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" The Arithmetic of Elliptic Curves "
by Joseph H. Silverman.
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BL
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| Record Number
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574554
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b403773
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| Main Entry
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Silverman, Joseph H.
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| Title & Author
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The Arithmetic of Elliptic Curves\ by Joseph H. Silverman.
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| Publication Statement
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New York, NY :: Springer New York :: Imprint: Springer,, 1986.
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| Series Statement
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Graduate Texts in Mathematics,; 106
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| ISBN
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9781475719208
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: 9781475719222
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| Contents
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I Algebraic Varieties -- II Algebraic Curves -- III The Geometry of Elliptic Curves -- IV The Formal Group of an Elliptic Curve -- V Elliptic Curves over Finite Fields -- VIElliptic Curves over ? -- VII Elliptic Curves over Local Fields -- VIII Elliptic Curves over Global Fields -- IX Integral Points on Elliptic Curves -- X Computing the Mordell-Weil Group -- Appendix A Elliptic Curves in Characteristics 2 and 3 -- {sect}1. Cohomology of Finite Groups -- {sect}2. Galois Cohomology -- {sect}3. Non-Abelian Cohomology -- Appendix C Further Topics: An Overview -- {sect}11. Complex Multiplication -- {sect}12. Modular Functions -- {sect}13. Modular Curves -- {sect}14. Tate Curves -- {sect}15. Néron Models and Tate's Algorithm -- {sect}17. Duality Theory -- {sect}18. Local Height Functions -- {sect}19. The Image of Galois -- {sect}20. Function Fields and Specialization Theorems -- Notes on Exercises -- List of Notation.
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| Abstract
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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.
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Mathematics.
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Algebra.
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Number theory.
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SpringerLink (Online service)
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