Arithmetic Geometry and Number Theory

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ARITHMETIC

GEOMETRY AND

NUMBER THEORY

Editors Lin Weng & Iku Nakamura

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ARITHMETIC

GEOMETRY AND

NUMBER THEORY

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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

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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

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World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224

USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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ARITHMETIC GEOMETRY AND NUMBER THEORY

Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd.

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-256-814-X

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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 g a m m a 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 p u n c t u r e d Riemann surfaces using Deligne pair-ings and an arithmetic Riemann-Roch isometry; Werner's paper ex-plains her recent works with Deninger on vector bundles on curves over Cp; Yoshida's paper exposes his beautiful theory on C M

peri-ods; and Yu's paper studies the transcendence of special values for zetas over finite fields. All these well-prepared articles t h e n bring us to the uppermost frontiers of the current researches in Arith-metic Geometry and Number Theory. Secondly, t h e lecture notes of Weng explains basic ideas and m e t h o d s behind the fundamental yet famously difficult work of Langlands on the Eisenstein series and spectral decompositions. T h e reader will find these notes invalu-able in understanding the original theory. Finally, Weng's paper of Geometric Arithmetic outlines a P r o g r a m for understanding global arithmetic using algebraic a n d / o r analytic methods based on geo-metric 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 A r t h u r ' 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 p a p e r was presented at the Conference on L-Functions (February 18-23, 2006, Fukuoka). Partial contents of the papers of

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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, immedi-ately 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, be-tween October and November, 2005. Finally, the Program paper, of which the first version was circulated around the turn of the millen-nium, is revised significantly for this publication and is indeed the driving force for the whole project1.

The Editors

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Contents

Foreword v Preface vii On Local 7-Factors 1

D. H. Jiang

Deligne P a i r i n g s 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 29 K. Obitsu, W.-K. To and L. Weng

Vector B u n d l e s on C u r v e s over Cp 47

A. Werner

A b s o l u t e C M - p e r i o d s — C o m p l e x a n d p-Adic . . . . 65 H. Yoshida

Special Z e t a Values in Positive C h a r a c t e r i s t i c . . . . 103 J. Yu A u t o m o r p h i c F o r m s , E i s e n s t e i n Series a n d S p e c t r a l D e c o m p o s i t i o n s 123 L. Weng G e o m e t r i c A r i t h m e t i c : A P r o g r a m 211 L. Weng IX

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Dihua J I A N G

Contents

1 Introduction 1 2 Basic Properties of Local 7-Factors 5

2.1 Multiplicativity 6

2.2 Stability 6 2.3 Remarks 7

3 Local Converse Theorems 7

3.1 The case of GL„(F) 7 3.2 A conjectural LOT 9

3.3 The case of S02n+i(F) 12

4 Poles of Local 7-Factors 15

4.1 The case of G = S02„+i 17

4.2 Other classical groups 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 Gv

and the absolute Galois group r > = G a l ( F / F ) . Consider continuous homomorphisms (j) from the Weil-Deligne group Wf x SL2(C) to the Langlands dual group LG, which is admissible in the sense of

[B79]. The Gv-conjugacy class of such a homomorphism <p is called

a local Langlands parameter. The set of local Langlands parameters

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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 Lang-lands 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 inter-esting to characterize the local L-packets in general. The most well-known approach to characterize local L-packets is in terms of stabil-ity 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 irre-ducible admissible representations of G(F). They yield information about the classification theory and the functorial structures of irre-ducible admissible representations of G(F).

First, we recall the local Langlands conjecture for GLn over F,

which is proved by Harris-Taylor [HT01] and by Henniart [H00].

Theorem 1.1 ([HT01], [H00], [H93]). There is a unique

collec-tion of biseccollec-tions

recF : 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),

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and

e(s,7Ti x TT2,ip) = e(s,recF(wi) ( g i r e c F ^ ) , ^ )

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 <g> ( 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), Tecf{nv) = recir(7r)v, where V denotes the contragredient.

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 GLn(F). The local factors on the GLn(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 the-ory of local coefficients defines the local L-, e-, and 7-factors. It is expected that the local factors defined by Shahidi should be essen-tially 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

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4 D. Jiang

should be mentioned t h a t for nongeneric representations, there are cases where t h e local factors can be defined by t h e Rankin-Selberg m e t h o d ([GPSR97] and [LR05]), and also t h a t t h e work [FG99] has t h e potential t o define t h e local factors for nongeneric representa-tions, which can be viewed as the n a t u r a l extension of Shahidi's work. Of course, one may define t h e local factors by means of t h e conjectured local Langlands conjecture for G over F, and this defi-nition should b e consistent with all other defidefi-nitions.

