Elliptic Diophantine Equations

Nikos Tzanakis

Published by De Gruyter 2013

Elliptic Diophantine Equations

A Concrete Approach via the Elliptic Logarithm

Nikos Tzanakis

Volume 2 in the series De Gruyter Series in Discrete Mathematics and Applications

https://doi.org/10.1515/9783110281149

Overview
Contents

About this book

This book presents in a unified and concrete way the beautiful and deep mathematics - both theoretical and computational - on which the explicit solution of an elliptic Diophantine equation is based. It collects numerous results and methods that are scattered in the literature. Some results are hidden behind a number of routines in software packages, like Magma and Maple; professional mathematicians very often use these routines just as a black-box, having little idea about the mathematical treasure behind them. Almost 20 years have passed since the first publications on the explicit solution of elliptic Diophantine equations with the use of elliptic logarithms. The "art" of solving this type of equation has now reached its full maturity. The author is one of the main persons that contributed to the development of this art.

The monograph presents a well-balanced combination of

a variety of theoretical tools (from Diophantine geometry, algebraic number theory, theory of linear forms in logarithms of various forms - real/complex and p-adic elliptic - and classical complex analysis),
clever computational methods and techniques (LLL algorithm and de Weger's reduction technique, AGM algorithm, Zagier's technique for computing elliptic integrals),
ready-to-use computer packages.

A result is the solution in practice of a large general class of Diophantine equations.

Author information

Nikos Tzanakis, University of Crete, Heraklion, Greece.

Supplementary Materials

Contents

Topics

Frontmatter Publicly Available Download PDF	i
Preface Requires Authentication Licensed Download PDF	vii
Contents Publicly Available Download PDF	xv
Chapter 1 Elliptic curves and equations Requires Authentication Licensed Download PDF	1
Chapter 2 Heights Requires Authentication Licensed Download PDF	9
Chapter 3 Weierstrass equations over C and R Requires Authentication Licensed Download PDF	29
Chapter 4 The elliptic logarithm method Requires Authentication Licensed Download PDF	54
Chapter 5 Linear form for the Weierstrass equation Requires Authentication Licensed Download PDF	57
Chapter 6 Linear form for the quartic equation Requires Authentication Licensed Download PDF	60
Chapter 7 Linear form for simultaneous Pell equations Requires Authentication Licensed Download PDF	69
Chapter 8 Linear form for the general elliptic equation Requires Authentication Licensed Download PDF	78
Chapter 9 Bound for the coefficients of the linear form Requires Authentication Licensed Download PDF	98
Chapter 10 Reducing the bound obtained in Chapter 9 Requires Authentication Licensed Download PDF	121
Chapter 11 S-integer solutions of Weierstrass equations Requires Authentication Licensed Download PDF	137
List of symbols Requires Authentication Licensed Download PDF	165
Bibliography Requires Authentication Licensed Download PDF	173
Index Requires Authentication Licensed Download PDF	177

(Deutsch)

Language: English
Publisher: De Gruyter
Audience: Graduate and Advanced Undergraduate Students in Mathematics, Advanced Mathematicians; Academic Libraries
Pages
- Front matter: 16
- Main content: 179
Illustrations
- Illustrations: 2
- Tables: 8
Keywords: Elliptic Diophantine Equation; Diophantine Geometry; Algebraic Number Theory; Magma; Computational Method

eBook

Published: August 29, 2013
ISBN: 9783110281149

Hardcover

Published: August 19, 2013
ISBN: 9783110280913