Second descent and rational points on Kummer varieties

Abstract

A powerful method, pioneered by Swinnerton-Dyer, allows one to study rational points on pencils of curves of genus 1 by combining the fibration method with a sophisticated form of descent. A variant of this method, first used by Skorobogatov and Swinnerton-Dyer in 2005, can be applied to the study of rational points on Kummer varieties. In this paper we extend the method to include an additional step of second descent. Assuming finiteness of the relevant Tate-Shafarevich groups, we use the extended method to show that the Brauer-Manin obstruction is the only obstruction to the Hasse principle on Kummer varieties associated to abelian varieties with all rational 2-torsion, under relatively mild additional hypotheses. ContentsConjecture 1.1. Let X be a Kummer surface over k with associated exceptional divisor D ⊆ X. If the Brauer-Manin obstruction to the Hasse principle is the only one for all 2-coverings Y t associated to (X, D), then the same holds for X.Remark 1.2. In Conjecture 1.1 one may freely replace the Brauer-Manin obstruction by the analogous obstruction formed only by the 2-primary part of the Brauer group. This is because for 2-coverings of abelian varieties as well as for Kummer surfaces the latter obstruction is equivalent to the full Brauer-Manin obstruction, see [7, Theorem 1.2 and Theorem 1.7].Conjecture 1.1 combined with the Tate-Shafarevich conjecture together imply that the Brauer-Manin obstruction controls the existence of rational points on Kummer surfaces. We may therefore consider any instance of Conjecture 1.1 as giving support for this latter

Arithmetic of Rational Points and Zero-cycles on Products of Kummer Varieties and K3 Surfaces

“…The exploitation of the rich structure of abelian varieties is a common theme underlying much of the recent rapid progress in the study of rational points on the related K3 surfaces and Kummer varieties (see e.g. [ISZ11], [SZ12], [IS15], [New16], [HS16], [VAV17], [Har19], [CV17], [SZ17]). Our work in Section 4 on the transcendental part of the Brauer group of geometrically abelian varieties and geometrically Kummer varieties may be of independent interest.…”

Descent and étale-Brauer obstructions for 0-cycles

“…For some evidence towards a positive answer to Question 5.5, see for example [SSD05], [HS16], and [Har19].…”

Diagonal quartic surfaces with a Brauer-Manin obstruction

“…It was conjectured by Skorogobatov [Sko09] that the Brauer-Manin obstruction is the only obstruction for K3 surfaces. This conjecture is only known for certain Kummer varieties [HS16,Har19] when assuming the truth of some big conjectures in number theory (finiteness of Shafarevich-Tate groups) and wide open in general.…”