The Arithmetic of Elliptic Curves
The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic theory of elliptic curves in its modern formulation, through the use of basic algebrai
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Editorial Board S. Axler K.A. Ribet
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Joseph H. Silverman
The Arithmetic of Elliptic Curves Second Edition
With 14 Illustrations
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Joseph H. Silverman Department of Mathematics Brown University 151 Thayer St. Providence, RI 02912 USA [email protected]
Editorial Board S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA [email protected]
K.A. Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720-3840 USA [email protected]
ISBN 978-0-387-09493-9 e-ISBN 978-0-387-09494-6 DOI 10.1007/978-0-387-09494-6 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009926474 Mathematics Subject Classification (2000): 11-01, 11GXX, 14GXX, 11G05 c Springer Science+Business Media, LLC 2009, 1986 ° 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 is part of Springer Science+Business Media (www.springer.com)
Preface to the Second Edition In the preface to the first edition of this book I remarked on the paucity of introductory texts devoted to the arithmetic of elliptic curves. That unfortunate state of affairs has long since been remedied with the publication of many volumes, among which may be mentioned books by Cassels [43], Cremona [54], Husem¨oller [118], Knapp [127], McKean et. al [167], Milne [178], and Schmitt et. al [222] that highlight the arithmetic and modular theory, and books by Blake et. al [22], Cohen et. al [51], Hankerson et. al [107], and Washington [304] that concentrate on the use of elliptic curves in cryptography. However, even among this cornucopia of literature, I hope that this updated version of the original text will continue to be useful. The past two decades have witnessed tremendous progress in the study of elliptic curves. Among the many highlights are the proof by Merel [170] of uniform boundedness for torsion points on elliptic curves over number fields, results of Rubin [215] and Kolyvagin [130] on the finiteness of Shafarevich–Tate groups and on the conjecture of Birch and Swinnerton-Dyer, the work of Wiles [311] on the modularity of elliptic curves, and the proof by Elkies [77] that there exist infinitely many supersingular primes. Although this introductory volume is unable to include proofs of
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