Rational Points on Diagonal Cubic Surfaces

Abstract

We show under the assumption that the Tate-Shafarevich group of any elliptic curve over the rational numbers is finite that the cubic surface $x_1^3+p_1{p}_2{x}_2^3+p_2{p}_3{x}_3^3+p_3{p}_1{x}_4^3=0$ has a rational point, where $p_1,p_2$ and $p_3$ are rational primes congruent to $2$ or $5$ modulo $9$.