TY  - BOOK
AU  - Silverman,Joseph H.
AU  - Tate,John Torrence
TI  - Rational points on elliptic curves
T2  - Undergraduate texts in mathematics
SN  - 0387978259
AV  - QA567.2.E44 S55 1992
PY  - 1992///]
CY  - New York
PB  - Springer-Verlag
KW  - Curves, Elliptic
KW  - Rational points (Geometry)
KW  - Diophantine analysis
N1  - Includes bibliographical references (pages [259]-262) and index; Ch. I. Geometry and Arithmetic; -- 1. Rational Points on Conics; -- 2. The Geometry of Cubic Curves; -- 3. Weierstrass Normal Form; -- 4. Explicit Formulas for the Group Law; -- Ch. II. Points of Finite Order; -- 1. Points of Order Two and Three; -- 2. Real and Complex Points on Cubic Curves; -- 3. The Discriminant; -- 4. Points of Finite Order Have Integer Coordinates; -- 5. The Nagell-Lutz Theorem and Further Developments; -- Ch. III. The Group of Rational Points; -- 1. Heights and Descent; -- 2. The Height of P + P[subscript 0]; -- 3. The Height of 2P; -- 4. A Useful Homomorphism; -- 5. Mordell's Theorem; -- 6. Examples and Further Developments; -- 7. Singular Cubic Curves; -- Ch. IV. Cubic Curves over Finite Fields; -- 1. Rational Points over Finite Fields; -- 2. A Theorem of Gauss; -- 3. Points of Finite Order Revisited; -- 4. A Factorization Algorithm Using Elliptic Curves; -- Ch. V. Integer Points on Cubic Curves; -- 1. How Many Integer Points?; -- 2. Taxicabs and Sums of Two Cubes; -- 3. Thue's Theorem and Diophantine Approximation; -- 4. Construction of an Auxiliary Polynomial; -- 5. The Auxiliary Polynomial Is Small; -- 6. The Auxiliary Polynomial Does Not Vanish; -- 7. Proof of the Diophantine Approximation Theorem; -- 8. Further Developments; -- Ch. VI. Complex Multiplication; -- 1. Abelian Extensions of Q; -- 2. Algebraic Points on Cubic Curves; -- 3. A Galois Representation; -- 4. Complex Multiplication; -- 5. Abelian Extensions of Q(i). Appendix A: Projective Geometry; -- 1. Homogeneous Coordinates and the Projective Plane; -- 2. Curves in the Projective Plane; -- 3. Intersections of Projective Curves; -- 4. Intersection Multiplicities and a Proof of Bezout's Theorem; -- 5. Reduction Modulo pages
ER  -