We recall t h e local Langlands functoriality principle. Let G and

H be reductive algebraic groups defined over F. For an admissible

homomorphism Lp ([B79]) from t h e Langlands dual group LH t o t h e

Langlands dual group LG, there should be a functorial transfer p from

U{H/F) t o n ( G / F ) , which takes L-packets of H(F) t o L-packets of G(F), and satisfies t h e following conditions.

1. For any local Langlands parameter <pjj € $(H/F), Lpo<pH is a

local Langlands parameter in <&(G/F), such t h a t t h e functorial transfer p takes t h e local L-packet I I ( < £ H ) t o t h e local L-packet

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 Lp),

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 t h e

formula-tion of t h e local Langlands conjecture for GLn(F) (Theorem 1.1), t h e

functorial transfer should be characterized by t h e 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

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for all irreducible supercuspidal representations r of GLn(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 func-toriality 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 dis-cussions in this aspect.

For a given ir € H(G/F), we have two collections of local 7-factors: one is

(1.3)

{-y(s, re x r, ip) I for all r G I I ( G Ln/ 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 repre-sentations 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 7-factors.

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6 D. Jiang

2.1 Multiplicativity

For the twisted local 7-factors, one expects the multiplicativity holds. More precisely, it can be stated as follows. For an irreducible ad-missible 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 dp^ {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 = GLr x H, then one may write a = rr ® a',

and expect

(2.2) 7(0- x r, s, ip) — 7(rr x r, s, •*/>) • 7(0-' x r, s, ip).

Properties (2.1) and (2.2) are called the Multiplicativity of the lo-cal 7-factors. F. Shahidi proved in [Sh90b] the multiplicativity for irreducible generic representations n of all F-quasisplit reductive al-gebraic 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 represen-tations 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 GLm x GLn.

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

2.3 R e m a r k s

It is important to mention that E. Lapid and S. Rallis ([LR05]) deter-mines 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 usu-ally 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 sub-collection 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 super-cuspidal representation

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8 D. Jiang of G Ln i( F ) x • • • x GLnr(F) such that the representation -K can be

realized as a subrepresentation of the (normalized) induced represen-tation

(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 repre-sentations 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 ir-reducible admissible representations up to the equivalence of super-cuspidal data.

We first consider the case of irreducible supercuspidal representa-tions of GLn(F). The first local converse theorem (LCT) for GLn(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

super-cuspidal 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) with I = 1,2,..., n — 1, then 7Ti and KI are equivalent.

It follows that an irreducible supercuspidal representation ir can be determined up to equivalence by the subcollection of twisted local 7-factors

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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

irre-ducible 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) with I = 1,2,..., n — 2, then TT\ and -K2 are equivalent.

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 supercuspi-dal 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) with I = 1,2,..., [§], then -K\ and 7T2 are equivalent.

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 ex-pect 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 col-lection of twisted local 7-factors 7(3, IT X r, ip) is expected to deter-mine the irreducible supercuspidal representation IT up to the local

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10 D. Jiang L-packet. Note that all irreducible supercuspidal representations of GLn(F) are generic, i.e. have nonzero Whittaker models. It is

nat-ural 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 reduc-tive 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 quo-tient 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

non-trivial 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 ^ ) * H o mG ( F )( K , I n d ^ ( ^ ) )

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 as-sociated 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-1iti). If ip is generic, then t o ip is generic for all t G T(F). Also it is clear that

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for all t G T(F). It follows that if an irreducible admissible repre-sentation 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 SLn(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 con-jecture.

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 -T7^) are disjoint and the union of T(F)-orbits

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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 ver-sion of the local converse theorem.

Conjecture 3.7 (LCT). For any irreducible admissible generic

rep-resentations 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 GLn. We have

no strong evidence about this claim. In the next section we discuss the author's joint work with David Soudry for S02n

+i-3.3 T h e c a s e of S02n+i(F)

We review briefly here the joint work with Soudry ([JS03]) on LCT for S02

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n+i-Theorem 3.8 (LCT for S02 n+i [JS03]). Let TT and ir' be ir-reducible 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 The-orem 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 I Pm( S 02 r i+ i / A ) be the set of all equivalence classes of irreducible generic cuspidal automor-phic representations of S02n+i(A) and IIa(GL2n/A) be the set of

all equivalence classes of irreducible automorphic representations of GL2n (A). Then the Langlands functorial lifting from II9 c a (S02n+i/A) to IIa(GL2n/A) is an injective map.

The second global application is to determine the generic cuspi-dal data for an irreducible cuspicuspi-dal automorphic representations. It is a well-known theorem that every irreducible cuspidal automorphic representation of GL„(A) is generic (i.e. having nonzero Whittaker-Fourier coefficients). This follows from the Whittaker-Whittaker-Fourier ex-pansion of cuspidal automorphic forms of GLn(A). In general, we

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14 D. Jiang consider a cuspidal datum (P, a), where P is a parabolic subgroup of GLn 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 representa-tions I n d p ^ | (a) and I n d ^ / ^ (r) share the same irreducible un-ramified 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 in-duced representations Indp (^? (a) and IndQ/^? (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 dp^ ?+ (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;<r) is an invariant for irre-ducible automorphic representations o/S02n+i(A) up to near equivalence.

