Elliptic Diophantine Equations : : A Concrete Approach via the Elliptic Logarithm / / Nikos Tzanakis.

• Superior document: Title is part of eBook package: De Gruyter DGBA Backlist Complete English Language 2000-2014 PART1
• VerfasserIn: Tzanakis, Nikos,
• Place / Publishing House: Berlin ;, Boston : : De Gruyter, , [2013]; ©2013
• Year of Publication: 2013
• Language: English
• Series: De Gruyter Series in Discrete Mathematics and Applications , 2
• Physical Description: 1 online resource (179 p.)

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.

Table of Contents

Frontmatter -- Preface -- Contents -- Chapter 1 Elliptic curves and equations -- Chapter 2 Heights -- Chapter 3 Weierstrass equations over C and R -- Chapter 4 The elliptic logarithm method -- Chapter 5 Linear form for the Weierstrass equation -- Chapter 6 Linear form for the quartic equation -- Chapter 7 Linear form for simultaneous Pell equations -- Chapter 8 Linear form for the general elliptic equation -- Chapter 9 Bound for the coefficients of the linear form -- Chapter 10 Reducing the bound obtained in Chapter 9 -- Chapter 11 S-integer solutions of Weierstrass equations -- List of symbols -- Bibliography -- Index