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Counting imaginary quadratic points via universal torsors

Published online by Cambridge University Press:  25 July 2014


Ulrich Derenthal
Affiliation:
Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany email derenthal@math.uni-hannover.de
Christopher Frei
Affiliation:
Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany email frei@math.tugraz.at
Corresponding

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 ${x}_{0}{x}_{1}-{x}_{2}{x}_{3}={x}_{0}{x}_{3}+{x}_{1}{x}_{3}+{x}_{2}{x}_{4}=0$ .


Type
Research Article
Copyright
© The Author(s) 2014 

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