Elliptic Curves Diophantine Analysis
It is possible to write endlessly on elliptic curves. (This is not a threat.) We deal here with diophantine problems, and we lay the foundations, especially for the theory of integral points. We review briefly the analytic theory of the Weierstrass functi
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Editors S. S. Chern J. L. Doob J. A. Grothendieck E. Heinz S. Mac Lane W. Magnus W. Schmidt D. S. Scott K. Stein J. Tits B. L. van
Managing Editors B. Eckmann
J. K. Moser
Douglas, jr. F. Hirzebruch E. Hopf M. M. Postnikov der Waerden
Serge Lang
Elliptic Curves Diophantine Analysis
Springer-Verlag Berlin Heidelberg GmbH 1978
Serge Lang Department of Mathematics, Yale University, New Haven, CT 06520, U.S.A.
AMS Subject Classification (1970): 10 B 45, 10 F 99, 14 G 25, 14 H 25 ISBN 978-3-642-05717-5 ISBN 978-3-662-07010-9 (eBook) DOI 10.1007/978-3-662-07010-9 Library of Congress Cataloging in Publication Data. Lang. Serge, 1927-. Elliptic curves (Grundlehren der mathernatischen Wi,senschaflen: 231). Bibliography: p. Includes index. I. Diophantine analysis. 2. Curves. Elliptic. L Title. II. Series: Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen: 231. QA242.L234. 512'.74. 77-21139. 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, fe-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 1978
Originally published by Springer-Verlag Berlin Heidelberg New York in 1978
Typesetting: William Clowes & Sons Limited, London, Beccles and Colchester. 2141/314(}-543210
Foreword
It is possible to write endlessly on elliptic curves. (This is not a threat.) We deal here with diophantine problems, and we lay the foundations, especially for the theory of integral points. We review briefly the analytic theory of the Weierstrass function, and then deal with the arithmetic aspects of the addition formula, over complete fields and over number fields, giving rise to the theory of the height and its quadraticity. We apply this to integral points, covering the inequalities of diophantine approximation both on the multiplicative group and on the elliptic curve directly. Thus the book splits naturally in two parts. The first part deals with the ordinary arithmetic of the elliptic curve: The transcendental parametrization, the p-adic parametrization, points of finite order and the group of rational points, and the reduction of certain diophantine problems by the theory of heights to diophantine inequalities involving logarithms. The second part deals with the proofs of selected inequalities, at least strong enough to obtain the finiteness of integral points. The historical development is such that the first part represents a relatively mature state of the subject, whereas the second part is in a state offlux (due in large measure to the Baker method), so that no current account can be regarded as in any way definitive. The selection of which theorems and which methods to include was based on emphasiz
