Counting rational points over number fields on a singular cubic surface. (English) Zbl 1343.11043

Frei, Christopher

Algebra Number Theory 7, No. 6, 1451-1479 (2013).

Summary: 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 \(x_0^3=x_1x_2x_3\).

MSC:

11D45	Counting solutions of Diophantine equations
14G05	Rational points

Keywords:

Manin’s conjecture; number fields; rational points; singular cubic surface