The Arithmetic of Elliptic Curves

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Springer Science & Business Media, Apr 20, 2009 - Mathematics - 513 pages
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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 approach in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. Following a brief discussion of the necessary algebro-geometric results, the book proceeds with an exposition of the geometry and the formal group of elliptic curves, elliptic curves over finite fields, the complex numbers, local fields, and global fields. Final chapters deal with integral and rational points, including Siegels theorem and explicit computations for the curve Y = X + DX, while three appendices conclude the whole: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and an overview of more advanced topics.
 

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Contents

I Algebraic Varieties
1
II Algebraic Curves
17
III The Geometry of Elliptic Curves
41
IV The Formal Group of an Elliptic Curve
115
V Elliptic Curves over Finite Fields
137
VI Elliptic Curves over C
157
VII Elliptic Curves over Local Fields
184
VIII Elliptic Curves over Global Fields
207
XI Algorithmic Aspects of Elliptic Curves
362
A Elliptic Curves in Characteristics 2 and 3
409
B Group Cohomology H0 and H1
415
An Overview
425
Notes on Exercises
458
List of Notation
467
References
472
Index
489

IX Integral Points on Elliptic Curves
268
X Computing the MordellWeil Group
309

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About the author (2009)

Dr. Joseph Silverman is a professor at Brown University and has been an instructor or professors since 1982. He was the Chair of the Brown Mathematics department from 2001-2004. He has received numerous fellowships, grants and awards, as well as being a frequently invited lecturer. He is currently a member of the Council of the American Mathematical Society. His research areas of interest are number theory, arithmetic geometry, elliptic curves, dynamical systems and cryptography. He has co-authored over 120 publications and has had over 20 doctoral students under his tutelage. He has published 9 highly successful books with Springer, including the recently released, An Introduction to Mathematical Cryptography, for Undergraduate Texts in Mathematics.

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