semi-stable, then there should exist a parabolic subgroup which takes responsibility, in the view of the so-called HilbertMumford criterion. This is confirmed by Kempf. (In Kempf's study, as suggested by Mumford, the rationality problem is also treated successfully, at least when constant fields are perfect.) Kempf's result then motivates Ramanan and Ramanathan to show that even though, for the original action, the corresponding point may possibly be not semi-stable, but if a certain type well-controlled modification is allowed (with the aim to cancel the instability contribution in the original action), a new (yet well-associated) point could be constructed such that with respect to the natural induced action this new point becomes semi-stable. As a direct consequence, in the case for semi-stable vector bundles without parabolic structures, since the new well-associated action may be associated to the intersection stability condition for bundles naturally, we can then obtain an algebraic proof of (2.a) for bundles. R e m a r k . Over complex numbers, the rationality is not a serious problem. However, over finite fields, this turns to be a difficult one. We then need to consider the Probenius, or better pn reduction, to tackle the rationality problem. Therefore, to make our non-abelian CFT for function fields over complex numbers pure, we need to find a more general version, what we call the KR2-tiick, of the above result of Kempf, and Ramanan and Ramanathan, which works for parabolic bundles. For this purpose, we may follow a more down-to-earth approach of Faltings and Totaro ([FW] and [To]).

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A.2.4. Non-Abelian CFT for Function Fields over Complex Numbers A.2.4.1. Micro Reciprocity Law, Tannakian Duality and the Reciprocity Map Let M be a compact Riemann surface of genus g with marks P i , . . . , P A T - Set M° := M\{Pi,...,PN}. Then, it is well known that the fundamental group 7Ti(M°) is generated by 2g hyperboilic generators Ai,Bi,...,Ag,Bg and N parabolic generators S\,...,SN which satisfy one single relation [A\, B\] • • • [Ag, Bg] • S\ • • • SN = 1Moreover, with respect to an effective divisor D — X)£Li eiPi-> w e have a Tannakian category U(M; D) consisting of equivalent classes of unitary representations [p : 7Ti(M°) —> Aut(Vp)] of 7Ti(M°) such that (i) p(Si) - di&g[exp(2ni fin),... ,exv(2niPa)) and

for alH = 1 , . . . , N;

(ii) there exist integers 7ij > 0 and 6ij > 0 such that (a) 7ij|ej; (b) (-yij,Sij) = 1; and (c) fa = ^

for all i = 1 , . . . ,N and j = 1 , . . . ,1.

Now in each equivalence class [p], fix a unitary representation, denoted also by p, once and hence for all. Clearly, p induces a unitary representation of the group ni(M°)/J(D), where J(D) denotes the normal subgroup of 7Ti(M°) generated by 5 ^ , . . . , 5 ^ . Call this latest representation p as well. Then, for any element g 6 T T I ( M ° ) / J ( Z ) ) , p{g) induces for each object [p : T T ^ M 0 ) -> Aut(y p )] in U(M; D) an automorphism of Vp. As a direct consequence, we obtain a natural group morphism from 7ri(M°)/J(D) to the automorphism group of the corresponding fiber functor U(M; D) —> VeccOn the other hand, for the Tannakian category M(M; D) of geoari bundles on M along D together with the fiber functor UJ(M; D) : M(M; D) —* Verc, by Tannakian Duality, we conclude that u(M; D) is equivalent to the Tannakian category of the representations of Aut®w. Therefore, by the Narasimhan-Seshadri correspondence (and the Harder-Narasimhan correspondence), the so-called micro reci-

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procity law, we obtain a canonical group morphism Q(D) : 7n(M°)/J(D) -> Aut®(w(M;D)). We will call U(D) the reciprocity map associated with (M, £)). A.2.4.2. Non-Abelian CFT By definition, a subcategory S of a Tannakian category with respect to a fiber functor UJ is called a finitely completed Tannakian subcategory, if (1) it is a completed Tannakian subcategory; (2) there exist finitely many objects which generated S as an abelian tensor subcategory; and (3) Aut®(u|s) is a finite group. With this, by using the van Kampen completeness theorem and the above reciprocity map, we then can manage to obtain the following fundamental theorem on non-abelian class field theory for function fields over complex numbers. Non-Abelian C F T for Riemann Surfaces. (Weng [We5]) (1) (Existence and Conductor Theorem) There is a natural one-to-one correspondence UJM.D between {S : finitely completed Tannakian subcategory of

M(M;D)}

and {TT : M' —> M : finite Galois covering branched at most at D};

(2) (Reciprocity Law) There is a natural group isomorphism Aut®(w(M;L>)| s ) ~ Gal(w M ,z>(S)). We end this discussion by pointing out that, as an application, one may use this fundamental theorem to solve the geometric inverse Galois problem.

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A.2.5. Classical (Abelian) CFT: An Example of Kwada-Tata and Kawada A.2.5.1. Class Formation Classical abelian class field theory was first formulated axiomatically in Artin-Tate seminar in terms of cohomology of groups. As an example of this formulation, later Kawada-Tate [KT] and Kawada [Ka] studied function fields over complex numbers. To make a comparison between this classical approach of CFT and what we outlined above, next we recall the works of Kawada-Tate and Kawada. Let ko be a given ground field and Q a fixed infinite separable normal algebraic extension of ko. Let 1Z be the set of all finite extensions of ko in Q,. By definition we call a collection {E(K) : K € TZ] of abelian groups E(K) a formulation if the following conditions are satisfied: F l . If k C K then there is an injective morphism (frh/K '• E(k) <—> E(K); F2. If k C I C K, then <j>l/K o k/l = (j>k/K;

F3. If K/k is normal and G = Gal(K/k) is its Galois group, then G acts on E(K) and (f>k/K(E(k)) = E(K)G; and F4. If k C L C K and L/k, K/k are both normal, then the Galois group F = Gal(L/k) is a quotient group of G = Ga\(K/k). Furthermore, denote by XQ/F '• G —> F the canonical quotient map. Then, 0" ° 4>L/KU)

= 4>L/K ° (AG/F<^)(/)

for

any a EG

and / €

E{L).

Moreover a formation is called a class formation if it satisfies the following additional conditions on group cohomology: CI: Hl{G,E(K)) = Q-and 2 C2: H (G,E{K))~[K :k0]Z. By a theorem of Tate, CI and C2 imply and hence are equivalent to the following stronger condition C. In a class formation, for all r € Z, Hr(G,E(K)) ~ Hr-2(G,Z) for all r. In particular, if a 2-cocycle a{K) generates the cyclic group H2(G,E(K)^), then the above isomorphisms are all induced by the cup-product g — i > a(K) U g. Note that by definition, H'2(G,Z)

= Gah := G/[G,G\, and the

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0-th cohomology group H°(G, E(K)) is nothing but E(K)G/TGE(K), a a (where TGa = Y^aeG ( ))- So for a class formation, we obtain then the reciprocity law ab

E{k)/
Furthermore, for a class formation, with respect to the so-called res (restriction), infi (inflation) and ver (Verlagerung) operations of group cohomology, we have the follows; Case 1. For k C I C K with K/k normal, G := G&l(K/k) and H := Gal{K/l), in H2(G,E(K)),verH/G o r e s G / H = [G : H] • 1. Consequently, res G /# is surjective and ver#/c? is injective; Case 2. For k C L C K with L/fc, K/k normal, and G := G&\(K/k), H := Gal(K/L), we have the exact sequences 0 -> tf 2 (F, £(L)) ^ tf

2

(G, £ ( * ) ) ^ s H2 (H, E{KJ).

Therefore, there exist 2-cocycles {a(K)j of G(K/k) over E(K) such that Dl. a(K/k) are generators of the cyclic groups H2(G, E(K)); D2. In Case 1, resG/H(a(K/k))

~ /(K/Z);

ver„ / G (a(K/fc)) ~ [G : H} • a(K/k);

D3. In Case 2, inflp/Ga:(L/fc) ~ [K : L] • a(K/k). Here ~ means co homologous. We call such a system {a(K/k)} the canonical 2-cocycle of G(K/k) over E(K), with which we can write down the reciprocity law as follows: Introduce (-~-) £ E(k), where a e G := Gal(K/k) for normal extension K/k by

^r):= **}K{II «*/*(**)) 7

v

peG

mod

OrG*wfc))-

7

Then by a result of Nakayama, the symbol (a, K/k) e G ab , a e £?(fc) defined by (a, K/k) := o- mod [G,G] when ( ^

J = a mod ^ . ( T G E C K " ) ) ,

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satisfies all properties of the norm residue symbol in number theory. Let k € V, be fixed and Q,a(k) be the maximal abelian extension of k in Q. Set TZa(k) be the collection of finite abelian extensions of k (in D,a). Then for K G Ka(k) define a subgroup A(K/k) of £(fc) to be (j>-}K(TGaX(K/k)E(K)). By definition, a subgroup F of E(fc) is called admissible if F = A[K/k) for some K G Ka(k). Denote the set of all admissible subgroups of E{k) by U{E{k)~). Then one checks, by the properties of norm residue symbol, that (1) (Combination Axiom) A((Ki • K2)/k) = A(Ki/k) 0 A(K2/k) and A{{Ki n K2)/k) = A{Ki/k) • A(K2/k); (2) (Ordering Axiom) A{Ki/k) D A(K2/k) if and only if Kx C K2; (3) (Uniqueness Axiom) A{K\/k) = A(K2/k) if and only if K\ = K2 E Ua{k). Also, let F(k) be the compact Galois group Qa(k)/k, then T(k) is the inverse limit group of {G(K/k) : K G 1Za(k)}. Moreover, for a G E(k), the limit, called the generalized norm residue symbol, (a, &) := limKena{k) (a, K/k) G T(k) exists. Set T(fc) := (E(k),k) c r(fc) and 7?.(A;) := {(a G -E(fc) : (a,k) = l } c S(fc), then the mapping a— i > (a, fc) induces an isomorphism E(k)/TZ(k) ~ T(fc) C T(fc) with T(k) dense in T(A:). For examples, classical (abelian) CFTs for global fields are all class formations: (A) For a number field k, we may take E(k) to be the idele class group Cfc and 4>k/K the natural inclusion. In particular, TZ(k) is then the connected component of the unity of Ck, r(fc) = T(fc), and U(E{k)) is the set of all open subgroup of E(k) of finite index; (B) For a function field k of one variable over a finite field, we may take E(k) to be the idele class group Ck and 4>k/K t n e natural inclusion. In particular, TZ(k) = 1, T(k)/T(k) is a uniquely divisible group isomorphic to Z/Z, and U{E{k)) is the set of all open subgroup of E(k) of finite index. A.2.5.2. The Work of Kawada and Tate Choose fco to be an algebraic function field of one variable over complex numbers with H the algebraic closure of ko. Let D{k) be

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the group of all fractional divisors Y\v Pl%', where rv G Q, and rv = 0 for almost all v, and P(k) the group of all principle divisors. Let D(Jfe) = D(k)/P(k). Define £(£;) := D(A;)V = Hom(D(fc),R/Z), the group of characters of D(fc), then {E(k)} with the conorm map (pk/K IS a class formation such that the norm residue map $^ : E(k) —> A(k) is surjective but F(k) := Ker <&£ 7^ 0. Consequently, if TD(fc) denotes the torsion subgroup of D(fc), {#(&)* = E(k)/F(k) = (TD(fc)) V } gives also a class formation. On the other hand, clearly, (i) the character of ko is zero; (ii) fco contains all the roots of unity; and (iii) for any finite normal extension K/k with k D ko and K/k finite, NK/kK = k. So, by applying the so-called Kummer theory, E(k) := (k* ® Q/Z) , and the conorm k/K : E(k) -» E(K), i.e., Q / Z ~ TD(fc). Moreover, if Q
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related with our approach to the CFT. It goes as follows: For ko C k, let S(k) denote the set of prime divisors of k which are extensions of a prime divisor of ko contained in S. Let R(k) be the Riemann surface of k and Rs(k) = R(k)\S(k). Let E(k) be the one-dimensional integral homology group Hi(Rs(k), Z) of Rs(k), and define k/K '• E(k) —> E(K) by 7 1—> Vj where V7 is the covering path of 7 € Rs(k) on the unramified covering surface Rs(K) of Rs(k). Then {E(k)} forms again a class formation for ts with Im$fc dense in A(k) and Ker$fc = 0. Indeed, one may obtain this by looking at the canonical pairing, the micro reciprocity law in this context, H±(Rs{k),Z) x TGs(k) -> S 1 C C defined by (j],A)s 1—> exp( f dlogA) for 1-cycle 77 on Rs(k) and A € TDg(fc), where dlogA denotes the abelian differential of the third kind on Rs(k) corresponding to a divisor A. On the other hand, over a compact Riemann surface M with punctures P i , . . . , PJV with respect to an effective divisor D = X) ej-Pj> we may introduce the group Div^ (M; 2?) of degree zero Q-divisors along D on M by collecting all degree zero Q-divisors of the form ^Zi^rPj + E with Oj 6 Z and E an ordinary integral divisor on M. Denote the induced (rational equivalence) divisor class group ClJ (M; D). Then, CI? (M; D) is simply the collection of all geo-ari bundles of rank 1 introduced in 2.2.2. Hence Theorem 2.4.2 then implies the following Theorem. There is a one-to-one and onto correspondence between the set of all isomorphism classes of finite abelian coverings ir : X —> M branched at most at D and the set of all finite subgroups S of CIQ (M; D). Moreover, the correspondence n t-> S(n) satisfies that (i) (Reciprocity Law) S(ir) ~ G^; and (ii) (Ordering Theorem) ir < n' if and only if S(ir) C S(ir'). Therefore, it seems to be very crucial to understand the precise relation between Kawada-Tate's results and this latest theorem, in order to develop a (non-abelian) CFT for global fields.

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A.3. Towards Non-Abelian CFT for Global Fields A.3.1. Weil-Narasimhan-Seshadri Type Correspondence A.3.1.1. Geometric Representations From what discussed above, in order to develop a non-abelian CFT for local and global fields, the first step should be the one to establish a micro reciprocity law, i.e., a Weil-Narasimhan-Seshadri type correspondence. Therefore, we are supposed to (1) introduce suitable classes of representations of Galois groups; (2) find corresponding classes for bundles in terms of intersection; (3) establish natural correspondences between classes in (1) and (2). Hence, it is then more practical to divide the problem into two. Namely, a general one in the sense of Weil Correspondence, and a refined one in the sense of Narasimhan-Seshadri correspondence. We start with the Weil Correspondence. Here, we are then supposed to first introduce a general notion for representation of Galois groups such that, naturally, associated to such representations are special vector bundles together with additional structure. As an example, let us explain what we have in mind in the case for number fields. So, let F be a number field with a finite subset S of places of F, including all Archimedean places. Denote the corresponding Galois group by GF,S- Naturally, by a representation, it should be first a continuous group homolorphism p : GF,S —* G L „ ( A . F , S ) , where Ap,s denotes the ring of 5-adeles. Moreover, among others, we should assume that (a) for all places v of F, the induced representations pv : GF,S -> GLn{Fv) are unramified almost everywhere; (b) for a fixed places p of Q, there should be a compatibility condition for all places v above p. In addition, it also seems to be very natural to assume that (c) for all induced pv : Gv —> GLn(Fv), there are invariant 0„-lattices Mv of Fv which are induced from a global C?F-lattice M over F; and (d) at places v € S, there should be naturally a certain weighted filtration induced by the action of Frobenius. Thus in particular, associated to such a representation, is natu-

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rally a well-defined vector bundle equipped with a parabolic structure. We are expecting that geometric bundles are of Arakelov degree zero. With Weil type correspondence, we are able to enter the level of Narasimhan-Seshadri type correspondence. Then we are expect to overcome certain essential difficulties. Chiefly, what should be a natural analog of being unitary? A suitable candidate seems to be that of Fountaine's semi-stability at finite places and (unitary) at infinite places. However, we are not very sure about this, as somehow we believe that the condition of semi-stable representation is too restricted. For this reason, we propose the follows; (e) for all places v, the images of the induced representation pv : Gv —> GLn(Fv) are contained in maximal compact subgroups; moreover, certain compatibility conditions are satisfied by pv for all v over a fixed place of Q; (f) there should be a natural deformation theory for these representations such that (i) the size of all equivalence classes of these representations can be controlled; (ii) a certain completeness holds. Here the reader may find the work of Rapoport and Zink [RZ] and the lecture notes of Tilouine [Ti] very useful. The cases for local fields and function fields may be similarly discussed. If successful, we call such a representation a geometric representation. A.3.1.2. Semi-Stability in Terms of Intersection This is the algebraic side of the micro reciprocity law. We here only study what happens for function fields over finite fields, while leave a detailed definition for number fields in Part (B). With this restriction of fields, the situation becomes much simpler: We may use the existing stability condition of Seshadri for parabolic bundles. That is to say, what we care of here are the so-called parabolic semi-stable bundles of parabolic degree zero defined on algebraic curves over finite fields, introduced by Seshadri. However, in doing so, we are afraid that our program leads to only a non-abelian

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CFT with tame ramifications. So it seems that, to deal with wild ramifications, additional works are needed. Therefore, in general, for this algebraic aspect, we propose the following: (0) Once with intersection semi-stability, we should be able to establish (i) an analog of the existence and uniqueness of HarderNarasimhan type filtration; (ii) the existence of Jordan-Holder type filtration and the uniqueness of the associated graded Jordal-Holder objects; and (iii) morphisms of stable objects are either zero or isomorphisms; (1) The so-called Langton's Completeness principle should hold for such intersection semi-stabilities; (2) The intersection semi-stability should be naturally related with a GIT stability. As a direct consequence, then moduli spaces can be formed naturally, and by (1), are indeed compact; (3) With the help of the Frobenius, or better, modulo ^"-reduction for all n, we should be able to define a special subset (of points) of moduli spaces, tensor products of whose corresponding bundles are represented again by points in this special subset. If successful, we will call these objects geometric bundles. A.3.1.3. Weil-Narasimhan-Seshadri Type Correspondence For a Weil type correspondence, from the classical proof, say, the one given in Gunning's Princeton lecture notes, we should develop an analog of the de Rham, the Dolbeault cohomologies as well as a Hodge type theory. While, in general, it is out of reach, note that our base is of dimension one, it is still hopeful. For example, for curves over finite fields, we understand that all this is known to experts. Next, let us consider a Narasimhan-Seshadri type correspondence. Here, we should establish a natural correspondence between equivalence classes of geometric representations and (Seshadri) equivalence classes of geometric bundles. Hence, the key points are the follows: (1) By definition, a geometric representation should naturally give a geometric bundle; (2) A Weil type correspondence holds. In particular, this implies that, by definition, geometric bundles come naturally from representations of Galois groups;

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(3) Representations resulting from geometric bundles in (2) should be geometric. To establish this, we should use the deformation theories of geometric representations and geometric bundles as proposed in 3.1.1 and 3.1.2 above. For example, the final justification should be based on the compactness of both geometric representations and geometric bundles, via a direct counting. (In geometry, the count is possible since we know the structure of the fundamental group. But in arithmetic, this is quite difficult. Here the work of Fried and Volklein [FV] seems to be quite helpful.) A.3.2. Harder-Narasimhan Correspondence This is designed to select special subclasses of geometric representations and geometric bundles so as to get what we call geo-ari representations and geo-ari bundles. There are two main reasons for doing this. The first is based on the facts that for function fields, there are examples of stable bundles whose tensor products are no longer semi-stable; and more importantly that if the bundles and their associated Frobenius twists are all semi-stable, then, so are their tensor products. The second comes from our construction of non-abelian zeta functions in Part (B). To define such non-abelian zeta functions, we use only rational points (over constant fields) of the moduli spaces, based on a result of Harder and Narasimhan, which guarantees the coincidence of the rationality of moduli points and the rationality of the corresponding bundles. Thus, for example, for function fields over finite fields, a geoari bundle is defined to a geometric bundle which satisfies not only all the conditions for geo-ari bundles over Riemann surfaces, but an additional one, which says that all its Frobenius twists are semi-stable as well. So for our more general purpose, the key points are as follows; (1) to give a proper definition of geo-ari representations so that they form a natural abelian category; (2) to give a suitable definition of geo-ari bundles so that they are closed under tensor product; and (3) to establish a natural correspondence between (1) and (2). If successful, we call such a result a Harder-Narasimhan type correspondence, even we here substantially board the meaning of [HN].

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A.3.3. KR2-Trick This is specially designed to offer an algebraic proof of the following key statement appeared as 3.2(2): tensor products of geo-ari bundles are again geo-ari bundles. We start with geo-ari bundles over function fields. When no parabolic structures is involved, this is solved by Kempf and RamananRamanthan. In their proof, the key points are as follows. (1) Rationality and uniqueness of instability flags, where the Frobenius twists are used; (2) Existence of a GIT stable modification associated to any non semi-stable one; and (3) Existence of GIT points for bundles such that GIT stability implies intersection stability by definition; (4) Relation between the GIT modification in (2) and the intersection stability, where the intersection stability for each component of the tensor product plays a key rule. More precisely, if the tensor were not stable in terms of intersection, by (3), the associated GIT point would not be GIT stable. Thus from (2) there exists a GIT stable modification which is well understood by (1). Therefore, finally, by (4), i.e., the intersection stability of each components, we may finally conclude the intersection stability for the tensor product with the help of (2). Thus, the problem left here is to see how the work of RamananRananthan can be developed to deal with parabolic structure. In theory, this may be done by working on product of varieties instead of just a single one. As this process is closed related to the construction of moduli space of parabolic semi-stable bundles, an expert should be able to carry the details out. However, there is a more elementary approach, essentially given in the supplementary works of Faltings and Totaro. It goes as follows. First, as what Faltings does, write parabolic subbundles for the tensor in terms of weighted nitrations on the fibers (supported over points which are disjoint from punctures); then introduce a GIT stability with respect to weighted nitrations. So there are two possibilities: if the weighted filtration resulted from the parabolic subbundle is GIT stable, then by (3) above, we get the intersection stability of

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the tensor product. Otherwise, by (1) and (2) above, we may obtain a well-associated modification which is then GIT stable. This then by applying (4) completes the proof. We would like to draw reader's attention to a relation between the stabilities of generic and special points due to Mumford: GIT stability for objects over the generic point implies that over almost all but finitely many special points, the associated points are GIT semi-stable. All in all the basic principle here is that if an object is not stable, then there should be a parabolic subgroup which takes responsibility. In this sense, we have already had a very satisfied result for lattices. For details, please see the global bridge between algebraic and analytic truncations in B.4. A.3.4. Tannakian Category Theory over Arbitrary Bases To establish a non-abelian CFT for function fields over complex numbers, we apply a standard theory of Tannakian category. However, for global fields, a much more general theory is needed. Say for number fields, a theory of Tannakian categories over the ring of integers should be developed. Key points for this new type of Tannakian categories are supposed to satisfy the follows. (1) Automorphisms of fiber functors should give us a group. Particularly, whether fiber functors are to categories of vector spaces is not really important. Moreover, fiber functors should be faithful exact and tensor, among others; (2) An analog of Tannakian duality holds; and (3) Tannakian category should contain the so-called finitely completed Tannakian subcategories with respect to which an analog of van Kampen Completeness Theorem holds. If successful, by (1) and (2), using Narasimhan-Seshadri and Harder-Narasimhan correspondences, we get naturally a reciprocity map. This then by (3), leads to a completed non-abelian CFT. A.3.5. Non-Abelian C F T for Local and Global Fields All in all, from the above discussion, what we then expect is the following conjecture, or better,

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Working Hypothesis. For local and global fields, (1) there are well-defined geometric representations and geometric bundles such that a Weil-Narasimhan-Seshadri type correspondence holds; (2) there are refined well-defined geo-ari representations and geo-ari bundles such that a Harder-Narasimhan type correspondence holds; (3) there is well-defined GIT type stability such that the intersection stability as appeared in the definition of geo-ari bundles could be understood in terms of this new GIT stability. Moreover, an analog of KR2-trick works; (4) there is a well-established Tannakian type category theory, for which a Tannaka type duality and van Kampen type completeness theorem hold; moreover the category of geo-ari objects is naturally such a Tannakian type category. Consequently, the fundamental results in non-abelian class field theory, such as existence theorem, conductor theorem and reciprocity law, hold. A.3.6. Non-Commutative Euler Product In the classical CFT for number fields, reciprocity law can be used to define L-functions in terms of Euler product formalism. Thus, we expect that non-abelian reciprocity law will help us to understand non-abelian L-functions. However, there is a difficulty in doing so: in abelian theory, we do not really care about the order of Frobenius appearing in the Euler product since Galois groups involved are commutative. This then leads naturally to the consideration of non-abelian Euler products. It is for this purpose, motivated by the non-abelian CFT for function fields over complex numbers, we propose to study algebraic relations among images of the Frobenius classes of absolute Galois groups in finite (subquotient) groups in E r . Here E r is defined to be the collection of all finite groups G satisfying (1) all irreducible unitary representations of G are of rank < r; (2) G admits at least one rank r irreducible unitary representation. E x a m p l e s . (1) When r = 1, E r is simply the collection of all finite

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abelian groups and hence the algebraic relations we are seeking for are simply the commutative law; (2) Groups G in S r may be characterized by the following property: For any 2r elements x\,X2,.-.,%2r, in the group ring Z[G], ^2

sgn(a) • x(r(1)X(T(2) • • • xC(2r) = 0,

where Sjr denotes the symmetric groups on 2r symbols, and sgn(cr) denotes the sign of a. (In particular, if r = 1 we get the commutative law X1X2 = x<2X\.)

As such, then non-abelian Euler product is supposed to be a formalism which exposes intrinsic relations for algebraic structures among Galois groups in S r whose specializations give non-abelian L- functions.

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L-Functions

B . l . Non-Abelian Zeta Functions for Curves B . l . l . Local Non-Abelian Zeta Functions B . l . 1 . 1 . Artin Zeta Functions for Curves We start this part by recalling the construction and basic properties of classical Artin zeta functions for curves defined over finite fields. Let C be a projective irreducible reduced regular curve of genus g defined over a finite field k := Fq with q elements. Then the arithmetic degree d(P) of a closed point P on C is defined to be d(P) := [k(P) : k], where k(P) denotes the residue class field of C at P. So g d ( p ) is nothing but the number N(P) of elements in k(P). Extending this to all divisors, we define the Artin zeta function (c(s) for curve C over k by setting

Cc(s) := I I ! _ MP)-* = £ ^ ) " 5 = E (TS)d(D\ Ms) > 1, V

P

'

D>0

D>0

where the sum is taken over all effective divisors D on C. As usual, set t := q~s and Zc(t) := ^ ^ ( D ) D>0

Note that the number of positive divisors which are rational equivalent to D is

q

1 t

,

cc(t) = E r r = E E E r ) d D>0 nh°(C,V)

d>0

V

d>0 i

V

D€V

H

where the sum YLv ^s taken over all rational divisor classes of degree d. Hence by the duality, the Riemann-Roch and a vanishing theorem, we have the following

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Theorem, (see e.g. [Me]) The Artin zeta function Cc{s) is welldefined forRe s > 1, and admits a meromorphic continuation to the whole complex s-plane. Moreover, (c(s) = N(Kc)1/2-s<;c(l

~ s),

and there exists a polynomial Pcif) € Z[T] of degree 2g such that

Zc(t) =

Pc(t)

(l-t)(l-qty

B . l . 1 . 2 . Two Different Generalizations: Weil Zeta Functions and A New Approach Based on the so-called reciprocity law, Artin zeta function may be written as

^)-(E^) where Nm := #C(Fqm) denotes the number of F ? m-rational points of C. This then leads to a far reaching generalization to the socalled Weil zeta functions of higher dimensional varieties defined over finite fields, the study of which dominates what we call Arithmetic Geometry in the second half of 20th century. On the other hand, Artin zeta functions may also be naturally interpreted in a different way which leads to a non-abelian theory for zeta functions, based on the follows; (1) Yl,v m a y D e v i e w e d as a sum over degree d Picard group of the curve; (2) Picard groups are moduli spaces of (semi-stable) line bundles; (3) All the terms appeared in Artin zeta functions, such as h° and degree, make sense for vector bundles as well; and (4) q — 1 in the denominator is the order of the automorphic group of a line bundle. In other words, the Artin zeta function Cc(s) may be rewritten as d>0LePic d (C)

#Aut(L)

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As such, these abelian zetas are ready to be naturally extended to their non-abelian counterparts. B . l . 1 . 3 . Moduli Spaces of Semi-Stable Bundles Assume now that C is defined over an algebraically closed field k. Then according to Mumford [M], a vector bundle V on C is called semi-stable (resp. stable) if for any proper subbundle V of V,

Here d denotes the degree and r denotes the rank. Proposition 1. Let V be a vector bundle over C. Then (a) ([HN]) there exists a unique filtration of subbundles of V, the so-called Harder-Narasimhan filtration of V, {0} = VQ c Vi C V2 C • • • C Vs-i CVS = V such that Vi/Vi-i is semi-stable and /x(V^/Vi_i) > /x(V^+i/Vi); (b) (see e.g. [Se2]) if moreover V is semi-stable, there exists a filtration of subbundles of V, a Jordan-Holder filtration of V, {0} = Vt+l C V1 c • • • C V1 C V° = V s.t. Vi/Vi+1 is stable and /i(V*/V i+l ) = y.{V). Moreover, Gr(V) := l+1 ®\=QV% /V , the associated (Jordan-Holder) graded bundle ofV, is determined uniquely by V. Following Seshadri, two semi-stable vector bundles V and W are called 5-equivalent, if their associated Jordan-Holder graded bundles are isomorphic, i.e., Gr(V) ~ Gr(W). Applying Mumford's general result on geometric invariant theory ([M]), Narasimhan and Seshadri establish the following Theorem 2. ([NS], [Sel]) Let C be a regular, reduced, irreducible projective curve defined over an algebraically closed field. Then over the set M.c,r(d) of S-equivalence classes ofrankr and degree d semistable vector bundles over C, there is a natural normal, projective algebraic variety structure.