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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 L-functions 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

Lang-lands dual group ([B79]). Let n be an irreducible cuspidal auto-morphic representation of G(A). Langlands denned autoauto-morphic L-functions L(s, IT, p) for all finite-dimensional complex representation p of LG ([B79]). It is a theorem of Langlands that the

automor-phic 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 sub-group Ji-K of LG such that for every finite-dimensional complex

rep-resentation 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

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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 repre-sentations of the complex dual group Sp2n(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 Lang-lands problem and the local analogy of [J05]. For simplicity of dis-cussions below, it is assumed that all reductive algebraic groups con-sidered below are F-split, and the Langlands dual group LG is taken

to be the complex dual group GV(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 GV(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 def-inition 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 al-gebraic subgroup H^ o/Gv(C) such that if p is a finite-dimensional complex representation o/Gv(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)

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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 02 n+ i

Let G = S 02 n+ i be the F-split odd special orthogonal group. Then

its complex dual group is Sp2 n(C). The fundamental representations

of Sp2n(C) are the finite-dimensional irreducible complex

represen-tations associated to the fundamental weights. They can be con-structed by the following split exact sequence

(4.3) 0 - • Vgn) - • Aa(C2 n) -+ Aa~2(C2 n) -» 0,

where Aa(C2 n) denotes the a-th exterior power of C2™, the

contrac-tion map from Aa(C2 n) onto Aa"2(C2") 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

Sp2n(C) with the a-th fundamental weight. Let t be the natural

embedding of Sp2„(C) into GL2 n(C). Let A2 be the exterior square

representation of GL2n(C) on the vector space A2(C2 n), which has

dimension 2n2 — n. The composition A2 o i of A2 with t is a complex

representation of Sp2n(C). By (2.1) and by complete reducibility of

representations of Sp2n(C), we obtain

(4.4) A2 o i = P2 © lS p 2 n

where p2 is the second fundamental complex representation of the group Sp2n(C), which is irreducible and has dimension 2n2 — n — 1,

and lsp2 n is the trivial representation of Sp2 n(C).

Let r = T(TT) be the image of -K under the Langlands functo-rial transfer from S02n+i(-F1) to GL2 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) =

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18 D. Jiang and

7(s,r(7r),Aa,</0

(4-6) 7(8, 7T, pa, Ip) =

7 ( s , r ( 7 r ) , Aa-2, ^ ) '

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

repre-sentation of S02n+i{F) and pi be the second fundamental complex representation of Sp2 n(C). Then the second fundamental local 7-factors 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\ <g> • • • <g> TXr Of

S C W i ^ x . x S C W i ^ )

-2. The partition [ni • • • nr] is uniquely determined by the

irre-ducible 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 irre-ducible supercuspidal representation r o / G Lm( 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

rep-resentations of S02m+i(F) is completely determined by the ir-reducible generic supercuspidal representation IT, up to equiva-lence, namely, it is the set of irreducible generic supercuspidal

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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 02 n +i . Let n = n\ + n<i with n i , n2 > 0. Take a semisimple

element

° "\

hn2 e SP 2 n(C).

-W

Then the centralizer of sn i )„2 in Sp2n(C) is given by

H[num] = CentS p 2 n ( c )(S r i l,n 2) = Sp2 n i(C) x Sp2 n 2(C).

The standard elliptic endoscopy group associated to the partition n = n\ + n2 is

H[ni,n2] = S 02 n i +i X S 02„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 admis-sible representations of G(F), which is characterized by the stability

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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 gener-icity 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 pack-ets (L-packpack-ets), the endoscopy transfer should take the distinguished member of a local Arthur packet to its image of the Langlands func-torial 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+i-As in the global case considered in [J05], we can discuss the or-der 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 S02n+i, i-e. complete determination of the set

{ni,n2, ...,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.

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28 D. Jiang Dihua J I A N G School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA Email: dhjiang@math.umn.edu

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of P u n c t u r e d 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 (<?, N). Each point of Tg^ is a Riemann surface M° of type

(g, N), i.e., M° = M \ { P i , . . . , PAT}, where M is a compact Riemann surface of genus g, and the punctures P i , . . . , PN of M ° are N distinct points in M. We will always assume that 2g — 2 + N > 0. The Teichmuller space Tg^ is naturally a complex manifold of dimension

3g - 3 + N.

