Abstract
We study Kummer varieties attached to 2-coverings of abelian varieties of arbitrary dimension. Over a number field we show that the subgroup of odd order elements of the Brauer group does not obstruct the Hasse principle. Sufficient conditions for the triviality of the Brauer group are given, which allow us to give an example of a Kummer K3 surface of geometric Picard rank 17 over the rationals with trivial Brauer group. We establish the non-emptyness of the Brauer--Manin set of everywhere locally soluble Kummer varieties attached to 2-coverings of products of hyperelliptic Jacobians with large Galois action on 2-torsion.
Highlights
In [12, 13] Yu.I
In this paper we study the Brauer–Manin obstruction on Kummer varieties, which are higher-dimensional generalisations of classical Kummer K3 surfaces
In the characteristic zero case we describe a natural isomorphism of Galois modules between the geometric Brauer group of a Kummer variety and the geometric Brauer group of the corresponding abelian variety (Proposition 2.7)
Summary
In [12, 13] Yu.I. Manin introduced what is called the Brauer–Manin obstruction. That the canonical class of a Kummer variety of dimension g ≥ 3 is represented by an effective divisor (Proposition 2.6), higher-dimensional Kummer varieties are not Calabi–Yau. Yonatan Harpaz asked if this could be relevant for the tension which exists, in the light of the result of Holmes and Pannekoek, between the heuristics for the ranks of elliptic curves over Q [18] and the conjecture that Q-points of K3 surfaces are dense in the Brauer–Manin set [22, p. If the Kummer variety X attached to a 2-covering of A is everywhere locally soluble, X(Ak)Br = ∅ This explains the absence of the Brauer group from the statements of the Hasse principle for K3 surfaces in Theorems A and B of [6]. We are grateful to Tatiana Bandman and the referees for their comments
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
