# Rational points on elliptic curves (Record no. 214853)

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000 -LEADER | |
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fixed length control field | aam a22 4500 |

008 - FIXED-LENGTH DATA ELEMENTS--GENERAL INFORMATION | |

fixed length control field | 200311b 2015 ||||| |||| 00| 0 eng d |

020 ## - INTERNATIONAL STANDARD BOOK NUMBER | |

International Standard Book Number | 9783319185873 |

082 ## - DEWEY DECIMAL CLASSIFICATION NUMBER | |

Classification number | 516.352 |

Item number | S4R2 |

100 ## - MAIN ENTRY--PERSONAL NAME | |

Personal name | Silverman, Joseph |

9 (RLIN) | 394816 |

245 ## - TITLE STATEMENT | |

Title | Rational points on elliptic curves |

250 ## - EDITION STATEMENT | |

Edition statement | 2nd |

260 ## - PUBLICATION, DISTRIBUTION, ETC. (IMPRINT) | |

Name of publisher, distributor, etc | Springer International Publishing |

Date of publication, distribution, etc | 2015 |

Place of publication, distribution, etc | Cham |

300 ## - PHYSICAL DESCRIPTION | |

Extent | xxii, 332 p. |

Other physical details | Includes bibliographical references and index |

440 ## - SERIES STATEMENT/ADDED ENTRY--TITLE | |

Title | Undergraduate texts in mathematics |

9 (RLIN) | 394092 |

504 ## - BIBLIOGRAPHY, ETC. NOTE | |

Bibliography, etc | Table of Contents<br/><br/>1.Geometry and Arithmetic<br/>1.1.Rational Points on Conies<br/>1.2.The Geometry of Cubic Curves<br/>1.3.Weierstrass Normal Form<br/>1.4.Explicit Formulas for the Group Law<br/>Exercises<br/>2.Points of Finite Order<br/>2.1.Points of Order Two and Three<br/>2.2.Real and Complex Points on Cubic Curves<br/>2.3.The Discriminant<br/>2.4.Points of Finite Order Have Integer Coordinates<br/>2.5.The Nagell<br/>Lutz Theorem and Further Developments<br/>3.The Group of Rational Points<br/>3.1.Heights and Descent<br/>3.2.The Height of P + P0<br/>3.3.The Height of 2P<br/>3.4.A Useful Homomorphism<br/>3.5.Mordell's Theorem<br/>3.6.Examples and Further Developments<br/>3.7.Singular Cubic Curves<br/>Exercises Ill<br/>4.Cubic Curves over Finite Fields<br/>4.1.Rational Points over Finite Fields<br/>4.2.A Theorem of Gauss<br/>4.3.Points of Finite Order Revisited<br/>4.4.A Factorization Algorithm Using Elliptic Curves<br/>4.5.Elliptic Curve Cryptography<br/>Contents note continued: 5.Integer Points on Cubic Curves<br/>5.1.How Many Integer Points?<br/>5.2.Taxicabs and Sums of Two Cubes<br/>5.3.Thue's Theorem and Diophantine Approximation<br/>5.4.Construction of an Auxiliary Polynomial<br/>5.5.The Auxiliary Polynomial Is Small<br/>5.6.The Auxiliary Polynomial Does Not Vanish<br/>5.7.Proof of the Diophantine Approximation Theorem<br/>5.8.Further Developments<br/>6.Complex Multiplication<br/>6.1.Abelian Extensions of Q<br/>6.2.Algebraic Points on Cubic Curves<br/>6.3.A Galois Representation<br/>6.4.Complex Multiplication<br/>6.5.Abelian Extensions of Q(i)<br/>6.6.Elliptic Curves and Fermat's Last Theorem<br/>A.Projective Geometry<br/>A.1.Homogeneous Coordinates and the Projective Plane<br/>A.2.Curves in the Projective Plane<br/>A.3.Intersections of Projective Curves<br/>A.4.Intersection Multiplicities and a Proof of Bezout's Theorem<br/>A.5.Reduction Modulo p<br/>B.Transformation to Weierstrass Form. |

520 ## - SUMMARY, ETC. | |

Summary, etc | In 1961 the second author deliv1lred a series of lectures at Haverford Col lege on the subject of "Rational Points on Cubic Curves. " These lectures, intended for junior and senior mathematics majors, were recorded, tran scribed, and printed in mimeograph form. Since that time they have been widely distributed as photocopies of ever decreasing legibility, and por tions have appeared in various textbooks (Husemoller [1], Chahal [1]), but they have never appeared in their entirety. In view of the recent inter est in the theory of elliptic curves for subjects ranging from cryptogra phy (Lenstra [1], Koblitz [2]) to physics (Luck-Moussa-Waldschmidt [1]), as well as the tremendous purely mathematical activity in this area, it seems a propitious time to publish an expanded version of those original notes suitable for presentation to an advanced undergraduate audience. We have attempted to maintain much of the informality of the orig inal Haverford lectures. Our main goal in doing this has been to write a textbook in a technically difficult field which is "readable" by the average undergraduate mathematics major. We hope we have succeeded in this goal. The most obvious drawback to such an approach is that we have not been entirely rigorous in all of our proofs. In particular, much of the foundational material on elliptic curves presented in Chapter I is meant to explain and convince, rather than to rigorously prove.<br/><br/>https://link.springer.com/book/10.1007/978-1-4757-4252-7#about |

650 ## - SUBJECT ADDED ENTRY--TOPICAL TERM | |

Topical term or geographic name as entry element | Curves, Elliptic |

9 (RLIN) | 394817 |

650 ## - SUBJECT ADDED ENTRY--TOPICAL TERM | |

Topical term or geographic name as entry element | Diophantine analysis |

9 (RLIN) | 394818 |

650 ## - SUBJECT ADDED ENTRY--TOPICAL TERM | |

Topical term or geographic name as entry element | Rational points - Geometry |

9 (RLIN) | 394819 |

650 ## - SUBJECT ADDED ENTRY--TOPICAL TERM | |

Topical term or geographic name as entry element | Mathematics |

9 (RLIN) | 394820 |

650 ## - SUBJECT ADDED ENTRY--TOPICAL TERM | |

Topical term or geographic name as entry element | Number theory |

9 (RLIN) | 394821 |

650 ## - SUBJECT ADDED ENTRY--TOPICAL TERM | |

Topical term or geographic name as entry element | Data structures - Computer science |

9 (RLIN) | 394822 |

700 ## - ADDED ENTRY--PERSONAL NAME | |

Personal name | Tate, John |

Relator term | Co-author |

9 (RLIN) | 394823 |

942 ## - ADDED ENTRY ELEMENTS (KOHA) | |

Source of classification or shelving scheme | |

Item type | Books |

Withdrawn status | Lost status | Source of classification or shelving scheme | Damaged status | Not for loan | Collection code | Permanent location | Current location | Shelving location | Date acquired | Source of acquisition | Cost, normal purchase price | Item location | Total Checkouts | Total Renewals | Full call number | Barcode | Date last seen | Date last borrowed | Cost, replacement price | Koha item type |
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Non-fiction | Vikram Sarabhai Library | Vikram Sarabhai Library | General Stacks | 11/03/2020 | 22 | 2.00 | Slot 1384 (0 Floor, East Wing) | 1 | 3 | 516.352 S4R2 | 201624 | 27/03/2021 | 12/03/2020 | 3666.68 | Books |