Automorphic Forms and Zeta Functions: Proceedings of the Conference in Memory of Tsuneo Arakawa Rikkyo University PDF
Preview Automorphic Forms and Zeta Functions: Proceedings of the Conference in Memory of Tsuneo Arakawa Rikkyo University
Proceedings of the Conference in Memory ofHisuneo tXrakawa Siegfried Bocherer Tomoyoshi Ibukiyama Masanobu Kaneko Fumihiro Sato World Scientific Proceedings of the Conference in Memory ofl'suneo drakawa AUTOMORPHIC FORMS AND ZETA FUNCTIONS This page is intentionally left blank Proceedings of the Conference in Memory ofrfsuneo tArakawa AUTOMORPHIC FORMS AND ZETA FUNCTIONS Rikkyo University, Japan 4-7 September 2004 Editors Siegfried Bocherer Universitdt Mannheim, Germany Tomoyoshi Ibukiyama Osaka University, Japan Masanobu Kaneko Kyushu University, Japan Fumihiro Sato Rikkyo University, Japan \[p World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONGKONG • 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. AUTOMORPHIC FORMS AND ZETA FUNCTIONS Proceedings of the Conference in Memory of Tsuneo Arakawa 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 981-256-632-5 Printed in Singapore by World Scientific Printers (S) Pte Ltd In memory of Tsuneo Arakawa (1949-2003) This page is intentionally left blank PREFACE Tsuneo Arakawa, an eminent researcher in modular forms in several vari ables and zeta functions, passed away suddenly on October 3, 2003 by rup ture of a cerebral aneurysm. This is the proceedings of the "Conference on Automorphic Forms and Zeta Functions" in memory of Tsuneo Arakawa, which was held at Rikkyo University, Tokyo, on September 4-7, 2004. This volume is dedicated to his memory. Most of the papers are based on the lectures given at the conference. Some of the authors, such as Don Zagier and Aloys Krieg, who could not take part in the conference, contributed their papers at our solicitation. Arakawa's works are reviewed mainly in the first article of this volume. The articles of S. Hayashida and H. Narita may also serve as reviews of Arakawa's achievements on skew holomorphic Jacobi forms and automor phic forms on Sp(l,q), respectively, since they report new results of their own obtained on the basis of Arakawa's results. Many of the other papers are also more or less related to Arakawa's works. The example most notable in this respect is the article of Ibukiyama and Katsurada. The Koecher- Maass type Dirichlet series considered in that paper was first introduced by Arakawa almost 30 years ago, and attracted attention only very recently. We believe that this collection of papers illustrates both the fruitfulness of Arakawa's works and current trends in modular forms in several variables and related zeta functions. The conference was supported financially by the Grant in Aid for Sci entific Research from JSPS (Japan Society of Promotion of Science) (B) No. 16340012 (Principal Researcher F. Sato) and partly by the same JSPS Grant (A) No. 13304002 (Principal Researcher T. Ibukiyama). We thank JSPS for the support. We also thank all the speakers at the conference and the more than 100 participants. September, 2005 The Editors: Siegfried Bocherer (Universitat Mannheim) Tomoyoshi Ibukiyama (Osaka University) Masanobu Kaneko (Kyushu University) Fumihiro Sato (Rikkyo University) vu This page is intentionally left blank CONTENTS Preface v Tsuneo Arakawa and His Works 1 Estimate of the Dimensions of Hilbert Modular Forms by Means of Differential Operators 20 Hiroki Aoki Marsden-Weinstein Reduction, Orbits and Representations of the Jacobi Group 29 Rolf Berndt On Eisenstein Series of Degree Two for Squarefree Levels and the Genus Version of the Basis Problem I 43 Siegfried Bocherer Double Zeta Values and Modular Forms 71 Herbert Gangl, Masanobu Kaneko and Don Zagier Type Numbers and Linear Relations of Theta Series for Some General Orders of Quaternion Algebras 107 Ki-Ichiro Hashimoto Skew-Holomorphic Jacobi Forms of Higher Degree 130 Shuichi Hayashida A Hermitian Analog of the Schottky Form 140 M. Hentschel and A. Krieg The Siegel Series and Spherical Functions on 0(2n)/(0(n) x 0(n)) 150 Yumiko Hironaka and Fumihiro Sato IX
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