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Now assume that C is defined over a finite field k. Then there is a notion of fc-rational bundles over C, i.e., bundles which are defined over k. Moreover, from geometric invariant theory, projective varieties Mc,r(d) are defined over a certain finite extension of k. Thus it also makes sense to talk about fc-rational points of these moduli spaces too. The relation between these two types of rationality is given by Harder and Narasimhan based on a discussion about Brauer groups: Proposition 3. ([HN]) There exists a finite field Fq such that for any d, the subset of Fq-rational points of M.c,r{d) consists exactly of all S-equivalence classes of Fq-rational bundles in M.c,r{d)Prom now on, without loss of generality, we always assume that finite fields Fq (with q elements) satisfy the property in this proposition. Also for simplicity, we write Mc,r(d) for Mc,r(d)(Fq), the subset of Fg-rational points, and call them moduli spaces by an abuse of notation. B.1.1.4. Local Non-Abelian Z e t a Functions For C a regular, reduced, irreducible projective curve over Fq, define its rank r non-abelian zeta function Cc,r,F,(s) by setting „h°(C,V)

ve[v]eMc,r(d),d>o ^

_ 1 K

'

Clearly, we have the following Lemma. Cc,i,F,(s) is nothing but the classical Artin zeta function for curve C. R e m a r k . In the definition of ( above, the sum is taken over all elements in Seshadri classes. This may be modified by taking only one representative for each class. Another modification, as suggested by Deninger, is to consider only stable objects. We leave the details to the reader.

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B . l . 1 . 5 . Basic Properties for Non-Abelian Zeta Functions Many basic properties for classical Artin zeta functions are satisfied by our non-abelian zeta functions as well. More precisely, by using a vanishing theorem, Serre duality, and the Riemann-Roch, we obtain the following Theorem. ([We4]) (1) The non-abelian zeta function (c,r,Fq(s) is well-defined for Re(s) > 1, and admits a meromorphic continuation to the whole complex s-plane; (2) (Rationality) Set t := q~s and introduce the non-abelian Zfunction of C by

) =: Z0lP,P,(t) :=

£

nh°(C,V) g

_ i ( )

-t^\

\t\ < 1.

V£[V}eMC,r(d)>d>°

Then there exists a polynomial Pc,r,Fq{s) S Q[t] such that Zc,r,Fq(t) =

Pc,r,Fq(t) (l-tr){l-qrtr))

(3) (Functional Equation) Set the rank r non-abelian &7,r,F,(s) by

^-function

Then &7,r,F,(«)=&7,r,F,(l-s)-

B.1.2. Global Non-Abelian Zeta Functions for Curves B. 1.2.1. Definition Let C b e a regular, reduced, irreducible projective curve of genus g defined over a number field F. Let Sbad be the collection of all infinite places and these finite places of F at which C does not have good reduction. As usual, a place v of F is called good if v $ 5badFor any good place v of F, the v-reduction of C, denoted as Cv, gives a regular, reduced, irreducible projective curve defined over the

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residue field F(v) of F at v. Denote the cardinal number of F(v) by qv. Then, by 1.1, we obtain the associated rank r non-abelian zeta function Cc„,r,F„„(s)- Moreover, from the rationality of Cc„,r,F,„(s), there exists a degree Irg polynomial Pev,r,FQv(t) € Q[t] such that y

(ft

Pcv,r,Fqv{t) (I - tr){l -

Cv,r,Fqv{ )

qrtr)'

Clearly, PCv,r,Fqv ( 0 ) ^ 0 . Set ^C v ,r,F(t;)(*)

:=

•Pc„,r,F(i;)(*) Pc v,r,F(v)(Q)

Now by definition, the rank r non-abelian zeta function C over F is the following Euler product Cc,r,Hs):= I I

5

v:good'

PC

l

-T^~V

r F

'" ' ^(-^ ^

£C,T-,.F(S)

of

Re(S)»0.

Clearly, when r = 1, Cc,r,^(s) coincides with the classical HasseWeil zeta function for C over F. Conjecture. For a regular, reduced, irreducible projective curve C of genus g defined over a number field F, its associated rank r global non-abelian zeta function Cc,r,p(s) admits a meromorphic continuation to the whole complex s-plane. Recall that even when r = 1, i.e., for the classical Hasse-Weil zeta functions, this statement, as a part of a series of high profile conjectures is still open. On the other hand, using a result of (HarderNarasimhan) Siegel on Tamagawa numbers of SLr, the ugly yet very precise formula for local zeta function in 1.2.1, Clifford Lemma for semi-stable bundles, and Weil's theorem on the Riemann hypothesis for Artin zeta functions, we have the following Proposition. ([We4]) WhenRe(s) converges.

> l + g + (r2-r)(g-l),

CC,V,F(S)

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B . l . 2 . 2 . Working Hypothesis Like in the theory for abelian zeta functions, we want to use our non-abelian zeta functions to study non-abelian aspect of arithmetic of curves. For doing so, we then also need to introduce local factors for 'bad' places. This may be done as follows. For T-factors, we take these coming from the functional equation for CF(TS) • CF(?"( S ~ 1))) where CF(S) denotes the standard Dedekind zeta function for F; while for finite bad factors, first, use the semi-stable reduction for curves to find a semi-stable model for C, then use Seshadri's moduli spaces of parabolic bundles to construct polynomials for singular fibers, which usually have degree lower than 2rg. With all this being done, we then can introduce a completed rank r non-abelian zeta function for C over F, or better, the completed rank r non-abelian zeta function Oc,r,o F ( s ) f° r a semi-stable model X —> Spec((9p) of C. Here OF denotes the ring of integers of F. (If necessary, we take a finite extension of F.) Working Hypothesis. £x,r,oF{s) a ^™* s a unique meromorphic continuation to the whole complex s -plane and satisfies the functional equation {,X,r,0F{s)

= E • ix,r,0F

( 1+ - - S j

with |ej = 1. Moreover, we expect that for certain classes of curves, the inverse Mellin transform of our non-abelian zeta functions are naturally associated to certain modular forms of weight 1 + £. To end this general discussion, we remark that as they stand, these global zeta functions are expected to reflect arithmetic aspect of moduli spaces of semi-stable hermitian vector sheaves over arithmetic surfaces. (The definition of semi-stability is clear, as we have the notation of arithmetic ample line bundle and hermitian vector sheaves - in general we do have problem to develop an Arakelov theory for coherent sheaves since there is no Bott-Chern theory for coherent sheaves over complex manifolds, but for our limited purpose

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here we can restrict ourselves to hermitian vector bundles at infinity. That is to say, over integral models, coherent sheaves are allowed, but at infinite places, these sheaves are supposed to be locally free.) B.1.3. Refined Brill-Noether Locus: Towards a Reciprocity Law B. 1.3.1 Results of Atiyah Let E be an elliptic curve defined over F g . Recall that a vector bundle V on E is called indecomposable if V is not a direct sum of two proper subbundles, and that every vector bundle on E may be written as a direct sum of indecomposable bundles, whose summands and multiplicities are uniquely determined. Thus to understand vector bundles, it suffices to study indecomposable ones. To this end, we have the following result of Atiyah. In the sequel, for simplicity, we always assume that the characteristic of Fq is strictly bigger than the rank of V. Theorem 1. ([At]) (a) For any r > 1, there is a unique indecomposable vector bundle Ir of rank r over E, all of whose Jordan-Holder constituents are isomorphic to OE- Moreover, the bundle Ir has a canonical filtration {0} C F 1 C • • • C Fr = Ir with Fi = h and Fi+1/Fi = OE; (b) For any r > 1 and any integer a, relative prime to r and each line bundle X over E of degree a, there exists up to isomorphism a unique indecomposable bundle Wr(a;X) over E of rank r with X the determinant; (c) The bundle Ir [Wr> (a; A)) = IT <8> Wr* (a; A) is indecomposable and every indecomposable bundle is isomorphic to Ir (Wri(a; X)) for a suitable choice of r,r',X. Every bundle V over E is a direct sum of vector bundles of the form Iri {Wr',(af, A^)), for suitable choices of ri,r'i and Aj. Moreover, the triples (ri,r^,Ai) are uniquely specified up to permutation by the isomorphism type of V. Here note in particular that Wr (0, A) ~ A, and that indeed Ir(Wr>(a; A)) is the unique indecomposable bundle of rank rr' such

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that all of whose successive quotients in the Jordan-Holder filtration are isomorphic to Wr>(a;A). Clearly, Ir(Wr>(a; A)) is semi-stable with n(lr(Wr>(a;X))) = a/r'. Theorem 2. (a) (Atiyah) Every bundle V over E is isomorphic to a direct sum ®iVi of semi-stable bundles, where n(Vi) > /i(Vi+i); (b) (Atiyah) Let V be a semi-stable bundle over E with slop ^{V) = a/r' where r' is a positive integer and a is an integer relatively prime to r'. Then V is a direct sum of bundles of the form Ir{Wr'{a; A)), where X is a line bundle of degree a; (c) (Atiyah, Mumford-Seshadri) There exists a natural projective algebraic variety structure on M.E,r(X) :— {V : semi — stable, detF = A, rank(F) = r } / ~ s . Moreover, if A G Pic°(.E), then ME,T{X) space P^~ .

is simply the projective

B . l . 3 . 2 . Refined Brill-Noether Locus Note that if V is semi-stable with strictly positive degree d, then h°(E, V) = d. Hence standard Brill-Noether locus is either the whole space or empty. In this way, we are led to the case d = 0. For this, recall that for A € Pic 0 (£), ME,r(X)

= iV '• s e m i - stable, rank(F) = r,det(V) = A}/ ~ s

is identified with {V = ®ri=lLi : ®iL{ = A, L{ e Pic°{E),i = 1 , . . . , r } / ~ i s o ^ P ^ 1 where / ~i s o means modulo isomorphisms. Now introduce the standard Brill-Noether locus W%,r(X) := {[V] E ME,r{\)

: h°(E,gv(V))

> a}

and its 'stratification' Wjj r (A)° := {[V] E WE,r(X) : h°(E,gv(V))

= a} = Wlr(X)\Ub>a+1

WE^T{X).

260

L. Weng

One checks easily that W|>r(A) ~ P ^ - 0 ) - 1 , W ^ S - i W - Wg,r(A),

and

W g ^ A ) 0 ~ Wg>r(A)°.

The Brill-Noether theory is based on the consideration of h°. But in the case for elliptic curves, for arithmetic consideration, such a theory is not fine enough: not only h° plays a crucial role, the automorphism groups are important as well. Based on this, we introduce, for a fixed (k + l)-tuple non-negative integers (ao; ai, • • •, a&), the subvariety of W^°r by setting W^ a i --° f c (A) :={[V] G W%r(\) : gr(V) = C?Jo) © © L i i f ° , (git Lfai = A, ^ G Pic°(£), 1 < i < kX. Moreover, we define the associated 'stratification' by setting wa0,ai,..,ak{x)0

.= |[y]

€ W g;«i,...,a fc(A)>

#{QE, Llt . . . , Lk} = k+l}.

Easily we have

w^+.1iai--a*(A)~w^;0l'-,0fc(A), w

^°i1iaiv'Mafc(A)0-w£°;ai

afc

(*)°>

and A^, r (A) = U a o ; a i ,..., a f c Wg a i -- a *(A)°. Now recall that for elliptic curves i£, (1) The quotient space E^/Sn is isomorphic to the P n _ 1 -bundle over E; and (2) The quotient of E^'^/Sn is isomorphic to P ^ " 1 ) . Here we embed E^n~l> as a subspace of E^ under the map ( x i , . . . , x n _i) i-> ( x i , . . . , x n _ ! , x n ) with x n = A - (xi + x 2 H hxn_i). Proposition. ([We3]) Regroup (ao; o i , . . . , a*,) as (ao; &i , . . . , of) where we assume that b\ > 62 > • • • > h and s\,S2,..-,si G Z>o, then

(1) i/bt = 1, w^;oi--a*(A) ~ n t i p r 1 >i,wg ai, -' afc (A)= i n!=iPS-

Geo-Arithmetic: B. Non-Abelian

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261

Remark. When A = OE, refined Brill-Noether loci WE°;ai'-'ak(0E) are isomorphic to products of (copies of) projective bundles over E and (copies of) projective spaces, which are special subvarieties in M.E,T{OE) = P r _ 1 - It appears that intersections among these refined Brill-Noether loci are quite interesting. So define the BrillNoether tautological ring BN£ i r (Og) to be the subring generated by all the associated refined Brill-Noether loci (in the corresponding Chow ring). What can we say about it? Examples. (1) If r = 2, then this ring consists of only two elements: the O-dimensional one WE2(OE) = {[OE®OE\] and the whole space 1 P . So everything is simple; (2) If r = 3, then (generators of) this ring contains five elements: 2 of O-dimensional objects: WE'°3(OE) = {[0{E]]} and WE'23(OE) = {[OE © ^2 ] : T
qh (E V)

^

° '

Ve[v}eME,rW ,ao+l;ai,...,ak / \\ K) a E'r ' paEOp,..,ak{x)

;=

V^ *-> Ve[y]e<° r +liQ1

£

„h°(E,V)

ak

J* #Aut(V)' W° 1 #Aut(F)'

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L. Weng

Before going further, we remark that above, we write V £ [V] in the summation. This is because in each 5-equivalence class [V], there are usually more than one vector bundles V. For example, [0$] consists of 0{£\ OP © h, h® h, OE © h, and J4. Theorem. ([HN] & [DR]) For all A, A' € Picd(E), pE,r(x) = PEAX')Moreover, T

NvfcrW = -^ -flCE® Yl

IIfan(di)

Ei< ,(rjd._r.d,}

•

Sfr<=r,Si*=d,|j->->^,fc>2 #ere iVi denotes # £ ( : =

#E(Fq)).

Thus, we are lead to introduce the 7-series invariants 7£,r(A) and 7^°+ 1;ai '-' afc (A) by setting 7 := a - (3. That is to say,

1EA)

"

r

A

, , #Aut(F) ' 0 fc°(£,v)

£ r

'

^

J

""

ft ve[v]ew^°;ai

_ 1

a #Aut(V) • v y °fc(A)° ^

Based on the above theorem and lower rank examples using the stratification just introduced, we make the following Conjecture. For all A G Pic°(.E), a.E,rW = a;.E,r(C>£;)- Hence also lE,r{X) = lE,r(0E)-

Appendix to B.l:

Weierstrass G r o u p s

Motivated by a certain construction of Euler systems for elliptic curves in terms of elements in K2 using torsion points, we here introduce what we call Weierstrass groups using Weierstrass divisors for curves.

Geo-Arithmetic: B. Non-Abelian L-Functions

263

1. Weierstrass Divisors (1.1) Let M be a compact Riemann surface of genus g > 2. Denote its degree d Picard variety by Pic d (M). Fix a Poincare line bundle Vd on M x Pic d (M). (One checks easily that our constructions do not depend on a particular choice of the Poincare line bundle.) Let 0 be the theta divisor of Pic s _ 1 (M), i.e., the image of the natural map Ma-1 -> Pic^-^M) denned by ( P i , . . . , P 9 _i) y-* [OM(PI + - • •+ Pg-i)]. Here [•] denotes the class denned by •. We will view the theta divisor as a pair (0Picg-i/M\(@), l e ) with l e the defining section of 0 via the structure exact sequence 0 —> 0Picg-i,M\ —> C Pic <,-i(^)(0)• Denote by pi the i-th. projection of M x M to M, i = 1, 2. Then for any degree d — g — 1 + n line bundle on M, we get a line bundle p\L{—nA) on M x M which has relative p2~degree g — 1. Here, A denotes the diagonal divisor on M x M. Hence, we get a classifying map 4>L : M —• Pic 9 _ 1 (M) which makes the following diagram commute: MxM -» M x P i c 9 - 1 ^ ) P2 I M

| 7T

^4

PicO-^M).

One checks that there are canonical isomorphisms

and Ap a (plL(-nA)) ~ ^ O P i c 9 - i ( M ) ( - 0 ) . Here, A,,, (resp. AP2) denotes the Grothendieck-Mumford cohomology determinant with respect to TV (resp. p2}- (See e.g., [L].) Thus, (^^l© gives a canonical holomorphic section of the dual of the line bundle \P2(p\L(—nA)), which in turn gives an effective divisor WL{M) on M, the so-called Weierstrass divisor associated to L. Example. With the same notation as above, take L ~ K^71 with KM the canonical line bundle of M and m £ Z . Then we get an effective divisor WK®m(M) on M, which will be called the m-th Weier-

264

L. Weng

strass divisor associated to M. For simplicity, denote (resp. $K®rn) by Wm(M) (resp. <j)m).

Ww®m(M)

One checks easily that the degree of Wm(M) is g(g — l ) 2 ( 2 m - 1 ) 2 m and we have an isomorphism OM{Wm(M)) ~ KjJ1 ~ • Thus, in particular,

_ (cie) 0(2 "- 1)2 Jm,n

•— (* l e ) ® ( 2 m - l ) 2

gives a canonical meromorphic function on M for all m, n € Z. Remark. We may also assume that m € ^Z. Furthermore, this construction has a relative version as well, for which we assume that / : X —> B is a semi-stable family of curves of genus g > 2. In that case, we get an effective divisor (Ox{Wm(f)), lvv m (/)) a n d canonical isomorphism (Ox{Wm(f)),lWm(f)) * (Ox(Wi(f)),

l W l (/))® ( a m - 1 ) a ® ( C W W i ( / ) ) , l W l (fl)* 4 ** 1 "" 0 .

The proof may be given by using Deligne-Riemann-Roch theorem, which in general, implies that we have the following canonical isomorphism: (Ox(WL(f)),

1WLU))

® /*A/(L) ~ L®" ® ^ « ( « - D / 2 .

(See e.g. [Bu].) To allow m be a half integer, we then should assume that / has a spin structure. Certainly, without using spin structure, a modified canonical isomorphism, valid for integers, can be given. 2. K-Groups (2.1) Let M be a compact Riemann surface of genus g > 2. Then by the localization theorem, we get the following exact sequence for K-groups K2(M) A K2(C(M))

Upe dp

^

]l

c;.

p&M

Note that the middle term may also be written as K2(C(M\S)) for any finite subset 5 of M, we see that naturally by a theorem of

Geo-Arithmetic: B. Non-Abelian

L-Functions

265

Matsumoto, the Steinberg symbol {/m,n, fm',n'} gives a well-defined element in K2(C(M)). Denote the subgroup generated by all these {fm,n,fm',n'} with m,n,m',n' e Z > 0 in K2(C(M)) as S(M). Definition. With the same notation as above, the first Weierstrass group W[(M) of M is defined to be the A-pull-back of £ ( M ) , i.e., the subgroup A- 1 (S(M)) of K2(M). (2.2) For simplicity, now let C be a regular projective irreducible curve of genus g > 2 defined over Q. Assume that C has a semistable regular module X over Z as well. Then we have a natural morphism K2{X) -£ K2(M).

Here M := C(C).

Conjecture I. With the same notation as above, (j)(K2(X)}Q W>(Af)Q.

=

3. Generalized Jacobians (3.1) Let C be a projective, regular, irreducible curve. Then for any effective divisor D, one may canonically construct the so-called generalized Jacobian JD(C) together with a rational map / D : C —> JD(C).

More precisely, let Co be the group of classes of divisors prime to D modulo these which can be written as div(/). Let CD be the subgroup of CJJ which consists of all elements of degree zero. For each pi in the support of D, the invertible elements modulo those congruent to 1 (mod D) form an algebraic group RD,pt of dimension rij, where ni is the multiplicity of pi in D. Let RD be the product of these RD,pi- O n e checks easily that G m , the multiplicative group of constants naturally embeds into RD. It is a classical result that we then have the short exact sequence 0 - RD/Gm

-^C°D^J^0

where J denotes the standard Jacobian of C. (See e.g., [S].) Denote RD/Gm simply by R D . Now the map fo extends naturally to a bijection from CD to JD- In this way the commutative algebraic group JD becomes an

266

L. Weng

extension as algebraic groups of the standard Jacobian by the group Example. Take the field of constants as C and D = Wm(M), the m-th Weierstrass divisor of a compact Riemann surface M of genus g > 2. By (1.1), Wm(M) is effective. So we get the associated generalized Jacobian Jwm(M)- Denote it by WJm{M) and call it the m-th Weierstras-Jacobian of M. For example, if m = 0, then WJQ(M) = J(M) is the standard Jacobian of M. Moreover, one knows that the dimension of Rwm{M),p i s a t most g(g + l ) / 2 . For later use denote R.vvm(M) simply by Rm. (3.2) The above construction works on any base field as well. We leave the detail to the reader while point out that if the curve is denned over a field F, then its associated m-th Weierstrass divisor is rational over the same field as well. (Obviously, this is not true for the so-called Weierstrass points, which behavior in a rather random way.) As a consequence, by the construction of the generalized Jacobian, we see that the m-th Weierstrass-Jacobians are also defined over F. (See e.g., [S].) 4. Galois Cohomology Groups (4.1) Let K be a perfect field, K be an algebraic closure of K and G-^iK be the Galois group of K over K. Then for any G^iKmodule M, we have the Galois cohomology groups H°(G^,K,M) l and H (Gj£iK)M) such that if 0 -» Mi -> M 2 - • M 3 -> 0 is an exact sequence of G^; K -modules, then we get a natural long exact sequence 0 - H°(GK/K,

MI)

- H\GJZ/K,M{)

-> H°(Gx/K, 1

M 2 ) - H°(GK/K,

-> H (G1^/K,M2)

-

M3) &(&«,«,M3).

Moreover, if G is a subgroup of GJ?IK of finite index or a finite subgroup, then M is naturally a G-module. This leads a restriction map on cohomology res : H1(Gj^,K,M) —> Hl(G,M).

Geo-Arithmetic: B. Non-Abelian

L-Functions

267

(4.2) Now let C be a projective, regular irreducible curve defined over a number field K. Then for each place p of K, fix an extension of p to K, which then gives an embedding K c Kp for the p-adic completion Kp of K and a decomposition group Gp C Gj^,K. Apply the construction in (3.1) to the short exact sequence 0 -.- R m - • WJm{C)

- • J(C) - • 0

over K. Then we have the following long exact sequence

- H\G«/K,

WJm{K))

^ H\GTZIK,

J{K)).

Similarly, for each place p of K, we have the following exact sequence 0 -+Rm(Kp)

- • WJm{Kp)

-> J(Kp)

-> ^(Gp.RmC/fp)) -> ^ ( G p , WJ m (Kp)) ^

H\GP,J{KP)).

Natural inclusions Gp C Gj^,K and K C ifp give restriction maps on cohomology, so we arrive at a natural morphism $rn:il>(H\aR/K,WJm{.K)))-*

J]

$p(H\Gp,WJm(Kp)j).

peMK

Here MK denotes the set of all places over K. Definition. The second Weierstrass group Wa(C) of C is defined to be the subgroup of Hl(G^,K, J(C)(K)) generated by all Ker $ m , the kernel of $ m , i.e., Wu(C) := ( K e r $ m : m € Z>o)zConjecture I I . The second Weierstrass group Wn(C)

is finite.

5. Deligne-Beilinson Cohomology (5.1) Let C be a projective regular curve of genus g. Let P be a finite set of C. For simplicity, assume that all of them are defined over R.

L. Weng

268

Then we have the associated Deligne-Belinsion cohomology group H^(C\P, R(l)) which leads to the following short exact sequence: 0 -> R -* H%,(C\P, R(l)) ^ R[P]° -* 0 where R[P]° denotes (the group of degree zero divisors with support on P ) R . The standard cup product on Deligne-Beilinson cohomology leads to a well-defined map: U : fl£(CYP,R(l)) x HUC\P,

R(l)) -> tf|>(C\P, R(2)).

Furthermore, by Hodge theory, there is a canonical short exact sequence H\C\P,R(1))

nF\C\P)

-> ij£(C\P,R(2)) -

Hl(C,R(2))

where F 1 denotes the F ^ t e r m of the Hodge filtration on H1^^, All this then leads to a well-defined morphism [;.}v : A 2 R[P]° -* Hl{C,R(2))

C).

= ^'(C^l))

which make the associated diagram coming from the above two short exact sequences commute. (See e.g. [Bel].) (5.2) Now applying the above construction with P being the union of the supports of Wi, Wm and Wn for m, n > 0. Thus for fixed m, n, in R[P]°, we get two elements div(/i )Tn ) and div(/i )7J ). This then gives [div(/i ;m ),div(/i ;n )]x> € if 1 (X, R ( l ) ) . Thus, by a simple argument using the Stokes formula, we obtain the following Lemma. For any holomorphic differential 1-form UJ on C, we have ([div(/i, m ),div(/i in )]2j,w) :

= -

2

rZl

J [div(/i, m ), div(/i,n)]x> A u

= -

2?r

/ = 1 / 5(div(/i, m ),z)rfflf(div(/i i „) ) z) Aw,

Here g(D,z) denotes the Green's function of D with respect to any fixed normalized (possibly singular) volume form of quasi-hyperbolic type. (See e.g., [Wei].)

Geo-Arithmetic; B. Non-Abelian

L-Functions

269

(5.3) With exactly the same notation as in (4.2), then in Hl(X, R(l)) we get a collection of elements [div(/i !Tn ), div(/i:„)]x> for m, n G Z>oDefinition. For a curve C defined over Z, define the - first quasiWeierstrass group W'_j(C) of C as the subgroup of Hl(X, R(l)) generated by [div(/ 1>m ), div(/iifl)]x> for all m, n G Z > 0 and call W 1 J ( C ) Q the - /ir.s£ Weiersirass group W_/(C) of C. That is to say, W_/(C) := ([div(/i, m ),div(/i >n )]x) : m,n G Z > 0 ) Q . Conjecture III. W_i(C) R = i f ^ R ^ ) ) . That is, Weierstrass divisors should give a new rational structure for iJ 1 (X, R(l)), and hence the corresponding regulator should give the leading coefficient of the L-function of C at s = 0, up to rationals.

B.2. New Non-Abelian Zeta Functions for Number Fields B.2.1. Iwasawa's ICM Talk on Dedekind Zeta Functions As for function fields, here we start with a discussion on abelian zeta functions for number fields, i.e., Dedekind zeta functions. However, we will not adopt the classical approach, rather we would like to recall Iwasawa's interpretation in our language. (Iwasawa's original choice of certain auxiliary functions do not naturally lead to any meaningful cohomology, we make some subtle changes here.) Let F be a number field. Denote by 5 the collection of all (unequivalent) normalized places of F. Set S^ to be the collection of all Archimedean places of F and 5fin := S\S'ooDenote 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: Tfl

={a = (av) e I : deg(a) = 0},

F* ={a = ( c ) G l ° : a , = a G F\{0},Vu G 5 } , U ={a = (av) E 1° : \av\v = lVu G 5 } , Ifln —{a = (av) e I : av = l\fv G Soo}, ={a = (o„) G I : av — 1W G 5fin}-

270

L. Weng

Set £/nn := £/TlIfin. 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) {/ <-> I is compact; (3) E/fin w Ifln is both open and compact. Moreover, the morphism I : [a = (av)] l—> 1(a) '•= Flues ^ induces an isomorphism between Ifin/Cfin and the ideal group of F, where Pv denotes the maximal ideal of the ring of integers OF corresponding to the place v; and, N(a) = Uves \av\vv:=[Fv'Qp] = ^ U ( a ) ) " 1 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 • aoo with afin € Ifln and doo S loo respectively. In particular, if dfi(a) denotes the normalized Haar measure on I as say in Weil's Basic Number Theory, we have d/j,(a) = dfj,(a&n)

•

d^a^)

corresponding to the decomposition I = I n n x I ^ . Set 1, if /(ofin) C OF; e(a fin ) := 0, otherwise, e(floo) = e x p ( -w^al v.K

-27r^|a„|2 J v.C

and e(a nn • aoo) = e(afin) • e(aoo), for afin G Ifin and a

oo € loo, Denote by A f (the absolute values of) the discriminant of F, r\ and r 1,

tF(s):=AU2^r(S-))ri((2n)-°T(s)Y2

£

N{1)-

Geo-Arithmetic: B. Non-Abelian L-Functions

271

0//COf

H\tv\svexp(-nJ2t2v-2n^2\tv\2)l[d*tv

f JtveFv,veSoo = A|:A

\

v N I

Y,

( )~"

V.-R

v.c

/

v

N(a00)se(a00)du.(a00)

f

^ ' v o l ( t /1f i n ) /

Ar(afin)se(afin)d/x(ofln) • /

•'Ifin

VOl(C/fin)

• /

iV(a 00 ) s e(a 00 )d^(a 00 )

•'loo

A2 (Ar(afin)iV(a00)J

[N(a)se(a)d^a).

