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Rational points on elliptic curves

Author: Joseph H Silverman; John Torrence Tate
Publisher: New York : Springer-Verlag, ©1992.
Series: Undergraduate texts in mathematics.
Edition/Format:   Book : EnglishView all editions and formats
Database:WorldCat
Summary:

The theory of elliptic curves involves a blend of algebra, geometry, analysis, and number theory. This book stresses this interplay as it develops the basic theory, providing an opportunity for  Read more...

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Additional Physical Format: Online version:
Silverman, Joseph H., 1955-
Rational points on elliptic curves.
New York : Springer-Verlag, c1992
(OCoLC)876656004
Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Joseph H Silverman; John Torrence Tate
ISBN: 0387978259 9780387978253 3540978259 9783540978251
OCLC Number: 25409071
Description: x, 281 p. : ill. ; 24 cm.
Contents: 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 p.
Series Title: Undergraduate texts in mathematics.
Responsibility: Joseph H. Silverman, John Tate.
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