Rational Points on Diagonal Cubic Surfaces

TL;DR

This paper proves the existence of rational points on certain diagonal cubic surfaces over the rationals, assuming the finiteness of the Tate-Shafarevich group for elliptic curves, linking deep conjectures to explicit surface solutions.

Contribution

It establishes a conditional result on rational points for specific diagonal cubic surfaces based on a major unproven conjecture in number theory.

Findings

Conditional proof of rational points existence on specified cubic surfaces

Links the Tate-Shafarevich group's finiteness to Diophantine solutions

Provides explicit criteria involving primes modulo 9

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 $x_1^3 + p_1p_2x_2^3 + p_2p_3x_3^3 + p_3p_1x_4^3 = 0$ has a rational point, where $p_1, p_2$ and $p_3$ are rational primes congruent to $2$ or $5$ modulo $9$.