=- j ^ n ' 4 ' VO^UfinJ

fe(a fin )e(a 00 )jfd/x(a fin )d/i(a 00 )J

Ji

Now denote by d//([a]) the induced Haar measure on the Picard group Pic(F) := l/F*. Note that ,

fe(aoo), 10,

A

if i"(afin) c C F , otherwise,

and that by (1) above, F* <-» I is discrete, (hence taking integration over F* means taking summation), we get

VOl(tffin).A/Vfr.(s) = /

( I

= f

(

N{aa)se{aa)da\d^{[a\)

£e(aa)Vtf([a])'d/i([a])

(by the product formula)

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L. Weng

N = / ([°])Sdv([a\) Jl/F*

= f

• ( Y\ e(aa fin )e(aooo) \aeF,

N([a]yd»([a}) • C

I/F

V

*

= [


Y,

N([a]yd»([a])-(

Y

Jl/F*

J

aeF*,/(aafln)cOF

c(««oo))-

\' aeF* ,aav cOv, VveSRn

For an idele L = (av), define the 0-th algebraic cohomology group of the idele L~l by H°(F,L-X)

:= {a G F*,aav C Ov,Vv G S fin } = / ( L " 1 ) ;

moreover for the associated idele class, introduce its associated geometric 0-th geo-ari cohomology via h\F,L~x) :=logf

expf -7r^|^a| 2 -27r^|^a| 2 J j .

Y ^aeH°(F,L-!)\{0}

^

v.R

v.C

' '

With this, then easily, we have

= /

ex

Y) WF

W)

-^Y\9v^\2-2irY\9va\2)

P[ V

^(w-W

«R

«C • N{L) dn{L) s

±.(eh°lF'L-1)

= [

-i)N(L)°dn(L) J

JPic(F) WF ^

= —• W

F

(eh°™-l)N(L)-ad»(L)

I 'Pic(F) V Jvic(F)

J

eh°{F,L) _ x

'Pic(F)

Aut(I)

{^"ML).

/

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273

Here wp denotes the number of units of F, and Aut(L)(= U>F) denotes the number of automorphisms of L. It is very clear that, formally, this version of (the completed) Dedekind zeta functions for number fields stands exactly the same as our interpretation of Artin zeta functions for function fields. So to introduce a non-abelian zeta function for number fields, the key points are as follows: (1) A suitable stability in terms of intersection for bundles over number fields should be introduced; (2) Stable bundles over number fields should form moduli spaces, over which there exist natural measures; and (3) There should be a geo-ari cohomology such that duality and Riemann-Roch type results hold. B.2.2. A Geo-Arithmetical Cohomology B.2.2.1. Cohomology Groups For the number field F, denote by A = Ap the ring of adeles of F, by GL r (A) the rank r general linear group over A, and write A := Afin © AQO and GL r (A) := GLr(A)fln x GLr(A)oo according to finite and infinite partition. Also for simplicity, we often use v (resp. a) to denote elements in Sfin (resp. Soo)For any g = (#fin : g^) = {gv;ga) € GL r (A), define an injective morphism i(g) := i{gj) : Fr -> A r by / t-> (f\ga • / ) . Let Fr(g) := Im(i(g)) and set

Ar(g) := (i) 9v{av) € Orv, Vu; (av;aa) 6 A r : (ii) 3 / <E Fr such that (ii.a) gv(f) € Orv, Vv;

(ii.b) (/; aa) =

i(9oo)(f)

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L. Weng

Then we have the following 9-diagram with exact columns and rows: 0 Ar(g)nFr(g) Fr(g) Fr(g)/Ar(g)nFr(g)

0 ~ >-» ~

0

0

Ar(g)

-»

Ar

-»

Ar/Ar(g)

-»

0

Ar(g)/Ar(g)

n Fr(g)

Ar/Fr(g) A'/A1" (c?) + F r ( 5 ) 0.

Guided by C.4 of this paper, which itself is motivated by Weil's adelic cohomology theory for divisors over algebraic curves, (see e.g, [S]), we introduce the following Definition. ([We6]) For any g € GL r (A), define its 0-th and 1-st geo-arithmetical cohomology groups by H°(AF,g)

:= Ar(g) n Fr(g),

and H\AF,g)

:= Ar/Ar(g)

+ Fr(g).

Proposition 1. ([We6]) (Serre Duality = Pontrjagin Duality) As locally compact groups,

ifere rep denotes an idelic dualizing element of F, and "• the Pontrjagin dual. In particular, H° is discrete and Hl is compact. Remark. For v E San, denote by dv the local different of Fv, the ^-completion of F, and by Ov the valuation ring with TTV a local parameter. Then dv =: jtf**^ . Ov. We call KF ~ (d°rdv{dv); 1) 6 IF := GLi(A) an idelic dualizing element of F. B.2.2.2 Geo-Arithmetical Counts Motivated by the Pontrjagin 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:

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275

Counting Axiom. / / # g a counts a certain class of locally compact groups G, then # g a (G) = # g a ( G ) . Practically, our counts of arithmetic cohomology groups are based on the 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 / := n « fv • ILr U, where (i) fv is the characteristic function of Orv; (ii.a) fa{xc) •— e-7rlx
:=#jH\AF,gyJ,dt) V

'

:= /

\f(x)\2dx;

:= /

|/(0|2#-

JW-{KF,g)

Here dx denotes (the restriction of) the standard Haar measure on A, d£, (the induced quotient measure from) the dual measure (with respect to x), and f the corresponding Fourier transform of f; (2) The 0-th and the 1-st geo-arithmetical cohomologies of g € GL r (A) are h°(AF,g) :=log ( # g a ( t f ° ( A F ,
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L. Weng

B.2.2.3. Serre Duality and Riemann-Roch For the arithmetic cohomologies just introduced, we have the following Theorem. ([We6]) For any g e GL r (F)\GL r (A), (1) (Serre Duality) / ^ ( A ^ g ) = h°(AF,KF g-1); (2) (Riemann-Roch Theorem) h°(AF,g)

- h^A^g)

Often, we also write eh^kF'^

= deg(g) - V- -log|A F |. := ^ a ( F , g ) , i = 1,2.

R e m a r k s . (1) Our work here is motivated by the works of Weil [W4], Tate [T], Iwasawa [Iwl,2], van der Geer-Schoof [GS], and Li [Li], as well as the works of Lang [LI,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 .^-functions as done in Tate's Thesis. But we later would take a substantially different approach. (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 SF x R to get a much more refined geo-arithmetical cohomology theory. To understand the above theory, we suggest the reader use the language of lattices (whose details will be recalled in the following section) : Denote by OF the ring of integers of F. Then for an C^-lattice A of rank r, its geo-arithmetical cohomology groups are defined to be the locally compact groups H°(F, A) := A Consequently, then

and

H\F, A) := (R ri x C 2 ) r / A .

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277

(i) H°(F,A) is discrete while Hl(F, A) is compact; (ii) The Serre duality coincides with the Pontrajin duality, say in the case F = Q,

((Mri xC r 2 )7A) V ~A; (iii) the 0-th cohomology is then simply given by

h°(F, A) := log (J2

ex

P ( - ^V^IJ] llxll, -

2TTV^1 £

||x||. >

a definition due to van der Geer and Schoof [GS]. B.2.3. Non-Abelian Zeta Functions B.2.3.1. Space of Ox-Lattices B.2.3.1.1. Projective OK-modules Let K be an algebraic number field. Denote by OK the ring of integers of K. Then an Ox-module M is called projective if there exists an Ox-module N such that M(&N is a free Ox-module. Easily, for a fractional ideal a, Pa : = Pr;a ;— Oj(

© CI

is a rank r projective Ox-module. The nice thing is that such types of projective Ox-modules, up to isomorphism, give all rank r projective O^-modules. Proposition. (1) For fractional ideals a and b, P r ; 0 ~ Pr;(, if and only if o ~ b; (2) For a rank r projective OK-module P, there exists a fractional ideal a such that P ~ Pa. B.2.3.1.2. Ox-Lattices Let a be an Archimedean place of K, and Ka be the cr-completion of K. It is well known that Ka is equal to either R, or C. Accordingly, we call a (to be) real or complex, write sometimes in terms of a : R or a : C accordingly. Recall also that a finite dimensional i^-vector space Va is called a metrized space if it is equipped with an inner product.

278

L. Weng

By definition, an Ox-lattice A consists of (1) a projective C^-module P = P(A) of finite rank; and (2) an inner product on the vector space Va := P ®oK Ka for each of the Archmidean place a of K. Set V = P (g>z M, then V — YlceS ^ » where S^ denotes the collection of all (inequivalent) Archimedean places of K. Indeed, this is a direct consequence of the fact that as a Z-module, an O^-ideal is of rank n = r\ + 2r 2 where n = [K : Q], ri denotes the number of real places and r 2 denotes the number of complex places (in Sx,). B.2.3.1.3. Space of (^-Lattices Let P be a rank r projective O^-module. Denote by GL(P) := Auto K (P). Let A := A(P) be the space of C^-lattices A whose underlying 0/f-module is P. For a € Soo, let Aa be the space of inner products on Va; if a basis is chosen for Va as a real or a complex vector space according to whether a is real or complex, Aa may be realized as an open set of a real or complex vector space. We have A = Ylcres ^-
•=

(A^1

• u, A~l

•

W)A,C-

This defines an action of GL(P) on A from the left. Clearly, then the map v — i > Av gives an isometry A = A • A of the lattices. (By an isometry here, we mean an isomorphism of C^-modules for the underlying C^-modules subjecting the condition that the isomorphism also keeps the inner product unchanged.) Conversely, suppose that A : A\ = A2 is an isometry of O^-lattices, each of which is in A. Then, A defines an element, also denoted by A, of GL(P). Clearly A2 = A-A\. Therefore, the orbit set GL(P)\A(P) can be regarded as the set of isometry classes of (^-lattices whose underlying C/f-modules a r e isomorphic to P.

Geo-Arithmetic: B. Non-Abelian

L-Functions

279

B.2.3.2. Geometric Truncation B.2.3.2.1. Semi-Stable Lattices Let A be an Ojtf-lattice with underlying O^-module P . Then any submodule Pi C P can be made into an (^-lattice by restricting the inner product on each Va to the subspace Vi><7 := Pi ®xKa. Call the resulting ©^-lattice Ai := A n Pi and write Ai C A. If moreover, P / P i is projective, we say that Ai is a sublattice of A. The orthogonal projections -Ka : Va —> V^~a to the orthogonal complement V^a of V\:a in Va provide isomorphisms (P/Pi) ®oK Ka ~ Vi^, which can be used to make P/P\ into an O^-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 of scalars which makes an 0K-lattice into a standard Z-lattice. Recall that V = A z R — Y[<j£S Vv Define an inner product on the real vector space V by (it, 10)00 := ^2(ucr, wa)a + ^2 Re(ua, <J:R

wa)a.


Let Res^/njA denote the Z-lattice obtained by equipped P , regarding as a Z-module, with this inner product (at the unique infinite place ooof Q). We let rk(A) denote the (D^-module rank of P (or of A), and define the Lebesgue volume of A, denoted by VolLeb(A), to be the (co)volume of the lattice Res^-/QA inside its inner product space V. Clearly, if P' is a submodule of finite index in P , then Vol Leb (A') = [P : P']Vol Leb (A), where A' = A n P is the lattice induced from P'. Examples. Take P = OK and for each place
280

L. Weng

where Ap denotes the absolute value of the discriminant of K. More generally, take P = a as a fractional idea of K and equip the same inner product as above on Va. Then a becomes an OKlattice a = (a, 1) in a natural way with rk(a) = 1. It is a well-known fact, see e.g., [Ne], that Vol L e b (a) = 2 " ^ • (N(a) • y/A~^j, where N(a) denote the norm of a. Due to the appearence of the factor 2 _ r 2 , we also define the canonical volume of A, denoted by Volcan(A) or simply by Vol(A), to be 2 r 2 rk(A) VolLeb ( A ) S o i n particular,

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(A!) rk ( A) > (resp. >) Vol(A)rk(Al>. Clearly the last inequality is equivalent to Vol Leb (A 1 ) rk ( A ) > Vol Leb (A) rk ( A i). So it does not matter which volume, the canonical one or the Lebesgue one, we use. Remark. The canonical measures has an advantage theoretically. For example, we have the following Arakelov-Riemann-Roch Formula: For an 0ft--lattice A of rank r, - log (vol(A)) = deg(A) - r- log AK. (For the reader who does not know the definition of the Arakelov degree, he or she may simply take this relation as the definition.)

Geo-Arithmetic: B. Non-Abelian

L-Functions

281

B.2.3.2.2. Canonical Filtration Based on stability, we may introduce a more general geometric truncation for the space of lattices. For this, we start with the following well-known Lemma. For a fixed O-lattice A, {Vol(Ai) : Ai c A} c R>0 is discrete and bounded from below. Sketch of 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. ([We2]) 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, Aj/Aj_i is semi-stable; and (ii) for allj =

l,...,s-l,

(VoKAj+i/Aj)]

> (VoKA./A^OJ

;

(2) (Jordan-Holder Filtration) / / moreover, A is semi-stable, then there exists a filtration of proper sublattices, 0 = A t + 1 c A* c • • • c A0 = A such that (i) for all k = 0 , . . . ,t, Afc/Afc+1 is stable; and (ii) for all I = 1 , . . . , £, /

,

, , , \rk(A'-VA')

(VO1(A7A' +1 ))

/

, ,

, N rk(A'/A'+ 1 )

= (VO1(A'-7A Z ))

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L. Weng

Furthermore, up to isometry, the lattice Gr(A) := ®tk-0Ak/Ak+1 uniquely determined by A.

is

Sketch of Proof. Existence is clear and the uniqueness of JordanHolder graded lattice is fairly standard. Let us look at the uniqueness of canonical filtration, an analogue of the well-known HarderNarasimhan filtration for vector bundles, a bit carefully: Existence of two such filtrations will lead to a contradiction by applying fact that for any two sublattices Ai, A2 of A, Vol(Ai/(Ai n A 2 )) > Vol((Ai + A 2 )/A 2 ) • B.2.3.2.3. C o m p a c t n e s s For a lattice A € A with the associated canonical filtration 0 = A0 C Ai C • • • C As = A define the associated canonical polygone p^ : [0, r] —> R by the following conditions: (1) p A ( 0 ) = p A ( r ) = 0; (2) pA is affine over the closed interval [rkAj,rkAj+i]; and (3) p A (rkAi) = deg(Ai) - rk(Ki) • ^ A ) . 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 Ox-lattice called A[T] by multiplying each of the inner products on A, or better, on Aa for a G S ^ , by T 2 . Obviously, the [reconstruction changes volumes of lattices in the following way Vol(A[T]) = Trk
Geo-Arithmetic: B. Non-Abelian

L-Functions

283

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 : pA < p > is compact in A. Sketch of 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] G A : pA < p \ is compact. B.2.3.2.4. Canonical Polygons and Geo-Arithmetical Truncation: Adelic Version Let X = Spec OF . If E is a vector sheaf of rank r over X, i.e, a locally free Op-sheaf of rank r, denote by Ep 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 G Sf := 5fin, set EQV •= H°(SpecOFV,E) a free CVmodule of rank r. In particular, we have a canonical isomorphism: can„ : Fv ®ov E0v ^ Fv ®F EF. Now if E is a vector sheaf of rank r over X equipped with a basis ap : Fr ~ Ep of its generic fiber and a basis otov • Orv ~ EQV for any v G Sf, the elements gv := (Fv ®p ap)-1 o can,, o (Fv ®ov otov) G GLr(Fv) for all v G Sf define an element
L. Weng

284

(E;aF o r _ 1 ; (aov ° kv)v&sf) onto rg&k. Therefore the above bijection induces a bijection between the set of isomorphism classes of vector sheaves of rank r on Spec OF and the double coset space GLr(F)\GLr(Af)/GLr(0F). More generally, let r = r\ + • • • + rs be a partition / = ( r i , . . . , rs) of r and let Pj be the corresponding standard parabolic subgroup of GLr. Then we have a natural bijection from the set of isomorphism classes of triple (E*;a*iF : (a*tov)veSf) onto Pr(A/), where E* := ((0) = Eo C £1 C • • • C £ s ) is a filtration of vector sheaves of rank (j*i,ri+7"2, • • • , r i + ^ H \-rs = r) over X, (i.e, each Ej is a vector sheaf of rank r\ + ri + • • • + Tj over X and each quotient Ej/Ej-i is torsion free,) which is equipped with an isomorphism of filtrations of F-vector spaces a„, F : ((0) = F 0 C F r i C • • • C im+r 2 +-+r.=r) „

(

^)

F )

and with an isomorphism of filtrations of free CVmodules

for every v G 5 / . Moreover this bijection induces a bijection between the set of isomorphism classes of the filtrations of vector sheaves of rank (r\,ri+r2, •• • , r\+r2 + - • -+rs = r) over X and the double coset space P / ( F ) \ P / ( A / ) / P / ( C ? F ) - The natural embedding Pj(A/) *-> Pr(A) (resp. canonical projections P/(A/) —> M/(A/) —» GLrj(Af) for j = 1 , . . . , s, where M/ denotes the standard Levi of P/) admits the modular interpretation (K*;a*„F : (a*,o„)i;eS/) ^ (Es;as,F

: (a S ) o„) u e s / )

(resp. (F*;a*,F : (a*,oJ«eS,) ^

(g r j( £! *);gr i (Q;*,F),grj(a;* i oJ^5 / ),

where gr^F*) := Ej/Ej-i, gVj(a^F) : Fri ~ g r ^ K ^ F and g r ^ a ^ o j : (D/-7' ~ g r ^ F * ) ^ , v E Sf are induced by a*,F and «*,£>„ respectively.) Moreover, any g = (gf,goo) G GLr(Af) x GLr(Aoo) = GLr(A) gives first a rank r vector sheaf Eg on Spec Op, which via the embedding EF <—> (R ri x C r 2 ) r gives a discrete subgroup, a free rank r

Geo-Arithmetic: B. Non-Abelian

L-Functions

285

d?p-module. In particular, #00 = (ga) then induces a natural metric on Eg by twisting the standard one on (R n x C r 2 ) r via the linear transformation induced from g^. As a direct consequence, see e.g, [L3],

deg(Eg,pg)

=

-log(N{detg))

with N : GL\(AF) ~ IF —* K>o the standard norm map of the idelic group of F. With this, for g = ( R by the following 3 conditions: (i)P S o (0)=P 9 Q (r) = 0; (ii) PQ is affine on the interval [v&xikEf_(,rankEf'^]; and (iii) for all indices i, p9Q(vankE['Q) = d e g ( £ f Q , p f Q ) - " ^ 1

.

deg(Eg,pg).

Then, there is a unique convex polygon p9, the canonical polygon associated with g, which bounds all p% from above for all parabolic subgroups Q for GLr. Moreover there exists a parabolic subgroup Q9 such that p j 9 = p 9 . Hence we have the following fundamental Main Lemma. For any fixed polygon p : [0, r] —* R and any d € R, the subset {g e GLr(F)\GLr(A)

: deg 5 = d,p> < p}

is compact. Similarly yet more generally, for a fixed parabolic subgroup P of GLr and g G GLr(A), there is a unique maximal element p9p among all PQ, where Q runs over all parabolic subgroups of GLr which are contained in P. And we have

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L. Weng

Main Lemma'. For any fixed polygon p : [0, r] —> R, d € R and any standard parabolic subgroup P ofGLr, the subset {g G GLr(F)\GLr(A)

: degg = d, fP < p,p% > -p}

is compact. B.2.3.3. Non-Abelian Zeta Functions for Number Fields For the number field F with (absolute value of) discriminant A^, denote by MF,V the moduli space of semi-stable 0^-lattices of rank r. Then MF,r : = U T G M + MF,r[T\ w h e r e MFAT\ •= MF,rQ°& T) d e " notes the volume T part. For an adele element g € GL r (A), denote its associated Op-lattice of rank r by II (g). Clearly, II factors through GL r (F)\GL r (A). Denote by MAF,r C GL r (F)\GL r (A) the inverse image of M.F,T- It is well known that fibers of II : M.AF,r —> M.F,r are all compact. Hence so is M hF,r\T\ '•— n _ 1 (A^F i r [T] J. Consequently, we obtain natural measures dfi on MF,V and -^AF,r[T] induced from that on GL r (A). In particular, being compact, MAF,r[AF] is of finite volume. For any A € MAF,T, set hz(F,A) :— hl(AF,g) where g is an element of GL r (A) whose associated O^-lattice coincides with A. This is well-defined since for any a G GL r (F), /i l (A^, a-g) = hl(AF,g) by definition. Indeed, as said earlier, h°(F, A) = log ( £

exp ( -

^xGA

^

TTV^I

J2 W*W* ~ W = l E H x 0 ) • cr:R

T:C

' '

Define the (completed) rank r non-abelian zeta function ^F,r(s) of F by frM

•= ( 4 V v

J

f JAeMF:r

(eh°{F'A) ~ l ) ( ^ S ) d 6 S ( A ) • dfi, Ms)

v

> I-

'

Following 2.1, i.e., Iwasawa's interpretation of Dedekind zeta functions, we see that £F,I(S) is essentially the completed Dedekind zeta function £F(S) for F. R e m a r k s . (1) The terms appeared above, such as degree, 0-th cohomology, moduli spaces, and the Tamagawa type measure are all

Geo-Arithmetic: B. Non-Abelian L-Functions

287

canonically associated to number fields. So non-abelian zeta functions are naturally there. By exposing them, we hope that then non-abelian arithmetic properties of number fields can be obtained systematically. (2) The map II : M.xF,r{d) —• M.F,r(d) may be understood as a kind of moment map. Thus non-abelian zetas may be understood as a kind of algebraic version of Feynman type integral. On the other hand, in Part A, we propose a Weil-Narasimhan-Seshadri type correspondence, a micro reciprocity law. So it is not unreasonable to expect that these zetas cab be written in terms of analytic Feynman type integrals, i.e., via global Galois representations. B.2.3.4. Basic Properties Just as Dedekind zeta functions, non-abelian zetas are canonical as well. Theorem. ([We6]) (1) Non-abelian zeta function

tFA') ••= (4)'

[

(eW>-l)(e-)de^.^

converges absolutely and uniformly when Re(s) > 1+5 for any 5 > 0. Moreover, £F I7 -(S) admits a unique meromorphic continuation to the whole complex s-plane; (2) The extended ^F,r(s) satisfies the functional equation ^F,r(s) = £F,T-(1 - s);

and

(3) The extended ^F,r{s) has two singularities, simple poles at s = 01 with Res s = 0 ^F,r(s) = Res s=0 £F,r(s) = Vol(yWF,r[A£]J. Remarks. (1) With respect to fixed real constants A, B, C, a and /3, introduce a formal zeta integration ZF,r;A,B,C;aAs) f JE€MAFtr

h

E

f(e °^ Y ^

:=

{1^1^)*• (edeg(£))^+C _

( e d e g ^ ^ W s ) . J

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L. Weng

Formally, from the duality and Riemann-Roch, we obtain a functional equation ZF,r;A,B,C;a,/3(s)

=

Z

F,r;A,B,C;a,0

A + 2C B

Moreover, with A > 0 and the compatibility conditions: a = B and P = C, these integrals are convergent. Thus, by a change of variables, we may further assume that B = 1 and C = 0 without loss of generality to get ZF,T;A(S) := ^F,r;i4,-i,0;-i,o(s)- This is a version of two variables non-abelian zetas, and £F,T(S) = ^F,r;i(s)In particular, the above theorem holds for ZF,r-,A(s),A > 0. (2) The global non-abelian invariant Vol (-Mjr,r [A|.] 1 for F can be calculated in terms of special values £ F ( 2 ) , £ F ( 3 ) , . . . , £F(»") °f the Dedekind zeta. For details, see B.5.6. B.2.4. Examples B.2.4.1. Epstein Zeta Functions and Non-Abelian Zeta Functions Recall that the rank r non-abelian zeta function £
(eh°^

- l ) • ( e - ) * * ^ d/*(A),

Re( S ) > 1.

Here as before, h°(Q, A) := log ( J ^ e x p ( - TT|X|2) ] , deg(A) = - l o g Vol(R r /A). \xeA ) Theorem', (i) £Q,I(S) coincides with the (completed) Riemann-zeta function; (ii) ^Q,r(s) can be meromorphically extended to the whole complex plane; (hi) £Q,T-(S) satisfies the functional equation £Q,T-(S) = £
Geo-Arithmetic: B. Non-Abelian

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289

Decomposite according to their volumes, Mq^r = UT>O-MQ,T[T], and there is a natural morphism M.qtr[T] —•> .MQ )T .[1], A w T r -A. Consequently, (eft°«-A)-l)-(e-)deg(A)^(A)

fc,r(*)=/

(efc°-iWA).

=fVf / JO

1

JMQ,r[l]

V

/

But h°(Q,T$ • A) = log ( E x e A e x P ( - ^ W 2 • T^)) 00

-

dT

1

„ '-""^T-i-^-nBi

/ Jo

and

forB

^°'

we h a v e n s ) = §-TT §*r(§ s ) - / ^ ^ ^ , ( E X G A \ { O } M r s ) ^Mi(A). Accordingly, introduce the completed Epstein zeta function for A by x£A\{0}

Proposition. (Eisenstein series and Zeta Functions) T

•(s) =

-l

1

'MQ,r[l] JMO.MI

E(k/-s)dm{k). l

B.2.4.2. Rankin-Selberg Method: An Example with SL2 Consider the action of SL(2, Z) on the upper half plane 7i. Then a standard 'fundamental domain' is given by D — {z = x + iy € 1i '• \x\ < 5 ) y > 0 ) a ; 2 + y 2 > ! } • Recall also the completed standard Eisenstein series E(z;s):=7r-Sr(s)-

V -t "£—&-. ^—' rn^ + n 2s (m,n)ez2\{(0,0)} ' '

Naturally, we are led to consider the integral jDE(z,s)J^fL. However, this integration diverges. Indeed, near the only cusp y — GO, E(z, s) has the Fourier expansion 00

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with an[V,S)

U(28)ya + Z(2-2s)y1-, \2|n|s-^1_2s(|n|)Vy^_l(27r|n|y),

ifn = 0; if n ± 0,

where £(s) is the completed Riemann zeta function, as{n) := ^2d\n ds, and Ks(y) over,

:= ± f™ e - ^ + i ) / 2 i s f is the K-Bessel function. More-

\Ks{y)\<e-y'2KKe{s){2),

if y > 4,

and

KS = K_S.

So an^o(j/)S) decay exponentially, and the problematic term comes from ao(y, s), which is of slow growth. Therefore, to make the original integration meaningful, we need to cut off 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 €

DT = {z = x + iy € D:y
Then D = DT U DT. Introduce a well-defined integration dxdy I^°(s):= f E(z,s) DT ' ' y2 JDT (b) Analytic Truncation Define a truncated Eisenstein series ET(Z;S) £T(,;S):=(^S)' \E(z,3)~a0(y,3),

tt

"*T> ily>T.

Introduce a well-defined integration l£™(s):=

[ JD

ET{z-s)

by

dxdy y2

D:y>T}.

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With this, from the Rankin-Selberg method, one checks that we have the following: Proposition. Rank 2) I^°(s)

(Analytic Truncation = Geometric Truncation in

= £ ( 2 s ) ^ - - £(2* - 1 ) — = l£™(s). S

X.

S

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 TC 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 idealistic if we could at the same time keep the Eisenstein series unchanged, while offer some integration domains which appear naturally in certain moduli problems. Guided by this, in B.3, we will introduce non-abelian L-functions using integrations of Eisenstein series over generalized moduli spaces. (c) Algebraic Truncation Now we explain why the above discussion and Rankin-Selberg method have anything to do with our non-abelian zeta functions. For this, we introduce yet another truncation, the algebraic one. 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 G A such that its length gives the first Minkowski minimum Ai of A. Then via rotation, we may assume that xi = (Ai,0). Further, from the reduction theory -^A may be viewed as the lattice of the volume A^2 = yo which is generated by (1,0) and UJ — XQ + iyo € D. That is to say, the points in DT are in one-to-one correspondence with the rank two lattices of volume one whose first Minkowski minimum A^2 < T, i.e, Ai > T~2. Set M.Q \ [1] be the moduli space of rank 2 lattices A of volume 1 over Q whose sublattices Ai of rank 1 have degrees < \ logT. As a direct consequence, we have the following

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Proposition. (Geometric Truncation = Algebraic Truncation) There is a natural one-to-one, onto morphism M$l°sT[l]

~ DT.

In particular, M^2[l]

=

MQ,2[l]^D1.

Consequently, we have the following Example in Rank 2. £Q, 2 (S) = ~r

-

^~L_

Remark. This explicit expression proves to be very useful. (i) Using it, Suzuki and Lagarias show that all zeros of £Q,2(S) lie on the critical line Re(s) = 1/2; (For details, see C.4.) (ii) Special values C,{2n) and ((2n — 1) of the Riemann zeta function are naturally related by our rank two zeta functions. As Eisenstein series may be studied independently, our formula indicates that nonabelian zetas could be used to understand abelian zetas; and (iii) The volume of DT may be evaluated from this formula via a residue argument; Moreover, the volume of D1 also have an interpretation in terms of the special values of the Riemann zeta.

B.3. Non-Abelian L-Functions for Number Fields B.3.1. Automorphic Forms and Eisenstein Series To facilitate our ensuing discussion, we make the following preparation. For details, see e.g., [MW] and [We8]. Fix a connected reduction group G defined over F, denote by ZQ its center. Fix a minimal parabolic subgroup Fo of G. Then PQ = MQIIQ, where as usual we fix once and for all the Levi Mo and the unipotent radical UQ. A parabolic subgroup P is G is called standard if P D Fo- For such groups write P = MU with Mo C M the standard Levi and U the unipotent radical. Denote by Rat(M) the group of rational characters of M, i.e, the morphism M —> Gm where Gm denotes the multiplicative group. Set a*M := Rat(M) ®% C,

aM •= Hom z (Rat(M), C),

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and R e a ^ := Rat(M) ®% R,

Rea M := Hom z (Rat(M), R).

