**ARITHMETIC **

**GEOMETRY AND **

**NUMBER THEORY **

### Editors Lin Weng & Iku Nakamura

**ARITHMETIC **

**GEOMETRY AND **

**NUMBER THEORY **

**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

### 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 **

**\[p World Scientific**

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

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A catalogue record for this book is available from the British Library.

**ARITHMETIC GEOMETRY AND NUMBER THEORY **

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

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ISBN 981-256-814-X

### 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 *

### 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

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

### 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 **

**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

**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 TT*2* € U(GLn2/F), *

*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

**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

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 L*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
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.

**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 = 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 *
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.

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

**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 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 **

*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

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,ir**2 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 , 7T*2 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 *

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

*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 -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 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 , 7T*2* 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 S02n

**+i-3.3 T h e c a s e of S0**2n+i(F)

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

**n+i-Theorem 3.8 (LCT for S0**2 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 P*m( 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 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 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

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 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;<r) is an invariant for *
*irre-ducible automorphic representations o/S02n+i(A) up to near *
*equivalence. *

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

**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/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) *

*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 = S 0*2 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 A*2* o i of A*2* 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) = *

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 Sp*2 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> TX****r**** 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 L*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,..., 7r*r*} 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 *

*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 GL*2*j (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 , n*2* > 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] = Cent*S 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\ + n*2 is

*H[ni,n2] = S 0*2 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 *

**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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Math. Conf. P r o c , 3, Weizmann, Jerusalem, 1990, 279-289.

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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

### 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, } / ~is*0*. 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 7mTSi*AT. Then Mo is a Riemann surface with N punctures *

*P i , . . . , PJV and m nodes Q i , . . . , Qm, and Mfi := M*0*\ { 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

**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 tf*7l* ^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. *

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.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)<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 g*W 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 *

**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/} =: 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,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 **

*curve 7r: C9}N -> M9,N, together with N sections P;'s of 7r. Hence, *

*for x = [ ( M ; P i , . . . , Pjy)] € JW*5|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 C*9)./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 *

AW*p := (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;

**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 M**9 = Mg,Q,

*\8 + -)\i>A*

*bdy*

*; *

**( F R III) (ii) (Weng [Wel,2]) On M**g,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 - 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 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)] € M**9,N, we may

*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 PSL*2(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 Th*yp 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