Arithmetic Geometry And Number Theory

411 Pages·2006·14.06 MB·English

by  Lin Weg

Series on Number Theoiy and Its Applications Vol. 1
ARITHMETIC
GEOMETRY AND
NUMBER THEORY
Editors Lin Weng & Iku Nakamura
World Scientific
ARITHMETIC
GEOMETRY AND
NUMBER THEORY
Series on Number Theory and Its Applications ISSN 1793-3161
Series Editor: Shigeru Kanemitsu (Kinki University, Japan)
Editorial Board Members:
V. N. Chubarikov (Moscow State University, Russian Federation)
Christopher Deninger (Universitat Munster, Germany)
Chaohua Jia (Chinese Academy of Sciences, PR China)
H. Niederreiter (National University of Singapore, Singapore)
M. Waldschmidt (Universite Pierre et Marie Curie, France)
Advisory Board:
K. Ramachandra (Tata Institute of Fundamental Research, India (retired))
A. Schinzel (Polish Academy of Sciences, Poland)
Vol. 1 Arithmetic Geometry and Number Theory
edited by Lin Weng & Iku Nakamura
Series on Number Theory and Its Applications Vol. 1
ARITHMETIC
GEOMETRY AND
NUMBER THEORY
Editors
Lin Weng
Kyushu University, Japan
Iku Nakamura
Hokkaido University, Japan
\[p World Scientific
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ARITHMETIC GEOMETRY AND NUMBER THEORY
Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd.
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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
v
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Preface
Mathematics is a part of our culture. As such, the works presented
here serve the purposes of developing branches involved, popularizing
existing theories, and guiding our future explorations.
Accordingly, the collection of this volume may be roughly divided
into three categories. More precisely, first, Jiang's paper deals with
local gamma factors that appeared in the theory of automorphic
representations; Obitsu-To-Weng's paper investigates the intrinsic
relations between Weil-Petersson and Takhtajan-Zograf metrics on
moduli spaces of punctured Riemann surfaces using Deligne pair
ings and an arithmetic Riemann-Roch isometry; Werner's paper ex
plains her recent works with Deninger on vector bundles on curves
over C ; Yoshida's paper exposes his beautiful theory on CM peri
p
ods; and Yu's paper studies the transcendence of special values for
zetas over finite fields. All these well-prepared articles then bring
us to the uppermost frontiers of the current researches in Arith
metic Geometry and Number Theory. Secondly, the lecture notes
of Weng explains basic ideas and methods behind the fundamental
yet famously difficult work of Langlands on the Eisenstein series and
spectral decompositions. The reader will find these notes invalu
able in understanding the original theory. Finally, Weng's paper of
Geometric Arithmetic outlines a Program for understanding global
arithmetic using algebraic and/or 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 Arthur's periods;
and an intersection approach to the Riemann Hypothesis.
While various important topics are selected, all papers share
common themes such as the Eisenstein series, stability and zeta
functions.
Jiang's paper was presented at the Conference on L-Functions
(February 18-23, 2006, Fukuoka). Partial contents of the papers of
vn
viii Preface
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
lrrhis project is partially supported by JSPS.
Contents
Foreword v
Preface vii
On Local 7-Factors 1
D. H. Jiang
Deligne Pairings over Moduli Spaces of Punctured
Riemann Surfaces 29
K. Obitsu, W.-K. To and L. Weng
Vector Bundles on Curves over C 47
p
A. Werner
Absolute CM-periods — Complex and p-Adic . . .. 65
H. Yoshida
Special Zeta Values in Positive Characteristic . . .. 103
J. Yu
Automorphic Forms, Eisenstein Series and Spectral
Decompositions 123
L. Weng
Geometric Arithmetic: A Program 211
L. Weng
IX