For any x € Rat(M), we obtain a (real) character \x\ • M(A) —> R* defined by m = (mv) — i > m)x< := ri^esl 77 ^!*" with | • |„ the vabsolute values. Set then M(A) 1 := n x G R a t ( M )Ker|x|, which is a normal subgroup of M(A). Set XM to be the group of complex characters which are trivial on M(A) 1 . Denote by HM '•= l°gM : M(A) —» a M the map such that Vx G Rat(M) c a^, (x, log M (m)) := log(m' x '). Clearly, M(A) 1 = Ker(log M );

log M (M(A)/M(A) 1 ) ~ Rea M -

Hence in particular there is a natural isomorphism K : aM ~ A"A/. Set Re^M := /t(Rea^f), I m X M := «(z • Reo^). Moreover define our working space X ^ to be the subgroup of XM consisting of complex characters of M(A)/M(A) 1 which are trivial o n ZG{h) •

Fix a maximal compact subgroup K such that for all standard parabolic subgroups P = MU as above, P(A)nK = M(A)PiIK-[/(A)n K. Hence we get the Langlands decomposition G(A) = M(A) • U(A) • K. Denote by mP : G(A) -»• M(A)/M(A) 1 the map g = m-n-k^ 1 M(A) • m where g G G(A),m G M(A), n G U{A) and fe G K. Fix Haar measures on Mo (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 compact.) (2) the induced measure on Uo(F) is the counting measure and the volume of U(F)\UQ(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 log M to the spaces a-M0, aM , 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

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ZM0 of Mo, then / »

/

/(mn*)**™-*.*,

J M0(A)-U0(A)-K

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 &M, a*Mi -P(A), etc, for all parabolic subgroups P. In particular, one checks that the following compatibility condition holds f(mnk)dkdnm~2podm

/ JMo(A)-J/0(A)-K JM{A)-U(A)-K

for all continuous functions / with compact supports on G(A), where PP denotes the half of the sum of the 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 = Ap 0 . Fix an isomorphism T 0 ~ G%- Embed R+ by the map t i-> (1;£). Then we obtain a natural injection (R+) fl <-^ To(A) which splits. Denote by AMO(A) the unique connected subgroup of TQ(A) which projects onto (IR^)^. More generally, for a standard parabolic subgroup P = MU, set AM(A) '•= ^Mo(A) H ZM(A) where as used above Z* denotes the center of the group *. Clearly, M(A) = J4M(A) • M(A) 1 . For later use, set also A ^ ( A ) := {a 6 ^M(A) : l°gG a = 0}Then AM{A) = AG{A) © A%{A). Note that K, M ( F ) \ M ( A ) 1 and U(F)\U(A) are all compact, thus with the Langlands decomposition G(A) = U(A)M(A)K in mind, the reduction theory for G(F)\G(A) or more generally P(F)\G(A) is reduced to that for AM(A) s m c e ZQ{F) n -^G(A)\^G(A) H G(A) 1 is compact as well. As such for to € M 0 (A) set ^M0(A)(*o) := {a 6 AMo{A)

: aa > t^Va € A 0 } .

Then, for a fixed compact subset w C Po(A), we have the corresponding Siegel set S(u;t0)

:= {p-a-k

:p€uj,a€

^M0(A)(*o),fc € K}.

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In particular, for big enough ui and small enough to, i.e, tg *S v e r y close to 0 for all a G An, the classical reduction theory may be restated as G(A) = G(F) • S(ui;to). More generally set A

M0(A)(*o) := {a G AMo{A)

: aa > ffla G Ap},

and Sp{u;tQ)

:= {p-a-k:p£u,aE

A^ o(A) (t 0 ),fc G K}.

Then similarly as above for big enough u and small enough to, G(A) = P ( F ) • 5 p (w; to). (Here A^ denotes the set of positive roots for ( P 0 n M , To).) Fix an embedding IQ : G <—• 5 L n sending g to (<7JJ). Introducing a height function on G(A) by setting ||g|| := nueSsuP{ISijlt> '• ^hj}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 G R such that \f(g)\ < c • \\g\\r for all g € 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 G M, A G KeXM0 such that for any a G Aw (A) > k G K, m G M(A)lr\Sp(u;t0), \f(amk)\

< c • ||a|| r • mp0(m)

.

Also a function / : G(A) —> C is said to be smooth if for any g = 9f9oo 6 G(Af) x G(Aoo), there exist open neighborhoods V* of g* in G(A) and a G°°-function / ' : V^ -» C such that / ( ^ • <4) = / ' ( ^ J for all ^ G Vf and p^, G V^. By contrast, a function / : S(ui;to) —• C is said to be rapidly decreasing if there exists r > 0 and for all A G Re.Xjvf0 there exists c > 0 such that for a G ^4M(A)>3 G G(A) 1 n S(u;t0), \4>{ag)\ < c • \\a\\ • mp0 {g)x. And a function / : G(F)\G(A) -» C is said to be rapidly decreasing if /|s( w;to ) is so. By definition, a function 0 : U(A)M(F)\G(A) -> C is called automorphic if

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(i) 4> has moderate growth; (ii) 4> is smooth; (iii) is K-finite, i.e, the C-span of all fak\ • * • A^) parametrized by (fci, fo) £ I x K is finite dimensional; and (iv) (f> is 3-finite, i.e, the C-span of all 5{X)(f) parametrized by all X € 3 is finite dimensional. Here 3 denotes the center of the universal enveloping algebra u := U(LieG(A00)) of the Lie algebra of G(Aoo) and S(X) denotes the derivative of along X. For such a function G ^4(C/(A)M(F)\G(A)) is called a cusp form if for any standard parabolic subgroup P' properly contained in P, = 0. Denote by A0(U(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 (•, •) : A0(U(A)M(F)\G(A))

x A(U(A)M(F)\G{A))

-> C

defined by (,) := JZMWU(A)M(F)\G(A) ^(g)4>(g)dg. Moreover, for a (complex) character £ : ^M(A) ~* C* of ^M(A) A(U(A)M(F)\G(A))z

se

:= { 6 A([/(A)M(F)\G(A)) :


e G(A)}

and Ao(U(A)M(F)\G(A))s := A 0 ([/(A)M(F)\G(A)) n A(L/(A)M(F)\G(A)) e . Set now A(U(A)M(F)\G(A))Z

:=

£ A([/(A)M(F)\G(A))^ £eHom(ZM(A),C*)

*

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297

and AQ(U(A)M(F)\G(A))Z

:=

£

A0(U(A)M(F)\G(A))^.

£€Hom(ZM(A),C*)

One checks that the natural morphism C[Rea M ] ® A(t/(A)M(F)\G(A)) Z -> A(tf (A)Af (F)\G(A)) denned by (Q, ^») — i > (^ — i >• Q(logM(mp(g))) • (f>(g) is an isomorphism, using the special structure of Aw (A) -finite functions and the Fourier analysis over the compact space ^Af(A)\^M(A)- Consequently, we also obtain a natural isomorphism C[Rea M ] ® A0(U(A)M(F)\G(A))Z

->

A0(U(A)M(F)\G(A))i.

Set also IIo(M(A))£ be isomorphism classes of irreducible representations of M(A) occurring in the space Ao(M(F)\M(A))^, and n 0 (M(A) := Ue6Hom(zM(A),c*)no(M(A))?. (More precisely, we should use M(A/) x (M(A)nK, Lie(M(Aoo)) R C)) instead of M(A).) For any TT € n 0 (M(A)) 4 set v4 0 (M(F)\M(A) 7r to be the isotypic component of type TT of AQ(M(F)\M(A)^, i.e, the set of cusp forms of M(A) generating a semi-simple isotypic M(A/) x (M(A) fl K, Lie(M(Aoo)) ® R C))-module of type TT. Set ^ 0 (f/(A)M(F)\G'(A)) 7r := { e A 0 (t/(A)M(F)\G(A)) : <\>k E AQ(M(F)\M(A))^k

e X}.

Clearly A0(U(A)M(F)\G(A))^

= ©^n 0 (M(A))^o(C/(A)M(F)\G(A)) 7r .

More generally, let V C A(M(F)\M(A)) be an irreducible M(A/) x (M(A)nK, Lie(M(Aoo)) ®RRC)). Then we call ir0 an automorphic representation of M(A). Denote by A(M(F)\M(A)n{

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the isotypic subquotient module of type 7To of A(M(F)\M(A). checks that V <8> Hom M ( A / ) x ( M ( A ) n K i L i e ( M ( A o o ) ) ( g ) R C ) ) (V',

One

A(M(F)\M(A)))

~A(M(F)\M(A)) W 0 . Set A(U(A)M(F)\G(A))7rQ := {0 G A(U(A)M(F)\G(A))

: <£fe G A(M(F)\M(A)) W 0 , VA: G K}.

Moreover if A(M(F)\M(A)) 7 r o C AQ(M(F)\M(A)), we call TT0 a cuspidal representation. Two automorphic representations 7r and 7To of M(A) are said to be equivalent if there exists A G X^ such that 7r ~ 7To® A. This, in practice, means that A(M(F)\M(A))n = A • A(M(F)\M(A))7Vo. That is for any wo(m). Consequently, A(C^(A)M(F)\G(A)) T = (A o mP) •

A(U(A)M(F)\G(A))no.

Denote by ^3 := [no] the equivalence class of IT®. Then ty is an X^-principal homogeneous space, hence admits a natural complex structure. Usually we call (M, *}3) a cuspidal datum of G if 7ro is cuspidal. Also for IT G ^3 set Re7r := Rex^ = Iwl € ReXjvf, where Xn is the central character of w, and Im7r := IT ® (—Re7r). Now fix an irreducible automorphic representation it of M(A) and an automorphic form 4> G J 4(l7(A)M(i ? )\G(A)) 7r , define the associated Eisenstein series E(<j>, IT) : G(F)\G(A) —> C by E(<j>,n)(g):=

]£

<j>(8g).

6eP(F)\G{F)

Then one checks that there is an open cone C C ReXj^ such that if Re7r € C, E(X • X)(g) converges uniformly for g in a compact subset of G(A) and A in an open neighborhood of 0 in X^- For example, if ^3 = [IT] is cuspidal, we may even take C to be the cone {A G ReXj^ : (A — pp, a v ) > 0, Va G Ap}. As a direct consequence, then E((j>,7r) G A(G(F)\G(A)). That is, it is an automorphic form.

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299

B.3.2. Non-Abelian L-Functions 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 2.3.2, we obtain compact moduli spaces M%[tfp]

:= {g E GLr(F)\GLr(A)

: d e g 5 = 0,j? < p}

for a fixed convex polygon p : [0,r] —• R. For example, .Ms°[l] = (the adelic inverse image of) the moduli space of rank r semi-stable Z-lattices of volume 1. More generally, for the standard parabolic subgroup P of GLr, we introduce the moduli spaces A^Q ]7 .[1],

KflAh

•= {9 e P(F)\GLr(A)

:degg = 0,p9P < p,p9P >

-p}.

By 2.3.2, these moduli spaces Mp^p[AF] are all compact. As usual, we fix the minimal parabolic subgroup PQ corresponding to the partition ( 1 , . . . , 1) with Mo consisting of diagonal matrices. Then P = Pj = UIMJ corresponds to a certain partition I = ( n , . . . ,T\p\) of r with Mi the standard Levi and Ui the unipotent radical. Now for a fixed irreducible automorphic representation n of M/(A), choose <$> €

A(Ui(A)MI(F)\G(A))7,nL2(UI(A)MI(F)\G(A))

:=A2(UI(A)MI(F)\G(A))7n with L2(Ui(A)Mi(F)\G(A)) the space of L2-functions on the quotient space Z G ( A )[//(A)M/(F)\G(A). Denote the associated Eisenstein series by E{(j>, TV) e A(G{F)\G{A)). Then, we define the rank r non-abelian L-function LpPr((f),ir) for the number field F associated to the L 2 -automorphic form e A2(UI(A)MI(F)\G{A))7r to be the

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integration Lfr(ct>,n) := [

T

E(cf>,7r)(g)dg,

Reyr G C.

More generally, for any standard parabolic subgroup Pj = UJMJ Z> Pi (SO that the partition J is a refinement of 7), we obtain a relative Eisenstein series £/(, n)(g):=

£

<(>(6g),

Mg €

Pj(F)\G(A).

6ePi(F)\Pj(F)

There is an open cone Cj in ReX M J s.t. if Re7r € C/, then £/(, 7r) € A(Pj(F)\G(A)), where X j g is defined similarly as XM with G replaced by P j . As such, we are able to define the associated nonabelian L-function by

L^ftf, n):= [

r

EJW, *)(g) dg,

Ren € CJ.

Remark. Here when defining non-abelian L-functions we assume that 4> comes from a single irreducible automorphic representations. But this restriction is rather artificial and can be removed easily: such a restriction only serves the purpose of giving the constructions and results in a very neat form. We end this section by pointing out that the discussion for rank r non-abelian L-functions Lfpr holds for general non-abelian L-functions L i / r ' _ p as well. So from now on, we will leave the discussion on general L i / r '- p to the reader, while we concentrate ourselves on the special case LpPr. B.3.3. Meromorphic Extension and Functional Equations With the same notation as above, set ^3 = [71-]. For w € W the Weyl group of G = GLr, fix once and for all representative w e G(F) of w. Set M' := wMw~l and denote the associated parabolic subgroup by P' = U'M'. W acts naturally on the automorphic representations, from which we obtain an equivalence classes w^ of automorphic representations of M'(A). As usual, define the associated

Geo-Arithmetic: B. Non-Abelian

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301

intertwining operator M(w, TT) by (M(w,ir)){g) := f

1

^(W1n''g)dn!',

V3 G G(A).

JU'{F)rnvU{F)w- \U'{A)

One checks that if (Re7r, a v ) > 0, Va € A^, (i) for a fixed automorphic 0, M(io, 7r)0 depends only on the double coset M'(F)wM(F). So M(w,ir)<j) is well-defined for w € W; (ii) the above integral converges absolutely and uniformly for g varying in a compact subset of G(A); (hi) M(w, ir)4> € A(U'(A)M'(F)\G(A))W7T; and if is L 2 , which from now on we always assume, so is M(w, ir)<j>. Basic Facts of Non-Abelian L-Functions. ([We6]) (I) (Meromorphic Continuation) LpPr(<j>,ir) for Re7r G C is welldefined and admits a unique meromorphic continuation to the whole space ty; (II) (Functional Equation) As meromorphic functions on ^3, L%{4>, TT) = Lfpr(M(w,7r)0,

WIT),

VW e W.

This is a direct consequence of the fundamental results of Langlands on Eisenstein series and spectrum decompositions and explains why only L 2 -automorphic forms are used in the definition of non-abelian LB. (See e.g, [Arl], [Lai], [MW] and/or [We8]). B.3.4. Holomorphicity and Singularities Let TT £ %S and a € A ^ . Define the function h : *P —• C by TT A t-> (A, a v ),VA G X^j — a^. Here as usual, a v denotes the coroot associated to a. Set iJ := {TT' £

Z>o zero almost everywhere such that 7r' I—> (ji-HeDhH^')711'^) • fW) is holomorphic at 7r';

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(ii) the singularities of / are said to be without multiplicity at n if n« € {0,1}; (iii) D is said to be locally finite, if for any compact subset C Cty, {H ED-.HnC / 0} is finite. Basic Facts of Non-Abelian L-Functions. ([We6]) (III) (Holomorphicity) (i) When Reir € C, LpPr(,Tr) is holomorphic; (ii) LfPr(,Tr) is holomorphic at ir where Keir — 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 of LjPr (, TT) are carried out by D; (ii) The singularities of LjPr(,ix) are without multiplicities at TT if (Re7r,a v ) > 0 , V a € A ^ ; (iii) There are only finitely many of singular hyperplanes ofLpPr((j), TT) which intersect {TT € ^3 : (Re7r,a v ) > 0,Vo; € A ^ } . 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 [We8]).

B.4. Geometric and Analytic Truncations: A Bridge B.4.1. Geometric Truncation: Revised B.4.1.1. Slopes, Canonical Filtrations Following Lafforgue [Laf], we call an abelian category A together with two additive morphisms rk : A -+ N,

deg : A -> E

a category with slope structure. In particular, for non-zero A £ A, (1) define the slope of A by fi(A) := ^ g ? ; (2) If 0 = A) C Ai c • • • C A[ = A is a filtration of A in A with rk(j4o) < rk(^4i) < • • • < vk(Ai), define the associated polygon to be the function [0, rk^4] —> M. such that (i) its values at 0 and rk(^4) are 0;

Geo-Arithmetic: B. Non-Abelian

L-Functions

(ii) it is affine on the intervals [rk(^4j_i), rk(-Aj)] with slope

303

/j,(Ai/Ai-i)

(3) If a is a collection of subobjects of A in A, then a is said to be nice if (i) a is stable under intersection and finite summation; (ii) a is Noetherian, i.e., every increasing chain of elements in a has a maximal element in o; (iii) if A\ G a then A\ ± 0 if and only if rk(-Ai) ^ 0; and (iv) for Ai,A2 G a with rk(Ai) = rk(^42)- Then A\ C A2 is proper implies that deg(Ai) < deg(A2); (4) For any nice o, set fj,+ (A) :=sup{ M (^i) : Ax e o,rk(^i) > l } , li-{A) :=inf{ M (^/Ai) : Ax € o,rk(Ai) < rk(A)}. Then we say (A, a) is semi-stable if n+(A) = n(A) = n~(A). over if rk(^4) = 0, set also fJ-+(A) = — oo and fi~(A) — +oo.

More-

Proposition 1. ([Laf]) Let A be a category with slope structure, A an object in A and a a nice family of subobjects of A in A. Then (1) (Canonical Filtration) A admits a unique filtration 0 = AQ C AI 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 (iii) ii(Ai/Ao) > n(A~2/Ai >•••> ii(Ak/~A~k-i); (2) (Boundness) All polygons of filtrations of A with elements in a are bounded from above by p, where p := pA is the associated polygon for the canonical filtration in (1); (3) For any Ai G a,rk(Ai) > 1 implies fi(Ai) < fi(A) + g( r ffiffl; (4) The polygon p is convex with maximal slope fi+(A) — fi(A) and minimal slope ^~(A) — fJ>(A); (5) If (A1, a') is another pair, and u : A —> A' is a homomorphism such that Ker(u) G a and lm(u) G a'. Then yT(A) > n+{A') implies that u = 0.

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This results from a Harder-Narasimhan type nitration consideration. A detailed proof may be found at pp. 87-88 in [Laf]. As an example, we have the following Proposition 2. ([We6]) Let F be a number field. Then (1) the abelian category of hermitian vector sheaves on SpecOp together with the natural rank and the Arakelov degree is a category with slopes; (2) For any hermitian vector sheaf (E, p), a consisting of pairs {E\, p\) with Ei sub vector sheaves of E and p\ the restrictions of p, forms a nice family. Indeed, (1) is obvious, while (2) is a direct consequence of the following standard facts: (i) For a fixed (E,p), <deg(Ei, pi) : (E\,pi) € o > is discrete subset of R; and (ii) for any two sublattices Ai, A2 of A, Vol(Ai/(A 1 nA 2 )) >Vol((Ai + A 2 )/A 2 ). Thus we get the canonical filtration of Harder-Narasimhan type for hermitian vector sheaves over Spec Op. Recall that hermitian vector sheaves over Spec OF are Op-lattices in (R n x c r 2 ) r = r k ( £ ) in the language of Arakelov theory: Say, corresponding 0p-lattices are induced from their H° via the natural embedding Fr •=—• (R ri x C r 2 ) r where r\ (resp. r 2 ) denotes the real (resp. complex) embeddings of F. B.4.1.2. Micro-Global Relation for Geo-Ari Truncations 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 MicroGlobal Principle holds. (Recall that in geometry, for vector bundles, intersection stability and GIT stability are equivalent. But we do

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not know whether this is true say in geometry for manifolds and in arithmetic for lattices. It is our belief that intersection stability in application should be equally important. Say in the case for manifolds it claims that if a manifold is stable in terms of intersection, then certain kinds of submanifolds should not exist, a far better statement than merely claiming that the Hilbert or Chow points of the manifold in question are GIT stability. This is why we when working in geometry always try to convince the colleagues there to find a proper definition for intersection stability. Surely, we also understand that for theoretical purpose such as building up the moduli GIT stability plays a central role.) Let A = A9 be a rank r lattice associated to g e GL r (A) and P a parabolic subgroup. Denote the sublattices filtration associated to Pby 0 = A0 C Ai c A2 c • • • C A| P | = A. Assume that P corresponds to the partition J = (di, d^, • • •, dn=.\P\). Consequently, we have rk(Aj) = 7-j := di + d2 H

h dt,

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 p, if qifi) — p(ri) > 0 for all z = l , . . . , | P | — 1. Introduce also the characteristic function l(p* < p)

10,

otherwise.

Recall that for a parabolic subgroup P, p9p denotes the polygon induced by P for (the lattice corresponding to) the element g £ G(A). Fundamental Relation. ([We9]) For a fixed convex polygon p : [0, r] —• K such that p(0) = p(r) — 0, we have

I&
E P: standard parabolic

(-1)|p|_1

E

HP5P9>PP)-

5eP(F)\G(F)

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This is an analogue for lattices of a result of Lafforgue [La] for vector bundles. B.4.2. Arthur's Analytic Truncation B.4.2.1. 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 Po of G with its unipotent radical No = Np0 and fix a P-Levi subgroup Mo = MpQ of Po so as to have a Levi decomposition PQ = MoiVo. An P-parabolic subgroup P is called standard if it contains PQ. For such parabolic subgroups P , there exists a unique Levi subgroup M — Mp containing Ma which we call the standard Levi subgroup of P . Let N = Np be the unipotent radical. Let us agree to use the term parabolic subgroups and Levi subgroups to denote standard P-parabolic subgroups and standard Levi subgroups respectively, unless otherwise is stated. Let P 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 dp := X*(Tp) R and denote its real dimension by rf(P), where X*(T) is the lattice of 1-parameter subgroups in the torus T. Then it is known that aP = X*(T'p) R as well. The two descriptions of ap show that if Q C P is a parabolic subgroup, then there is a canonical injection ap <^-> OQ and a natural surjection &Q -» op. We thus obtain a canonical decomposition OQ = <*Q © ap for a certain subspace a^ of OQ. In particular, a© is a summand of o = ap for all P . Set ap := OP/OG and OQ := dg/oc. Then we have *Q = *Q © dp and ap is canonically identified as a subspace of OQ. Set ao := ap0 and OQ" = cip then we also have ao = a^ © ap for all P . B.4.2.2. Logarithmic M a p For a real vector space V, write V* its dual space over R. Then dually we have the spaces OQ, ap, ( a f )

and hence the decomposi-

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tions «•$= ( o o ) * © ( o Q ) * © a p So ap = X(MP)®R with X ( M P ) the group Hom F (.Mp, GL(1)) i.e., collection of characters on Afp. It is known that op = X(Ap) ® K 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 ap also naturally induce an injection aP <—> a*Q and a surjection OQ -» op, compactible with the decomposition

a£=(ag)*0ap. Every x = ^2siXi m a p c : = a p ® ^ determines a morphism P(A) —» C* by p H-> p x := IllXi(p)| Si - Consequently, we have a natural logarithmic map Hp : -P(A) —> op defined by (HP(p),x)=Px,

Vxeop. 1

The kernel of # p is denoted by P(A) and we set Mp(A) 1 := P(A) 1 n M P (A). Let also A+ be the set of a 6 ^4p(A) such that (1) av = 1 for all finite places v oi F; and (2) x(a, and define the dual basis of Ap by Ap.

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More generally, if Q c P , write A Q to denote the subset a € A Q appearing in the action of TQ in the unipotent radical of Q n Mp. (Indeed, MpflQ is a parabolic subgroup of Mp with nilpotent radical -/VQ := NQC\MP. Thus A Q is simply the set of roots of the parabolic subgroup (MpC\Q, AQ). And one checks that the map P <—>• A Q gives a natural bijection between parabolic subgroups P containing Q and subsets of A Q . ) Then dp is the subspace of CIQ annihilated by A Q . Denote by (A v )g the dual of Ag. Let (Ag) v := | a v : a € A g } and denote by A Q the dual of (Ag) v . B.4.2.4. Positive Cone and Positive Chamber Let Q C P be two parabolic subgroups of G. We extend the linear functionals in A Q and A Q to elements of the dual space OQ by means of the canonical projection from oo to CIQ given by the decomposition do = (*Q © OQ © op. Let TQ be the characteristic function of the positive chamber { t f e a o : (a,H) > 0 Va G A ^ } = c$ © [H € ag : (a, fl") > 0 for all a € A g } © o P and let TQ be the characteristic function of the positive cone { f f e a o : (w,H) > 0 V t u € A g } = a? © hi € og : (zo, H)>0

for all ro G A § } © a P .

Note that elements in A Q are non-negative linear combinations of elements in A Q , we have

B.4.2.5. Partial Truncation and First Estimations Denote Tp and fp simply by Tp and rp. Basic Estimation. (Arthur) Suppose that we are given a parabolic subgroup P, and a Euclidean norm \\ • \\ on op. Then there are

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constants c and N such that for all x € G(A) 1 and X € ap,

J2

TP(H{SX) ~X)<

c(||x||e"xll)iV.

6eP(F)\G(F)

Moreover, the sum is finite. As a direct consequence, we have the following Corollary. Suppose that T E OQ and N > 0. Then there exist constants c' and N' such that for any function on P(F)\G(A)1, and x, y € G(A) 1 ,

£

\(6x)\-TP(H(6x)-H(y)-x)

8€P(F)\G(F)

is bounded by c'\\x\\N' • \\y\\N' • sup ueG(A)

(\4>(u)\.\\u\\-N). 1 v

'

B.4.2.6. Langlands' Combinatorial Lemma In this section, following Arthur, we recall Langlands' combinatorial lemma. If Pi C P 2 , following Arthur [Ar2], set ol(H) := o% :=

J2

(~l)dim{A3/A2)rhH)-r3(H),

P3:P20P2

for H € do- Then we have L e m m a 1. If Pi C P2, o\ is a characteristic function of the subset of H € 01 such that (i) a(H) > 0 for all a E Aj; (ii) a{H) < 0 for all a G Ai\A?; and (hi) w(H) > 0 for all w € A 2 . As a special case, with P\ = P%, we get the following important consequence:

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Langlands' Combinatorial Lemma. If Q C P are parabolic subgroups, then for all H € do,

(-l) dim(ARMp) r£(tf)f£(tf) =5QP;

E R-.QcRcP

(-^)dim{AQ/AR)r§(H)ri(H)

£

=6QP.

R-.QcRcP

We next give an application of this combinatorial lemma, which will be used in 4.2.8. Suppose that Q C P are parabolic subgroups. Fix a vector A G O Q . Let 4 ( A ) := (_l)#{«eA5:A(aV)<0}j and let 4>%(A,H),

tfeao,

be the characteristic function of the set f„

{^

Gao:

w(H)>0, TOU)<0;

ifA(av)<0w

A P

1

ifA(aV);0'VaeA£}-

L e m m a 2. With the same notation as above,

B.4.2.7. 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 a; is a compact subset of No(A)Mo(A)1 and that To € — OQ. For any parabolic subgroup Pi, introduce the associated Siegel set sPl(To,u>) as the collection of pak,

p £ w , o £ A0(R)°,

k € K,

where a(Ho(a) — To) is positive for each a e A j . Then from classical reduction theory, we conclude that for sufficiently big UJ and sufficiently small T0, G(A) = Pi(F)s P l (T 0 ,w).

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Suppose now that Pi is given. Let $Pl (T0, T, u>) be the set of x in s Pl (7b,u;) such that W(HQ(X) - TJ < 0 for each w e Aj. Let FPl(x,T) := Fl(x,T) be the characteristic function of the set of a; € (7(A) such that
A?uP2:=APinMP2{k)\

Then the logarithmic map Hp1 maps .Af°2 isomorphically onto of, the orthogonal complement of 02 in ai. If To and T are points in 00, set Af2(T0,T) to be the set {a E A^2 : a ( ^ i ( a ) - r ) > 0, a € A?; w ( # 1 ( a ) - r ) < 0, 07 G A ? } , where Af := Ap in M 2 a n ( I Af := Ap in M 2 . In particular, for To such that —To is suitably regular, F(x, T) is the characteristic function of the compact subset of G(F)\G(A)1 obtained by projecting AT0(A) • Mo(A) 1 • Af0tG(T0,T)

•K

onto G(F)\G{k)1. All in all, by Lemma 2 of 4.2.7, we arrive at the following Arthur's Partition. ([We9]) Fix P and let T be any suitably point inT0 + a£. Then J2 PI-.POCPICP

J2 SePi{F)\G(F)

F1{8X)-TF(H0(5X)-T^=1

VXGG(A).

