Elliptic Curves: A Computational Approach (Google eBook)
The basics of the theory of elliptic curves should be known to everybody, be he (or she) a mathematician or a computer scientist. Especially everybody concerned with cryptography should know the elements of this theory.
The purpose of the present textbook is to give an elementary introduction to elliptic curves. Since this branch of number theory is particularly accessible to computer-assisted calculations, the authors make use of it by approaching the theory under a computational point of view. Specifically, the computer-algebra package SIMATH can be applied on several occasions. However, the book can be read also by those not interested in any computations. Of course, the theory of elliptic curves is very comprehensive and becomes correspondingly sophisticated. That is why the authors made a choice of the topics treated. Topics covered include the determination of torsion groups, computations regarding the Mordell-Weil group, height calculations, S-integral points. The contents is kept as elementary as possible. In this way it becomes obvious in which respect the book differs from the numerous textbooks on elliptic curves nowadays available.
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Chapter 2 Elliptic curves over the complex numbers
Chapter 3 Elliptic curves over finite fields
Chapter 4 Elliptic curves over local fields
Chapter 5 The MordellWeil theorem and heights
Chapter 6 Torsion group
Chapter 7 The rank
Chapter 8 Basis
Chapter 9 Sintegral points
Appendix A Algorithmic theory of diophantine equations
Appendix B Multiquadratic number fields