Introduction to Siegel Modular Forms and Dirichlet Series
Introduction to Siegel Modular Forms and Dirichlet Series gives a concise and self-contained introduction to the multiplicative theory of Siegel modular forms, Hecke operators, and zeta functions, including the classical case of modular forms in one varia
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Anatoli Andrianov
Introduction to Siegel Modular Forms and Dirichlet Series
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Anatoli Andrianov Russian Academy of Sciences Steklov Institute of Mathematics Fontanka 27 191023 St. Petersburg Russia [email protected]
Editorial board: Sheldon Axler, San Francisco State University, San Francisco, CA, USA Vincenzo Capasso, University of Milan, Milan, Italy Carles Casacuberta, Universitat de Barcelona, Barcelona, Spain Angus MacIntyre, Queen Mary, University of London, London, UK Kenneth Ribet, University of California, Berkeley, CA, USA Claude Sabbah, Ecole Polytechnique, Palaiseau, France Endre Süli, Oxford University, Oxford, UK Wojbor Woyczynski, Case Western Reserve University, Cleveland, OH, USA
ISBN 978-0-387-78752-7 DOI 10.1007/978-0-387-78753-4
e-ISBN 978-0-387-78753-4
Library of Congress Control Number: 2008938066 Mathematics Subject Classification (2000): 11Fxx, 11F66 This is a translation of the Dutch, Meetkunde, originally published by Epsilon–Uitgaven, 2000. c Springer Science+Business Media, LLC 2009 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 springer.com
To Goro Shimura and to my granddaughter Sasha
Preface
Several years ago I was invited to an American university to give one-term graduate course on Siegel modular forms, Hecke operators, and related zeta functions. The idea to present in a concise but basically complete and self-contained form an introduction to an important and developing area based partly on my own work attracted me. I accepted the invitation and started to prepare the course. Unfortunately, the visit was not realized. But the idea of such a course continued to be alive till after a number of years this book was finally completed. I hope that this short book will serve to attract young researchers to this beautiful field, and that it will simplify and make more pleasant the initial steps. No special knowledge is presupposed for reading this book beyond standard courses in algebra and calculus (one and several variables), although some skill in working with mathematical texts would be helpful. The reader will judge whether the result was worth the effort. Dedications. The ideas of Goro Shimura exerted a deep influence on the number theory of the second half of the twentieth century in general a
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