Elementary Dirichlet Series and Modular Forms
The main topics of the book are the critical values of Dirichlet L-functions and Hecke L-functions of an imaginary quadratic field, and various problems on elliptic modular forms. As to the values of Dirichlet L-functions, all previous papers and books re
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Goro Shimura
EElementary Dirichlet Series
and Modular Forms
Goro Shimura Department of Mathematics Princeton University Princeton, New Jersey 08544-1000 [email protected]
ISBN: 978-0-387-72473-7
eISBN: 978-0-387-72474-4
Library of Congress Control N umber: 2007931368 M athematics Subject Classification (2000): 11F 11, 11F67, 11G05, 11G10, 11G15, 11R29, 11M06 © 2007 Springer Science +Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper. 987654321 springer .com
PREFACE
A book on any mathematical subject above textbook level is not of much value unless it contains new ideas and new perspectives. Also, the author may be encouraged to include new results, provided that they help the reader gain new insights and are presented along with known old results in a clear exposition. It is with this philosophy that I write this volume. The two subjects, Dirichlet series and modular forms, are traditional, but I treat them in both orthodox and unorthodox ways. However, I try to make the book accessible to those who are not familiar with such topics, by including plenty of expository material. More specific descriptions of the contents will be given in the Introduction. To some extent, this book has a supplementary nature to my previous book Introduction to the Arithmetic Theory of Automorphic Functions, published by Princeton University Press in 1971, though I do not write the present book with that intent. While the 1971 book grew out of my lectures in various places, the essential points of this new book have never been presented publicly or privately. I hope that it will draw an audience as large as that of the previous book. Princeton March 2007
Goro Shimura
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TABLE OF CONTENTS
Preface
v
Introduction
1
Chapter I. Preliminaries on Modular Forms and Dirichlet Series 1. Basic symbols and the definition of modular forms 2. Elementary Fourier analysis 3. The functional equation of a Dirichlet series
5 5 13 19
Chapter II. Critical Values of Dirichlet L-functions 4. The values of elementary Dirichlet series at integers 5. The class number of a cyclotomic field 6. Some more formulas for L(k, χ)
25 25 39 45
Chapter III. The Case of Imaginary Quadratic Fields and Nearly Holomorphic Modular Forms 7. Dirichlet series associated with an imaginary quadratic field 8. Nearly holomorphic modular forms Chapt
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