#### Vol. 7, No. 6, 2013

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Counting rational points over number fields on a singular cubic surface

### Christopher Frei

Vol. 7 (2013), No. 6, 1451–1479
##### 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 $ℚ$. 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 ${x}_{0}^{3}={x}_{1}{x}_{2}{x}_{3}$.

##### Keywords
Manin's conjecture, number fields, rational points, singular cubic surface
Primary: 11D45
Secondary: 14G05