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Counting imaginary quadratic points via universal torsors

Part of: Algebraic number theory: global fields Arithmetic problems. Diophantine geometry Diophantine equations

Published online by Cambridge University Press:  25 July 2014

Ulrich Derenthal  and
Christopher Frei
Ulrich Derenthal
Affiliation:
Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany email [email protected]
Christopher Frei
Affiliation:
Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany email [email protected]
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Abstract

A conjecture of Manin predicts the distribution of rational points on Fano varieties. We provide a framework for proofs of Manin’s conjecture for del Pezzo surfaces over imaginary quadratic fields, using universal torsors. Some of our tools are formulated over arbitrary number fields. As an application, we prove Manin’s conjecture over imaginary quadratic fields $K$ for the quartic del Pezzo surface $S$ of singularity type ${\boldsymbol{A}}_{3}$ with five lines given in ${\mathbb{P}}_{K}^{4}$ by the equations ${x}_{0}{x}_{1}-{x}_{2}{x}_{3}={x}_{0}{x}_{3}+{x}_{1}{x}_{3}+{x}_{2}{x}_{4}=0$.

Keywords

Manin’s conjectureimaginary quadratic fieldsuniversal torsorsdel Pezzo surfaces

MSC classification

Secondary: 11D45: Counting solutions of Diophantine equations 14G05: Rational points 11R11: Quadratic extensions
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 10 , October 2014 , pp. 1631 - 1678
Copyright
© The Author(s) 2014 

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