Counting rational points over number fields on a singular cubic surface

TL;DR

This paper proves Manin's conjecture for a specific singular cubic surface over arbitrary number fields by combining universal torsors and Schanuel's techniques, extending previous results beyond the rational numbers.

Contribution

It extends the proof of Manin's conjecture to arbitrary number fields for a singular cubic surface using a novel combination of methods.

Findings

Confirmed Manin's conjecture for the surface over any number field

Developed a new approach combining universal torsors and Schanuel's techniques

Established distribution of rational points on the surface

Abstract

A conjecture of Manin predicts the distribution of K-rational points on certain algebraic varieties defined over a number field K. In recent years, a method using universal torsors has been successfully applied to several hard special cases of Manin's conjecture over the field Q of rational numbers. Combining this method with techniques developed by Schanuel, we give a proof of Manin's conjecture over arbitrary number fields for the singular cubic surface S given by the equation w^3 = x y z.