Undergraduate Texts in Mathematics

By: Joseph H. Silverman, John Tate

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Hardcover | 18 November 1994

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300 Pages

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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.

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"The authors' goal has been to write a textbook in a technically difficult field which is accessible to the average undergraduate mathematics major, and it seems that they have succeeded admirably..."--MATHEMATICAL REVIEWS

"This is a very leisurely introduction to the theory of elliptic curves, concentrating on an algebraic and number-theoretic viewpoint. It is pitched at an undergraduate level and simplifies the work by proving the main theorems with additional hypotheses or by only proving special cases. ... The examples really pull together the material and make it clear. ... a great book for a first introduction to the subject of elliptic curves. ... very clearly written and you will understand a lot when you are done." (Allen Stenger, The Mathematical Association of America, August, 2008)

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Non-Fiction Mathematics Number Theory Geometry Algebraic Geometry Algebra

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Published: 1st December 2010

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Preface
Computer Packages
Acknowledgments
Introduction	p. 1
Geometry and Arithmetic	p. 9
Rational Points on Conics	p. 9
The Geometry of Cubic Curves	p. 15
Weierstrass Normal Form	p. 22
Explicit Formulas for the Group Law	p. 28
Exercises	p. 32
Points of Finite Order	p. 38
Points of Order Two and Three	p. 38
Real and Complex Points on Cubic Curves	p. 41
The Discriminant	p. 47
Points of Finite Order Have Integer Coordinates	p. 49
The Nagell-Lutz Theorem and Further Developments	p. 56
Exercises	p. 58
The Group of Rational Points	p. 63
Heights and Descent	p. 63
The Height of P + P[subscript 0]	p. 68
The Height of 2P	p. 71
A Useful Homomorphism	p. 76
Mordell's Theorem	p. 83
Examples and Further Developments	p. 89
Singular Cubic Curves	p. 99
Exercises	p. 102
Cubic Curves over Finite Fields	p. 107
Rational Points over Finite Fields	p. 107
A Theorem of Gauss	p. 110
Points of Finite Order Revisited	p. 121
A Factorization Algorithm Using Elliptic Curves	p. 125
Exercises	p. 138
Integer Points on Cubic Curves	p. 145
How Many Integer Points?	p. 145
Taxicabs and Sums of Two Cubes	p. 147
Thue's Theorem and Diophantine Approximation	p. 152
Construction of an Auxiliary Polynomial	p. 157
The Auxiliary Polynomial Is Small	p. 165
The Auxiliary Polynomial Does Not Vanish	p. 168
Proof of the Diophantine Approximation Theorem	p. 171
Further Developments	p. 174
Exercises	p. 177
Complex Multiplication	p. 180
Abelian Extensions of Q	p. 180
Algebraic Points on Cubic Curves	p. 185
A Galois Representation	p. 193
Complex Multiplication	p. 199
Abelian Extensions of Q(i)	p. 205
Exercises	p. 213
Appendix A: Projective Geometry	p. 220
Homogeneous Coordinates and the Projective Plane	p. 220
Curves in the Projective Plane	p. 225
Intersections of Projective Curves	p. 233
Intersection Multiplicities and a Proof of Bezout's Theorem	p. 242
Reduction Modulo p	p. 251
Exercises	p. 254
Bibliography	p. 259
List of Notation	p. 263
Index	p. 267
Table of Contents provided by Blackwell. All Rights Reserved.

Undergraduate Texts in Mathematics

At a Glance

Hardcover

Industry Reviews

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How to return your order

Defective items

You Can Find This Book In

Other Editions and Formats

Paperback

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