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B.4.2.8. Arthur's Analytic Truncation Definition. Fix a suitably regular point T 6 u j . If 0 is a continuous function on G(F)\G(A)1, define Arthur's analytic trunction (AT(p J (re) to be the function (A T ^)(x) := £(_i)dim(A/z) P

£

P(8x)-fp(H(6x)-T),

SeP(F)\G(F)

where 4>p{x) := I

4>{nx) dn

JN(F)\N(A)

denotes the constant term of <j> along P, and the sum is over all (standard) parabolic subgroups. The main purpose for introducing analytic truncation is to give a natural way to construct integrable functions: even from the example of GL2, we know that automorphic forms are generally not integrable over the total fundamental domain G(F)\G(A)1 mainly due to the fact that in the Fourier expansions of such functions, constant terms are only of moderate growth (hence not integrable). Thus in order to naturally obtain integrable functions, we should truncate the original function along the cuspidal regions by removing constant terms. Simply put, Arthur's analytic truncation is a well-designed device in which constant terms are tackled in such a way that different levels of parabolic subgroups are suitably counted at the corresponding cuspidal region so that the whole truncation will not be overdone while there will be no parabolic subgroups left untackled. Note that all parabolic subgroups of G can be obtained from standard parabolic subgroups by taking conjugations with elements from P(F)\G(F). So we have: (a) ( A ^ x ) = J ] ( - l ) d i m ( A / z V p ( x ) • TP(H{X)

- T ) , where the

p

sum is over all, both standard and non-standard, parabolic subgroups; (b) If = ; This is because by definition, all constant terms along proper P : P ^ G are zero. Moreover, it is a direct consequence of the Basic

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Estimation for partial truncation, (see e.g. the Corollary there), we have (c) / / (x)\ < c\\x\\N for all x € G(A), then so is KT. Recall that an element T € a j is called sufficiently regular, if for any a € Ao, ot(T) ^> 0. Fundamental properties of Arthur's analytic truncation may be summarized as follows: Theorem. ([Ar2,3]) For sufficiently regular T in do, (1) Let <j>: G(F)\G(A) -> C be a locally L1 function. Then ATAT cl>(g) = KT4>{g) for almost all g. If is also locally bounded, then the above is true for all g; (2) Let fa, fa be two locally L1 functions on G(F)\G(A). Suppose that fa is of moderate growth and fa is rapidly decreasing. Then f

ATfa(g).fa(g)dg=

JZG(A)G(F)\G(A)

fad)-kTfa{g)dg-

f JzG{k)G(F)\G(A)

(3) Let Kf be an open compact subgroup of G(Af), and r, r' are two positive real numbers.

Then there exists a finite subset \ X{ : i =

1, 2 , . . . , N > c W , 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 G J4G(A)> 9 G G(A)1 D S. Then ATcf>(ag) < Wgirj^sup^SiX^ag'K

\\g'\\-r' = 9 e G(A) 1 },

i=l

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 ofG(F)\G(A)\ 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 dis-

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cussion in 4.2.6-8. As for (4), which has been considered to be only secondary, please see 4.3 where it plays a central role. B.4.3. Arthur's Periods B.4.3.1. Arthur's Period Fix a sufficiently regular T E OQ and let cj> be an automorphic form of G. Then by Theorem 4.2.9, A T is rapidly decreasing, and hence integrable. Hence, the integration

A(<j>;T):=

AT{g)dg

f JG(F)\G(A) G(F)\G(A)

makes sense. We claim that A((f>; T) can be written as an integration of the original automorphic form over a certain compact subset. To start with, note that for Arthur's analytic truncation A T , we have A T o A T = A T . Hence,

ATcj>d^g)

A{4>;T)=I JZG{A)G(F)\G(A)

AT(ATA(g)d^g).

= f

V

JZG(A)G(F)\G(A)

'

Moreover, by the self-adjoint property, for the constant function 1 on G(A),

1(g)-AT(ATA

/

V

J ZG(A)G(F)\G(A)

(g)dn(g) >

(ATl)(g).(AT(l))(g)dfi(g)

= [ JZG(A)G(F)\G(A)

V

/

V

/

AT(ATl){g)-{g)dn{g),

= / I ZG(A)G(F)\G(A) JZr?,^G(F)\G(A)

v

y

since AT<j> and A r l are rapidly decreasing. Therefore, using AT o A T = A T again, we arrive at ATl(g)-(f>(g)d^g).

A(
ZGWG(F)\G(A)

(*)

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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 S(T) := (zG(A)G(F)\G(A)\ of the space 1 G(F)\G(A) by S(T) := (ZG{A)G(F)\G(A))T

:= [g E ZG(A)G(F)\G(A)

: ATl(g)

= l}.

We claim that S(T) or the same (Z G ^G(F)\G(A)j , is compact. In fact, much stronger result is correct. Namely, we have the following Basic Fact. ([Ar4]) For sufficiently regular T € OQ , A T l(x) = F(x,T). That is to say, A : l is the characteristic function of the compact subset S(T) of G(F)\G(A)1 obtained by projecting iVo(A) • Mo(A) 1 • A%tG(T0,T) • K onto G(F)\G(A)\ Thus, by the equation (*) and the Basic Fact above,

L

AT<j>(g)d»(g)

ZG(A)G(F)\G(k)

ATl{g)-{g)diJ,{g)

= / J

ZGWG(F)\G(A)

JT,(T)

That is to say, we have obtained the following very beautiful relation: Arthur's Periods. For a sufficiently regular T € ao, and an automorphic form 0 on G(F)\G(A), f JT;(T)

ATcj>(g)d^g).

{g)dn(g)= [ JG(F)\G(Ay

The importance of this fundamental relation can hardly be overestimated. Say, when taking 4> to be Eisenstein series associated with an L 2 -automorphic forms, (i) (Analytic Evaluation for RHS) The right hand side can be evaluated using certain analytic methods. For example, in the case for

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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], see also the end of B.4 where certain new yet fundamental relations among special values of zeta functions are exposed. B.4.3.2. A x l is a Characteristic Function The 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 £ ao, level P-Arthur's analytic truncation of is defined by the formula Ar^G,):=

£

(-1)<*(*)-^)

R-.RCP

£

4>R{S9)^(H{Sg)-T).

SeR(F)\P(F)

Then A r stands simply for A T,G . Thus, together with Langlands' combinatorial lemma, we have the following:

(a) kTm

:= £ ( - l ) < ^ ) - ^ ) R

£

M69)

• ?R(H(S9) - T);

SeR(F)\G(F)

and (b) (Inversion Formula) For a G(F) -invariant function 4>,

M

= E P

E

^Wg)

• rP (H{69) - T) .

5eP(F)\G(F)

(c) Assume that T G Cp and <j) is automorphic on N(A)M(F)\G(A). TP 7 1 Then A ' (f)(m) is rapidly decreasing on M(i ')\M(A) . In particular, the integration JMIF^AKA)1 A/r,p
G(F)\G(A),

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we have £ P

Fp (fa, T ) • r P (tf P (fa) - r ) = 1

]T 5eP(F)\G(F)

where rp is the characteristic function of (H

G a0 : a ( # ) > 0, a G A P J

and Fp(nmk,T)

= FMp(m,T),

n € NP(A),m

e MP(A),ke

K.

On the other hand, by the inversion formula, applying to the constant function 1, we have E P

E

(AT'Pl)(Sx)-rP(HP(6x)-T)=l,

6eP(F)\G(F)

where A ' is the partial analytic truncation operator defined above. With this, the desired result is immediately obtained by induction. B.4.4. Geometric and Analytic Truncations: A Bridge B.4.4.1. 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 the sublattices filtration associated to a parabolic subgroup P by 0 = A0 c Ai c A2 C • • • C A| P | = A. Assume that P corresponds to the partition I = (d\, di, • •., d|p|)- 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 Mo by the condition that ai(T) = \p(i) - p{i - 1)1 - \p(i + 1) - p(i)

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for alH = 1,2,..., r — 1. Then one checks that T(p) = ( p ( l ) , p ( 2 ) - p ( l ) , . . . ,p(i)-p(i-l),

.• .,p(r-l)-p(r-2), -p(r-l)).

Now take g = g(A) e G(A). Denote its lattice by A9, and its induced filtration from P by 0 = A 9 ' P c A f ' P C - - - c A j ^ p = A9. Consequently, the polygon p9p = pp9 : [0, r] —•> E is characterized by (l)Pp(0)=pp(r) = 0; (2) pp is affine on [r-j, ri+i], i = 1,2,..., | P | — 1; and (3)/P(ri) Note that equivalent (3)' pp(n)

= d e g ( A f p ) - r i . ^ M , i = 1 , 2 , . . . , | P | - 1. the volume of A is assumed to be one, therefore (3) is to = 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 € Np(A),m € Mp(A) 1 , a E A+ Write andkeK := UpSL(.0Qp) x SO(r)a = a(g) = diag[aildl,a2ld2,

• • •, a| P |/ d | p | J

where r = di + d2 + • • • + d\p\ is the partition corresponding to P. Then it is a standard fact that i

i

deg(Afp)=-log(j]a^) = -^^logaJ-, 3=1

z = l,...,|P|.

3=1

Set now l(p*P >p p) to be the characteristic function of the subset of g's such that pp >p p. Micro Bridge. For a fixed convex normalized polygon p : [0,r] —> E, and g € SLr(A), with respect to any parabolic subgroup P, we have Tp(K-HG{g)-T(p))=l(fp>Pp).

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B.4.4.2. 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. Global Bridge. For a fixed normalized convex polygon p : [0, r] —> R, let T{p) :=

(p(l),p(2)-p(l),... , p ( i ) - p ( i - l ) , • • • , p ( r - l ) - p ( r - 2 ) , - p ( r - l ) ) &e t/ie associated vector in do. IfT{p)

is sufficiently positive, then

l(p9
B.5. Rankin-Selberg Method Global bridge between geometric and analytic truncations in B.4 shows that at least for the polygons p whose associated T{p) € Oo are sufficiently positive, the corresponding non-abelian L-functions may be viewed as a special kinds of Arthur's periods. Therefore, to understand our non-abelian 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 obtained for a sufficiently regular T € oo. Simply put, the aim of this chapter is to show that a special kind of Eisenstein periods and hence also certain classes of non-abelian Ls can be evaluated via an advanced version of Rankin-Selberg method. B.5.1. Regularized Integration: 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

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320

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

/(x) = j ^ e ^ P i O r ) t=i

where the Aj are distinct elements of V* and the Pi{x) 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:= \x£V

: (m,x) > 0 V i }

where {/ij} 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 6 V* is negative (resp. non-degenerate) with respect to C if Re (A, ej) < 0 (resp. (A, ej) ^ 0) for each j = 1 , . . . , r. (2) The function f(s) = X)[=i e^Ai'^Pj(a;) is integrable over C if and only if Xi are negative with respect to C for all i. To define regularized integrals over C, we study the integral Jc(/;A):= f

f(x)e^dx.

It converges absolutely for A in the open set heV*

: Re(Ai - 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 an polynomial of one variable,

f Jo

(Dmp)

(0)

m>0

We then can use the right hand side to obtain the continuation.

Qeo-Arithmetic: B. Non-Abelian

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To deal with more general case, fix an index k and let C^ be the intersection of C with the hyperplane Vk := < x : (/zfc, x) = 0 >. Then, we can write x = {/j,k,x)ek + y with y E Vk- Thus, for a suitable choice of Haar measures,

*(/;A) = E E 7

S ^ r / *-{x-Xi'v)(PZ)PM dy-

•=im>o [ ( A - Aj,efe)J

-^

This formula then gives the analytic continuation of Ic[/; A) to the tube domain defined by Re(A, — A, e^) > 0 for j ' ^ k, 1 < i < n with singularities on hyperplanes H ^ := < A : (A, e^) = (A,, eu) >, 1 < i < n. Moreover, since this is true for all k = 1,2,..., r, the function ^c(/;AJ has analytic continuation to V* by Hartogs' lemma with (only) hyperplane singularities along Hik, 1 < k, < r, 1 < i < r. In particular, from such an induction, one checkes (3) Suppose that f is absolute integrable over C. Then Ic[f;X)

is

holomorphic at 0 and Ic ( / ; 0 I = L f(x) dx; and (4) The function Ic(f;Xj is holomorphic at 0 if and only if for all i, Xi are non-degenerate with respect to C, i.e.,(\i, e^) ^ 0 for all pairs (i,k). Denote the characteristic function of a set 7 by r Y . For A 6 V* such that Xi — X are negative with respect to C for alH = 1,2,..., n, set F(A;C,T) := f f(x) • rc{x - T) • e'^dx n

»

= V / Pi(x)TC(x 7^i Jy

= £ e-^-^h

T)e-<x~Xi'x)dx

(Pi{* + T)e^; A) .

i=i

The integrals are absolutely convergent and F( A;C,TJ extends to a

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meromorphic function on V*. With this, following [JLR], we call the function f(x) • TC(X — T) #-integrable if F(X;C,T) is holomorphic at A = 0. In this case, set

f

-TC(X-

f(x)

T)dx

:= F(0; C, T\ .

Jv

By (4), the ^-integral exists if and only if each exponent Xj is nondegenerate with respect to C. To go further, suppose that V = W\ © Wt is a decomposition of 7 a s a direct sum, and let Cj be a cone in Wj. Write T — T\ + T2 and x = x\ + X2 relative to this decomposition. If the exponents Xj of / are non-degenerate with respect to C = C\ + C2, then the function

r* W2H>

/

c

f(wi

+ 1 0 2 ) ^ l (w\ - Ti)

dw\

JW-i

is defined and is an exponential polynomial. And it follows by analytic continuation that (5) / Jv

f(x)TC(x-T)dx

= [ { f f(m + «;2)rf1 (wx - Tx) dwx ) • rf2 (w2 - T2) dw2; Jw2 \Jwi ) 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 W\ and consider functions of the form g{w\ — ti)rc(w2 — T2), which we call functions of type (C). It is clear that the integral F(X) := / f(x) • g(Wl - h)TC(w2 - T2) • Jv

e'^dx

converges absolutely for an open set of A whose restriction to W2 is negative with respect to C2- Furthermore, F(X) has a meromorphic continuation to V*. Following JLR, the function f{x)g{w\ —

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*I)T"C(U>2 —

T2) is called #-integrable if F(A) is holomorphic at A = 0. If so we define

r* / f(x) • g(Wl - h)TC{w2 - T2) dx := F(0). Jv Note that as a function of 11)2, Jw f(wi + ^2)5(^1 — ^1) dw\ is an exponential polynomial on W2, hence /•# / f(x) g(wi - ti)TC(w2 - T2) dx Jv =

( /

f{Wi+W2)g(wi-Ti)dwi)T2(W2-T2)dW2.

Jw2 yjwx

j

We end this discussion by the following explicit formulas: e^rc{x-T)dx

(6) /

= (-iyVol(ei!e2,...,er) • v

JV

'

* ' 1 l j = l \A) e j /

where C := I Y^i=iajej '• aj — Of an<^ Vol! ei,e2,. • • , e r ) is the volume of the associated parallelepiped { £ S = i ajej '• aj €[0,1]}. In particular, if V = R, we have /f°° e dt = JT

eXT

— and hence for A

any cone C c R ,

/

e^'xHl-TC(x-T))dx

=-

e^TC(x-T)dx

since 1 — r c is the characteristic function of the cone — C. B.5.2. JLR Periods of Automorphic Forms Now let us follow [JLR] to apply the above discussion on integration over cones to the theory of periods of automorphic forms. Let Tk(X) be a function of type (C) on dp that depends continuously on k G K, i.e., we assume that there is a decomposition ap — W\ © W2 such that % has the form 0fc(wi - Ti)Tc2k{w2 - T2)

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where the compactly supported function gk varies continuously in the L 1 -norm and linear inequalities defining the cone C%k vary continuously. Also with the application to automorphic forms in mind, let / be a function on M(F)N(A)\G(A) of the form s

f(namk)

= J ^ ( m , f c ) • a j (fl P (a),fc)e< A ' +w ' >Hp(o)> i=i

for n E N(A),a € AP, m € M(A) 1 and k € K, where, for all j , (a) j(m,k) is absolutely integrable on M(F)\M(A)1 x K; and (b) Xj € ap and ay (X, A;) is a continuous family of polynomials on ap such that, for all k € K, otj(X, k)e^'x^Tk(X) is #-integrable. As such, following [JLR], define the ^-integral / JP(F)\G(A)

f(g)-Tk(HP(g))dg:= V

M(F)\M(A)!

'

j{mk) dm) • I f J \Jap

aj{X,k)rk{X)dx)

dk. J

Remark. The reader should notice that in this definition there is no integration involving fN(F\\N(&) ^n : ^ n P r a c tice, / is supposed to be the constant term along P, so is iV(A)-invariant. Clearly, the set Sp(f) of distinct exponents {Aj} of / is uniquely determined by / , but the functions j and ay are not. So we need to show that the ^-integral is independent of the choices of these functions. Clearly, the function

X»

f

1

f(exmk)dm

IM(F)\M(A) JMiFKMfA)

is an exponential polynomial on ap and the above ^-integral is equal to f(exmk) M(F)\M(A)!

pp

dm

e-<

/

^Tk(X)

dX dk.

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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 Xj is negative with respect to Cik for all k £ K, then the ordinary integral

[

f(g)-rk(HP(g))dg

is absolutely convergent and equals to the ^-integral by Basic Property (3) of integration over cones. That is to say, we have the following Lemma. (1) The above #-integral / / ( F ) \ G ( A ) f{g) • Tk[HP(g)J dg is well-defined; and (2) If each of the exponents Xj of f is negative with respect to Cik for all k € K, then the ordinary integral

I

f(g)-rk(HP(g))dg

is absolutely convergent and equals to the ^-integral. Fix a sufficiently regular element T e OQ. Then the above construction applies to Arthur's analytic truncation AT'Pif!(g) and the characteristic function Tp(Hp(g) — T) where ^ 6 Ap(G), the space of level P automorphic forms. Indeed, with s

^(namk)

= Y J Qj(Hp(a))

• ipj(amk)

3=1

for n £ N(A),a £ Ap, m £ M(A) 1 and k £ K, where Qj are polynomials and ipj € Ap{G) satisfies

for some Xj £ a*P and all a £ Ap, JMrF\\M/A\ixK AT'Pi(jj(mk)dmdk T,p is well-defined, since the function m >—> A tpj(mk) is rapidly decreasing on M ( F ) \ M ( A ) 1 x K. Now Tp is the characteristic function

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of the cone spanned by the coweights AP, so AT'P*(g)-TP(HP(g)-T)dg

/

v

JP(F)\G(Ay

'

exists if and only if (Xj,wv)

^ 0

VroeA^

and

\j e

£P(V).

The same is true for AT'pV(g) • TP(HP(gx) ~ T) for x G G(A). Indeed, for x G G(A), let K(g) G if be an element such that gK(g)-1 € Po(A), then # P ( # X ) = HP(g) + HP(K(g)x) and hence r p ( X — T — ifp(.fir ((7)3;) J is the characteristic function of a cone depending continuously on g. Set then IpT{^>) := /

AT'P*(
For any automorphic form G -4(G), write £p(<j>)forthe set of exponents £P(cf)P). Set „4(G)* := <jy G .4(G) : (A,tu v ) ^ 0

Vrov G A P , A G SP{), P + G } .

If ! G .4(G)*, then IpT((f)P) exists for all F ; and define JLR's regularized period by

JG( G(F)\G(A)

p

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 G(Aj), let p(x) denote right translation by x: p(x)4>(g) — 4>(gx). The space -4(G) is stable under right translation by G(A/). Furthermore, p(x)<j) has the same set of exponents as 4>. Consequently, the space .4(G)* is invariant under right translation

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327

by G(Af). Indeed, for k e K, write the Iwasawa decomposition of kx as kx = n'a'm'K(kx) and write p in the form

Since amkx — n*aa!mm'K(kx) p f namk

for some n* G N(A), we have

•x J

j

This shows that I = £p(). Theorem. ([JLR]) (1) i g rfe/i nes a G(Af)1 -invariant linear functional on A(G)*; (2) IQ^) := /G(ir)\G(A)i € A(G)* and

I

{g)dg=

JG(F)\G(A)1

I

4>{g)dg.

^G(,F)\G(A)1

B.5.3. Arthur's Periods and JLR's Periods For / e AP{G) with l_

ffnamk)

= ^^(mk)

• aJY/f(a))e^+pp'H(a)

where n € iV(A), a e AP, m € M(A) 1 , k £ K, and for all j , CHJ(X) is a polynomial, and (j)j(g) is an automorphic form in Ap(G) such

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that 4>j(ag) =
f

f(g).TP(H(g)-T)dg

'•=/_\ \ ~[

I

(j)j(mk)dmdk

JK JM(F)\M(A) 1

f l

x ai{X)e^ \P{X

-T)dx\

This is well-defined provided that the following two conditions are satisfied: (i*) (fi, wv) ^ 0 VQ C P, zuv e (A v )g, n e £Q(<j>); and (ii*) (A, a v ) ^ 0 VQ G A P , A e €P((f>). Let .4(G)** be the space of ^ G -4(G) such that (i*) and hence also (ii*) are satisfied for any P. As a direct consequence of the theorem in 5.2, we have the following Theorem. ([JLR]) For 0 e A(G)**,

L

AT(g)dg

G(F)\G(A)i

= E(-i)d(PM(G) f

Mg)

• rP(H(g)

- T) dg.

B.5.4. Bernstein Principle With above, to understand non-abelian L-functions at least for T > 0, i.e., for T € aj" sufficiently positive, we need to study Eisenstein periods. For this, the following result plays a key role. Bernstein's Principle. ([JLR]) Let P = MN be a proper parabolic subgroup and let a be an irreducible cuspidal representation in the space L2(M(F)\M(A)1). Let E(g, , A) be an Eisenstein series associated to (ft 6 Ap(G)a- Then

ig(£(s,&A))=0 for all A such that E(g, cj), A) and l]^{E(g, , A)) are defined.

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329

We believe that this principle is the main reason behind special yet very important relations among multiple and ordinary zeta functions in literature. B.5.5. Eisenstein Periods Associated to Cusp Forms As a direct consequence of Bernstein principle, with the explicit formula for constant terms of Eisenstein series associated to cusp forms, (see e.g., [We8],) we have the following closed formula for Eisenstein periods associated with cusp forms: Theorem. ([JLR]) Fix a sufficiently positive T G O Q " , be a parabolic subgroup and a an irreducible cuspidal in L 2 ( M ( F ) \ M ( A ) 1 ) . Let E(g,
(MKA)^)(mfc)dmdfc>

Let P = MN representation series associ^-^G?) $•> A) d9

n J ^ X ^ )

if P = Po = M0NQ

is

minimal. Consequently, we have the following Theorem'. ([We9]) Fix a convex polygon p such that T(p) 6 ao is sufficiently positive. Then for a P-level cusp form
n

j

^

^

•fM0(F)\M0(AyxK{M(W>X)(f>)(rnk)dmdk> if P = Po = M0N0 is minimal. We remind the reader that this result is merely the beginning of our quantitive study about non-abelian L-functions. Say, (1) When the polygon p satisfies T(p) » 0, we still need to understand Lp;-P((j), A) where P' D P is a parabolic subgroup of G; (2) We have to weak the restriction that T(p) 3> 0; and (3) We need also to study non-abelian L-functions for principal lattices, i.e., that for general reductive groups.

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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) € a^; and for (3), no essentially difficulties appear despite its generality. For more details, see [We9]. As an application, we here recall a recent result of H. Kim and the author. Let then G be a split, semi-simple group of rank r. For any w € ft, let Jw := {« £ Ao : wa e Ao} and Pw be the standard parabolic subgroup corresponding to Jw. Accordingly, let rw := {a 6 Ao : wa < 0}, and for i > 1, m :=#{a

> 0 : (p,av) = i} - #{a > 0,: (p,av) = i - 1},

ni,w '•= # { « > 0, wa < 0 : (p, a v ) = i} - #{a >0,wa<0: (p, ay)=i-

1}.

As usual, denote by Ap 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 s = 1. Denote also by v the volume of

{£aeA 0 a <*a V : a ° € f 0 ' 1 )}" Theorem. ([KW]) For a sufficiently positive (G(F)\G(A) 1 ) is given by

TGOJ"

the volume of

v(2nrrApH^F(i)-^ »>i

, e^r-Pr, £<-D ,.„kp„, UaeAfrar-rifeiSf(•)"'-wJ (! - (^P>« )) V

w€Wo

0

w

Note that as T —> oo, only the term corresponding to w = 1 survives and the limit is exactly the volume of the fundamental domain given by Langlands [La2], namely, Corollary. ([La2]) If G is split and semi-stable, then the volume of a fundamental domain 5 for G(F)\G(A)1 is given by

Vol(ff) = v • (27r)^Ap • JlMi)""*•

Geo-Arithmetic: B. Non-Abelian

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In particular, this formula suggests that the volume of cuspidal regions can be expressed in terms of special values of Dedekind zeta functions. B.5.6. Fundamental Relations between Special Values of Zeta Functions By strengthening the results in 5.5, namely working with all T € OQ" instead of T » 0, we then can obtain the following result which reveals certain fundamental relations among special values of abelian and non-abelian zeta functions ([We9]). To state them, let £F,T-(1) := Res s= i£F, r (s). Level 1.

£F,I(1)

=

fr(l);

Level 2.

£F>2(1) = £ F (2)fr(l) - £ F (1)£ F (1);

Level 3.

£F>3(1) = ^ ( 3 ) ^ ( 2 ) ^ ( 1 )

-^F(2)fr(l) a - |fr(2)fr(l)a + Jfr(l)3; Level 4.

£F,4(1) = £ F ( 4 ) £ F ( 3 ) £ F ( 2 ) £ F ( 1 )

- J f r . ( 3 ) f r ( 2 ) M l ) a - ^ F ( 2 ) 2 ^ ( 1 ) 2 - ^F(3)eF(2)^F(l) 2 + ^F(2)2CF(1)2 + ^F(2)€F(1)2 + ^F(2)£F(1)3

Level r. The r-level relation can be described as follows. (1) £F,T-(1) is a degree homogenous polynomial in £F(*)> * = 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. For example, in the case r = r\ + r-i + • • • + r/v, the corresponding term consists of three parts, that is, (i) the sign ( - l ) ^ - 1 ; (ii) the product of zeta values: If Ti > 2, the product ^F(n)^F(n — 1 ) - - - £ F ( 2 ) does appear. Moreover, these subfactors together with certain powers of £ F ( 1 ) then gives all the product of zeta values (under the constrain that the final product is of degree r); and

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(iii) The coefficient

r-. (n + r2){r2 + r 3 ) • • • (rjv-i + rN) It is well known that following a result of Siegel, in order to read the special value £F(W), we have to go to rank n lattices. The above relations may be viewed as a refined version of this basic result. Roughly put, (i) among all rank n-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 £p(m)'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.

Geo-Arithmetic: C. The Riemann Hypothesis

333

C. Explicit Formula, Functional Equation and Geo-Ari Intersection C.l. The Riemann Hypothesis for Curves C . l . l . Weil's Explicit Formula: The Reciprocity Law Let C be a projective irreducible reduced regular curve of genus g defined over a finite field k := Fq. Denote by Cc(s) the associated Artin zeta function. Set t = q~s and Zc(t) := Cc(s)- Then by the rationality, there exists a polynomial Pc(t) of degree 2g such that

(1-()(!-
= Yl<*Zc(Ct) where the product runs

Moreover, by the Euler product,

#(*) = ££W" d(p) = £ (£^)V m C

n=l

m=l ^ P

P

'

where ^2P is taken over those closed points rational over Fq whose degree divides m. Hence,

(

°°

tm\

m=l

'

where 7Vm = £p,d(p)|m d(P). Clearly when m = 1, the sum X)p,d(P)|i * simply counts the number of closed points of C rational over Fq. Thus, by the reciprocity law above,

m=l

'

^m=l

^ C,

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This implies that Nn = Ni(Cn), i.e., Nn is the number of Fqn-rational points on C. Weil's Explicit Formula. Nn = qn + 1 - £ C c ( p ) = 0 / 3 n C.1.2. Geometric Version of Explicit Formula Let C be an algebraic curve defined over F p . Over C x C, for n G Z, introduce (micro) divisors A n (via algebraic correspondence) as follows: ' \(x,xPn):xeC\, A n := < \ K ^|(xP n , a ; ) : x € C J ,

ifn>0; if n < 0.

Then we have the following relations for the intersections among An s. (i) lin>m>0,

(An,Am)=pm(An-m,A0);

(ii) If m > n > 0, ( A , An) = p"(An-n, A ) ; (iii)

If m > 0 > n, ( A , An) = (^ n—m>

A);

(iv) If n < m < 0, p m ( A , An) = (An-n, A ) ; (v) If m < n < 0, p " ( A , An) = ( A - m , A ) ; (vi) If n > 0 > m , ( A , An) ~ \An—mi

AQ)

.

Hence, we have the following Lemma 1. (1) For all m,nEZ, (An,Am) = (Am,An); (2) For all m , n G Z ; (A-n,A-m) = (An, A ) ; (3) For alln>m>0 inZ, ( A , A m ) = pm(An^m,A0); (4) For alln>0> m in L, \An, Am) — (An-m, AQ). Obviously, (1) ~ (4) are equivalent to (i) ~ (vi) above. Therefore, in order to understand (An,Am) for all n, m G Z, we only need to know (AU,AQ) for n > 0. For this latest purpose, first, note that AQ is simply the diagonal. Hence, by definition, for n > 0, (An,Ao) is Nn(C), the number of Fpn -rational points of C;

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335

Secondly, by the functional equation of Cc(s)> (and the rationality of Cc(s)), we have the following Lemma 2. (Explicit Formula of Weil) Denote by F\ and F2 horizontal and vertical fibers in two directions of C x C respectively. For n € Z>o, (An,A0)

= (An,F1) + (An,F2)-

sn.

