Elliptic Diophantine Equations

Download or read book Elliptic Diophantine Equations written by Nikos Tzanakis and published by Walter de Gruyter. This book was released on 2013-08-29 with total page 196 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Elliptic Curves written by S. Lang and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 270 pages. Available in PDF, EPUB and Kindle. Book excerpt: It is possible to write endlessly on elliptic curves. (This is not a threat.) We deal here with diophantine problems, and we lay the foundations, especially for the theory of integral points. We review briefly the analytic theory of the Weierstrass function, and then deal with the arithmetic aspects of the addition formula, over complete fields and over number fields, giving rise to the theory of the height and its quadraticity. We apply this to integral points, covering the inequalities of diophantine approximation both on the multiplicative group and on the elliptic curve directly. Thus the book splits naturally in two parts. The first part deals with the ordinary arithmetic of the elliptic curve: The transcendental parametrization, the p-adic parametrization, points of finite order and the group of rational points, and the reduction of certain diophantine problems by the theory of heights to diophantine inequalities involving logarithms. The second part deals with the proofs of selected inequalities, at least strong enough to obtain the finiteness of integral points.

Download or read book Diophantus and Diophantine Equations written by Isabella Grigoryevna Bashmakova and published by American Mathematical Soc.. This book was released on 2019-01-29 with total page 90 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book tells the story of Diophantine analysis, a subject that, owing to its thematic proximity to algebraic geometry, became fashionable in the last half century and has remained so ever since. This new treatment of the methods of Diophantus--a person whose very existence has long been doubted by most historians of mathematics--will be accessible to readers who have taken some university mathematics. It includes the elementary facts of algebraic geometry indispensable for its understanding. The heart of the book is a fascinating account of the development of Diophantine methods during the.

Download or read book Rational Points on Elliptic Curves written by Joseph H. Silverman and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of elliptic curves involves a blend of algebra, geometry, analysis, and number theory. This book stresses this interplay as it develops the basic theory, providing an opportunity for readers to appreciate the unity of modern mathematics. The book’s accessibility, the informal writing style, and a wealth of exercises make it an ideal introduction for those interested in learning about Diophantine equations and arithmetic geometry.

Download or read book Arithmetic and Geometry written by Michael Artin and published by Springer Science & Business Media. This book was released on 2013-11-11 with total page 366 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book The Algorithmic Resolution of Diophantine Equations written by Nigel P. Smart and published by Cambridge University Press. This book was released on 1998-11-12 with total page 264 pages. Available in PDF, EPUB and Kindle. Book excerpt: A coherent account of the computational methods used to solve diophantine equations.

Download or read book Elliptic Tales written by Avner Ash and published by Princeton University Press. This book was released on 2012 with total page 277 pages. Available in PDF, EPUB and Kindle. Book excerpt: Describes the latest developments in number theory by looking at the Birch and Swinnerton-Dyer Conjecture.