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Elliptic Curves - A Computational Approach

Name: Elliptic Curves - A Computational Approach
Brand: De Gruyter
Price: 323.22 AUD

By: Susanne Schmitt, Horst G. Zimmer, Attila Pethoe (Contribution by)

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Hardcover | 1 December 2002

At a Glance

Hardcover
376 Pages

Age
22+

Dimensions(cm)
24.77 x 17.78 x 1.91

Hardcover

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The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist.

The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level.

The series de Gruyter Studies in Mathematics was founded ca. 30 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics.
While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community.

Please submit any book proposals to Niels Jacob.

Industry Reviews

"This is a most welcome addition to the existing literature on elliptic curves."Ch. Baxa in: Monatshefte fur Mathematik 4/2004

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You Can Find This Book In

Non-Fiction Mathematics Calculus & Mathematical Analysis Geometry Algebraic Geometry Analytic Geometry Algebra

Preface	p. v
Elliptic curves	p. 1
Normal forms	p. 1
The addition law	p. 11
Multiplication formulas	p. 19
Factorization and primality test	p. 24
Isogenies and endomorphisms of elliptic curves	p. 27
Exercises	p. 30
Elliptic curves over the complex numbers	p. 33
Lattices	p. 33
Weierstrass [Weierstrass p]-function	p. 36
Periods of elliptic curves	p. 52
Complex multiplication	p. 55
Exercises	p. 62
Elliptic curves over finite fields	p. 63
Frobenius endomorphism and supersingular curves	p. 63
Computing the number of points	p. 65
Construction of elliptic curves with given group order	p. 74
Elliptic curves in cryptography	p. 79
The discrete logarithm problem on elliptic curves	p. 83
Exercises	p. 85
Elliptic curves over local fields	p. 87
Reduction	p. 87
The filtration	p. 93
The theorem of Nagell, Lutz, and Cassels	p. 98
Exercises	p. 102
The Mordell-Weil theorem and heights	p. 103
Theorem of Mordell and Weil	p. 103
Heights	p. 116
Computation of the heights	p. 128
Points of bounded height	p. 133
The differences between the heights	p. 136
Exercises	p. 145
Torsion group	p. 147
Structure of the torsion group	p. 147
Elliptic curves with integral j-invariant	p. 151
The theorem of Nagell, Lutz, and Cassels	p. 177
Reduction	p. 182
Computation of the torsion group	p. 185
Examples	p. 187
Exercises	p. 196
The rank	p. 198
L-series	p. 198
The coefficients of the L-series	p. 202
Continuation of the L-series	p. 207
Conjectures concerning the rank	p. 214
The Selmer and the Tate-Shafarevich group	p. 216
2-descent	p. 228
The rank in field extensions	p. 233
Exercises	p. 240
Basis	p. 242
Linearly independent points	p. 242
Computation of a basis	p. 247
Examples	p. 251
Heegner point method	p. 254
Exercises	p. 260
S-integral points	p. 263
Overview	p. 263
Elliptic logarithms	p. 265
S-integral points over Q	p. 272
Proof of the theorem	p. 278
Example	p. 287
Exercises	p. 292
Algorithmic theory of diophantine equations	p. 294
Hilbert's 10th problem	p. 294
Introduction to Baker's method	p. 295
S-unit equations	p. 298
Thue equations	p. 303
Small collection of other results	p. 305
Lower bounds for linear forms in logarithms	p. 308
LLL-algorithm	p. 309
Reduction of the large bound	p. 311
Multiquadratic number fields	p. 316
Multiquadratic fields and Galois groups	p. 316
Discriminants	p. 317
Integral Bases	p. 321
Decomposition Law	p. 324
Biquadratic number fields	p. 330
Totally real and totally complex biquadratic fields	p. 341
Exercises	p. 349
Bibliography	p. 351
Index	p. 365
Table of Contents provided by Rittenhouse. All Rights Reserved.

Elliptic Curves - A Computational Approach

At a Glance

Hardcover

Industry Reviews

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How to return your order

Defective items

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