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 NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI Published by 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 British Library Cataloguing-in-Publication Data 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. 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. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 981-256-814-X Printed in Singapore by B & JO Enterprise 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 This page is intentionally left blank 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