Rational Points on Elliptic Curves

Joseph H. Silverman

282 pages, parution le 01/01/1994

Ajouter à une liste

Indisponible

Résumé

The theory of elliptic curves involves a pleasing blend of algebra, geometry, analysis, and number theory. "Rational Points on Elliptic Curves" streses 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.

Preface Computer Packages Acknowledgments Introduction 1

Ch. I Geometry and Arithmetic 9 1 Rational Points on Conics 9 2 The Geometry of Cubic Curves 15 3 Weierstrass Normal Form 22 4 Explicit Formulas for the Group Law 28 Exercises 32

Ch. II Points of Finite Order 38 1 Points of Order Two and Three 38 2 Real and Complex Points on Cubic Curves 41 3 The Discriminant 47 4 Points of Finite Order Have Integer Coordinates 49 5 The Nagell-Lutz Theorem and Further Developments 56 Exercises 58

Ch. III The Group of Rational Points 63 1 Heights and Descent 63 2 The Height of P + P[subscript 0] 68 3 The Height of 2P 71 4 A Useful Homomorphism 76 5 Mordell's Theorem 83 6 Examples and Further Developments 89 7 Singular Cubic Curves 99 Exercises 102

Ch. IV Cubic Curves over Finite Fields 107 Rational Points over Finite Fields 107 2 A Theorem of Gauss 110 3 Points of Finite Order Revisited 121 4 A Factorization Algorithm Using Elliptic Curves 125 Exercises 138

Ch. V Integer Points on Cubic Curves 145 1 How Many Integer Points? 145 2 Taxicabs and Sums of Two Cubes 147 3 Thue's Theorem and Diophantine Approximation 152 4 Construction of an Auxiliary Polynomial 157 5 The Auxiliary Polynomial Is Small 165 6 The Auxiliary Polynomial Does Not Vanish 168 7 Proof of the Diophantine Approximation Theorem 171 8 Further Developments 174 Exercises 177

Ch. VI Complex Multiplication 180 1 Abelian Extensions of Q 180 2 Algebraic Points on Cubic Curves 185 3 A Galois Representation 193 4 Complex Multiplication 199 5 Abelian Extensions of Q(i) 205 Exercises 213

Appendix A: Projective Geometry 220 1 Homogeneous Coordinates and the Projective Plane 220 2 Curves in the Projective Plane 225 3 Intersections of Projective Curves 233 4 Intersection Multiplicities and a Proof of Bezout's Theorem 242 5 Reduction Modulo p 251 Exercises 254

Bibliography 259 List of Notation 263 Index 267

Caractéristiques techniques

	PAPIER
Éditeur(s)	Springer
Auteur(s)	Joseph H. Silverman
Parution	01/01/1994
Nb. de pages	282
Couverture	Relié
Intérieur	Noir et Blanc
EAN13	9780387978253
ISBN13	978-3-540-97825-1