Hecke's Theory of Modular Forms and Dirichlet Series PDF
Preview Hecke's Theory of Modular Forms and Dirichlet Series
Hecke’s Theory of Modular Forms and Dirichlet Series 6438 tp.indd 1 11/19/07 3:19:37 PM TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk 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 • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI 6438 tp.indd 2 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 November17,2007 11:23 WSPC/BookTrimSizefor9inx6in bcb In Memory of Hans Rademacher, the Father of our Mathematical Family v November17,2007 11:23 WSPC/BookTrimSizefor9inx6in bcb vi Hecke’sTheory of Modular Forms and DirichletSeries Erich Hecke November17,2007 11:23 WSPC/BookTrimSizefor9inx6in bcb Preface in Two Acts with a Prelude, Interlude, and Postlude Prelude Thirty-seven years have elapsed between the (cid:12)rst version and the present versionof this monograph. We begin with the (cid:12)rst author’s slightly edited preface from his (cid:12)rst 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 (cid:12)rst author at the University of Illinois at Urbana-Champaignin the springof 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 for those who are unfamiliarwiththisbeautifultheory. ReadersalreadyfamiliarwithHecke’s theory will (cid:12)nd little that is new here. The (cid:12)rst author is especially grateful to Ronald J. Evans for providing anewproofof afundamentalregionforHecke’smodulargroups,whichwe present here. We express our thanks also to Elmer Hayashi for a detailed reading of the manuscript and to Harold Diamond for several suggestions. Bruce Berndt, May, 1970 & May, 2007 Interlude The(cid:12)rstauthormailedacopyofhisnotesonHecke’stheoryofmodular forms and Dirichlet series to Dr. Ju(cid:127)rgen Elstrodt, who at that time was at Universita(cid:127)t Mu(cid:127)nchen. He responded with about a dozen pages of detailed vii November17,2007 11:23 WSPC/BookTrimSizefor9inx6in bcb viii Hecke’sTheory of Modular Forms and DirichletSeries comments, which, after an undeservedly quick reading, were deposited in the (cid:12)rst author’s(cid:12)le cabinet forapproximatelythirty-(cid:12)ve years,until they weredusted o(cid:11) and sent to the second author forincorporationin the new version. We hope that it is not too late to thank Elstrodt for his kind suggestions and patience. The Second Preface In the spring of 1971, I received the following letter, dated June 17. Since it is brief, I quote it in full. Underseparatecover,Iamsendingyouacopyofsomelec- turenotes,\Hecke’stheoryofmodularformsandDirichlet series."Iwouldappreciateanycomments,corrections,crit- icisms,orsuggestionsthat youmayhave. Thankyouvery much. Most sincerely, (signed) Bruce Toestablishthecontextofthisletter, Irecallthatinthespringof1938 Erich Hecke gave an important series of lectures at the Institute for Ad- vanced Study, Princeton, on his correspondence theory published in 1936. The notes from these lectures, taken by Hyman Serbin and produced in planographed form by Edwards Brothers of Ann Arbor, received only lim- ited circulation. To my knowledge there are only a few copies extant in mathematics libraries (for example, the University of Illinois at Urbana- Champaign) and private collections of professional mathematicians. In1970BerndtproducedasetoflecturenotesbaseduponHecke’snotes, but with the addition of many details omitted from Hecke’s originalnotes. The more extensive notes, too, had only limited circulation. For the past thirty-(cid:12)ve years I have employed both sets of notes to in- troduce graduate students to the Hecke theory and the broader theory of modular/automorphicforms. DuringthistimemyPh.D.studentsandoth- ersfrequentlyaskedwhyBerndt’snoteshadneverbeenpublished. Because we areconvincedthat the reactionsof these students re(cid:13)ect agenuine use- fulnessofthesenotestothemathematicalcommunity,wehaveundertaken the task of publishing this book based upon them, corrected and modi(cid:12)ed where necessary, and expanded to include some of the many new develop- mentsinthetheoryduringthepastdecades,aswellasrelevantearlierwork not previously included. We stress that the Hecke correspondence theory hasremainedanactivefeatureofresearchinnumbertheorysincethe1930s November17,2007 11:23 WSPC/BookTrimSizefor9inx6in bcb Preface ix and,infact, itsimportanceisperhapsbetterunderstoodtodaythanitwas in 1936. The (cid:12)rst six chapters of this book follow the organization of Berndt’s originalnotes, hence that of the (cid:12)rst partof Hecke’snotes aswell. Beyond this, we have added two completely new chapters based upon work done since1970anduponearlierworknotoriginallyunderstoodtoliewithinthe circle of ideas surrounding Hecke’s correspondence theorem. Chapter7featuresBochner’simportantgeneralizationofHecke’scorre- spondence theorem and some closely related results. Chapter 8 is devoted to the great variety of identities related to the Hecke correspondence the- ory (but not explicitly present in that theory) that have been developed over the years. Among others, these identities are due to S. Ramanujan, N. S. Koshliakov, G. N. Watson, A. P. Guinand, K. Chandrasekharan, R.Narasimhan,andBerndt. SomeantedateHecke’swork,whileothersare more recent. Marvin Knopp, April, 2007 Postlude We are grateful for the comments made by our students over the past several decades. More recently, Shigeru Kanemitsu and Yoshio Tanigawa o(cid:11)eredseveraladditionalremarksandreferences. WethankHildaBrittfor expertly typing most of our manuscript and Tim Huber for his graphical expertise.
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