Computing all integer solutions of a genus 1 equation

Abstract

The Elliptic Logarithm Method has been applied with great successto the problem of computing all integer solutions of equations ofdegree 3 and 4 defining elliptic curves. We extend this methodto include any equation f(u,v)=0 that defines a curve of genus 1.Here f is a polynomial with integer coefficients and irreducible overthe algebraic closure of the rationals, but is otherwise of arbitrary shape and degree.We give a detailed description of the general features of our approach,and conclude with two rather unusual examples corresponding to equationsof degree 5 and degree 9.Elliptic curve;Elliptic logarithm;Dophantine equation