Preface to the Second Edition.- Preface to the First Edition.- Introduction.- Algebraic Varieties.- Algebraic Curves.- The Geometry of Elliptic Curves.- The Formal Group of an Elliptic Curve.- Elliptic Curves over Finite Fields.- Elliptic Curves over C.- Elliptic Curves over Local Fields.- Elliptic Curves over Global Fields.- Integral Points on Elliptic Curves.- Computing the Mordell Weil Group.- Algorithmic Aspects of Elliptic Curves.- Appendix A: Elliptic Curves in Characteristics 2 and 3.- Appendix B: Group Cohomology (H 0 and H 1 ).- Appendix C: Further Topics: An Overview.- Notes on Exercises.- Bibliography.- List of Notation.- References.- Index.
Über den Autor
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.
Algebraic Varieties.- Algebraic Curves.- The Geometry of Elliptic Curves.- The Formal Group of an Elliptic Curve.- Elliptic Curves over Finite Fields.- Elliptic Curves over C.- Elliptic Curves over Local Fields.- Elliptic Curves over Global Fields.- Integral Points on Elliptic Curves.- Computing the Mordell#x2013;Weil Group.- Algorithmic Aspects of Elliptic Curves.
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.