Rational points on elliptic curves (Q1814504)

For a non-zero integer \(A\) consider the elliptic curve given by the equation \(x^ 3+y^ 3=Az^ 3\). This curve was studied by D. K. Fadeev in 1934 who gave estimates of its rank and found informations about the group of its rational points. The aim of the present paper is to take up the study of the arithmetic of this curve from a modern point of view. In particular, the author proves that Mazur's conjecture for this curve holds true.