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In 1938, at the Institute for Advanced Study, E Hecke gave a series of lectures on his theory of correspondence between modular forms and Dirichlet series. Since then, the Hecke correspondence has remained an active feature of number theory and, indeed, it is more important today than it was in 1936 when Hecke published his original papers. This book is an amplified and up-to-date version of the former author's lectures at the University of Illinois at Urbana-Champaign, based on Hecke's notes. Providing many details omitted from Hecke's notes, it includes various new and important developments in recent years. In particular, several generalizations and analogues of the original Hecke theory are briefly described in this concise volume.
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Hecke’s Theory of Modular Forms and Dirichlet Series 6438 tp.indd 1 11/19/07 3:19:37 PM This page intentionally left blank Hecke’s Theory of Modular Forms and Dirichlet Series Bruce C Berndt University of Illinois at Urbana-Champaign,USA Marvin I Knopp Temple University,USA World Scientific NEW JERSEY 6438 tp.indd 2 • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TA I P E I • CHENNAI 11/19/07 3:19:37 PM 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. Photo of Erich Hecke (page vi) courtesy of Vandenhoeck & Ruprecht. HECKE’S THEORY OF MODULAR FORMS AND DIRICHLET SERIES Copyright © 2008 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-13 978-981-270-635-5 ISBN-10 981-270-635-6 Printed in Singapore. LaiFun - Hecke's Theory of modular.pmd 1 1/30/2008, 1:25 PM November 17, 2007 11:23 WSPC/Book Trim Size for 9in x 6in In Memory of Hans Rademacher, the Father of our Mathematical Family v bcb November 17, 2007 vi 11:23 WSPC/Book Trim Size for 9in x 6in Hecke’s Theory of Modular Forms and Dirichlet Series Erich Hecke bcb November 17, 2007 11:23 WSPC/Book Trim Size for 9in x 6in Preface in Two Acts with a Prelude, Interlude, and Postlude Prelude Thirty-seven years have elapsed between the first version and the present version of this monograph. We begin with the first author’s slightly edited preface from his first version. We then provide a lengthier second preface composed by the second author. The Original Preface These notes are part of a course on modular forms and applications to analytic number theory given by the first author at the University of Illinois at Urbana-Champaign in the spring of 1970. The existing accounts [47], [48], [87] of Hecke’s theory of modular forms and Dirichlet series are somewhat concise. Therefore, it has been our intention to present a more detailed account of a major portion of this material f