Elliptic Diophantine equations. A concrete approach via the elliptic logarithm (Q357892): Difference between revisions

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 \(K=\mathbb{Q}\). \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.'' The 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. The 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 \(\mathbb{C}/\Lambda\) (with suitably chosen \(\Lambda\)). Next, a precise description of the set \(E(\mathbb{R})\) of real points on \(E\) (defined over \(\mathbb{R}\)) with the help of \(\psi\) is presented. Finally, the elliptic logarithm \(L(P)\) of the point \(P\) lying on \(E\) is introduced and a practical method of computing \(L(P)\) is presented. The fourth chapter deals with the description of the elliptic logarithm method which in later chapters is used for solving various Diophantine problems of elliptic type. In chapters five and six this method is adjusted for elliptic equations given by Weierstraß\ and quartic equations, respectively. Then chapter VII is devoted to the study of the elliptic logarithm method in case of simultaneous Pell equations. Finally, in the eighth chapter we can find a description of the method in the general case of elliptic equation. Chapter IX is devoted to the description of the lower bound for linear forms in elliptic logarithms. In particular, the result of \textit{S. David} is presented [Mém. Soc. Math. Fr., Nouv. Sér. 62, 143 p. (1995; Zbl 0859.11048)]. In the tenth chapter the author presents how the LLL-algorithm can be used in order to reduce the bound obtained in chapter IX. This is very important in applications. In this chapter we can also find real applications of the presented method. More precisely, the author shows how one can find all integral points on each of the curves \(C_{i}:\;g_{i}(u,v)=0\), for \(i=1, 2, 3\), where \[ g_{1}(u,v)=v^2+uv+v-(u^3+u^2-71u-196), \quad g_{2}(u,v)=v^2-\tfrac52 u^4-\tfrac12 u^3-u^2-1, \] and \[ g_{3}(u,v)=3v^5+3uv^3-271uv-3u^2. \] Finally the method of finding all integral solutions of simultaneous Pell equations given by \[ U^2-11V^2=14,\quad W^2-7V^2=-6 \] is presented. The last chapter of the book is devoted to the application of the elliptic logarithm method to the problem of finding all \(S\)-integral points on elliptic curves given by the Weierstraß\ equation. As an example, the author presents a characterization of all \(\{2,3,5,7,\infty\}\)-integral points on the curve \(C\colon v^2+v=u^3-7u+6\). The book is relatively short and written in a clear and accessible style and can be seen as a welcome addition to the literature devoted to Diophantine equations. I think that each researcher working in this area should have this book on the bookshelf.