On a certain non-split cubic surface

TL;DR

This paper proves an asymptotic formula with a power-saving error term for counting rational points on a specific singular cubic surface, confirming the Manin-Peyre conjectures for this case.

Contribution

It provides the first verification of the Manin-Peyre conjectures for this particular non-split cubic surface by establishing an asymptotic count of rational points.

Findings

Asymptotic formula for rational points established

Power-saving error term achieved

Verification of Manin-Peyre conjectures for the surface

Abstract

In this note, we establish an asymptotic formula for the number of rational points of bounded height on the singular cubic surface $$ x_0(x_1^2 + x_2^2)=x_3^3 $$ with a power-saving error term, which verifies the Manin-Peyre conjectures for this surface.