£ Cc(s)=o

C.1.3. Riemann Hypothesis for Function Fields Following Weil again, (see e.g., [Hal]), we now prove the (Artin-) Riemann hypothesis. Let / : pz —> Z be a function with finite supports. Define its Mellin transform via /(a) := E „ / ( P n ) P " S - Clearly, if f*{pn) := f(p~n)p~n, /*(«) = / ( I - s); moreover,

if (/* S)(pn) := J2f(Pm)9(Pn'm),

f^9(s)

= f(s)-g(s).

m

In particular, / * g*(s) = f(s) • g(l — s). With / , we may use the divisors A n 's above to define a (global) Q-divisor D1 on C x C as follows: D

f := E f(pn)A*+E n>0

f(p-n)p-nAn.

n>0

Theorem. (Weil) (1) (Relative Degrees) (Df-,Fi) = / ( l ) ; (Df~,F2) =

/(o); (2) (Fixed Points Formula) (D:,D§) = (Drpr, Diag), where Diag denotes the diagonal; (3) (Explicit Formula) /(0) + / ( l ) - £ c ( s ) = 0 / ( a ) = (DpDiag). Indeed, by definition,

(Df, F,) = £ /(p n ) • p" + £ /(p-")p- n • 1 = /(l); n>0

n>0

(£>;, F2) = £ /(p") + £ f(P~n)p-n • Pn = /(0). ra>0

n>0

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This gives (1). (2) is a direct consequence of the relations (i)-(vi) for the intersections of A n 's, and hence comes from Lemma 1 in 2.1. Finally, (3) is simply Lemma 2 of 1.2 by definition. Since (F1+F2, Dj-f(l)F2-f(Q)Fi) = 0, by the two-dimensional intersection theory, in particular, the Hodge Index Theorem, {Df - f(l)F2 - f(0)Fi,Df

- f{l)F2 - / ( O ) ^ } < 0.

That is,

/ ( 0 ) - / ( l ) > i <£>;,£);). Thus by the theorem above, this last equality is equivalent to

J2

f(s)-f(l-s)>0.

C(s)=0

From this, easily, we get the following Riemann Hypothesis for Artin Zetas. (Hasse-Weil) If({s) = 0, then Re(s) = \.

C.2. Geo-Ari Intersection: Model

A Mathematics

C.2.1. Motivation from Cramer's Formula In the above discussion, the summation J2rc(s)=o sH P^&YS a ^ e y role in understanding Artin-Riemann Hypothesis, via the so-called micro explicit formula of Weil. So naturally, we want to know whether this approach works for number fields, and hence are led to study the formal summation 2_2

xP

i

forxe[l,oo).

(*)

£Q(P)=O

Here £ Q ( S ) denotes the completed Riemann zeta, that is to say, £ Q ( S ) : = 7 r _ 5 r ( | ) C Q ( S ) with r(s) the standard gamma function and CQ(S) the Riemann zeta function.

Geo-Arithmetic: C. The Riemann Hypothesis

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The reader at this point certainly would reject (*), since the summation does not make any sense. How could we write such a monster down?! Well, do not panic!!! After all, in the study of the prime distributions, conditional convergent summation £ \ (p)=o ~ does appear. Recall that we have the following Riemann-von Mangoldt formula

^-^(O)-Ilog(l-* 2 ),

£logp = z - £

which itself motivates and hence stands as a special form of the more general form of explicit formulas. See e.g., Jorgenson and Lang's lecture notes on Explicit Formulas. More generally, in various discussions about prime distributions, we do use the summations such as X^Q(p)=o p(Pp±i)' * n ^ s s e n s e » *^e P ro blem is not whether we should introduce (*), rather, it should be how to justify it. Anyway, let us make a change of variables x := e* with t > 0. Then (*) becomes

V(t):=

Yl

<*"•

(**)

£Q(P)=O

So, following Riemann, we may further view V(t) as a function of complex variable z, i.e., eZP

V(z) := Y,

-

(**')

£Q(P)=O

Now, we claim that there is a nice way to regularize (**') hence also (**) and (*). To explain this in a simpler form, which in fact would not really make our life any easier, we instead consider the partial sum V+(z) defined by

V+(z) :=

Yl

£q(p)=0,Re(p)>0

Then we have the following result, dated in 1919.

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Theorem. (Cramer) The function V+(z) converges absolutely for Im(z) > 0. Moreover,

wvw - ^~ has a meromorphic continuation to C, with simple poles at the points ±7iV—In for all integers n, and at the points ± logp m for all powers of primes. We claim that this theorem actually offers us a natural analytic way to normalize the formal summation ]T\ (P)=o xP f ° r x ^ [1> °°)» by using [C], [JL1,2,3], and [DS]. In a sense, this is in a similar way as what we do when normalizing oo!: by the Stirling formula

nl

= y/2t.y/n-(£j . e x p(^

for n sufficient large with \0n\ < 1, we set oo! = V27r. However, in this article, we do it very differently - we are going to introduce a mathematics model to normalize this formal summation geometrically. C.2.2. Micro Divisors We will not use Grothendieck's scheme language. Instead, formally, we call (the set theoretical product) S := Spec(Z) x Spec(Z) a geometric arithmetic base (surface). Similarly as in 1.2, associated to all x G [0, oo] are symbols Dx which will be called micro divisors; and associated to any two micro divisors Dx, Dy is the intersection number (Dx,Dy) 6 R. Assume that the following fundamental relations are satisfied by our micro intersections:

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339

Axiom 1. (Symmetry) For any x, y G [0, oo], (Dx,Dy)

=

(Dy,Dx)-

Axiom 2. (Mirrow Image) For any x, y € [0, oo], (Dx,Dy)

=

(Di,Di); X

y

Axiom 3. (Fixed Points 1) If 0 < x < y < 1, then {Dx,Dy)=y(Dz,D1); Axiom 4. (Fixed Points 2) If 0 < x < 1 < y < oo, then <£> x ,£g = (£>!,£>!}• Note that [0, oo] x [0, oo] is simply the union of {0 < x < y < 1}, {0 < y < x < 1}, {1 < x < y < oo}, {1 < y < x < oo}, {0 < x < 1,1 < y < oo} and {0 < y < 1,1 < x < oo}. Thus by the above relations, if we know the pricise intersection (DX,D\) for all x G [0,1], then we obtain all the intersections (Dx,Dy) for all x,y e [0,oo]. Axiom 5. (Micro Explicit Formula) Denote the completed Riemann zeta function by £ Q ( S ) . Then, for all x € [0,1], (Dx,Dl)

= (DQ,Dx) + (DO0,Dx)-

Y,

x

°-

£Q(*)=O

Here, as said in 2.1, we certainly encounter with the convergence problem of X^£Q(s)=o xS • Instead of solving it, in our model, we simply assume that our micro intersection offers a natural normalization of Y2fa(s)=oxS v * a t n e (P). It is in this sense we say that our model gives a geometric way to normalize ^c Q ( s ) = o x8 • The compatibility among these Axioms is guaranteed by the functional equation for Riemann zeta function. Moreover, if x > 1, by

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the Mirrow principle, we see that (Dx,Di)

= (D\,D\).

So, by us-

x

ing the Explicit Formula 1, i.e., Axiom 5 above, together with the Relations II below, we have

(DxM = i + l-

£ £Q(S)=0

Multiplying both sides by x, we get x{Di,Di)

= l + x-

]T

x1~a = l + x-

SQ(S)=O

^ xs, Cq(s)=o

by the functional equation. On the other hand, by the Fixed Point 1, i.e., Axiom 3 above, we get (Dx,D1)

=

x(Di,D1). x

That is to say, formally, for all x,

(Dx,D1) = l + x-

xs.

Y, £Q(S)=0

From Axioms 1-4, we may formally get the following relations. Relations I. (i) (D0,D0) = (Doo-Doo);

(ii) (A), A ) = (A»,A> = (A>, A»>. Note that Do and Doo are supposed to be the vertical and horizontal fibers of Spec(Z) x Spec(Z) over oo € Spec(Z), so we normalize our intersection further by the following Axiom 6. (Normalization) (Do, Do) = 0,

(DQ,DI)

= 1.

With this, formally we may further get the following Relations II. (i) For

(Do,Dx),

/n lx> (D0,Dn X) \ = < II,

ifx€

M''

if x€ [l, oo};

Geo-Arithmetic: C. The Riemann Hypothesis

341

(UjForiD^Dx),

[-,

if

xe[l,oo];

In particular, for x € [0,1],

(Dx,Di) = l+x-

xS

Yl

-

e Q ( S )=o Remark. In (iii) above, taking x = 1, we get l

{DUDX) = 2+ Y,

-

{Q(")=O

In other words, ^£ Q ( s )=o 1 ' s supposed to be related to the selfintersection of the diagonal divisor. C.2.3. Global Divisors and Their Intersections: Geometric Reciprocity Law Motivated by C.l, i.e., the discussion about Artin-Riemann Hypothesis, we start with a standard construction in function theory. Let / : R + —> R be a smooth, compactly supported function. Define its Mellin transform via r°°

f(s) := /

Jo

dr

/(*)*•-. x

Then, if/*(x) : = / ( £ ) • i,

T*(s) =

f(l-s);

and if (/ * g)(x) := /0°° f(y)g(^)-jf denotes the standard multiplicative convolution, f*g(s) = f(s) -g(s). In particular,

7^?(s) =

f(s)-g(l-a).

L. Weng

342

Next, we give a parallel construction for our divisors. Standard wisdom says that we should use linear combinations of generalized divisors Dx, x G [0, oo] to form new type of divisors. But as we clearly see that such a conventional way does not result sufficiently many divisors, we do it very differently. Starting from divisors Dx, for any might-be-interesting function / , formally define the associated global divisor D t by setting

Df1:=

/ f(x).Dx.

Jo

—+ / x

f(x)

Ji

xDx

dx x

• —.

Remark. In all formal discussions here, we pay no attention to the convergence problem. See however 2.5 below. In other words, we assume also in our model that global divisors do exist and their relations with micro divisors are given as above. With this definition, we may extend the intersection in 2.2 to D ?'s by linearity. For example, the intersection (D;,Di) is by definition given by (D^Di) 3

f1

dr

= / /(x) • <£>*,£>!) • xJo

f°°

+ / Ji

Hr

/(*).*<£>*, A ) • —• x

Then formally, we have the following Key Relations, (i) (Relative Degrees in Two Fiber Directions) deglDr.=

(D0,Df)

= f(l)

and deg2D;:=(Doo,Df)

= f(0).

(ii) (Fixed Point Formula) (Df,D§)

=

(Dj^,D1);

(Hi) (Global Explicit Formula)

(Df,D1) = f(0) + f(l)-

Y, €Q(«)=O

/»•

Geo-Arithmetic: C. The Riemann

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343

In fact, this may be formally checked as follows using axioms. First consider the Fixed Point Formula. Set D'x := D\. Then, by definix

tion,

By changing variables x' := %, the latest quantity is simply

/ f°°

tl

^dy f°°

. . „ dx

f°° t . .dy

fy

, , „ dx „ \

But for x > y, we may split the region into three parts, i.e., (1.1) 1 > x > 0,1 > y > 0 and x > y; (1.2) oo > x > 1, oo > y > 1 and x > y; (1.3) oo > x > 1 and 1 > y > 0. Similarly, the region of x < y may be split into three parts, i.e., (2.1) 1 > x > 0,1 > y > 0 and x < y; (2.2) o o > x > l , o o > y > l and x < y; (2.3) oo > y > 1 and 1 > x > 0. Hence the latest quantity is simply, by writing according to (2.1),(2.2) and (2.3) (resp. (1.1), (1.2) and (1.3)) for the first term (resp. second term),

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y y x \V./o y Jo x Ji vh + f / W ^ f9(x)VD,_t*) + ( /"/(„>* / W ^ v x 7i y Jo % J 1\Jo y Jy %

f°° -/ x ^ f°° , ^ r, dx f + / / y — / g(x)xDu—+f(y)— x Ji y Jy % Jo

, . .dy

f°°

y J\

, , „ ds\ g{x)xD*—) x x / £>i

= ( ( [1m-\[V9(x)yD,-+ WJo y Uo /

= t

/(y)—

+ / Jl

Jy

y Uo /(y)— / V Ul

v

f1g(x)x(Du,D x

g(x)y{D^,D1)—+ V • X

+ flf{y)^

Tg{x)x{D^Dl)^.

Jo

y Ji

v

dx

Jy

[\(x)y(Dx_,Dl)-

y Jo

.

g{x)xDii —

+ T/(y)Ji

I\{X)XDM.x x

/ 9(x)yDE—+

[Vg(x)y(D^D1)-+

f(y)-\

Jo

v x

Jy

g(x)x(D^,D1) *

— X

x

x

Now accordingly call each of the terms from the beginning as (3.1), (3.2), (4.1), (4.2), (5.1) and (5.2), we then see that for (3.1), (resp. (3.2), (4.1), (4.2), (5.1) and (5.2)) we have 0 < x < y < 1, (resp. 0 < y < x < l , l < x < y < o o and oo > x > y > 1,) hence by our axioms, for 0 < x < y < 1 y(Dz,D1)

=

(Dv,Dx),

345

Geo-Arithmetic: C. The Riemann Hypothesis (resp. for 0 < y < x < 1 x(D*,D1) = {Dx,Dy), for 1 < x < y < oo (Dz,D1) =

x(Dx,Dy),

(D]L,D1) =

y(Dx,Dv),

for oo > x > y > 1 X

iorO<x
and, forO
We see that the latest combination is simply

/ f(y)~ fy9(x)(Dy,Dx)^+ x Jo y Uo y f°° dv T F + / f(y)~ / Ji y LJI

dx g(x)(xy)(Dx,Dyx)-

+ ng(x)(xy){Dx,Dy)—] Jy

+ fftoT Jo

f g(x)(Dx,Dy)^] x Jy J

x

+ I™ f{y)^ J Jl V JO

f1g(x)y(Dy,Dx)-

r9(x)x(Dy,Dx)^ x

y Ji

that is,

flm% Jo

flg(xADy,Dx) x

y Jo

+ f '/(v)-

rg{x)--x{D y,Dx) x

+ Tm-f

g{x)--y(D y,Dx) x

Jo Ji

f°°

+ /

Ji

y Ji y Jo dv

f(y)~

f°°

y Ji

dx

g(x)--xy(D y,Dx). x

X

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L. Weng

Clearly, by defintion, this latest combination is simply (Df,Dg). This then completes the proof of the Fixed Point Formula. To see the relative degree relation, we have the following formal arguments. (Df,D0) =

Cl

1"°°

dr

f(x)(Dx,D0)—+x

dr

f(x)x(Dx,D0)

— Jo Ji x fl ,, , dx f°° ,. . dx f°° , . , dx ,. n , = / f(x)x— + / f(x)x— = / f(x)x— x = / 1 Jo x Ji x Jo

and <£>;,£>oo)

=

/•! Jo

f(x)(Dx,D00)—

P°°

dx x

+

f°° Ji

dx — x

f{x)x{Dx,DO0)

dr

= 1 /(*)f = /(o). (Here a standard regularization is needed. For details, see 2.5 below on Not So Serious Convergence Problems.) Finally, let see how global explicit formula is established. Here the functional equation plays a key role as in function fields case. Indeed, by definition, (Df,Di)=

f1

Jo

dr

f{x)(Dx,D1)—+x

f°°

Ji

dr

f(x)x(Dx,D1)

— . X

Now by the micro explicit formula, we have <£>;, A ) = f Jo

f{x) f(Dx, DQ) + (Dx, £>«,) V

£ eQ(^)=o

xs) — J

X

+ J"f(x)x({Dx,D0) + (Dx,Doo)- Y, ^ _ 1 )f-

Geo-Arithmetic: C. The Riemann

347

Hypothesis

Recall that by the micro explicit formula, for x € [0, oo],

(Dx,D1) = (Dx,D0) + (Dx,Doo)-

xS

J2

-

6j(«)=o

Consequently, (D;,D1)=(D;,D0)

+ {D;,Doo)+

f(x) Jo

£ *QW=O

Xs — X

=/(o) + /(i)- £ ha). CQ(S)=O

C.2.4. The Riemann H y p o t h e s i s To finally relate our intersection with the Riemann Hypothesis, we should have a certain positivity. That is to say, we need an analog of the so-called Hodge Index Theorem. A Weak Version of Hodge Index Theorem. Self-intersection of the global divisor L; := Di — / ( l ) ! ) , ^ — f(0)Do is non-positive, i.e., (Lf,Lf)<0.

(*)

Remark. We call (*) a weak version of Hodge Index Theorem, since {Lj, D0 + Doc) = 0. Prom now on, let us assume that (*) holds. Then by the Key Relations in the previous section, with a direct calculation, we certainly will arrive at

Y,

f(s)-f(l-s)>0.

&j(«)=o

This is very nice, since then, following Weil, we may get the Riemann Hypothesis from this latest inequality. (See e.g., page 342 of the second edition of Lang's Algebraic Number Theory for more details.)

L. Weng

348 C.2.5. N o t So Serious Convergence Problem

We add some remarks on the formal calculation appeared in 2.3. Roughly speaking, to justify them, what we meet is a certain regularization process. This may be done as what Jorgenson and Lang do in their lecture notes on Basic Analysis of Regularized Series and Product. More precisely, motivated by the definitions of hyperbolic Green's functions (of Selberg, Hejhal, Gross and Zagier,) Ray-Singer's analytic torsions, we may first introduce imaginary divisors D;by setting

for certain type of suitable functions / and for s whose real parts are sufficiently large; then assume that in our model Dt has a meromorphic continuation to the half space Re(s) > — e with e > 0, from which we could finally get a well-defined D i after removing the singularity at s = 0. C.2.6. Weil's Explicit Formula and Two-Dimensional GeoAri Intersections We still have not explain why we say the above intersection is indeed an intersection over the geometric arithmetic surface Spec(Z) x Spec(Z). To understand this, we have to use yet another fundamental result of Weil, the Weil Explicit Formula. Note that by the Key Relations, we obtain the following crucial formula:

(D/,D1> = /(0) + / ( l ) - £

f(s).

(*)

However, as an intersection over Spec(Z) x Spec(Z), (Dt, D\) should be counted locally over each point (p, q) G Spec(Z) x Spec(Z), i.e., we should have the decomposition

p,q
Geo-Arithmetic:

C. The Riemann

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349

Now note that D\ is the diagonal, so besides at the points (p,p), p < oo on the diagonal, (D:,Di)faq) is naturally zero. With this, then we would have

p
Clearly, this latest expression suggests that via (*) above

/(o)+/(i)- E / W = E W P ( / ) . £Q(S)=0

(**)

p
where for each places p of Q, i.e., the primes p and the Archimedean place oo. Without any mistake, it is then nothing but Weil's explicit formula. This interpretation naturally leads to a question about the explicit formula for the micro intersection. Recall that one of the key assumption for our micro intersection is that, for x € [0,1], (Dx,D1)

= (Dx,D0) + (Dx,Doo)-

E

*S-

£Q(S)=O

Therefore, if we believe that (•, •) is indeed a two-dimensional intersection on Spec(Z) x Spec(Z), then we similarly should have 1 + x — 2_) £Q(*)=0

xS=

'52wp(x},

for x € [0,1].

(* * *)

P<°°

We will call (* * *) the micro explicit formula of Cremer, motivated by Jorgenson and Lang's 'ladder principle'. Remark. There are also fundamental works of Deninger and Connes on the Riemann Hypothesis, based on certain cohomological consideration. While these approaches appear quite different, they share one common part, the Weil Explicit Formula.

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C.3. Towards a Geo-Ari Cohomology in Lower Dimensions C.3.1. Classical Approach in Dimension One It is clear that we should go beyond a geo-ari intersection: With our experiences from algebraic geometry and Arakelov theory, to establish an analog of Hodge index theorem, we need to develop a corresponding cohomology theory. It is our belief that a general yet well-behaved cohomology theory is at the present time beyond our reach. However this does not mean that we cannot do anything about it. After all, what we need is a practical yet uniform cohomology (and intersection) in dimensions one and two, which satisfies duality, adjunction formula and Riemann-Roch. Recall that in algebraic geometry, we do have a general cohomology theory, thanks to Master Grothendieck. While extremely powerful, this theory is still far short when dealing with global arithmetic problems. Say, at least, analysis is largely left behind. Thus, for global arithmetic, new approach has to be taken. For this, we suggest to go back to one generation before Grothendieck, i.e., that of Chevalley and Weil who once believed that adelic language will sevre us equally if not better. To give the reader a good test, let us start with cohomology in dimension one i.e, that for curves. A good reference is Serre's GTM on Algebraic Groups and Class Fields. Let C be a regular irreducible reduced projective curve of genus g defined over k, a field which may not be finite, with F its associated function fields. Fix a divisor D on C. Then main points of the corresponding cohomology theory may be summarized as follows. (1) by definition, the 0-th cohomology group of (the divisor class associated to) D is given by H°(C, D) := {/ e F : div(/) + D > 0}; (2) From the short exact sequence of sheaves 0 -> Oc(D) -*?->

T/Oc{D)

- 0,

where T denotes the constant sheaf on C associated to the function

Geo-Arithmetic:

C. The Riemann Hypothesis

351

field F, we get a long exact sequence 0 - H°(C,Oc(D))

-» F -+ H°(C,F/Oc(D))

-+ H\C,Oc(D))

-> 0.

Consequently, there are canonical isomorphisms H°(X, Oc(D)) ~ A(D) n F, and tf X(X, 0c(£>)) =* A/(A(£>) + F ) , where A denotes the adelic ring of F, and A(£>) := {(rp) € A : ord p (r p ) + ordp(L>) > 0}; (3) For any point p, from the structural exact sequence of sheaves 0 - • Oc(D) -* Oc(D + p) -> Oc(D)\p -» 0, we get a long exact sequence of cohomology 0 - H°(C, Oc(D)) -> tf °(C, O c P + P)) - fi°(C, Oc(^)lp) -> H\C,Oc{D))

- ^ ( C O c P + p)) - 0;

(4) By studying the residue pairing, we get a canonical isomorphism A/(A(Z>) + F) ~ (A(tf c - L>) n F ) v which in particular implies the Serre duality; (5) Cohomology groups H° and H1 are all finite dimensional vector spaces. Thus in particular, h°(C, D) - hl(C, D) = h°(C, D + p)-

hl{C, D +

p)-l.

This then implies the Riemann-Roch h°(C, D) - h\C, D) = d(D) -(g-

1).

As it stands, at least for the algebraic part, i.e., the part on cohomology groups, the above theory may be completely understood in terms of adelic language. (1') For any divisor D on C, define its associated cohomology groups by H°(X,Oc(D))

= A(D)r\F,

and

H\C,D)

:= A/A(D)

+ F;

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(2') There is the following commutative 9-diagram T,(D), whose rows and colums are all exact: 0

0

1 0 -»

A(£>)nF

0 -»

F

1 -•

A(D)

->

A

1

i

I 0 -•

F/A(D)HF

0

i -»

A(D)/A(L>) n F

->

A/F

1 ->

A/A{D) -+

1

-* 0 -» 0

i A/A(£>) + F

i

i

1

0

0

0.

-•

0

So, we get the exact sequences 0 —» if0\C,Oc(D))^F^F/A(D)r F - -> 0 and 0 -» F/A(L>) n F -» A / A ( D ) -^Hl{C,Oc(D))^^ (3') For any p £ C, there exists a natural morphism from S(D) to £ ( D + p). Thus, by the five-lemma, based on the fact that F and A are the same in these two 9-diagrams, we obtain the exact sequences 0 -» A(£>) n F -> A(£> + p) n F -» A(F> + p) n F/A(D)

n F -> 0

A(F»+p)nF/A(F>)nF >-> A(Z?+p)/A(D) -» A ( F > + p ) + F / A ( D ) + F 0 -» A(L>+p)+F/A(£>)+F -» A/A(£>)+F -» A / A ( D + p ) + F -> 0; (4') Residue pairing works in terms of adelic language as well by the self-dual property of A. Thus, what is left is to count the cohomology groups in order to have a cohomology theory using only adelic language (which satisfies duality and the Riemann-Roch). Generally speaking, to find a reasonable way to count is very difficult. In Grothendieck's pure algebraic approach, this is based on the fact that all coherent sheaves are locally finitely generated. With adelic language, in general dimensions, we do not know how to do it. However for certain special kinds of adelic ring A, particular for

Geo-Arithmetic: C. The Riemann

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353

curves, we do have a limited success. To explain it, as an example, consider the simplest case for which the constant field k is a finite field say F g . Recall that with respect to the natural topology on A, F is discrete and A(D) is compact. Thus in particular, H°(C,Oc{D)) = A(D) n F is finite. Similarly, since A(D) is compact and A/F is compact so A(D) + F/F is again compact. But k = Fq is a finite fields, so compactness implies finiteness, and the number of elements in a finite dimensional space is simply q to the power of the corresponding dimension. Thus, with repsect to the natural Haar measures on the associated groups induced from that on A, Vol(A(L>)) - qh°(c,oc(D)) . V o l ( A (£>)

+

F

/F)

and Vol(A/F) = ) +

F/F).

Therefore, q

h?(C,Oc{D))-h^C,Oc(D))

=

VolA(-P) Vol(A/F)'

Easily by definition, VolA(L>) = qd^D\

Vol(A/F) = q9~\

so we obtain an analytic proof of the Riemann-Roch theorem h\C, Oc(D)) - hl(C, Oc{D)) = d{D) -{g-

1).

C.3.2. Chevalley's Linear Compacity The above discussion for curves over finite fields is not valid for curves over general fields, since even the space is finite dimensional, it is not finite itself. To overcome this difficult, Chevalley introduced his notion of linear compacity. Next, we indicate how Chevalley's idea works. (Instead of recalling all the details, we indicate what are the essential points involved. For details, we recommend the reader to consult Iwasawa's Princeton lecture notes.) (0) There exists the 9-diagram as above;

L. Weng

354

(1) (Additive Structure) Among subquotient groups of the adelic ring are topological spaces called discrete objects and linearly compact objects. Moreover, all groups used in our 9-diagram are supposed to be locally linearly compact, i.e., they are either extensions of discrete objects by linearly compact objects, or extensions of linearly compact objects by discrete objects, or simply generated by finitely many discrete objects and linearly compact objects. In particular, A is selfdual and locally linearly compact; (2) Discrete objects and linearly compact objects are dual to each other. Thus, if an object is both locally linearly compact and discrete, it is then isomorphic to a finite dimensional k vector space, and hence whose dimension makes sense; (3) (Multiplicative Structure) Under the multiplication, A is selfdual. As a direct consequence, we get the duality. (4) The count may be proceeded as follows to offer the RiemannRoch: (i) By definition, dim(A(0) f~l F) = 1 and dim(A/A(0) + F) - g; (ii) By counting local contribution, [A(D + p) : A(D)] — d(D + p) — d(D) = 1, despite the fact that A(D) and A(D + p) themselves cannot be counted; (iii) By definition, [A(D + p) : A(D)} = dim(A(L> +

p)/A{DJ)

(iv) The fundamental theorem of isomorphisms for groups implies that dim(A(Z> + p)/A(I>)) = dim(A(£> + p) + F/A{D) + F) + dim(A(D + p) n F/A(D)

D F)

= dim(A/A(D) + F) - dim(A/A(D + p) + F) + dim(A(D + p) n F) - dim(A(L>) n F) = (h°(C,D + p) - h\C,D

+ p)) - {h°{C,D) -

h\C,D)).

C.3.3. Adelic Approach in Geometric Dimension Two Now let us consider the situation in dimension two. Here in Grothendieck's theory, with a proper definition of cohomology groups

Geo-Arithmetic:

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355

and a suitable way of count, we can prove a (weak) Riemann-Roch theorem and duality using the adjunction formula, a one-dimensional Riemann-Roch and a long exact sequence of cohomology groups resulting from a short exact sequence of sheaves 0 -» 0{D)

-» 0(D + C)^

0{D)\C

-> 0

for a divisor D and a regular curve C on the surface, and a residual discussion. (See e.g. [Har].) On the other hand, just as for curves, there is a two-dimensional cohomology theory in adelic language. There are two parts: (I) Algebraic construction of cohomology groups and some associated structural exact sequences; (II) Analytic Count of cohomology groups in (I). We start with algebraic construction. In algebraic geometry, this is essentially given by Parshin as a by-product of his study on residues. (We now also have its higher dimensional generalization of Beilinson.) As above, we here only indicate the main points. The interested reader may consult Parshin's original paper for the details. So for a surface S with function field F , denote by A its ring of adeles. Inside A are two subrings, which we denote by Ao and A i , respectively. Similarly, for a divisor D on S, introduce its associated subgroup A(D) as in curve case. Then we obtain t h r e e 9-diagrams: (a) 0

0

i A(D)n A 0 n A i I Ao n Ai 1 A0nAi/A(D)nA0nAi

i >-. A(D) i >-> A i >-> A/A(D)

0

i -» -» -»

A(D)/A(D)n

A 0 n Ai I A/Ao n Ai i A/A(£>) + Ao n Ai

i

i

I

0

0

0

for which isomorphisms A(D)/A(D)

n ( A 0 n A i ) ~ A(£>) + A 0 n A i / A 0 n A i

Ao n A i / A ( D ) n Ao n A i ~ A ( D ) + ( A 0 n A i ) / A ( D )

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L. Weng

are used; (b)

A(D)n

0

0

I

4 A(D) n Ai

A0nAi

> — •

I

0

1 —»

A(D)nAi A(D)n A 0 n A !

