Elliptic Diophantine equations. A concrete approach via the elliptic logarithm
type: Item
instance of: scholarly article
title: Elliptic Diophantine equations. A concrete approach via the elliptic logarithm
zbMATH Open document ID: 1311.11023
published in: De Gruyter Series in Discrete Mathematics and Applications
publication date: 2013-08-14T00:00:00Z
reviewed by: Maciej Ulas
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 11D41
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11Y50
zbMATH DE Number: 6198404
zbMATH Keywords: elliptic Diophantine equations
zbMATH Keywords: rational points
zbMATH Keywords: integral points
zbMATH Keywords: elliptic logarithm
author: Nikos Tzanakis
describes a project that uses: Magma
describes a project that uses: Maple
MaRDI profile type: Publication
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review text: The book under review is devoted to the study of integral solutions of elliptic Diophantine equations. Let us recall that the equation \(f(x,y)=0\), with \(f\in K[x,y]\), where \(K\) is a number field, is called elliptic Diophantine equation if the corresponding algebraic curve \(C: f(x,y)=0\) is of genus one, and has at least one \(K\)-rational point, i.e. the point \([x:y:z]\in\mathbb{P}^{2}(K)\) satisfying \(F(x,y,z)=0\), where \(F(x,y,z)=z^{\deg F}f(x/z,y/z)\) is the homogenization of the polynomial \(f\). In the book the author concentrates on the classical case \(K=\mathbb{Q}\).NEWLINENEWLINE\quad From the description of the 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. [...] 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.''NEWLINENEWLINENEWLINEThe book has eleven chapters. Let us describe the contents of the chapters in more detail. The first chapter deals with some introductory material on elliptic curves. The second chapter is devoted to the study of the notion of heights on projective space and elliptic curves. In particular, it contains a discussion concerning properties of the canonical height of points on elliptic curves given by the Weierstraß\ equation.NEWLINENEWLINENEWLINEThe third chapter is devoted to the study of Weierstraß\ equations over \(\mathbb{C}\) and \(\mathbb{R}\). The author introduces the Weierstraß\ \(\wp\)-function corresponding to the lattice \(\Lambda\). Next, he describes the use of the function \(\wp\) in order to obtain the analytic isomorphism \(\psi\) between the set \(E(\mathbb{C})\) of all complex points on the elliptic curve \(E\) and the quotient \(\mathbb{C}/\Lambda\)…
description: scientific article; zbMATH DE number 6198404