The Brauer–Manin Obstruction for Zero-Cycles on K3 Surfaces

Abstract

We study local-global principles for zero-cycles on K3 surfaces defined over number fields. We follow an idea of Liang to use the trivial fibration over the projective line.

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

“…FB thanks the Max-Planck-Institut für Mathematik for the financial support and for providing excellent working conditions. The authors are grateful to Alexei Skorobogatov and Damián Gvirtz for useful conversations, to Daniel Loughran for pointing out the recent preprint [Ier18], and to Evis Ieronymou for his interest in this work. They are also indebted to the anonymous referee whose helpful comments improved this paper and its exposition.…”

“…After obtaining our results for Kummer varieties, we were made aware of a recent preprint of Ieronymou ([Ier18]) in which he uses Liang's approach together with work of Orr and Skorobogatov in [OS18] to prove analogues of Liang's results in [Lia13] for K3 surfaces. Since Kummer surfaces are both K3 surfaces and Kummer varieties of dimension 2, there is an overlap between our work and that of Ieronymou (note that Ieronymou's results do not require that the degree of the 0-cycle is odd).…”

Applications of the fibration method for zero-cycles to the Brauer-Manin obstruction to the existence of zero-cycles on certain varieties

“…Now, Br(X) contains finitely many elements of order less than C and each one of them is stabilized by a subgroup of Gal(k/k) of finite index, which corresponds to a finite extension of k. Taking the Galois closure of the finitely many such extensions we produce a finite Galois extension M/k. We now continue just like in the proof of [I,Thm. 1.2] with the definition of H and the application of [HW,Thm.…”

“…Assume 3(II). In this case in the proof of [I,Thm. 1.2] we need to define the Hilbert subset, H, of P 1 differently.…”

“…In order to conclude in this case we need to show that the image of Br(X) in Br(X k ′ )/Br 0 (X k ′ ) contains the image of Br(X k ′ ){n}, where k ′ = k(P) and P is a closed point of H with [k ′ : k] = δ . By our constructions we have the following commutative diagram with exact rows (this is similar to the diagram appearing in in the proof of [I,Thm. 1.2], the difference being that in this case we consider the n-primary part of the various groups).…”