Classics in Mathematics Andre Weil Elliptic Functions according to Eisenstein and Kronecker

Transcription

1 Classics in Mathematics Andre Weil Elliptic Functions according to Eisenstein and Kronecker

2 Andre Weil was horn on May 6, 1906 in Paris. After studying mathematics at the Ecole Normale Superieure and receiving a doctoral degree from the University of Paris in 1928, he held professorial positions in India, France, the United States and Brazii before being appointed to the Institute for Advanced Study, Princeton in 1958, where he remained until he died on August 6, Andre Weil's work laid the foundation for abstract algebraic geometry and the modern theory of abelian varieties. A great deal of his work was directed towards establishing the links between number theory and algebraic geometry and devising modern methods in analytic number theory. Weil was one of the founders, around 1934, of the group that published, under the collective name of N. Bourbaki, the highly influential multivolume treatise Elements de mathematique.

3 Andre Weil Elliptic Functions according to Eisenstein and Kronecker Reprint of the 1976 Edition Springer

4 Originally published as Vol. 88 of the Ergebnisse der Mathematik und ihrer Grenzgebiete Mathematics Subject Classification {1991}: llg Library of Congress Cataloging-in-Publication Data Weil,Andr~,1906- EIIiptic functions according to Eisenstein and Kronecker 1 Andr~ Weil. p. cm. -- ( Classics in mathematics, JSSN ) Originally published: Berlin; New York: Springer-Verlag,1976, in series: Ergebnisse der Mathematik und ihrer Grenzgebiete; 88. Jncludes bibliographical references and index. ISBN (soft: alk. paper) 1. Elliptic functions.!. Title. Il. Series. QA343.W '.983--dc21 Photograph of Andre Weil by kind permission of The lnamori Foundation, Kyoto ISSN ISBN Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of iuustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 1999 The use of general descriptive names, registered names, trademarks etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective Iaws and regulations and therefore free for general use. SPIN / O - Printed on acid-free pa per

5 Andre Weil Elliptic Functions according to Eisenstein and Kronecker Springer-Verlag Berlin Heidelberg New York 1976

6 Prof. Andre Weil The Institute for Advanced Study, School of Mathematics, Princeton, N.J (USA) AMS Subject Classification (1970): 33 A 25 ISBN DOI / ISBN (ebook) Library of Congress Cataloging in Publication Data. Weil, Andre, Elliptic functions according to Eisenstein& Kronecker. (Ergebnisse der Mathematik und ihrer Grenzgebiete; 88). Bibliography: p. Includes index. 1. Functions, Elliptic. 1. Title. II. Series. QA343.W45 515' This work is subject to copyright. Ali 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 the publisher, the amount of the fee to be determined by agreement with the publisher. by Springer-Verlag Berlin Heidelberg 1976 Softcover reprint of the hardcover Ist edition 1976

7 Editorial Preface It is a great pleasure to my colleagues on the editorial board of the series Ergebnisse der Mathematik and myself to welcome this work, Elliptic Functions according to Eisenstein and Kronecker, by Andre Weil. However, some readers may be surprised to find in this series a text which appears at first sight tobe in essence a contribution to the history of mathematics, and which would therefore seem to be very untypical of the books in this series. However, the editors are strongly of the opinion that, while this text undoubtedly contributes notably to the history of our science, it is also of great value to contemporary mathematical research. Thus we had no hesitation in asking the permission of Professor Weil to publish his text in our series; and we are delighted to have had his agreement to do so. PETER HILTON Chairman, Editorial Board, Ergebnisse der Mathematik Battelle Seattle Research Center, August, 1975

8 Foreword "When kings are building", says the German poet, "carters have work to do". Kronecker quoted this, in his letter to Cantor of September 1891, only to add, thinking of himself no doubt, that each mathematician has to be king and carter at the same time. But carters need roads. Not seldom, in the history of our science, has it happened that a king opened up a new road into the promised land and that his successors, intent upon their own paths, allowed it to be overrun by brambles and become unfit for transit. To help clean up such a road is the purpose of this little book, arising out of lectures given at the Institute for Advanced Study in the Fali of I am grateful to Melvyn Nathanson for having put at my disposal his notes from those lectures. Where the road will lead remains in large part to be seen, but indications are not lacking that fertile country lies ahead. On the other hand, since much of the material in this volume seems suitable for inclusion in elementary courses, it may not be superfluous to point out that it is almost entirely self-contained. Even the basic facts about trigonometric functions are treated ah initio in Chapter II, according to Eisenstein's method. It would have been both logical and convenient to treat the gamma-function similarly in Chapter VII; for the sake of brevity, this has not been done, and a knowledge of some elementary properties of r (s) has been assumed. One further prerequisite in Part II is Dirichlet's theorem on Fourier series, together with the method of Poisson summation which is only a special case of that theorem; in the case under consideration ( essentially no more than the transformation formula for the theta-function) this presupposes the calculation of some classical integrals. Distributions occur only in 10 of Chapter VII and of Chapter VIII; as these are disjoint from the rest of the book, they can be skipped without damage, if perhaps not without loss. In Chapter VIII, the Bessel function K. occurs inevitably, as a definite integral, and I have of course adopted for it the usual notation; no property of that function, or rather of that integral, is tised beyond the most obvious ones. As to the final chapter, it concems applications to number-theory, and there, of course, some number-theory is needed. Princeton, the 21"t of March ANDREWEIL

9 Table of Contents Part I EISENSTEIN Chapter I Introduction.... Chapter II Trigonometric functions Chapter III The basic elliptic functions Chapter IV Basic relations and infinite products Chapter V Variation I Chapter VI Variation II Part II KRONECKER Chapter VII Prelude to Kronecker. Chapter VIII Kronecker's double series. Chapter IX Finale: Allegro con brio (Pell's equation and the Chowla-Selberg formula) Index of Notations