I

4

A(D)nA0 i

> — •

A(D)nA0 A(D)n A0nAj

>—>

A{D)f]

Ao + Ai 4

—»

A(D)n

AQ+AX

A(D)nA0

4

A(D)n Ao+Ai A(D)nAi

—»

A(D)n Ao+Ai A(D)nA0+A(D)nAi

i

4

i

0

0

0

for which isomorphisms A(L>)nAi/A(£>)n(A 0 nAi) ~ A(L>)nAo + A(C)nAi/A(L>)nA 0 ,

A(D)nA0/A(L>)n(A0nAi) ~ A(L>)nA0+A(D)nA1/A(i?)nAi are used; and (c) 0

0

1

1

A(£>)n A o + A i

4

Ao + Ai

)—>

A(D)

>—>

A

> — •

A/A(D)

1

4

0

4

—w

A(D)/A(D)n Ao + Ai

—»

A/ A 0 + Ai

—+>

A / A ( D ) + Ao + Ai

1

Ao + Ai /A(£»)n Ao + Ax

1

1

0

0

4

4 i

0

for which isomorphisms A(D)/A(D)

n (A 0 + A x ) ~ A(D) + (A 0 + A i ) / ( A o + Ax)

(Ao + Ai)/A(I>) n (Ao + Ax) ~ A(D) + (A 0 + Ai)/A(£>) are used. Now, define the cohomology groups Hl, i = 0,1,2 by setting H°(S,D):=A(D)n(A0nA1); Hl{S, D) := A(D) n (Ao + A i ) / A ( D ) n A 0 + A(D) n A i ;

Geo-Arithmetic:

C. The Riemann

Hypothesis

357

and H2(S, D) := A/A(£>) + (A 0 + A i ) . Moreover, by working with D + C for regular curve C on S indtead of D, we get another set of three 9-diagrams, to which there are natural morphisms from the three 9-diagrams for D above to those for D + C. As a direct consequence, by a snake chasing, we arrive at the following long exact sequence of cohomologies, which as stated above, plays a key role in the induction process: 0 -> H°(S, D) -> H°(S, D + C)^ - » H \ S , D ) -> H\S,D 2

2

H°{C, D + C\c)

+ C) -> H\C,D

-»H (S, D) -» H (S, D +

+ C\c)

C)^0.

In this way, provided that a good counting process is available, we then are able to give an adelic approach to the weak Riemann-Roch in dimension two by using the Riemann-Roch for curves with the help of the adjunction formula. Thus the problem left is how to count. However we fail badly at this point. Say, in Parshin-Beilinson, this problem is 'solved' by claiming that adelic cohomology groups are the same as these resulting from Gerothendieck's theory and consequently we can count them. Motivated by Chevalley's theory on linear compacity for adelic rings of algebraic curves, we suggest to study the terms used in the above algebraic discussion. The key is the self-dual property of A. In fact, we have the following canonical isomorphisms: (a) Afr ~ A i , A f ~ A 0 , and A(Z?) ± = A(KS - D), where Ks denotes a canonical divisor of S. In particular, (Ao + A ^ - A ^ A a ,

(AJOAJ)

1

^

A0 + Ai.

(b) Duality between H°(S, D) and H2(S, Ks - D) (A(£>) n A 0 n A i ) v ~ A/(A(£>) n A 0 n A i ) X ~ A/AiD)1-

+ A^ + Ai ~ A/A(KS

-D)

+ (A0 + A1),

358

L. Weng

(c) Note that A 0 D A i = F is nothing but the function field of S. Moreover, algebrically A(£>) n (A 0 + Ai)/A(Z?) n A 0 + A(Z>) n A x ~ A 0 n (A(D) + Ai)/(A(JD) n A 0 + A 0 n Ai) ~ A x n (A(D) + A 0 ) / ( A ( D ) n A ! + A 0 n A i ) ; (d) Duality between if 1 (5, £ ) and if 1 (5,

Ks-D):

( A 0 n (A(£>) + Ai)/(A(£>) n A 0 + A 0 n A i ) ) v ~ (A(£>) n A 0 + A 0 n A 1 ) ± / ( A 0 n (A(D) +

Ax))^

~ (A(tf s - D) + Ai) n (A 0 + A i ) / ( A i + A ( K 5 - D) n Ai)

~ A(/srs - £>) n (A0 + Ai)/A(# s - P) n A0 + A(i>) n Ai. Thus, to attack (II), the analytic structure aiming at a reasonable count, we should introduce a gec-ari theory for surfaces which is compactible with (a), (b), (c) and (d) above, similar to that of linear compactness of Chevalley used in proving the Riemann-Roch in dimension one. For examples, there should be a notion of geo-ari compactness, and if a space is both discrete and geo-ari compact, it should be of finite dimensional; moreover, the duality should transform discrete spaces to geo-ari compact speces and vice versa. R e m a r k . The above program proves to be very valuable. Say along with the proposal outlined above, we successfully use Tate's thesis to provide a beautiful geo-ari cohomology theory over 1+0 dimensional bases, i.e., over number fields. For details, please refer to B.2 of this program. Also we expect that an adelic cohomology for 1+1 dimensional bases will be established shortly, while we have confidence on the theory over 2+0 dimensional bases up to Riemann-Roch. (Keep in mind that the main part of the Program was created around 19992000 and that the present one is a revised version in which important new developments are added.)

Geo-Arithmetic: C. The Riemann

Hypothesis

359

C.4. Riemann Hypothesis in Rank Two C.4.1. Non-Abelian Zeta Functions and Eisenstein Series C.4.1.1. Non-Abelian Zeta Functions for Number Fields Let K be an algebraic number field (offinitedegree n) with A x the absolute value of its discriminant. Denote by OK the ring of integers. For a fixed positive integer r G N, denote by MF,T the moduli space of semi-stable C?x-lattices of rank r with dp, the associated (Tamagawa type) measure (induced from that on GL). Recall that for each A E MK,TI the associated 0-th geo-arithmetical cohomology h°{K, A) is

(J2^p(-^\\x^l-2^\\x^l))

h\K,A) := log ^•x€A

^


CT:C

' '

where x = (x^aeSoo a n ( l (/9
(e*°<™>->) • ( 8 -»p V< " (A) MA), m . ) > 1,

&,(.) := /

or the same, ^K,r(s)

(«***>-.) • ( e - f *B 1.

:= (4)' • /

Theorem. ([We6]) (1) (Meromorphic Continuation) £x,r(s) *s well-defined when Re(s) > 1 and admits a meromorphic continuation, denoted also by £K,T{S), t° the whole complex s-plane; (2) (Functional Equation) £fir,r(l ~ s) = £ K > ( S ) I (3) (Singularities &: Residues) £,x,r(s) has two singularities, all simple poles, at s — 0,1, with the same residues

Vol(M.K,r([AK])),

r

where M.K,r {[&•%•]) denotes the moduli space of rank r semi-stable r

Ox-lattices whose volumes are fixed to be Aj^.

360

L. Weng

C.4.1.2. Space of O^-Lattices via Special Linear G r o u p s Recall that an O^-lattice A consists of a underlying projective Ox-module P and a metric structure on the space V = A ®z R = YlaeS V = CL(K) the class group of K. With this said, following Minkowski, we obtain a natural embedding for P: p •- 0%~1] © a -> K ( r ) w (M ri x C r 2 ) r Qi ( R r ) n X ( C ) ^ , which is simply the space V = A (gig R above. As a direct consequence, our lattice A then is determined by a metric structure on V = Fresco V°> o r t h e s a m e > o n ( R r ) n x (C r ) r 2 . It is well known that all these metrices are parametrized by the space (GL(r,R)/0(r))n x

(GL{r,C)/u{r))T2.

Next let us shift our discussion from GL to SL. For this, let us start with a local discussion on C^-lattice structures: For complex places r, by fixing a branch of the n-th root, we get GL(r,C)/u(r)

S (SL(r,C)/sC/(r)) x R;;

while for real places a, we have GL(r,E)/0(r)

2 (j5L(r,R)/SO(r))

x R^.

Consequently, metric structures on V — litres ^ — ( R r ) r i x (C r ) r 2 are parametrized by ( ( 5 L ( r , R ) / 5 0 ( r ) ) r i x (sL{r,C)/SU{r)X2

J x (R;)ri+r2.

Furthermore, when we work with C^-lattice strucures on P, i.e., with the space A = A(P), from above parametrized space of metric

Geo-Arithmetic: C. The Riemann structures o n F = Ilexes ^ ' w e automorphism group AutoK{0^~ to get

Hypothesis

361

nee(

i t o a ^ s o factor out GL(P), the ^ffia)ofO^~ ' © a a s C^-modules,

GL(P (SLir^/sO^y1

x ( 5 L ( r , C ) / 5 t / ( r ) y 2 j x (R^)\ri+r 2

Hence we want (a) to study the structure of the group Aut^ K {OK~ ' © a) in terms of SL and units; and (b) to see how this group acts on the space of metric structures ( S L ( V , R ) / S 0 ( r ) ) r i x ( S L ( r , C ) / S t / ( r ) ) r j x (R^)\r\+T2 View A u t o K ( 0 ^ Aut0jf(Of

_1)

' © a) as a subgroup of GL(r,K).

Then,

©fl) a\

/

= { A G GL(r, A-) n

0*

det

Va- 1 ••• a"1 Introduce subgroups GL+(r,R)

OK

AeUK

J

:= Ig 6 GL(r,R)

: detg >

OJ and 0 + ( r ) := IA e 0 ( r , R ) : det# > o | , and the subgroup AutJ (C?£-1^ © o) of A u t o K ( 0 ^ - 1 ^ © 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 A u t O K ( O j - 1 ) f f i a ) \ ( ( G L ( r , R ) / 0 ( r ) ) r i x (GL(r,C)/C/(r)) r 2 ) and A u t + x ( ^ - 1 ) f f i a ) \ ( ( G L + ( r , R ) / 0 + ( r ) ) r i x (GL(r,C)/t/(r))rz).

L. Weng

362

With all this, we are ready to shift further to the special linear group SL. It is here that Dirichlet's Unit Theorem plays a key role. Recall that for a unit e G UK> (a) diag(e,..., e) G A u t 0 i r ( 0 ^ _ 1 ) © a); and (b) det diag(e,..., e) = er G UrF := {e r : e 6 UK}So to begin with, note that to pass from GL to SL over K, we need to use the intermediate subgroup GL+. Accordingly, introduce a subgroup UK of UK by setting [/+ := {e G UK • ea > 0,V<7 real} so as to get a well-controlled subgroup UK := U^ fl UK: by Dirichlet's Unit Theorem, the quotient group 1^^/(1/^- fl UK) is finite. With this said, next we use U^ fl UK to decomposite the automorphism group AutoK(Ox ©a). Thus, choose elements m,. ..,u^rtK)

G UK

such that < [ui],..., [u^K)] f gives a complete representatives of the finite quotient group U^/\U^

fl UK), where fJ,(r,K) denotes the

cardinality of the group U^/(U^ fl UK). Set also SL{0%~1) © a) := SL(r, K) n GL(o£~ 1 } © a). Lemma. ([We7]) There exist A\,...,

A^r^

*n

GL+(oJ"1) © a)

suc/i £/ia£ (i) det Aj = Ui for i = 1 , . . . , fi(r, K); and (ii) < A\,...,

A^^x) \ is a completed representative of the quotient of _1)

Aut+ K ( o £ © o) modulo SL(0%~1) © a) x fe+ • d i a g ( l , . . . , 1) V That is to say, for automorphism groups, (a) A u t i (Oft © a) is naturally identified with the disjoint union u

i=l

-^

SL^-^OoJx^+.diagCl,...,!))!;

Geo-Arithmetic:

C. The Riemann

Hypothesis

363

and, consequently, (b) The Otf-lattice structures A(P) on the projective O^-module P — OK ® o are parametrized by the disjoint union U^M({SL{0%-

1 )

0 a)\((5L(r,M)/50(r))ri

x (sL(r,C)/SU(r))r2)^

x (\UK n ^ ( R ; ) '

1

" ^ ) J.

Therefore, to understand the space of O^-lattice structures, beyond the spaces SL(r,R)/SO(r) and SL(r,C)/SU(r), we further need to study (i) the quotient space \UK D J7^|\(R^ J

, which is more or less

standard (see e.g. [Ne] or [We7]); and more importantly, (ii) the quotient space 5 L ( 0 ^ " 1 ) e a ) \ M 5 L ( r , ] R ) / 5 ' ( 9 ( r ) ) r i x (sX(r,C)/S£/(r))

7-2

C.4.1.3. Non-Abelian Zeta Functions and Epstein Zeta Functions Choose integral C^-ideals ai = OK,fl2>• • •, Q-h such that the ideal class group CL(K) is given by \ [oi],..., [ah] >, and that any rank r projective OK-module P is isomorphic to P^ for a certain i, 1 < i < h. Here, P 0 := P r j 0 := 0%~ ' © a for a fractional O^-ideal a. Introduce the partial non-abelian zeta function ^K,r;a(s) associated to a by setting

&„(.):- f JMK,r(a)

(e^.'«- 1 ).( e -)- ,0£V< " (AI «A),ReW>l V

J

\

J

where M.K,r{&) denotes the part of the moduli space of semi-stable (^-lattices whose corresponding points lie in (sL{0%-l)®a)\((SL(r,R)/SO{r)y

x(\uknu^\\(R*+r^).

x

(SL{r,C)/SU{r)j2}

364

L. Weng

Accordingly, for Re(s) > 1, define the completed Epstein zeta function EK,r-,a(s) by EKMS)

:

=(-"

¥ r

(y))

n

T

• {M-rirs))'2

• [(iV(a) • A £ ) '

— \\x\\ r

xe(A\{o})/[/+ F

"*

With this, from the discussion in 4.1.2, by applying the Mellin transform and using the formula TOO

e

o

_AtMB

dt

._JL

1

„(

S

ts— = — • A B . r — t B \B

(whenever both sides make sense,) we obtain the following Proposition. ([We7]) (1) (Decomposition) The rankr non-abelian zeta function of K admits a natural decomposition h

i=l

(2) (Non-Abelian Zeta = Integration of Epstein Zeta) The partial rankr non-abelian zeta function ^F,r;a(s) of K associated to o is given by an integration of a completed Epstein type zeta function: ZF,TM

= O ^

2

•/

r EKtT,a{s) dfi,

Re(a) > 1.

C.4.2. Rank Two ©^-Lattices: Stability and Distance to Cusps C.4.2.1. Upper Half Plane On the upper half plane fi :— {z = x + iy E C,x E R,y E K+}, the natural hyperbolic metric is given by the line element ds2 := dx Xdy with volume form d\i := dx^V. Consequently, the associated hyperbolic Laplace-Beltrami operator can be written as 2

• d2 < ^ + ^ » -

2fd A : = y

Geo-Arithmetic: C. The Riemann

Hypothesis

365

The group SL(2, R) naturally acts on H via:

"*=-£T?

VM=(«5)6SK2,R), « K .

The stablizer of i = (0,1) € H is equal to 50(2) := {A e 0(2) : detA = 1}. Since this action is transitive, we can identify the quotient SL(2,R)/SO(2) with H with the quotient map induced from SL(2,R)-»K, g^g-i. H admits the real line R as its boundary. Consequently, to compactify it, we add on it the real projective line P*(R) with oo = Naturally, the above action of SL(2, R) also extends to PX(R) via a b c d

ax + by ex + dy

C.4.2.2. U p p e r Half Space The 3-dimensional hyperbolic space may be written as H : = C x ] 0 , o o [ = Uz,r) = {(x,y,r)

: z = x + iy € C,r 6 R + |

: x,y eR,r

eR*+Y

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 + r j

where 2 = x + iy, j = (0,0,1).

We equip H with the hyperbolic metric coming from the line element ds2 := x + \ + r with hyperbolic volume form dfx := dxAd^Adr ^ Consequently the hyperbolic Laplace-Beltrami operator associated to the hyperbolic metric ds2 is simply 0/ d2 A ~ r I 2 \dx

1

d2 d2\ d 1 2 2 I — r—. dy dr / dr

The natural action of SL(2, C) on H and on its boundary PX(C) may be described as follows: We represent an element of P 1 (C) by

366

L. Weng where x, y € C with (x, y) /

matrix M = (a

(0,0). Then the action of the

, J <S SX(2,C) on PX(C) is denned to be a b c d

ax + by ex + dy

Moreover, if we represent points P 6 H as quaternions whose fourth component equals zero, then the action of M on i is defined to be P ^ M P := (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) G H in SL(2,C) is equal to SU(2) := {A e 17(2) : deb A = 1}. Since the action of SL(2,C) on H is transitive, we obtain also a natural identification EI ~ SL(2, C)/SU(2) via the quotient map induced from SL(2, C) -» H, g^g-j. C.4.2.3. Rank Two C^-Lattices: Upper Half Space Model Identifying H with SL(2, R)/SO(2) and H with SL(2, using the discussion in C.4.1, we conclude that MK,2-AN(*)

• &K] - (SL(OK

e o)\ (wri x wrA)

C)/SU(2),

,

where as usual ss means the subset consisting of points corresponding to rank two semi-stable (9/<-lattices in SL(OK

0 a)\UsL(2,R)/SO(2)yi

x

(SL{2,C)/SU{2)Y

More precisely, if the metric on OK © a is given by g = (ga)aeSoo with ga G SL(2,Ka), then the corresponding points on the right hand side is g(ImJ) with ImJ := (i^Vl\j^), i.e., the point given by (g
Geo-Arithmetic: C. The Riemann

Hypothesis

367

C.4.2.4. Cusps As such, the working site becomes the space SL(OK®O)\ I F 2 ) . Here the action of SL{OK

\HTl x

© a) is via the action of SL(2, K)

on Hri x HP"2. More precisely, K2 admits natural embeddings K2 ^-> (w* x CrA

~ (R 2 ) r i x (C2)7"2 so that 0K®a naturally embeds into

(R2)ri x (C 2 ) r 2 as a rank two (^-lattice. As such, SL(OK © a) acts on the image of OK © a in ( R 2 ) n x (
:=

(oo( ri ),oo( r2 )). As usual, via fractional linear transformations, SL{2,R) acts on P^M), and 5L(2,C) acts on P ^ C ) , hence so does SL(2, K) on Being a discrete subgroup of SL(2, R)n x SL(2, C) r 2 , for the action of SL(OK®O) on P^-KT), we call the corresponding orbits (of SL{OK® a) on Fl(K)) the cusps. With this, we have the following fundamental result rooted back to Maa/3. Cusp and Ideal Class Correspondence. ([We7]) There is a natural bijection between the ideal class group CL(K) of K and the cusps Cv ofT = SL(OK © a) acting on HTx x W2 given by CT -

CL(K),

0Ka

+ a(3

Easily, one checks that the inverse map IT 1 is given as follows: For b, choose a^, /?(, € K such that OK • a& + a •/?{, = b; With this, then II

([b]) is simply the class of the point

in SL(2, OK ©

L. Weng

368 a ) \ IP1 (if).

Moreover, there always exists M

SL{2, K) such that M

a

• oo =

P_

a

a

P (3"

a

A

C.4.2.5. Stablizer Groups of Cusps Recall that under the Cusp-Ideal Class Correspondence, there are exactly h unequivalent cusps rji, i = 1, 2 , . . . , h. Moreover, if we write the cusp f] := rji = ""? for suitable o^, 0i € K, then the associated [Pi J ideal class is exactly the one for the fractional ideal OKOii + af3i =: bj. Denote the stablizer group of ry in SL(OK © <*) by Fv. L e m m a . ([We7]) The associated 'lattice' for the cusp r\ is given by ab~ 2 . Moreover, A-^A

=

Set r ; := J A (^

Tr, =

u 0

u

z 1 I : u € UK, Z € ab - 2

^ A-1 : z e a b ~ 2 j , Then

>T'„x^A H „°-Kl:"€[4

Note also that componentwise, I

_1 I z = - ^ = u2£- So, in

practice, what we really get is the following decomposition

r^ = r^ x uK with

Geo-Arithmetic:

C. The Riemann

Hypothesis

369

With this, we are ready to construct a fundamental domain for the action of I\, C SL(OK © a) on Hri x I F 2 . This is based on a construction of a fundamental domain for the action of r ^ on Hri x ET2.

More precisely, with an element A = ( "

a

) e

SL(2,K)

(always exists!), we have i) A • oo =

_ ; and

ii) The isotropy group of rj in A~1SL(OK © a)A is generated by -2 translations T I—> r + z with 2 € a b and by dilations r 1—> u r where « runs through the group C/^-. (Here, we use A, a, /?, b as running symbols for Ai, OL{, fa, bj := OKOH + afa.)

Consider then the map Hn x IF 2 ^ J M^0+r2 defined by (zi,...,

zn; Pu ..., Pr2) ^ (Im(zi),..., I m ( z n ) ; J {Pi),...,

J{Pr2))

where if z = x + iy & H resp. P = z + rj £ H, we set Im(z) = y resp. J(-P) = r. It induces a map (A-1

x H P ) -> ^ \ R ; i 0 + r a ,

• r„ • A)\(H^

which exhibits yA~l • Tv • Aj\ U

K\1SL>o"r2

with

fiber

the

\/Hri x IF 2 } as a torus bundle over

n = n + 2r 2 dimensional torus (w1

x

C " 2 ) / a b - 2 . Having factored out the action of the translations, we only have to construct a fundamental domain for the action of U\ on R>o~r2. For this, we look first at the action of U\ on the norm-one hypersurface S := iy € M>o"r2 : N(y) = 1 \. By taking logarithms, it is transformed bijectively into a trace-zero hyperplane which is isomorphic to the space R r i + r 2 - 1 S

l£f

y->

R

r1+r2-l .

=

f

(oi>

^

a n + r 2 )

(logyi,...,logj/ri+r2J,

e

R r 1+ r 2

. £

fl.

=

Q

\

370

L. Weng

where the action of UK on S is carried out over an action on R r i + r 2 - 1 by the translations aj — i * a,i + logs^. By Dirichlet's Unit Theorem, the logarithm transforms UK into a lattice in R r i + r 2 _ 1 . Accordingly, the exponential map transforms a fundamental domain, e.g., a fundamental parallelepiped, for this action back into a fundamental domain S j ^ for the action of UK on S. The cone over Sj/2 , that is, K>o • Sj/2 c R>o~r2, is then a fundamental domain for the action of 'K UK on Rl^f r 2 . Denote by T a fundamental domain for the action of the translations by elements of a b - 2 on R r i x C 2 , and set ReZ(zi,...,2ri;.Pi,...,Pr2J := (ite(si), • • •, Re(zri); Z(PX),...,

Z{Pr2))

with Re(z) := x resp. Z(P) := z if z — x+iy 6 H resp. P = z + rj E H, then what we have just said proves the following Theorem. ([We7]) A fundamental domain for the action of on 7iri x W2 is given by

A^T^A

E := IT E HT1 x W a : ReZ (r) 6 T, ImJ (r) e R > 0 • S ^ A For later use, we also set T^ := A~l • E. C.4.2.6. Fundamental Domain Guided by Siegel's discussion on totally real fields [S] and the discussion above, we are now ready to construct fundamental domains for SL(OK e a)\(nri x ET 2 ). As the first step, we generalize Siegel's 'distance to cusps'. For this, recall that for a cusp r\ = a € P 1 ^ ) . b y t h e Cusp-Ideal Class Correspondence, we obtain a natural ideal class associated to the fractional ideal b := OK • a + a • ft. Moreover, by assuming that a, f3 are all contained in OK, as we may, we know that the corresponding stablizer group T^ is given by

A - 1 • rv • A = | 7 = (H ^

er:ueUK,ze

ab- 2 },

Geo-Arithmetic:

C. The Riemann

Hypothesis

371

where A E SL(2, K) satisfying Aoo = r] which may be further chosen in the form A =

a

a

€ SL(2, K) so that 0Kf3* + or1 a* = b~\

Now for r = (zu ..., zn; Pi,..., N(T)

Pr2) E Hri x EF 2 , set

:=JvYlmJ(r)) ri

r2

= Hlm(Zi)

Then for all 7 = ( °

• J J J(Pj)2 = (yi • • • y r i ) • («i • • •«;

J ) € 5L(2, A"),

iV"(lmJ(7-T)) =

JV(ImJ(r)) ||JV(cr + d)||2'

(*)

(Note that here only the second row of 7 appears.) Moreover, define the reciprocal distance n(r), r) from the point r E Hri x IF 2 to the cusp r\ —

inF\K)

by

/ i ( r ? , r ) : = i v ( a - 1 - ( < ^ a + <^) 2 ) Im(z1)---Im(zri)-J(P1)2.-.J(Pr2)2 ' n i i i \{-f&>Zi + a « ) | 2 n - L i I K - ^ P i + aW))||= 1 JV(ImJ(r)) 2 A^(ob" ) ||7V(-/?r + a ) | | 2 ' Lemma 1. ([We7]) (i) \x is well-defined; (ii) /x is invariant under the action of SL{OK MT»7> 77")

= Kv, T)>

v

© a). ITiat is to say,

7 G <5L(0^ © a).

(iii) There exists a positive constant C depending only on K and a such that if H(T},T) > C and /i(ry',r) > C for r € Hri x HP"2 and rj, rf &¥l(K), thenr] = r)'.

372

L. Weng

(iv) There exists a positive real number T := T(K) depending only on K such that for r 6 TC1 x IfF2, there exists a cusp r\ such that KV,T)>T. Now for the cusp r\ — a

E P ^ i f ) , define the 'sphere of influ-

ence' of rj by Fv := { r € Hri x IF 2 : fi(rj, r) > /*(»/, r), VT/ € P 1 ^ ) } Lemma 2. ([We7]) T/ie aciion of SL(OK © a) ^ the interior F° o / i ^ reduces to that of the isotropy group T^ of r), i.e., if T and JT both belong to F®, then JT = r. Consequently, we arrive at the following way to decompose the orbit space SL{OK © o)\ [Hri x HT2 J into h pieces glued in some way along pants of their boundary. Theorem. (We]) Let iv : I\A Fv ^ denote the natural map. Then SL(OK

SL(OK

© a ) \ ( f t r i x HT2) =

© o)\ ( V 1 x

wA

Uvir,(rv\Fvy

where the union is taken over a set of h cusps representing the ideal classes of K. Each piece corresponds to an ideal class of K. Note that the action of Tv on Hri x IF 2 is free. Consequently, all fixed points of SL(OK © <*) on Hri x W2 lie on the boundaries of Fv. We can give a more precise description of the fundamental domain, based on our understanding of that for stabilizer groups of cusps. To state it, denote by rji, ..., rjh inequivalent cusps for the action of SL(OK © a) on Hri x HT2. Choose AVi e SL(2,K) such that A^oo = rji, i = l,2,...,h. Write S for the norm-one hyper1 7-2 surface S := < y e R^Q " : N(y) = 1 \, and SV2 for the action of UK on S. Denote by T a fundamental domain for the action of the

Geo-Arithmetic:

C. The Riemann

translations by elements of ab

2

Hypothesis

373

on W1 x C 2 , and

E := | r e Hri x HP : ReZ (r) G T, ImJ (r) € K > 0 • S ^ | a fundamental domain for the action of A^T^A^ on Hn x HF2. The intersections of E with ^ ( i ^ ) are connected. Consequently, we have the following T h e o r e m . ' ([We7]) (1) Dn :— A~xYi (~\ Fv is a fundamental domain for the action ofT^ on F^; (2) There exist ati, ..., a^ € SL(OK © a) such that U^ =1 a(£) % ) is connected and hence a fundamental domain for SL(OK © a). That is to say, a fundamental domain may be given as Sy U J'i(Fi) U • • • U ThiXh) with SY bounded, Ti{Yi) = At • T^Yi) and ^(Yi)

:= IT G W 1 x ET2 : ReZ(r) G S, ImJ(r) G R > T • ^u2K\

Moreover, all .Fi(Yi)'s are disjoint from each other when Yi are sufficiently large. C.4.2.7. Stability and Distances to Cusps Define now the distance of r to the cusp n by d(v,r\)

:= —.

r > 1.

Then we are ready to state the following fundamental result, which exposes a beautiful intrinsic relation between stability and the distance to cusps. T h e o r e m . ([We7]) The lattice A is semi-stable if and only if the distances of corresponding point TA G riri x W2 to all cusps are all bigger or equal to 1. C.4.2.8. Moduli of Rank Two Semi-Stable C^-Lattices For a rank two O^-lattice A, denote by TA G H1"1 X HP"2 the corresponding point. Then, by the previous subsection, A is semistable if and only if for all cusps n, d(r], T\) := ( lT •, are bigger than

374

L. Weng

or equal to 1. This then leads to the consideration of the following truncation of the fundamental domain V of SL(OK © a)\(Hri x wA:

For T > 1, denote by VT := IT E V : d(r),TA) > T ~ \ Vcusp 77}.

The space T>T may be precisely described in terms of V and certain neighborhood of cusps. To explain this, we first establish the following Lemma. For a cusp r\, denote by XV(T) := { r € Hri x W2 : rf(t/,r) < T " 1 } . TAen

forT>\,

xm(T)nxm(T)^m

&m = m-

With this, we are ready to state the following Theorem. ([We7]) There is a natural identification between (a) the moduli space of rank two semi-stable OK-lattices of volume N(a)Ax with underlying projective module OK © o; and (b) the truncated compact domain T>\ consisting of points in the fundamental domain T> whose distances to all cusps are bigger than 1. In other words, the truncated compact domain T>\ is obtained from the fundamental domain V of SL(OK © a ) \ (nn x wA by delecting the disjoint open neighborhoods U uf=1 Fi{\) associated to inequivalent cusps 771,7/2,... , ^ , where Fi(T) denotes the neighborhood of r\i consisting of r G V whose distance to rji is strictly less thanT-1. For later use, we set also VT := V\ U U? = i^i(r),

T > 1.

