Counting imaginary quadratic points via universal torsors

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 A 3 with five lines given in P 4 K by the equationsQ, and [FMT89, Appendix] and [PT01] for computational evidence in degree 3 over Q).The most important technique is the use of universal torsors, which were invented by Colliot-Thélène and Sansuc (see [CTS87], for example) and first applied to Manin's conjecture by Salberger (see [Pey98, Sal98]). The testing ground was a new proof in the case of split toric varieties over Q (see [Sal98]). 1 After submission of the present article, Loughran [Lou13] showed how to derive this over arbitrary number fields from the work of Skinner [Ski97].

Imaginary quadratic points on toric varieties via universal torsors

“…A similar trick works if k is an imaginary quadratic field. This explains why the first applications of the universal torsor method to number fields beyond Q [18][19][20][21] consider imaginary quadratic fields.…”

“…Only recently, a generalization of this method to other number fields was started [18][19][20][21]24,25].…”

O-minimality on twisted universal torsors and Manin's conjecture over number fields

“…(b) Counting these integral points of bounded height, essentially replacing sums by integrals and estimating the difference. A framework covering these parts in some generality was developed over Q in [Der09] and generalized in [DF14a] to imaginary quadratic fields.…”

“…The identification H [DF14a]. Under some additional technical conditions, we give an explicit construction of the twists σ π :…”

On Manin's Conjecture for a Certain Singular Cubic Surface Over Imaginary Quadratic Fields

“…The testing ground were certain del Pezzo surfaces of higher degree (just as over Q, where the investigation of Manin's conjecture beyond equivariant compactifications of algebraic groups and forms in many variables started with smooth quintic [Bre02] and singular quartic [BB07] del Pezzo surfaces). In [DF13a], we developed the necessary techniques over imaginary quadratic fields in some generality and applied them to a first example, and in [DF13b] we showed that they apply to some other singular quartic del Pezzo surfaces. While our general techniques apply in principle to del Pezzo surfaces of arbitrary degree, they do not provide sufficiently strong bounds for the error terms to prove Manin's conjecture for any cubic surface.…”