The moduli space M9<N of Riemann surfaces of type (g, N) is obtained as the quotient of T^JV by the Teichmuller modular group ModgiN, i.e., Mg<N ~ Tg,N/Modg<N := {(M; Pu F2, . . . ,PN) : M cpt

Riemann surface of genus g, Pi E M, Pi ^ Pj, } / ~is0. So M.9tN is

naturally endowed with the structure of a complex V-manifold. How-ever, M.9tN is not compact. The so-called Deligne-Mumford

com-pactification Mg^ is obtained by adding the so-called stable curves

M. Like M.g,N, M-g,N admits a ^-manifold structure. It is well

known that Abdy := M.9yN — -Mg,N is a normal crossing divisor.

The Riemann surfaces on the boundary may be understood as follows. Denote by tf7l imTg,N the boundary Teichmuller space of

Tg<N arising from pinching m distinct points. Take a point MQ G

^71 7mTSiAT. Then Mo is a Riemann surface with N punctures

P i , . . . , PJV and m nodes Q i , . . . , Qm, and Mfi := M0\ { Q i , . . . , Qm}

is a non-singular Riemann surface with N+2m punctures. Each node Qi corresponds to two punctures on MQ (other than P i , . . . , P ^ f ) . Denote the components of MQ by Ca, « = 1, 2 , . . . ,r. Each Ca is

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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 Tgai„Q, a - 1,2,..., r.

(1.2) W e i l - P e t e r s s o n M e t r i c and T a k h t a j a n - Z o g r a f 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 r1 • • • AgBga^Bg1 • 5 i 52 • • • SN = Id.

Let z\,..., ZN € R U {oo} be the fixed points of the parabolic trans-formations 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):= J2 I m( ^rV )s, zeH.

7G <s«>\r

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

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To describe the tangent and cotangent spaces at a point M of

T9)JV, we first denote by Q(M) the space of holomorphic quadratic

differentials (f> — 4>{z) dz2 on M with finite L1 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.sup2eM|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)<t>(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

pair-ing. 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 gW P and the Takhtajan-Zograf metric gTZ on Tgtw (the latter being introduced in [TZ1,2]) are defined as

follows: for X € Tg^ and p, v E HB(M), one has

. JV

3 W P( ^ , i / ) = / pup, and gTZ(p,v) = / V ^ ( - , 2 ) -/zi/p.

In particular, flTZ(M, i/) - ^ t i <?(i)(^, *) with 5W(M,«/) - JM £*(-,

2)-/iPp. We will call <jfW the Takhtajan-Zograf metric on T3}N

associ-ated to the cusp z% (or the puncture p{). It is well known that the Weil-Petersson metrics gWP is Kahlerian, non-complete ([Wol]) and

whose holomorphic sectional curvature is bounded from above by ~7r(2g-2+iv) ([W o 2l f o r N = ° a n d tW e l] for JV > 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

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N-32 K. Obitsu, W.-K. To & L. Weng

2. Line Bundles over Moduli Spaces

(2.1) Deligne Pairing Deligne pairing, a refined version of

inter-section, 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 P3. Then Cx • C2 = # C i n C2 = ^{P1,P2,P3} = 3.

Moreover, for any meromorphic function / on S, (C\ + d i v ( / ) J • C2 =

3 = C\ -C2, where div(/) is defined as the zeros minus the poles. Thus

if set {d + div(/) : V/} =: 0s( 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,1C2)B = 1Qi+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

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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

AWp := (Kn(¥1 + ••• + PJV), i ^ P i + • • • + PJV)>;

(ii) The Takhtajan-Zograf line bundle

AT 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

Grothendieck-Mumford determinant formalism and Serre duality. (See [We 1,2].)

3. Fundamental Relations on M

9

,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 ® Ab d y;

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34 K. Obitsu, W.-K. To & L. Weng ( F R II) (Weng [Wei,2])

A®N2 ^ A®(2<?-2+iV)2.

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

\8 + -)\i>A

bdy

;

( F R III) (ii) (Weng [Wel,2]) On Mg,N, N>1, 2 N \ i 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 - Ab 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 rela-tions 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 M

9

N- Arithmetic

Story

(4.1) Basic Relations For any [ ( M ; P i , . . . ,PN)] € M9,N, we may

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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 PSL2(R).

The Poincare metric on H given by ^ ^ is PSL2(R)-invariant,

and hence can be descended to T\H ~ M°. Denote this metric by

Ttyp-On 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

AWp = (#,r(Pi + • • • + PJV), - M P i + • • • +

PAT))Denote the resulting metrized line bundle by A W P -( F R IV) -(Wolpert [Wo2], Weng [Wel,2]) On Mg%N,

c i ( A w p j = — 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) ATZ-(Weng [Wei,2]) On Mg<N,

IK X 4

C I ( A T Z ) = r ^ T Z

Figure

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