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The Arithmetic of Elliptic Curves

  • Textbook
  • © 1986

Overview

Authors:
  1. Joseph H. Silverman

    1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, USA

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Part of the book series: Graduate Texts in Mathematics (GTM, volume 106)

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xii
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  2. Introduction

    • Joseph H. Silverman
    Pages 1-4

  3. Algebraic Varieties

    • Joseph H. Silverman
    Pages 5-20

  4. Algebraic Curves

    • Joseph H. Silverman
    Pages 21-44

  5. The Geometry of Elliptic Curves

    • Joseph H. Silverman
    Pages 45-109

  6. The Formal Group of an Elliptic Curve

    • Joseph H. Silverman
    Pages 110-129

  7. Elliptic Curves over Finite Fields

    • Joseph H. Silverman
    Pages 130-145

  8. Elliptic Curves over ℂ

    • Joseph H. Silverman
    Pages 146-170

  9. Elliptic Curves over Local Fields

    • Joseph H. Silverman
    Pages 171-188

  10. Elliptic Curves over Global Fields

    • Joseph H. Silverman
    Pages 189-240

  11. Integral Points on Elliptic Curves

    • Joseph H. Silverman
    Pages 241-275

  12. Computing the Mordell-Weil Group

    • Joseph H. Silverman
    Pages 276-323

  13. Back Matter

    Pages 324-402
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Keywords

About this book

The preface to a textbook frequently contains the author's justification for offering the public "another book" on the given subject. For our chosen topic, the arithmetic of elliptic curves, there is little need for such an apologia. Considering the vast amount of research currently being done in this area, the paucity of introductory texts is somewhat surprising. Parts of the theory are contained in various books of Lang (especially [La 3] and [La 5]); and there are books of Koblitz ([Kob]) and Robert ([Rob], now out of print) which concentrate mostly on the analytic and modular theory. In addition, survey articles have been written by Cassels ([Ca 7], really a short book) and Tate ([Ta 5J, which is beautifully written, but includes no proofs). Thus the author hopes that this volume will fill a real need, both for the serious student who wishes to learn the basic facts about the arithmetic of elliptic curves; and for the research mathematician who needs a reference source for those same basic facts. Our approach is more algebraic than that taken in, say, [La 3] or [La 5], where many of the basic theorems are derived using complex analytic methods and the Lefschetz principle. For this reason, we have had to rely somewhat more on techniques from algebraic geometry. However, the geom­ etry of (smooth) curves, which is essentially all that we use, does not require a great deal of machinery.

Reviews

"This well-written book covers the basic facts about the geometry and arithmetic of elliptic curves, and is sure to become the standard reference in the subject. It meets the needs of at least three groups of people: students interested in doing research in Diophantine geometry, mathematicians needing a reference for standard facts about elliptic curves, and computer scientists interested in algorithms and needing an introduction to elliptic curves..."-- MATHEMATICAL REVIEWS

Authors and Affiliations

  • Department of Mathematics, Massachusetts Institute of Technology, Cambridge, USA

    Joseph H. Silverman

Bibliographic Information

  • Book Title: The Arithmetic of Elliptic Curves

  • Authors: Joseph H. Silverman

  • Series Title: Graduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4757-1920-8

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1986

  • eBook ISBN: 978-1-4757-1920-8Published: 09 March 2013

  • Series ISSN: 0072-5285

  • Series E-ISSN: 2197-5612

  • Edition Number: 1

  • Number of Pages: XII, 402

  • Number of Illustrations: 12 b/w illustrations

  • Topics: Algebra, Number Theory

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