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

Abstract

Let k be a number field. In the spirit of a result by Yongqi Liang, we relate the arithmetic of rational points over finite extensions of k to that of zero-cycles over k for Kummer varieties over k. For example, for any Kummer variety X over k, we show that if the Brauer-Manin obstruction is the only obstruction to the Hasse principle for rational points on X over all finite extensions of k, then the (2-primary) Brauer-Manin obstruction is the only obstruction to the Hasse principle for zero-cycles of any given odd degree on X over k. We also obtain similar results for products of Kummer varieties, K3 surfaces and rationally connected varieties.

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

“…In this short note we refine the arguments appearing in [I] by using some recent results of Cadoret and Charles about uniform boundedness of Brauer groups [CC], in order to generalise the main results of [I] and [BN]. In particular, besides having somewhat wider applicability our results allow us to remove the restriction that δ is odd that appears in [BN,Thm. 1.1].…”

“…It suffices to show that Γ k and Γ k ′ have the same image in Aut(Br(X)[ℓ m ]) for any ℓ dividing n and any integer m. Fix ℓ dividing n. Since k ′ is linearly disjoint from M, it follows that Γ k and Γ k ′ have the same image in Aut(Br(X) [n]) and so a fortiori in Aut(Br(X) [ℓ]). We can now proceed as in the last part of the proof of [BN,Lem. 4.2], once we justify that ker(Aut(Br(X)[ℓ m ]) → Aut(Br(X)[ℓ])) is an ℓ-group.…”

Descent and étale-Brauer obstructions for 0-cycles

“…Remark 5.7. By using [BN21], one can also consider the more general case of torsors under arbitrary finite products of (twisted) Kummer varieties, K3 surfaces, and geometrically rationally connected varieties over some number field.…”