Geo-Arithmetic: C. The Riemann

Hypothesis

375

C.4.3. Epstein Zeta Functions and Their Fourier Expansions C.4.3.1. Epstein Zeta Function 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 -Er,a;A(s) associated to an Ojf-lattice A with underlying projective module Pa = OK~ © a

to be W * ) : = ( 7 r - ^ r ( ^ ) ) n ( 2 7 r — r ( r S ) ) r 2 • (iV(a)A K ,

A-

T

x \\rs

t—1 x€O^- 1) eo/(7+ ]r ,x^(0,...,0)

where UKr := {e r : e € UK, er G UK\

A

= UK n UrK. For example,

note that in the case r = 2, U^2 = UK, we have &,«A(>)

:=(>r-r(S))'''(2»-2'r(2«))''1 • (jV(a)AK)'

'xeo eo/Lf2., E ^(o,o) ||x|| " "

2s A '

K

e

x

Set for Re(s) > 1 also E2,a(r,s) := ( 7 r - T ( S ) ) r i ( 2 7 r - 2 T ( 2 S ) ) r 2 • (jV(a)A*)* y.

/Ar(ImJ(T A ))V

(^,!;)€OK®a/(/x,(i,!/)/(0,0) V I F - T + 2/|| / Lemma. For a ran& iw;o OK-lattice A = (OK © a, PA), denote by T\ the corresponding point in the moduli space I F 2 ) . Then £2,a;A(s) = E2,a(rA, s). C.4.3.2. Fourier Expansions For simplicity, introduce the standard Eisenstein series by setting

376

L. Weng

Then the completed one becomes E2Ar, s) = ( 7 r " s r ( S ) ) r i (27r- 2 s r(2s)) r 2 • (iV(a)A*) S • E2>a(r,s). (*) As before, for the cusp r\ = fa A = (R

a

P

, choose a (normalized) matrix

a*\ I 6 <SX(2, F) such that and that if b = OK a + a/?, then

+ aa* = b - 1 . Clearly, Aoo = 77, and moreover,

OKP*

^ ' ' V = { (J T)=" e -*-"}• Since E2,a(T, s), and thus £2,0(7") «)? is 5rL(C^:©a)-invariant, E2,a(T, s) is rjj C 5L(0/fffia)-rnvariant. Therefore E^a^ 7 ", s) is ob _2 -invariant, that is, E2,<X(AT, S) is invariant under parallel transforms by elements of a b - 2 . Consequently, we have the Fourier expansion E2,a(Ar, s)=

]T

(V (imj(r), s) • e ^ ^ M ) ,

w'G(ab- 2 ) v

where (ab~ 2 ) v denotes the dual lattice of a b - 2 . Thus, if we use Q to denote a fundamental parallolgram of ab~ 2 in R n x C 2 , then a

-'(ImJ(T)'s):=Vol(^) ^Q

\ II

~

II

S

/

CT.K

r

.c

T h e o r e m . ([We7]) We /taue the following Fourier expansion E2,a(Ar, s) =(({*-% +

2s) • Niab'1)'28

2 Vr.lCnh-2^ Vol(ob)

• iV(ImJ(r)) s /UC^2s "( 7r2 ) ' JV(c)

^Z ^ , eft c ao + a Tl

r(s)

'

/ V

n 2s

-!

^

iV(ImJ(r)) l - s

Geo-Arithmetic: C. The Riemann 1

x-^

Hypothesis

P 27ri(w',|>

377 / o-jr-s \

n

c dje*

(

o^sl

/ | 2 s - l \ ri

•n^a-i(2T|« , |rrr).

rUv

O.JB.

^

'

'

r:C

C.4.4. Explicit Formula for Rank Two Zetas: Rankin-Selberg & Zagier Method 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 by Zagier, 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

SSL

E2,a{T,s)d(J,(T).

Here T>T is the compact part obtained from the fundamental domain V for SL(OK®a)\('Hri xwA by cutting off the cusp neighborhoods defined by the conditions that the distance to cusps is less than T _ 1 . (Recall that, as such, T>i is simply the part corresponding to semistable lattices.) For doing so, first formulate the integration

JJf IT>T

where

(AKE2,a(T,s))d^T)

L. Weng

378 with a :

~ V°\dxl

+

dyl)>

T :

~

TT

\dx*

+

dy?

+

&r%)

*'8rT'

(For the time being, by an abuse of notation, we use AK to denote the hyperbolic Laplace operator for the space W1 x EP"2, not the absolute value of the discriminant of K which accordingly is changed to DK.) Note that Aa(y«)

= s{s - 1) • i£,

and

A r (r? s ) = 2s(2s - 2) - r2Ts

by the SL-invariance of the metrics, we conclude that AA-

(E2,a(r, *)) = ( n • (s(s - 1)) + r 2 • (2s(2s - 2 ) ) ) • £ 2 , 0 ( T , S).

Hence jjj

£ 2 , a (r, 8 ) dfx(tau) = ^

+

_ 4 ^ JjJ

AKE2^T,s)d^{r).

On the other hand, using Stokes' Formula, we have Iff J J JvT

= fJJv

AKE2
(AKE2,a(r,s))-ld^T)

- JJJ = JJJ^

=

^2,a(r, s) • ( A * I )

f ( A ^ 2 , , ( T , s)) • 1 - E2ta(r, s) • ( A * I ) J dfx(r)

rr

(dE2,a{T,s)

/•/•

dE2,a(T,8)

JJdD(T)

d^r)

9v

~

aA

Geo-Arithmetic: C. The Riemann

Hypothesis

379

where ^ is the outer normal derivative, dfi is the volume element of the boundary &DTTheorem. ([We7]) Up to a constant factor depending only on K,

Consequently, we have the following Theorem'. ([We7]) The rank two non-abelian zeta function of K is given by

ZKM = ^ A s—1

1

-^izilA? s

Re(s)

£K,2{S)

> x.

C.4.5. Zeros of Rank Two Non-Abelian Zeta Functions for Number Fields Let us first start with Suzuki's weak result and then give Lagarias' unconditional result [LS]. Meanwhile, for an independent parallel work, please go to Ki's beautiful paper [Ki]. Proposition. (Suzuki) If the Riemann Hypothesis for the Riemann zeta function holds, then all zeros of £Q,2(S) lie on the critical line Re(s) = \. Proof. Let F(z) = -z(^-+2iz)

with

Z{s) = s(l - s)f(s).

Lemma. (1) F(z + f) - F(z - \) = iz{\ + Az2) £ Q j 2 (i + iz). (2) Assume the RH, then all zeros of F(z + f) — F(z — \) are real. In particular, then the RH implies that £ Q , 2 ( | + zi) admits only real zeros. Proof. (1) Simple calculation. (2) Clearly, F{z) is an entire function of order 1, so there are constants A, B such that

/•w=.*"*. n ( I - J ) - « P ( | ) . p:F{p)=0

X

F /

V F /

380

L.

Weng

Note t h a t essentially, p are zeros of the completed Riemann zeta but transformed from z to \ + 2iz. Hence, by the RH, all p are real. Moreover, since for i g l , F(x) takes only real values. Hence, constants A and B are both real. Now let ZQ — XQ + iyo be a zero of

F z+

( i) - F v -1)= iz{1+Az2) ^2 (I+iz

Then ZQ — 0 a n d / o r ZQ is a zero of £ Q , 2 ( ^ + iz) since ^ , 2 ( 5 + iz) admits simple poles at z = ± ^ i . In any case, F(ZQ + f) = ^(^o — | ) - By taking absolute values on both sides, ,A+B(zQ+{) .

n

,A+B(z0-i)

ZQ +

• exp .

n

^0 -

I

ZQ + j

• exp

zo

Since S € R and pn e R (which is obtained by the RH as said above), we get

M ^ o - ^ + ^o + l) 2 ' Thus if j/o > 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, hence completes the proof. W i t h this in mind, note t h a t in the proof above, the RH was used to ensure t h a t p are real, which have the effect t h a t then in the calculation for the exponential factor exp

(¥)•

the ratio

exP(^i exp

*o-j

gives us the exact value 1. T h a t is to say, this factor of ratio of exp's does not contribute. Lagarias then offers his own unconditional result as a part of his understanding of de Branges's work. T h e trick is to use the functional equation. So instead of working on individual pn in the product, we

Geo-Arithmetic: C. The Riemann

Hypothesis

381

may equally use the functional equation to pair p 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

F(z) = eA+Bz- fl

(l-^\

p:F(p)=0

where ]\ means that p's are paired or grouped as above. Form here, it is an easy exercise to deduce the following result of (Suzuki and) Lagarias. Theorem. ([LS]) All zeros O/£Q,2(S) Zie on the line Re(s) = 5. Proof. Alternatively, as above, we have

,A+B{zv+i) . JT U _ ^±1) P

J

. e x p (*> + 2 \

P

^J JB- H<^ —» i-\i , .r nr (1\ _ £ ° _ L i ) . e x p ( £ » _ i ) Since J5 e R, so if we can take care of the factors exp ( 2°

4

J and

exp ( ^—— J 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 ^ + 4 =

,^

and

j ~ + j^jj = Ti- ft a r e a ^ r e a ^ s ' hence, the same prove as above works. As observed in [We7], the above method works for the functions £ Q 2 ( S ) as well, provided that T > 1. In particular, we then obtain the following RH in Rank Two. ([We7]) All zeros of rank two non-abelian zeta functions for number fields are on the critical line Re(s) = ^ •

382

L. Weng

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M. Rapoport &; Th. Zink, Period spaces for p-divisible groups. Annals of Mathematics Studies, 141. Princeton University Press, Princeton, NJ, 1996.

[Sch]

A.J. Scholl, An introduction to Kato's Euler systems, in Galois Representations in arithmetic algebraic geometry, London Math. Soc. Lecture Note Series 254, (1998) 379-460

[S]

J.-P. Serre, Algebraic Groups and Class Fields, GTM 117, Springer (1988)

[Sel]

C.S. Seshadri, Space of unitary vector bundles on a compact Riemann surface. Ann. of Math. (2) 85 (1967) 253-336.

[Se2]

C. S. Seshadri, Fibres vectoriels sur les courbes algebriques, Asterisque 96, 1982

[Sh]

I. Shafarevich, Lectures on minimal models and birational transformations of two dimensional schemes. Notes by C. P. Ramanujam. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, No. 37 (1966)

[Shi]

G. Shimura, Introduction to the arithmetic theory of automorphic functions, Iwanami, (1971)

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[Sie]

C.L. Siegel, Lectures on the geometry of numbers, notes by B. Friedman, rewritten by K. Chandrasekharan with the assistance of R. Suter, Springer-Verlag, 1989.

[Stl]

U. Stuhler, Eine Bemerkung zur Reduktionstheorie quadratischer Formen, Arch. Math. (Basel) 27 (1976), no. 6, 604-610

[St2]

U. Stuhler, Zur Reduktionstheorie der positiven quadratischen Formen. II, Arch. Math. (Basel) 28 (1977), no. 6, 611619

[T]

J. Tate, Fourier analysis in number fields and Hecke's zeta functions, Thesis, Princeton University, 1950

[Ti]

J. Tilouine, Deformations of Galois representations and Hecke algebras, Published for The Mehta Research Institute of Mathematics and Mathematical Physics, 1996.

[To]

B. Totaro, Tensor products in p-adic Hodge theory. Duke Math. J. 83 (1996), no. 1, 79-104.

[Wl]

A. Weil, Generalisation des fonctions abeliennes, J. Math Pures et Appl, 17, (1938) 47-87

[W2]

A. Weil, Sur les courbes algebriques et les varietes qui s'en deduisent, Herman, Paris (1948)

[W3]

A. Weil, Sur les formules explicites de la the'orie des nombres. (French) Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 3-18.

[W4]

A. Weil, Basic Number Theory, Springer-Verlag, 1973

[W5]

A. Weil, Adeles and algebraic groups. With appendices by M. Demazure and Takashi Ono. Progress in Math., 23. 1982.

[Wei]

L. Weng, f2-admissible theory II: Deligne pairings over moduli spaces of punctured Riemann surfaces, Math. Ann 320 (2001), 239-283

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[We2]

L. Weng, Stability and New Non-Abelian Zeta Functions, Number theoretic methods (Iizuka, 2001), 405-419, Dev. Math., 8, Kluwer Acad. Publ., 2002.

[We3]

L. Weng, Refined Brill-Noether Locus and Non-Abelian Zeta Functions for Elliptic Curves, Algebraic geometry in East Asia (Kyoto, 2001), 245-262, World Sci. 2002

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L. Weng, Non-abelian zeta functions for function fields, Amer. J. Math. 127 (2005), no. 5, 973-1017

[We5]

L. Weng, Non-abelian class field theory for Riemann surfaces, Appendix to this paper, this volume

[We6]

L. Weng, Non-Abelian L-Functions for number fields, available at http://xxx.lanl.gov/abs/math.NT/0412008, to appear

[We7]

L. Weng, Rank Two Non-Abelian Zeta and its Zeros, available at http://xxx.lanl.gov/abs/math.NT/0412009, submitted

[We8]

L. Weng, Automorphic Forms, Eisenstein Series and Spectral Decompositions, this volume

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L. Weng, Non-Abelian L-Functions for number fields II, in preperation

[Z]

Zagier, D. The Rankin-Selberg method for automorphic functions which are not of rapid decay. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28(3), 415-437 (1982)

Lin W E N G Graduate School of Mathematics Kyushu University Fukuoka 812-8581, Japan Email: [email protected]

L. Weng

392 Appendix:

Non-Abelian CFT for Function Fields over C In this appendix, using what we call a micro reciprocity law, we complete Weil's program [W] for non-abelian class field theory of Riemann surfaces. 1. Refined Structures for Tannakian Categories Let T be a Tannakian category with a fiber functor u : T —> Verc, where Verc denotes the category of finite dimensional C-vector spaces. An object t £ T is called decomposable if there exist nonzero objects x, y 6 T such that t — x © y. An object is called indecomposable if it is not decomposable. If moreover every object x of T can be written uniquely as a sum of irreducible objects x = xi © X2 © • • • © xn, then T is called a Tannakian category with unique factorization. Usually, we call Xj's the irreducible components of x. A Tannakian subcategory S of a unique factorization Tannakian category T is called completed if for i £ § , all its irreducible components X{ 's in T are also in S. § is called finitely generated if as an abelian category, it is generated by finitely many objects. Moreover, S is called finitely completed if (a) S is finitely generated; (b) § is completed; and (c) Aut®a>L is a finite group. 2. A n E x a m p l e Let M be a compact Riemann surface of genus g. Fix an effective divisor D = Yli=i ei^i with ei e ^ > 2 o n c e f° r all- For simplicity, in this note we always assume that (M; D) / (P 1 ; eiPi), or (P 1 ; e\P\ + e2-P2) with e\ ^ e^. (These cases may be easily treated.) By definition a parabolic semi-stable bundle ?Z:=(E=:E('2)]P1,...,PN;Fil(E\P1),...,Fil(E\pN); a n , • • • ,0-lri', • • • ',0,Ni, . . .

of parabolic degree 0 is called a GA bundle over M along D if (i) the parabolic weights are all rational, i.e., a^ € Q fl [0,1); (ii) there exist a^- e Z, fcj € Z F > 0 such that (a) (ay,/3y) = l;

,aNrN)

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393

(b) dij = ciij/Pij; and

(c) (3ij\eu for all i,j. Denote by [E] the Seshadri equivalence class associated with E. Moreover, for [E], define W.D([E]) as E(GT(E))\P, i.e., the fiber of the bundles associated with Jordan-Holder graded parabolic bundle of E at a fixed P € M° ~ M\\D\ = M\{Ph... ,PN}. Put M(M; D) := {[E] : E is a GA bundle over M along D}. Proposition. M(M;D) is a unique factorization Tannakian category and UJD : M(M; D) —> Vecc is a fiber functor. Proof. (1) By a result of Mehta-Seshadri [MS, Prop. 1.15], M{M; D) is an abelian category. Then from the unitary representation interpretation of a GA bundle, a fundamental result due to Seshadri, (see [MS, Thm 4.1], also in Step 3 of Section 5 below,) M(M; D) is closed under tensor product. The rigidity may be checked directly. (2) Since the Jordan-Holder graded bundle is a direct sum of stable and hence irreducible objects and is unique, (see [MS, Rm 1.16],) so, A4(M;D) is a unique factorization category. (3) By definition, we know that the functor ui is exact and tensor. So we should check whether it is faithful. This then is a direct consequence of the fact that M(M; D) is a unique factorization category and that any morphism between two irreducible objects is either zero or a constant multiple of the identity map.

3. Reciprocity Map In [E], choose its associated Jordan-Holder graded bundle Gr(E) as a representative. Then by the above mentioned fundamental result of Seshadri, Gr(E) corresponds to a unitary representation PGI(E) '• 7Ti(M°) -> C/(r E ), where r s denotes the rank of E(E) := E. For each element g £ 7ri(M°), we then obtain a C-isomorphism of J5(Gr(E))|p. Thus, in particular, we get a natural morphism W:7Ti(M°)-> A u t ' V p . Now note that ni(M°) is generated by 2g hyperbolic transformations Ai, Bi, ..., Ag, Bg and N parabolic transformations Si,..., SN

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satisfying a single relation A i B i A ^ B f 1 • • • AgBgA^B^Si

• • • SN = 1,

and that PGr(£)(<Sf) = 1 for a l i i = 1 , . . . , N. ([MS, §1].) Denote by J(D) the normal subgroup of 7Ti(M°) generated by S^1,..., S'jf. Then naturally we obtain the following reciprocity map W{D) : 7r1(M°)/J(L>) -» Aut®wD. 4. M a i n T h e o r e m As usual, a Galois covering it : M' —> M is called branched at most at D if (1) IT is branched at P i , . . . , PN] and (2) the ramification index e\ of points over P{ divides ej for alH = 1 , . . . , N. Clearly, by changing D, we get all finite Galois coverings of M. Non-Abelian CFT. (1) (Existence and Conductor Theorem) There is a natural one-to-one correspondence WJJ between < S : finitely completed Tannakian subcategory of M(M; D) \ and IT : M' —> M : finite Galois covering branched at most at D >; (2) (Reciprocity Law) The reciprocity map induces a natural group isomorphism Aut®(u^| s )~Gal(w.D(S)).

5. Proof Step 1: Galois Theory. By a result of Bungaard, Nielsen, Fox, M. Kato, and Namba, (see e.g., [Na, Thms 1.2.15 and 1.3.9],) we know that the assignment (n : M' —> M) H-> 7r*(7ri(M'\7r _1 {Pi,... ,P/v})) gives a one-to-one correspondence between isomorphism classes of finite Galois coverings TT : M' —> M branched at most at D and finite index (closed) normal subgroups K — K(ir) of 7Ti(M°) containing J(D). Moreover, we have a natural isomorphism Gal(7r) ~ ni(M°)/K(ir)(cz {TXI{M°)/J(D)) / (K(TT)/J(D))\ Thus the problem is transformed to the one for finite index normal subgroups of

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7Ti(M°) which contain J(D), or the same, finite index normal subgroups of G{D) := m(M0)/J(D). Step 2: Tannakian Category Theory: Geometric Side. Consider now the category T(D) of equivalence classes of unitary representations of G(D). Clearly, T(D) forms a unique factorization Tannakian category, whose fiber functor u>(D) may be defined to be the forget functor. Now fixed once for all a representative p-£ : G(D) —> U{r^) for each equivalence classes [E]. (The choice of the representative will not change the essentials below as the resulting groups are isomorphic to each other.) Let S be a finitely completed Tannakian subcategory of T(D). Then as in the definition of reciprocity map above, we have a natural morphism G(D) ^ Aut®w . Denote its kernel by K(§). Then, by definition, G(D)/K(S) is a finite group, and K(§) = npi e §ker/9£. Since S is finitely completed, there exists a finite subset of S, denoted by S = {[Si], [E2], • • •, [Et]}, which generates S as a completed Tannakian subcategory. Set [Eo] := ©*=1[Ej]. Then for any [E] € S, ker(pE0) C ker(pp]), since § is generated by S. Also, by definition, ker(p So ) = n* = 0 ker( P E J. Thus K(S) = ker(p Eo ). Therefore, for any [E] e S, px = p~z o II(D;S) where II(£>;S) : G(D) —> G(D)/K(§>) denotes the natural quotient map and p E is a suitable unitary representation of G(D)/K(S). Now set S := {[ps] '• [E] G S}. S is a finitely completed Tannakian subcategory in Uni(G(D)/K(S)), the category of equivalence classes of unitary representations of G(D)/K(§). In particular, since the unitary representation p^0 of G(D)/K(S) maps G(D)/K(S) injectively into its image, for any two elements gug2 E G(D)/K(§), fafa) + fe0(g2). With this, by applying the van Kampen Completeness Theorem ([Ka]), which claims that for any compact group G, if Z is a subset of the category Uni(G) such that for any two elements gi,g2, there exists a representation pgitg2 in Z such that pgi,g2(gi) 7^ pgi,32(^2)1 then the completed Tannakian subcategory generated by Z is the whole category Uni(G) itself, we conclude that S = Uni(G(D)/K(§)). But as categories, § is equivalent to S, thus by the Tannaka duality, (see e.g., [DM] or [Ta]) we obtain a natural isomorphism Aut®a;|§ ~

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G{D)/K(S). On the other hand, if K is a finite index (closed) normal subgroup of G(D). Set S := Uni{G(D)/K) with the fiber functor ug. Compositing with the natural quotient map II : G(D) —> G(D)/K we then obtain an equivalent category S consisting of corresponding unitary representations of G(D). S may also be viewed as a Tannakian subcategory of Uni(G(D)). We next show that indeed such an § is a finitely completed Tannakian subcategory. From definition, Aut®w|§ ~ Aut®o>g which by the Tannaka duality theorem is isomorphic to G(D)/K. So it then suffices to show that S is finitely generated. But this is then a direct consequence of the fact that for any finite group there always exists a unitary representation such that the group is injectively mapped into the unitary group. Step 3: A Micro Reciprocity Law. With Steps 1 and 2, the proof of the Main Theorem is then completed by the following Weil-Narasimhan-Seshadri Correspondence ([MS, Thm 4.1]) There is a natural one-to-one correspondence between isomorphism classes of unitary representations of fundamental groups of M° and Seshadri equivalence classes of semi-stable parabolic bundles over M° of parabolic degree zero. Indeed, with this theorem, the Seshadri equivalence classes of GA bundles over M along D correspond naturally in one-to-one to the equivalence classes of unitary representations of the group 7Ti(M°)/J(r>). Thus by Step 2, the finitely completed Tannakian subcategories of M.(M;D) are in one-to-one correspondence to the finite index closed normal subgroup of 7Ti(M0)/ J(D), which by Step 1 in one-to-one correspondence to the finite Galois coverings of M branched at most at D. This proves the theorem. 6. A n A p p l i c a t i o n t o Inverse Galois P r o b l e m As it stands, we may use our main theorem to see whether a finite group can be realized as a Galois group of certain coverings by constructing suitable finitely completed Tannakian subcategory of M(M;D). In particular, to relate with the so-called Inverse Galois Problem, or better, the Regular Inverse Galois Problem, via a fundamental result of Belyi, we only need to consider the finitely

Non-Abelian Class Field Theory over C

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completed Tannakian subcategories of .MO?1; A), for some effective divisors A = eoo • [oo] + e\ • [1] + eo • [0] supported on {oo, 1,0} C P 1 . An Example. By definition, a collection of objects of a finitely completed Tannakian subcategory in M(M;D) is called primitive generators if it is a smallest collection which generates the subcategory as an abelian category. Since ramification points are all fixed to be {oo, 1, 0}, to simplify the notation, set Sn:=(O;0;0;0); Eia:=(o(-l);0;l;i) S2i:=(o(-l)®0(-l);|,^,0;l,o). Then we may view E^- as parabolic vector bundles over P 1 with parabolic points oo, 1 and 0. That is to say, the bundle part is given by the line bundles or rank two vector bundle over P 1 , while the parabolic weights are given by (0; 0; 0), (0; \\ \) and ( | , | ; ^,0; ^,0) at points (oo; 1; 0). Here for rank two cases, the filtration is choosen to be the one such that Ep D Q(ei + e2) D {0} with the i-th factor gives 0{—\)p = C • ej. Proposition. Let 7£(£n, £12, £21) be the Tannakian subcategory generated by parabolic vector bundles E n , £12 and £21. Then the category7?.(£ii, Ei2,£2i) is generated by {En, £12, £21}- Moreover, Aut®w| ,

~ S3.

Remark. This proposition, via our non-abelian reciprocity law, reproves a well-known result that 53 may be realized as a Galois group of a branched covering of P 1 ramified at 00,1,0 with ramification index 3,2,2 respectively. Proof. We first need to show that all tensor products could be realized as extensions of E^-. For this, we check case by case as follows. (1) Clearly, E n ® £ y = E^;

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Weng

(2) E 1 2 ® £12 = E U . Indeed, £12 ® £12

= (
(0(-2); 0; 1; 1)

= ( © ( - 2 + 2 - l ) ; 0 ; l - l ; l - l ) = (C?;0;0;0) = £11; (3) £ 1 2 0 £ 2 i = £21- Indeed, £12 <8> £21

= ( O (-2)©O(-2 ) ; | + 0 > i + 0 ; i + i > i + 0;i + iI + 0) = (0(-2)e0(-2);|,i;l,|;l,i) = ( o ( - 2 + l)ffi0(-2 + l ) ; | , I ; I ; O ; I ; o ) = £21; (4) £ 2 i ® £21 = £21 © £12 © £11 • Indeed, £21 <8> £21

- (m o\®*> ~{

(

j

;

3

= (o(-2)e*;l

2

2 2

+

+

+

3'3

II

3'3

+

2

3'3

I II I -

+

3;2

+

2'2

+ U

'

| t l,l,| ; l,|,|,0;l,|,|,0)

= C0(-2 + l)®2 © 0 ( - 2 + 1) © 0 ( - 2 + 2); 1 + I - 1,

1-1,1-1, § ; l - l , ^ , 0 ; l - l , ± ± o ) = ^(-l) e2 ©(!?(-l)ffi(!?(0); 11,0,0; 1,1,0,0; I , I,0,o) = £2i©£i2©£n.

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Now note that the above decompositions for the tensor products are exactly the same as that for the irreducible representations of S3. So 7£(Eii, £12, £21) is equivalent to the category of finite dimensional complex representations of 53. As a direct consequence, we conclude that the structure group Aut®a;|^ ( s s s > is simply S3. This example shows that how the approach works. First we use the category of representations of a finite group to find how the tensor products of irreducible representations decompose. Then we find whether there exist corresponding semi-stable parabolic vector bundles which give an equivalent category. As such, the problem becomes a combinatorial one, since on P 1 , all vector bundles are direct sums of 0(n) for n G Z. Further examples with all dihedral groups may be easily constructed as well, since their irreducible representations are at most of rank 2. We leave this to the reader. Up to this point, naturally, we may ask to which extend this approach to the Inverse Galois Problem work over number fields K. To answer this, we come back to moduli spaces of parabolic vector bundles constructed by Narasimhan and Seshadri. Unlike existing approaches to this problem, we have enough rational points ready to use: the anti-canonical line bundles of these moduli spaces are (roughly speaking) positive. Therefore, for us, the essential problem is whether the categories generated by K-rational parabolic bundles correspond to coverings defined over K.

References [DM]

P. Deligne & J.S. Milne, Tannakian categories, in Hodge Cycles, Motives and Shimura Varieties, LNM 900, (1982), 101-228

[Ka]

E. van Kampen, Almost periodic functions and compact groups, Ann. of Math. 37 (1936), 78-91

[MS]

V.B. Mehta &; C.S. Seshadri, Moduli of vector bundles on curves with parabolic structures. Math. Ann. 248 (1980), no. 3, 205-239.

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[MFK] D. Mumford, J. Fogarty & F. Kirwan, Geometric Invariant Theory, Springer-Verlag, 1994 [Na]

M. Namba, Branched coverings and algebraic functions. Pitman Research Notes in Mathematics Series 161, Longman Scientific & Technical, 1987

[NS]

M.S. Narasimhan h C.S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface. Ann. of Math. 82 (1965), 540-567

[Sel]

C.S. Seshadri, Moduli of 7r-vector bundles over an algebraic curve. Questions on Algebraic Varieties, C.I.M.E., III, (1969) 139-260

[Se2]

C. S. Seshadri, Fibres vectoriels sur les courbes algebriques, Asterisque 96, 1982

[Ta]

T. Tannaka, Theory of topological groups, Iwanami, 1949 (in Japanese)

[W]

A. Weil, Generalisation des fonctions abeliennes, J. Math Pures et Appl, 17, (1938) 47-87

Lin W E N G Institute for Fundamental Research The L Academy

Series on Number Theory and Its Applications - Vol. 1

ARITHMETIC GEOMETRY AND NUMBER THEORY This invaluable collection aims to bring the reader to the frontier of current developments in arithmetic geometry and number theory through the works on vector bundles on curves overp-adic fields, on local gamma factors in automorphic representations, on Deligne pairings and Takhtajan-Zograf metrics, on CM-periods and on transcendence of special values of zetas over finite fields. In addition, there are lecture notes that explain basic ideas and reasons (not just the language and conclusions) behind Langlands' fundamental, yet notably difficult, works on the Eisenstein series and spectral decompositions. And finally, a brand new concept called the Geometric Arithmetic program that uses algebraic and/or analytic methods, based on geometric considerations, to develop the promising and yet to be cultivated land of global arithmetic that includes non-abelian Class Field Theory, Riemann Hypothesis and non-abelian Zeta and L Functions, etc.

6115 he ISBN 981-256-814 X

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