Counting rational points over number fields on a singular cubic surface

Frei, Christopher

doi:10.2140/ant.2013.7.1451

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Counting rational points over number fields on a singular cubic surface

Christopher Frei

Vol. 7 (2013), No. 6, 1451–1479

DOI: 10.2140/ant.2013.7.1451

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

Mathematical Subject Classification 2010

Primary: 11D45

Secondary: 14G05

Milestones

Received: 10 April 2012

Revised: 30 July 2012

Accepted: 7 September 2012

Published: 19 September 2013

Authors

Christopher Frei
Mathematisches Institut LMU München Theresienstr. 39 D-80333 München Germany
http://www.mathematik.uni-muenchen.de/~frei

Vol. 7, No. 6, 2013