Rational Points on Elliptic Curves
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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 advance 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 about Diophantine equations and arithmetic geometry. Booknews Based on a series of lectures for junior and senior mathematics majors. The theory of elliptic curves is used in a range of fields, from cryptography to physics. Annotation c. Book News, Inc., Portland, OR (booknews.com)
PrefaceComputer PackagesAcknowledgmentsIntroduction1Ch. IGeometry and Arithmetic91Rational Points on Conics92The Geometry of Cubic Curves153Weierstrass Normal Form224Explicit Formulas for the Group Law28Exercises32Ch. IIPoints of Finite Order381Points of Order Two and Three382Real and Complex Points on Cubic Curves413The Discriminant474Points of Finite Order Have Integer Coordinates495The Nagell-Lutz Theorem and Further Developments56Exercises58Ch. IIIThe Group of Rational Points631Heights and Descent632The Height of P + P[subscript 0]683The Height of 2P714A Useful Homomorphism765Mordell's Theorem836Examples and Further Developments897Singular Cubic Curves99Exercises102Ch. IVCubic Curves over Finite Fields1071Rational Points over Finite Fields1072A Theorem of Gauss1103Points of Finite Order Revisited1214A Factorization Algorithm Using Elliptic Curves125Exercises138Ch. VInteger Points on Cubic Curves1451How Many Integer Points?1452Taxicabs and Sums of Two Cubes1473Thue's Theorem and Diophantine Approximation1524Construction of an Auxiliary Polynomial1575The Auxiliary Polynomial Is Small1656The Auxiliary Polynomial Does Not Vanish1687Proof of the Diophantine Approximation Theorem1718Further Developments174Exercises177Ch. VIComplex Multiplication1801Abelian Extensions of Q1802Algebraic Points on Cubic Curves1853A Galois Representation1934Complex Multiplication1995Abelian Extensions of Q(i)205Exercises213Appendix A: Projective Geometry2201Homogeneous Coordinates and the Projective Plane2202Curves in the Projective Plane2253Intersections of Projective Curves2334Intersection Multiplicities and a Proof of Bezout's Theorem2425Reduction Modulo p251Exercises254Bibliography259List of Notation263Index267
