Elliptic Functions
This book has grown out of a course of lectures on elliptic functions, given in German, at the Swiss Federal Institute of Technology, Zurich, during the summer semester of 1982. Its aim is to give some idea of the theory of elliptic functions, and of its
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Editors M. Artin S. S. Chern 1. M. Frohlich E. Heinz H. Hironaka F. Hirzebruch L. Hormander S. Mac Lane W. Magnus C. C. Moore 1. K. Moser M. Nagata W. Schmidt D. S. Scott Ya. G. Sinai 1. Tits B. L. van der Waerden M. Waldschmidt S. Watanabe Managing Editors M. Berger B. Eckmann S. R. S. Varadhan
K. Chandrasekharan
Elliptic Functions
With 14 Figures
Springer-Verlag Berlin Heidelberg New York Tokyo
Komaravolu Chandrasekharan Professor of Mathematics Eidgenossische Technische Hochschule Zurich CH-8092 Zurich
Mathematics Subject Classification (1980): 10-xx, 30-xx, 33-xx ISBN 978-3-642-52246-8
ISBN 978-3-642-52244-4 (eBook)
DOI 10.1007/978-3-642-52244-4 Library of Congress Cataloging in Publication Data. Chandrasekharan, K. (Komaravolu), 1920- . Elliptic functions. (Grundlehren der mathematischen Wissenschaften; 281) Based on lectures given at the Swiss Federal Institute of Technology, ZUrich, during the summer semester of 1982. Includes bibliographies and index. I. Functions, Elliptic. I. Title. II. Series. QA343.C47 1985 515.9'83 85-9802 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1985 Softcover reprint of the hardcover I st edition 1985 2141/3140-543210
Dedicated to the memory of Heinz and Anja Hopf
When the familiar scene is suddenly strange Or the well known is what we have yet to learn, And two worlds meet, and intersect, and change; T. S. Eliot
Preface
This book has grown out of a course of lectures on elliptic functions, given in German, at the Swiss Federal Institute of Technology, Zurich, during the summer semester of 1982. Its aim is to give some idea of the theory of elliptic functions, and of its close connexion with theta-functions and modular functions, and to show how it provides an analytic approach to the solution of some classical problems in the theory of numbers. It comprises eleven chapters. The first seven are function-theoretic, and the next four concern arithmetical applications. There are Notes at the end of every chapter, which contain references to the literature, comments on the text, and on the ramifications, old and new, of the problems dealt with, some of them extending into cognate fields. The treatment is self-contained, and makes no special demand on the reader's knowledge beyond the elements of complex analysis in one variable, and of group theory. Professor Raghavan Narasimhan has read the definitive English version of the text, and made illuminating comments, as a result of which I have improved the presentation in several places. Dr. Anton Good has looked through the first German version, and spared the time for many useful discussions. Dr. Peter
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