Counting rational points over number fields on a singular cubic 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 $\mathbb{Q}$. 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 $x_0^3=x_1{x}_2{x}_3$.