Gaussian rational points on a singular cubic surface

TL;DR

This paper proves Manin's conjecture for a singular toric cubic surface over Gaussian rationals and certain imaginary quadratic fields, extending previous results beyond the rational number field.

Contribution

It provides the first proof of Manin's conjecture over imaginary quadratic fields for this specific singular cubic surface using universal torsors.

Findings

Manin's conjecture holds over Q(i) and other imaginary quadratic fields with class number 1.

The proof extends the universal torsor method to new number fields.

Results confirm the predicted asymptotic distribution of rational points.

Abstract

Manin's conjecture predicts the asymptotic behavior of the number of rational points of bounded height on algebraic varieties. For toric varieties, it was proved by Batyrev and Tschinkel via height zeta functions and an application of the Poisson formula. An alternative approach to Manin's conjecture via universal torsors was used so far mainly over the field Q of rational numbers. In this note, we give a proof of Manin's conjecture over the Gaussian rational numbers Q(i) and over other imaginary quadratic number fields with class number 1 for the singular toric cubic surface defined by t^3=xyz.