Counting imaginary quadratic points via universal torsors

2013/02/25 by Ulrich Derenthal, Derenthal, Ulrich, Christopher R. Frei +1
Mathematics · Social Sciences · #11D45 (Primary) 14G05 #12A25 (Secondary) #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Vietnamese History and Culture Studies

paper · doi:10.48550/arxiv.1302.6151

openalex publication_date 2013/02/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 A₃ with five lines given in P⁴ by the equations vw - xy = vy + wy + xz = 0.

Related

• O-minimality on twisted universal torsors and Manin's conjecture over number fields
• Universal torsors over Del Pezzo surfaces and rational points
• Counting integral points on universal torsors
• The split torsor method for Manin's conjecture
• Integral points on a del Pezzo surface over imaginary quadratic fields