The theory of elliptic curves involves a pleasing blend of algebra, geometry, analysis, and number theory. Rational Points on Elliptic Curves stresses this interplay as it develops the basic theory, thereby providing an opportunity for advanced undergraduates to appreciate the unity of modern mathematics. At the same time, every effort has been made to use only methods and results commonly included in the undergraduate curriculum. This accessibility, the informal writing style, and a wealth of exercises make Rational Points on Elliptic Curves an ideal introduction for students at all levels who are interested in learning abuot Diophantine equations and arithmetic geometry.
Most concretely, an elliptic curve is a set of zeros of a cubic polynomial in two variables. If the polynomial has rational coefficients, one can ask for a description of those zeros whose coordinates are either integers or rational numbers. It is this number theoretic question that is the main subject of Rational Points on Elliptic Curves. Topics covered include the geometry and group structure of elliptic curves, the Nagell–Lutz Theorem describing points of finite order, the Mordell–Weil theorem on the finite generation of the group of rational points, the Thue–Siegel theorem on the finiteness of the set of integer poitns, a result of Gauss counting the number of points with coordinates in a finite field, Lenstra's algorithm using elliptic curves to factor large integers, and a discussion of complex multiplication multiplication and the Galois representations associated to torsion points. Additional topics new to the second edition include an introduction to elliptic curve cryptography and a brief discussion of the stunning proof of Fermat's Last Theorem by Wiles et al. via the use of elliptic curves.
|Chapter I:||Geometry and Arithmetic|
|Chapter II:||Points of Finite Order|
|Chapter III:||The Group of Rational Points|
|Chapter IV:||Cubic Curves over Finite Fields|
|Chapter V:||Integer Points on Cubic Curves|
|Chapter VI:||Complex Multiplication|
|Appendix A:||Projective Geometry|
Supplementary Notes by Rich Schwartz
(1) A primer on complex analysis.
(2) A proof that the Weierstrass function gives an isomorphism from C/Λ to an elliptic curve.
(3) A proof that every elliptic curve over C has a Weierstrass parametrization.
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