# The Arithmetic of Elliptic Curves

• Joseph H. Silverman
Textbook

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

1. Front Matter
Pages i-xii
2. Joseph H. Silverman
Pages 1-4
3. Joseph H. Silverman
Pages 5-20
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Pages 21-44
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11. Joseph H. Silverman
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12. Joseph H. Silverman
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13. Back Matter
Pages 324-402

### Introduction

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.

### Keywords

Algebraic Cohomology Geometry Mordell-Weil algebra finite field number theory

#### Authors and affiliations

• Joseph H. Silverman
• 1
1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4757-1920-8
• Copyright Information Springer-Verlag New York 1986
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-1-4757-1922-2
• Online ISBN 978-1-4757-1920-8
• Series Print ISSN 0072-5285
• Buy this book on publisher's site