Rational points on certain del Pezzo surfaces of degree one

TL;DR

This paper investigates the density of rational points on certain degree one del Pezzo surfaces, establishing a link to the infinitude of rational points on associated elliptic curves.

Contribution

It proves that the infinitude of rational points on a specific elliptic curve implies the density of rational points on the del Pezzo surface of degree one.

Findings

Rational points on the elliptic curve imply density on the surface.

Conditions for infinitude of rational points are identified.

The result connects elliptic curve properties to surface rational points.

Abstract

Let $f(z)=z^5+az^3+bz^2+cz+d \in \Z[z]$ and let us consider a del Pezzo surface of degree one given by the equation $\cal{E}_{f}: x^2-y^3-f(z)=0$. In this note we prove that if the set of rational points on the curve $E_{a, b}:Y^2=X^3+135(2a-15)X-1350(5a+2b-26)$ is infinite, then the set of rational points on the surface $\cal{E}_{f}$ is dense in the Zariski topology.