Elliptic Curves, Modular Forms and Cryptography: Proceedings of the Advanced Instructional Workshop on Algebraic Number Theory PDF
Preview Elliptic Curves, Modular Forms and Cryptography: Proceedings of the Advanced Instructional Workshop on Algebraic Number Theory
Elliptic Curves, Modular Forms and Cryptography Proceedings of the Advanced Instructional Workshop on Aigebraic Number Theory Volumes containing conference proceedings, workshop lectures, and collection of invited articles in any area of mathematics are published in this series. The aim is to provide an overview of the development through surveys, recent trends, problems and their current status as weil as historical background. All contributions for publication should be thoroughly reviewed before submitting to uso We need manuscripts ready for direct reproduction. All articles must be prepared in a uniform format. Necessary instructions forformat etc., are available from the publisher. Already Published Volumes Algebra-Same Recent Advances Edited by I. B. S. Passi Number Theory Edited by R. P. Bambah, V. C. Oumir, R. J. Hans-Gill Number Theory and Discrete Mathematics Edited by A. K. Agarwal, Bruce C. Berndt, Christian F. Krattenthaler, Gray L. Mullen, K. Ramachandra and Michel Waldschmidt Current Trends in NumberTheory Edited by s.o. Adhikari, SA Katre, B. Ramakrishnan Advances in Algebra and Geometry Edited by C. Musili s. A Tribute to C. Seshadri: Perspectives in Geometry and Representation Theory Edited by V.lakshmibai, V. Balaji, V. B. Mehta, K. R. Nagaraj, K. Pranjape. P. Sankran and R. Sridharan Elliptic Curves, Modular Forms and Cryptography Proceedings of the Advanced Instructional Workshop on Aigebraic Number Theory Edited by A.K. Bhandari D.S. Nagaraj B. Ramakrishnan T.N. Venkataramana ~HINDUSTAN U LQj UBOOKAGENCY Editors: Ashwani K. Bhandari D.S. Nagaraj Centre for Advanced Study in Institute of Mathematical Sciences Mathematics C I T Campus, Taramani Panjab University Chennai 600 113, India Chandigarh 160 014, India e-mail: [email protected] e-mail: [email protected] T.N. Venkataramana B. Ramakrishnan School of Mathematics Harish-Chandra Research Institute Tata Institute of Fundamental Chhatnag Road Research Jhusi Dr. Homi Bhabha Road Allahabad 211 019, India Mumbai 400 005, India. e-mail: [email protected] e-mail: [email protected] Published by Hindustan Book Agency (India) P 19 Green Park Extension New Delhi 110016 India email: [email protected] http://www.hindbook.com Copyright © 2003 by Hindustan Book Agency (India) No part of the material protected by this copyright notice may be repro duced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner, who has also the sole right to grant licences for translation into other languages and publication thereof. All export rights for this edition vest exclusively with Hindustan Book Agency (India). Unauthorized export is a violation of Copyright Law and is subject to legal action. ISBN 978-81-85931-42-5 ISBN 978-93-86279-15-6 (eBook) DOI 10.1007/978-93-86279-15-6 Preface This volume represents the proceedings of the Advanced Instruc tional Workshop on Algebraic Number Theory held at Harish-Chandra Research Institute (HRI), (Formerly, Mehta Research Institute), Alla habad, during November 2000. The theme of the workshop was Aigebraic N umber Theory with spe cial emphasis on Elliptic Curves. The theory of Elliptic Curves has been the source ofnew approaches to classical problems in Number The ory. It has also found applications in Cryptography. The workshop also covered some aspects of Cryptography. During the workshop, several Mathematicians from India gathered at HRI and gave lectures on vari ous topics in these fields. This volume consists of articles prepared by some of the speakers of the workshop and we thank all the contributors to this volume. The volume is in three parts, the first part contains articles in the field of Elliptic Curves, the second contains articles on Modular Forms. Some basics as well as some advanced topics on Cryptography are pre sented in the third and final part of these proceedings. Each part contains an introduction, which, in some sense, gives the overall picture of the contents in that part. Most of the articles are presented in a self-contained style and they give a different flavour to the subject. Though some of the contents of a few articles are already contained in some text books, they are presented here (with due refer ences) in order to make this volume complete to some extent. We hope that the graduate students who want to pursue their research career in Number Theory will benefit from this volume. This workshop was followed by an International Conference on N um ber Theory, the proceedings ofwhich had already been published through Hindustan Book Agency, New Delhi, who also brings out the present one. We thank Mr. Jainendra K. Jain for agreeing to publish this volume and also for his patience in spite of the unexpected delay in finalising the proceedings. VI Preface It is our pleasure to express our sincere thanks to Professor S. D. Adhikari whose encouragement as weIl as help virtually brought out this volume. We thank Prof. H. S. Mani, the then director of HRI, and Prof. Ravi S. Kulkarni, Director of HRI for their constant support and encourage ment in our endeavour. We thank Professor S. A. Katre for his help in organising the workshop. Thanks are also due to the administrative staff of HRI for the hard work they put in to make the workshop a grand success. FinaIly, we acknowledge the generous financial support from HRI and the Department of Science and Technology, Govt. of India. April 2003 A. K. Bhandari D. S. Nagaraj B. Ramakrishnan T. N. Venkataramana Contents Preface v Part I. Elliptic Curves 1 An Overview D. S. NAGARAJ 3 A Quick Introduction to Algebraic Geometry and Elliptic Curves D.S. NAGARAJ AND B. SURY 5 Elliptic Curves over Finite Fields B. SURY 33 The Nagell-Lutz Theorem RAJAT TANDON 49 Weak Mordell-Weil Theorem C. S. RAJAN 63 The Mordell-Weil Theorem D.S. NAGARAJ AND B. SURY 73 Complex Multiplication EKNATH GHATE 85 The Main Theorem of Complex Multiplication DIPENDRA PRASAD 109 Approximations of Algebraic Numbers by Rationals: A Theorem of Thue T. N. SHOREY 119 viii Contents Siegel's Theorem: Finiteness of Integral Points S. D. ADHIKARI AND D. S. RAMANA 139 p-adic Theta Functions and Tate Curves ALEXANDER F. BROWN 151 t-adic Representation Attached to an Elliptic Curve over a Nu mber Field D. S. NAGARAJ 167 Arithmetic on Curves CHANDAN SINGH DALAWAT 193 Part 11. Modular Forms 201 Introduction B. RAMAKRISHNAN 203 Elliptic Functions PARVATI SHASTRI 205 An Introduction to Modular Forms and Hecke Operators M. MANICKAM AND B. RAMAKRISHNAN 223 L-Functions of Modular Forms C. S. YOGANANDA 247 On the Eichler-Shimura Congruence Relation T. N. VENKATARAMANA 255 Part 111. Cryptography 261 Cryptography ASHWANI K. BHANDARI 269 Classical Cryptosystems R. THANGADURAI 275 The Public Key Cryptography ASHWANI K. BHANDARI 287 Primality and Factoring AMORA N ONGKYNRIH 303 Elliptic Curves and Cryptography R. BALASUBRAMANIAN 325 Part I ELLIPTIC CURVES
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